\n+\n=\nD\n \n \nwhere \n)\nTHmedian\n(TH\nt\nit >\nD\n is an indicator variable that takes the value of 1 if the \nthreshold variable for a country is above the median value for all countries in that time \nperiod. \n \nThis approach sets the threshold exogenously and provides a simple way of testing if the \nlevel of a particular variable matters in terms of the quantitative effect of openness on \ngrowth outcomes. We also examine the impact of varying the high-low cut-off to check the \nappropriateness of the median approach.8 \n \nThe interpretation of reduced-form growth regressions is typically bedevilled by concerns \nabout endogeneity and the direction of causality. For instance, capital may flow \ndisproportionately to fast-growing economies, making financial integration dependent on \ngrowth rather than the reverse. Similarly, financial development and growth may both be \ndriven by common factors such as the legal or broader institutional frameworks. It is \ndifficult to come up with convincing and effective instruments to deal with these issues. \n \nHence, we use system generalized method of moments (GMM) techniques for dynamic \npanels to get around these problems. This involves estimating a system comprising a first-\ndifferenced equation to eliminate country fixed effects and an additional equation in levels. \nAppropriately lagged values of levels and first-differences, respectively, can then be used as \ninstruments in these equations to address endogeneity concerns. This approach is \nincreasingly being used in a variety of related contexts.9 In addition to the system GMM \nestimation we also provide basic fixed effects estimates as a consistency check. \n \n8 An alternative approach would be to use sample-splitting methodologies to endogenously \ndetermine the threshold (Hansen, 2000). Unfortunately, however, such models cannot be applied to \nthe dynamic panel approach that we employ. \n9 See Bond et al., 2001, for a detailed technical discussion of its application to empirical growth \nmodels. In related work, Chang et al. (2005) use this methodology to explore linear interaction \neffects of institutional features and trade openness. Aghion et al. (2005) look at interaction effects \nbetween financial development and the exchange rate regime. Roodman (2006, 2008) provides a \ndetailed review of the practical implementation of this methodology in a manner that obviates \npotential concerns related to its somewhat mechanical application and small sample problems. \n\n \n13\nB. Semi-parametric approaches \nNext, we turn to a nonparametric technique that allows us to model in a more flexible \nmanner the relationship between growth, on the one hand, and the financial openness and \nthreshold variables on the other. To keep the model tractable, we assume that the \nrelationship between growth and the standard controls plus fixed effects is linear as before. \nThe resulting semiparametric model is written as follows: \n \nit\nit\nit\nit\nt\ni\nit\n \n)\nTH\n,\nh(FO\nx'\n \ny\n!\n\"\n#\n$\n+\n+\n+\n+\n=\n%\n \nwhere we estimate the parametric coefficients and the nonparametric relationship \n)\nTH\n,\nh(FO\nit\nit\n. \n \nA few recent papers in the growth literature have used partial linear models to examine the \nrelationship between growth and a regressor of interest. For example, Banerjee and Duflo \n(2003) examine the nonparametric effects of inequality on growth while Imbs and Ranciere \n(2007) look at the relationship between external debt and growth. However, these papers \nfocus on the relationship between growth and a nonparametric function of a single variable \nrather than a function of two variables as is the case with the interaction effects we consider. \n \nYatchew (1998, 2003) provides a detailed guide to a variety of methods that can be \nemployed to estimate the parametric coefficients and the nonparametric function \n)\nTH\n,\nh(FO\nit\nit\n.10 In particular, as in Banerjee and Duflo (2003) and Imbs and Ranciere \n(2007), we focus on Robinson’s (1988) double residuals approach. This involves two \nstages. First, nonparametric regressions of growth and each of the other control variables on \nfinancial openness and the threshold variable are estimated to give \n)\nTH\n,\nFO\n | \ny\nE(\nit\nit\nit\n!\n and \n)\nTH\n,\nFO\n | \nE(\nit\nit\nitz\n where \nitz\n denotes the matrix of \nit\nx plus the fixed effects with \ncorresponding vector of coefficients ! . Various nonparametric estimation methodologies \ncan be employed, for example local regression or kernel estimation. The residuals from \nthese regressions are then used to estimate the parametric coefficients ! using an OLS \nregression: \n \n10 See also Yatchew and No (2001) for estimation of a partial linear model with two variables \nentering the nonparametric expression. We implement these partial linear estimations using S-plus \ncoding following the examples in Yatchew (2003). \n\n \n14\nit\nit\nit\nit\nit\nit\nit\nit\nit\nit\nit\nit\nit\nit\nit\n \n))'\nTH\n,\nFO\n | \nE(z\n -\n(\n)\nTH\n,\nh(FO\n- \n)'\nTH\n,\nFO\n | \nE(z\n -\ny\n \n )\nTH\n,\nFO\n | \ny\nE(\n- \ny\n!\n\"\n\"\n+\n=\n#\n=\n#\n#\nz\n \n \nThese OLS estimates of ˆ! can then be used to construct an expression for the residual \ngrowth with the estimated parametric effects removed: \n \nit\nit\nit\nit\nit\n \n)\nTH\n,\nh(FO\n \n ˆ\nz'\n- \ny\n!\n\"\n+\n#\n$\n \n \nThe nonparametric form of \n)\nTH\n,\nh(FO\nit\nit\n can be estimated using standard methods such as \nlocal regression. For details on the required assumptions and convergence properties, see \nRobinson (1988) and Yatchew (2003). We use OLS regressions in the different stages of the \npartial linear estimation, with time and country fixed effects included where appropriate.11 \n \nThe use of semi-parametric methods allows for a more flexible examination of the nature of \nthreshold effects in the relationship between financial openness and growth than is possible \nwith parametric approaches. However, there are trade-offs among different approaches. For \nexample, the flexibility of the semi-parametric estimates comes with other assumptions, \nsuch as that of a linear relationship for other control variables and the choice of the nature \nof the nonparametric estimation approach. More importantly, nonparametric relationships \nare somewhat more difficult to interpret and to translate into policy implications. \n \nA key issue concerns the significance and empirical content of the estimated thresholds. To \nhave policy relevance, our analysis requires more than just a demonstration of statistically \nsignificant conditional correlations between certain variables and growth. We need to \nconstruct confidence intervals around our estimates of the marginal effects of openness on \ngrowth, conditional on a particular level of a given threshold variable. We also need to \nknow if the magnitudes of the threshold effects are economically significant and if the \nestimated thresholds lie within the range of the sample used in the estimation (otherwise, \nthe thresholds would be of little practical value in terms of understanding differential \ngrowth outcomes). \n \n11 As discussed below, in the case of the non time-varying institutional quality index we do not \ninclude country dummies in the nonparametric estimation. \n\n \n15\nV. Basic Results \nWe motivate our empirical analysis by documenting a set of stylized facts for data averaged \nover the full sample period. We then present our baseline econometric results that rely on a \nfiner temporal breakdown of the data. As much of the existing literature has analyzed the \ninteraction between financial openness and financial development, we will focus our initial \nexposition on the latter as a threshold variable in order to illustrate our framework. \n \nV.1 Stylized facts \nWe begin by exploring if there are obvious threshold effects in the data. For this exercise, \nwe limit the sample to non-industrial countries split into two groups—emerging markets \n(EMs) and other developing countries (ODCs). Our interest is in whether, within each of \nthese groups, the levels of certain variables are associated with differences in average \ngrowth rates. Table 1 compares unconditional and conditional growth rates over the period \n1975-2004 for countries that are above or below the within-group sample medians for \ndifferent variables that have been posited as threshold variables. After sorting countries \nwithin each group by these group-specific thresholds, we then report cross-sectional \naverages within each cell. \n \nThere are three main results that can be gleaned from this table. First, EMs, which are more \nintegrated into international capital markets than ODCs, have a higher average growth rate \nthan ODCs over the period 1975-2004, but this effect becomes smaller when we control for \nother standard variables that influence growth. Second, unconditional growth rates in EMs \nare greater for those countries with higher (within-group above-median) levels of the \nillustrative threshold indicators for financial depth, trade openness, institutional quality, \nregulation and macro policies, although this difference is not always statistically significant. \nThese effects are less pronounced in ODCs, except that the institutional quality threshold is \neven more important for ODCs than for EMs. The picture is less clear when looking at \noverall development and financial openness as threshold variables. Growth rates are higher \nfor countries with lower initial GDP per capita, reflecting convergence effects. In both \ngroups, growth rates are higher for countries with lower relative financial openness. \nThird, for conditional growth rates the patterns are less pronounced, although the positive \nassociation of growth with higher values of certain threshold variables persists (e.g., private \ncredit, trade, reduced regulation and lower inflation variability among EMs). Table 1 also \n\n \n16\nsuggests that the difference between the growth rates of EMs and ODCs is generally more \npronounced at higher levels of the threshold variables (except for institutional quality, GDP \nper capita and financial openness). These stylized facts are suggestive of systematic \nthreshold or conditioning effects in the relationship between financial openness and growth. \nWe now turn to a more formal empirical analysis of these effects. \n \nV.2 Basic empirical analysis \nOur regression analysis is based on five-year averages of the underlying annual data. We \nbegin with a limited set of controls that have been identified in the literature as being \nrelatively robust determinants of long-term per capita GDP growth—initial income (at the \nstart of each five-year period), which picks up convergence effects; the level of investment \nto GDP; a proxy for human capital; and population growth. \n \nWe report the results of baseline growth regressions using these controls in the first panel of \nTable 2. The first column shows the results of OLS regressions with country fixed effects \n(FE). The population growth rate does not seem to matter for medium-term growth. \nHowever, when we switch to generalized method of moments (GMM) estimation to deal \nwith endogeneity issues (column 2), only the level of investment remains statistically \nsignificant. Nevertheless, we retain these four controls in the first stage of our analysis. FE \nand GMM are the two basic specifications that we will build upon in our further analysis.12 \n \nV.2.1 Financial depth as a threshold \nIn panel 2, we include a broad measure of de facto financial openness. As is typical in the \nliterature, we find that the correlation between financial integration and growth is weak or \neven slightly negative. This highlights the key discrepancy between theory and evidence on \nthe growth effects of financial integration. Consider a simple exercise where we look at \nwhether the correlation is different between countries with high and low levels of financial \ndepth (above or below the sample median). The third panel of Table 2 shows that there is a \nstriking difference. When we interact the indicator for a high degree of financial depth with \nthe financial openness variable, the coefficient on the interaction term is strongly positive \n \n12 Both specifications always include time effects to capture common factors affecting growth across \nall countries in each five-year period. \n\n \n17\nand nearly the same in magnitude as the negative coefficient on the financial openness \nvariable itself. In other words, the effect of financial openness is negative for economies \nwith comparatively low levels of financial depth and slightly positive but insignificant for \nthose with higher levels.13 Repeating the experiment using different percentiles of the \nfinancial depth variable rather than the median as the cutoff yields similar positive \nsignificant interaction coefficients for cutoffs from the 15th to the 60th percentile with FE \nestimates and from the 30th to the 65th percentile with GMM estimates (see Figure 2). \n \nIn panel 4, we allow for a linear interaction term between domestic financial depth and \nfinancial openness. Neither the coefficient on financial openness nor the one on the \ninteraction term is significantly different from zero. The level of financial depth does not \nseem to matter for the correlation between financial openness and growth. Could this non-\nresult be driven by the fact that, once a country has attained a certain level of financial \ndepth, further improvements don’t matter that much? \n \nIn panel 5, we allow for an additional interaction of financial openness with the square of \nthe financial depth variable. The coefficients on both the linear and quadratic interactions \nare now strongly significant in both the FE and GMM estimates, with the first coefficient \nbeing positive and the second negative in both cases. That is, greater financial depth leads to \nan improvement in the growth effects of financial integration but only up to a certain level \nof financial depth. \n \nWhere is the threshold and is it an economically reasonable one? We can calculate the level \nof the threshold, for a given level of credit to GDP, from the interaction terms. The overall \nfinancial openness coefficient in this case takes an inverted U-shape as the threshold \nvariable rises. It is thus possible to calculate the cutoffs at which its sign changes. Based on \nthe FE estimates, the threshold level below which the marginal effect of financial openness \non growth is negative corresponds to a credit to GDP ratio of 71 percent (-0.0825 + \n0.1761*0.71 – 0.0845*0.71^2 = 0). Above this level, the coefficient is positive before \nturning negative for credit to GDP above 137 percent. Based on the GMM estimates, the \ncorresponding threshold levels are credit to GDP ratios of 50 percent and 126 percent, \n \n13 The median levels of financial development that determine the high-low cutoffs are calculated \nseparately for each period. \n\n \n18\nrespectively. For reference, the median levels of credit to GDP for industrial countries, EMs \nand ODCs are 0.71, 0.32 and 0.19, respectively (calculated across all period-country \nobservations for each group). \n \nWith both estimation methods, the vast majority (over 90%) of ODC observations lie below \nthe lower threshold and have a negative financial openness coefficient. For emerging and \nindustrial economies, a much higher fraction of observations lie between the lower and \nupper thresholds and have a positive financial openness coefficient: about two-fifths for \nemerging economies and four-fifths for industrial countries (relative to the GMM-based \nthreshold). Thus, the threshold level seems plausible and of practical relevance for \ndeveloping countries contemplating capital account liberalization. In the remaining \ndiscussion, we focus on the lower threshold, which is the relevant one for developing and \nemerging economies.14 \n \nSince the threshold we have derived is static, it is interesting to see how different groups of \ncountries are doing relative to this threshold over time.15 In 1975-79, the proportion of \ncountries in each group above the GMM-based lower threshold (private credit to GDP ratio \nof 0.50) was as follows: industrial countries—62 percent; emerging markets—25 percent; \nand ODCs—2 percent. By 2000-04, the proportions had increased to 100 percent, 48 \npercent and 14 percent, respectively. Figure 3 shows how the credit to GDP ratio has \nchanged for each of the emerging market countries from 1985-89 to 2000-04, and how these \nlevels match up against the estimated FE and GMM thresholds. For most of the emerging \nmarkets, the data points lie above the 45-degree line, implying increases in financial depth \nover time by this measure. The fraction of emerging markets above the GMM threshold \nrises from 25% in 1975-79 to 48% in 2000-04, while the number above the FE threshold \ngoes from 0% to 38%. It is worth noting that a country like China comes out looking very \n \n14 The upper threshold is an artifact of the quadratic specification. We experimented with the \ninclusion of higher order polynomials of the threshold variable (and corresponding interactions with \nfinancial openness). The coefficients on the higher order terms were usually not statistically \nsignificant but their magnitudes generally showed a flattening out of (rather than a decline in) the \nimplied marginal effect of financial openness on growth at high levels of the threshold variable. This \nis another reason why we focus on the lower threshold. \n15 An important issue here is whether the thresholds themselves change over time. This is not an \neasy question to address in an empirical framework that uses cross-country data and, therefore, \ncomes up against obvious data limitations. We leave this for future work and note that our exercise \nhere is meant only to be illustrative of the empirical content of the thresholds concept. \n\n \n19\ngood by this measure despite the weaknesses in its financial sector, which is dominated by \nstate-owned banks. This is a useful reminder of the potential pitfalls of using a particular \nuni-dimensional measure of financial development. And of course the worldwide crisis that \nfirst hit the U.S. and then spread to other industrial countries has shown that financial depth \nis not equivalent to financial stability. \n \nV.2.2 Robustness of financial depth threshold \nWe test the sensitivity of our baseline results for the financial depth threshold in a number \nof ways. First, we use a different set of basic controls and redo the regressions in Table 2. \nWe retain log initial income and the education variable, and add the following controls—\ntrade openness, CPI inflation, and the logarithm of the number of phone lines per capita (a \nproxy for the level of infrastructure). We do not present the results here, but they were quite \nsimilar in terms of the signs and magnitudes of the coefficients of interest. The implied \nupper and lower thresholds from the FE specification with quadratic interactions are private \ncredit to GDP ratios of 63 percent and 148 percent, respectively (compared to 71 percent \nand 137 percent based on the results in Table 2). For the GMM specification the results are \nsuch that, while the estimated overall financial openness coefficient retains an inverted U-\nshape, it remains positive and does not cut the x-axis. \n \nSecond, we use an alternative measure of financial depth—the sum of private credit and \nstock market capitalization as a ratio to GDP. Unfortunately, given the absence of stock \nmarkets in many of the developing countries, especially in the early years of the sample, the \nsample drops to about half the original size. In the specification with quadratic interactions, \nthe estimated coefficients on the interaction terms have the same sign as in our baseline, but \nthey are smaller and not statistically significant. Given the low levels of stock market \ndevelopment in ODCs and, until recently, in emerging markets as well, this broader \nmeasure of financial depth does not seem to be useful for constructing thresholds. \n \nThird, we check if the results are driven by the choice of countries in our sample. We test \nfor robustness to the exclusion of three groups of countries (dropping one group at a time): \n(i) OPEC countries (Algeria, Ecuador, Indonesia, Iran, Kuwait, UAE and Venezuela); (ii) \noffshore financial centers (Ireland, Panama, Singapore); and (iii) countries hit by the Asian \nfinancial crisis (Indonesia, Korea, Malaysia, Philippines and Thailand). The results with the \n\n \n20\nhigh-low interactions and linear interactions were broadly similar when we excluded these \nsub-samples. Table 3 shows that the signs and magnitudes of the coefficients, as well as the \nimplied thresholds, are relatively stable when we drop each of these groups of countries, \nsuggesting that the results are not being driven by outliers or any specific group of \ncountries. \n \nFourth, we go back to the original financial depth variable but look at alternative measures \nof financial openness (FO). The threshold value of private credit to GDP is almost \nunchanged when we use the stock of gross external liabilities as a ratio to GDP—rather than \nthe sum of external assets and liabilities--as the measure of FO (0.51 in the GMM estimates, \nwhich is almost identical to the baseline result from Table 2). \n \nV.3 Breaking down the nature of financial integration \nThe literature on financial flows makes a distinction between FDI and portfolio equity \nflows, on the one hand, and debt on the other. It is generally believed that the former types \nof flows generate more of the indirect benefits of financial integration and also have fewer \nrisks than debt. Does the composition of external liabilities (or flows) influence the \nthreshold level of financial depth? Here we obtain a very interesting result (Table 4, Part A). \nWhen we measure FO as the stock of FDI plus portfolio equity liabilities, the threshold is \nlower (credit to GDP ratios of 58 percent and 34 percent for the FE and GMM estimates, \nrespectively). By contrast, when we use debt liabilities, the threshold is much higher (credit \nto GDP ratios of 75 percent and 55 percent for the FE and GMM estimates, respectively). \nThat is, the risks of financial integration seem to be lower when it takes the form of FDI or \nportfolio equity liabilities. When debt liabilities constitute the primary form of financial \nintegration, the level of financial depth necessary for financial integration to have growth \nbenefits is much higher. \n \nThe results with flows are more mixed (Table 4, Part B). When we use total inflows, the \nsigns of the interaction effects are such that the overall financial openness coefficient has a \nU-shape as credit to GDP rises, the reverse of the results with the stock measures of \nopenness. Again, there is a dramatic difference between the results when we use FDI plus \nportfolio equity inflows versus debt inflows. In the former case, the inverted U-shape of the \noverall financial openness coefficient remains (although insignificant with the GMM \n\n \n21\nestimates). By contrast, the results with debt inflows correspond to those for total inflows \n(as expected, given the high share of debt to total inflows over the sample period). In this \ncase, the impact of financial openness on growth is estimated to be positive for lower or \nparticularly high levels of financial depth but negative at intermediate levels. This result is \nconsistent with models of potential instability induced by greater capital inflows in \neconomies at an intermediate level of financial development (e.g., Aghion et al., 2004). \n \nVI. Alternative Thresholds \nOur focus has so far been on the financial depth threshold. We now examine threshold \neffects based on a range of other indicators suggested by the discussion of theoretical \nmodels in Section 2. We maintain the FE and GMM specifications as our benchmarks and \nfocus on the quadratic interaction specifications. \n \nThe first panel of Table 5 repeats the results for the financial depth variable. The second \npanel looks at a composite measure of institutional quality (IQ). Many authors have argued \nthat IQ is a crucial determinant of growth and volatility, especially crises (e.g., Acemoglu et \nal., 2003). There is indeed a clear threshold effect that we can identify; the interactions of \nfinancial openness with the level and squared level of the IQ variable are statistically \nsignificant. All of the industrial country observations (five-year averages) exceed the \nestimated threshold, while only 29 percent of emerging market observations and about 20 \npercent of ODC observations do. By this measure, most developing countries are below the \nlevel of IQ at which the marginal benefits of increasing financial openness become \napparent. \n \nWe also looked at some of the constituents of the composite measure of institutional \nquality—level of corruption, cost of enforcing debt contracts etc.—but could not identify \nany strong threshold effects based on these components of the IQ indicator (results not \nshown). The level of per capita income (on an internationally comparable basis) is often \nseen as a composite index that proxies for a variety of factors that have been found to boost \ngrowth. But there is no clear threshold effect based on this variable. \n \nWe can identify a threshold based on trade openness (the ratio of the sum of imports and \nexports to GDP) but the estimated threshold is so high that few countries meet this \nthreshold. We also experimented with a policy measure of trade openness (results not \n\n \n22\nreported here). The relevant interaction coefficients were significant in the FE regressions \nbut not in GMM. We also looked at thresholds based on a measure of structural policies—\nlabor market flexibility—and two measures of macro policies—inflation volatility and the \nratio of government revenues to expenditures. There are a number of significant interaction \nterms in the regressions with these variables, but they are in general not robust, so we \nchoose not to focus on the implied thresholds. \n \nTo visually examine how the estimated thresholds look for a few key variables, Figure 4 \nplots the overall (including interactions) financial openness coefficient estimates against \ndifferent values of the relevant threshold variable. Private credit and IQ illustrate the \ninverted U-shaped relationship, with the standard error bands often encompassing zero but \nstill leaving some empirical content in this threshold measure. When we use trade openness \nor the log of initial income, the threshold effects are essentially linear in the relevant range. \n \nThe analysis in this section suggests that, at a first pass, the results for financial and \ninstitutional development are more supportive of the presence of threshold effects. Other \nvariables we have looked at also hint at threshold effects, particularly for high/low \ninteractions, although the estimates from other specifications are less robust ant not always \nstatistically significant.16 \n \nVII. Results Based on Semi-parametric Approaches \nWe now explore the relationship between financial openness and growth using the semi-\nparametric methods outlined in Section IV. To illustrate these methods, we first start with a \nunivariate nonparametric specification in the partial linear setup. That is, we look at the \npotential nonlinear relationship between growth and financial openness itself. We then \nexamine interaction effects between financial openness and various threshold variables.17 \n \n16 We also experimented with using the de jure measure of financial openness as a threshold variable \nin place of the de facto measure. The coefficient on gross financial openness is positive at higher \nlevels of financial openness, although the coefficient is significant only in the FE estimates. \n17 To conserve space, we present only the key results in figures. Figures for all other results referred \nto in this section are in the Semi-Parametric Appendix of the web link to the working paper version \nof this paper. \n\n \n23\nVII.1 Semiparametric estimation of the effects of financial openness on growth \nThe regressions of growth against the baseline controls plus gross financial openness to \nGDP indicate an insignificant negative coefficient on the latter from both the FE and system \nGMM estimation (Table 4). However, unconditional plots suggest that the level and shape \nof the relationship between financial openness and growth vary by quintile of financial \nopenness. To investigate this in more detail, we employ the partial linear model with the \ngross financial openness variable alone entering the specification nonparametrically. \n \nFirst, we run a regression to eliminate the baseline parametric effects (including country and \ntime fixed effects) from the growth data.18 Figure 5 plots growth residuals from this \nregression against the gross financial openness variable. Next, we use nonparametric \nmethods to estimate the form of the relationship between these two variables. Specifically, \nwe employ the Robinson residual method, first using local regression with two different \nspans (the percentage of data points included in the local regression) and then a kernel \nestimator (with a triangular kernel) as the nonparametric technique. We also use an \nalternative “differencing approach” (for details, see the Semi-Parametric Appendix in the \nweb link to the working paper version). If we demean the growth estimates from the first-\nstage parametric regressions, we obtain “purged” or demeaned growth residual values that \nillustrate the nonparametric relationship at the mean of the parametric variables (Yatchew, \n2003). These different relationships are illustrated in the bottom panel of Figure 5. \n \nThese plots illustrate a similar pattern in the results from different approaches, with an \nincreasing relationship between growth and financial openness at low levels of the latter, \nwhich then turns negative and reverts to being positive at the highest levels of financial \nopenness. However, the estimated relationship becomes insignificant as financial openness \nrises. The plots also highlight the potential roles of outliers on financial openness in \ninfluencing the results and the relatively large confidence intervals attached to the point \nestimates. The variations in the effects across financial openness values may contribute to \nthe overall negative insignificant coefficient in the standard linear parametric estimation. \n \n \n18 Note that the baseline parametric effects exclude the indirect influence of the financial openness \non these variables. \n\n \n24\nWe replicated the above analysis for different measures of financial openness. As with the \nparametric results, there are marked differences across these measures. For example, the \nstock of FDI and portfolio equity liabilities, which has a positive but insignificant linear \ncoefficient in the parametric setup (see Table 4), has a relationship that is broadly flat at \npositive values of the demeaned growth residuals and then increases with the financial \nopenness measure. In contrast, the relationship of the debt measure with the demeaned \ngrowth residuals has a marked downward slope above a certain value of debt (Imbs and \nRanciere, 2007, discuss the external debt Laffer curve). \n \nVII.2 Semiparametric interactions between financial openness and threshold variables \nThe double residuals approach is applied in a similar manner when looking at interaction \neffects, i.e., when both financial openness and a threshold variable enter nonparametrically. \nAs before, we first obtain growth residuals by eliminating the baseline parametric effects. \nTo conduct the nonparametric smoothing, we then focus on the local regression estimator.19 \n \nUnconditional plots of growth against financial openness reveal patterns that vary by the \nlevel of credit to GDP. At low levels of credit to GDP, the relationship tends to be negative, \nthen moving towards a flat relationship at higher levels of credit to GDP. Using the double \nresidual approach with a local regression span of 0.75, the estimated nonparametric \nrelationship between growth residuals and financial openness is illustrated in Figure 6.20 \nThis figure is similar to Figure 6 but, rather than showing the univariate nonparametric \nrelationship between growth residuals and financial openness, it shows the multivariate \nrelationship of growth residuals with financial openness and the credit to GDP ratio. Thus, it \nrepresents one nonparametric approach to illustrating the interaction between financial \nopenness and a threshold variable in their relationship with growth residuals. For relatively \nlow levels of credit to GDP and low levels of financial openness, the estimated relationship \nbetween growth and financial openness is indeed negative. This is the range in which most \ncountry observations actually fall. \n \n \n19 This fits a local quadratic regression including the threshold and financial openness variables, \ntheir squares and cross-products. Insightful Corporation (2007) has details on local regression \nprocedures. \n20 The results were not greatly sensitive to alternative regression spans. \n\n \n25\nAn alternative way to examine this relationship is to look at how the relationship of the \ndemeaned growth residuals with financial openness varies with the level of the threshold \nvariable (and vice versa). Figure 7 shows such relationships and their confidence intervals \nfor different slices of the corresponding 3D plot. Figure 7a illustrates the negative \nrelationship between demeaned growth residuals and financial openness at low levels of \ncredit to GDP. Figure 7b shows that the inverted U-shaped relationship between these \nresiduals and credit-to-GDP tends to be more prevalent at higher levels of financial \nopenness. One point to note concerning these plots is that the slices are taken at equally \nspaced splits across the full range rather than at percentile values of the distribution of \nobservations. Thus, given the skewed distribution of both credit to GDP and financial \nopenness most country data points lie in the bottom and left-hand side plots. Again, these \nplots illustrate the wide confidence intervals around the estimated effects, which in many \ncases are not significantly different from zero. \n \nThis analysis can be repeated for different measures of financial openness. As with the \nparametric estimates, the results for total liabilities are similar to those for the gross \nmeasures. There are again marked differences between the estimates using FDI and \nportfolio equity liabilities versus debt liabilities (results not shown here). With the former, \nthe unconditional relationship between growth and financial openness is mostly flat or \nslightly positive throughout different sub-samples based on levels of credit to GDP. By \ncontrast, with debt liabilities the relationship with growth is downward sloping for half of \nthe sub-samples with lower levels of credit to GDP. \n \nTurning to the nonparametric model, Figures 8 and 9 compare the fitted nonparametric \ninteraction effects and the demeaned growth residuals. For low to medium levels of credit to \nGDP, the relationship between growth and the financial openness measure based on FDI \nand portfolio equity liabilities is flat or increasing. However, at these low levels of credit to \nGDP, the relationship between growth and debt liabilities is negative. Again, when \nanalyzing these results it is important to note that the confidence intervals around these \nestimates tend to be relatively large and that most observations lie at lower levels of \nfinancial openness and credit to GDP. \n \n\n \n26\nWe now apply this methodology to a few other threshold variables.21 Unconditional growth \nplots illustrate that the relationship between growth and financial openness is negative for \nsamples with lower trade openness ratios. This effect disappears once we control for other \ngrowth determinants and fixed effects in estimating the nonparametric interaction \nrelationship with the relationship between residual growth and financial openness broadly \nflat at different levels of trade. \n \nTurning to institutional quality, again unconditional plots indicate a negative relationship \nbetween growth and financial openness at lower levels of the threshold variable. At low \nlevels of institutional quality, the relationship between gross financial openness and growth \nis U-shaped. However, at higher levels of institutional quality the relationship becomes \nmore linear. In line with the quadratic parametric estimation, for a given level of financial \nopenness, residual growth increases with institutional quality at a decreasing rate. Once \nagain, the interpretation of these results is subject to caveats on the size of confidence \nintervals and also on the actual distribution of observations by institutional quality and \nfinancial openness.22 \n \nVIII. Summary and Implications \nRecent advances in the theoretical and empirical literatures indicate that the benefits of \nfinancial integration may be far subtler than had been presumed earlier. A new framework \nfor analyzing financial globalization highlights the tension between the indirect benefits of \nfinancial integration and the potential risks if a country opens up to capital flows without \nthe right initial conditions in place. From a practical policy perspective, however, a \nreasonable evaluation of the cost-benefit trade-off requires a better understanding of what \nthese initial conditions are and how exactly they matter. This is an essential component of \n \n21 See Semi-Parametric Appendix Figures 14-15 and 16-19 for trade openness and institutional \nquality as the threshold variables, respectively (this appendix is in the web link to the working paper \nversion). \n22 The double residual estimation process is complicated in this case by the non time-varying nature \nof the threshold variable. In the first stage nonparametric estimation we have been conducting \nnonparametric regression of each of the baseline controls, including country dummy variables, on \nthe threshold and financial openness variables. Applying this technique with institutional quality \nwould mean that the country dummy variables are regressed on institutional quality, which is also a \ncountry-specific time invariant variable. This leads to a singular regressor matrix in the second stage \nregression. To get around this problem, we remove fixed effects from the first stage regression. We \nthen estimate the second stage nonparametric interaction effects also without the country dummy \nvariables (although we obtain similar results if we then include them). \n\n \n27\nan analytical framework that can take account of country-specific features and initial \nconditions in designing a pragmatic approach to capital account liberalization (Prasad and \nRajan, 2008). \n \nIn this paper, we have tried to put some empirical structure on the concept of threshold \nconditions in order to give policymakers guidance on this issue. For instance, our results \nsupport the widely held conjecture that FDI and portfolio equity flows are safer than debt \nflows at low levels of financial and institutional development. We do not claim to have \nidentified definitive thresholds. Our main contribution, instead, has been to develop an \nempirical structure to address this issue and frame it in a more concrete and tractable \nmanner. Our analysis has already generated a number of interesting findings, which we now \nbriefly summarize before discussing what policymakers should make of them. \n \nBased on different methodologies and different definitions of thresholds, we conclude that \nthere are threshold levels of certain variables that are important determinants of the \nrelationship between financial integration and growth. In our empirical work, we have \nfocused on a few variables motivated by the existing theoretical literature. These include \ndomestic financial market development (in particular, the depth of credit markets), \ninstitutional quality, trade openness, labor market rigidities, and the overall level of \ndevelopment. All of these seem to be relevant threshold variables, with varying degrees of \nimportance—the most clearly-defined thresholds are based on the financial depth and \ninstitutional quality variables. We find that many of these thresholds are much lower when \nwe measure financial integration by the stocks of FDI and portfolio equity liabilities rather \nthan debt liabilities. \n \nThe confidence intervals around some of the estimated thresholds are large, but in many \ncases the estimated coefficients yield reasonably tight estimates of the threshold conditions. \nDo the thresholds have empirical content? Our results generally indicate that the estimated \nthresholds are reasonable and well within the ranges of the data samples. For instance, most \nindustrial countries and a few emerging markets are above the estimated threshold levels of \nfinancial depth, while a majority of emerging markets and nearly all other developing \ncountries are below them. This result is consistent with observed differences in growth \noutcomes associated with financial integration across these groups of countries. Of course, \n\n \n28\nthe recent global crisis shows that financial depth is not a reliable measure of financial \nstability, which should also take into account regulatory and supervisory structures. \n \nIndeed, there is a rich research agenda that comes out of our work. Future theoretical studies \nin this area should focus on the precise nature of the threshold relationship and provide \ntestable predictions in the context of reduced form solutions. On the empirical front, our \nresults show that focusing on individual threshold variables could lead to misleading \nconclusions. Some of the open questions prompted by our analysis are as follows. Are there \ntrade-offs among different threshold conditions, such that a high level of one variable can \nlower the threshold on another variable?23 If the level of financial integration itself acts as a \nthreshold, how can it be integrated into the framework based on other thresholds laid out in \nthis paper? Have the levels of different thresholds been changing over time as virtually all \ncountries become more financially open in de facto terms, irrespective of their capital \ncontrol regimes? How do circumstances in global financial markets affect the thresholds? \n \n23 We find preliminary evidence that financial depth matters less in countries that have high IQ \nlevels. We also checked if a simple composite measure derived from the different threshold \nvariables in our analysis could serve as a composite threshold indicator. Preliminary analysis \nsuggests that there are indeed threshold effects in the data based on this composite indicator. We \nhave not, however, developed a procedure to find the optimal composite indicator that captures the \ncomplementarity and substitutability among different threshold conditions and leave that for future \nwork. \n\n \n29\nReferences \nAcemoglu, D., S. Johnson, J. A. Robinson and Y. 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Thresholds in the Process of Financial Integration \n \n \n \nKk \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nSource: Kose, Prasad, Rogoff and Wei (2009) \n \n \nFinancial \nGlobalization \n \nThreshold Conditions \nFinancial market development \nInstitutional Quality, G overnance \nMacroeconomic policies \nTrade integ ration \nGDP / TFP growth \n \nRisks of Crises \nGDP / TFP growth \n \nRisks of Crises \nX \nAbove Thresholds \nBelow Thresholds \n? \nFinancial globalization leads to better macroeconomic outcomes when certain threshold \nconditions are met. \n \n\n \n34\n-0.1\n-0.05\n0\n0.05\n0.1\n0.15\n0\n10\n20\n30\n40\n50\n60\n70\n80\n90\n100\nCutoff percentile\nHigh/low interaction coefficient plus 95% CI\n-0.2\n-0.15\n-0.1\n-0.05\n0\n0.05\n0.1\n0.15\n0.2\n0.25\n0\n10\n20\n30\n40\n50\n60\n70\n80\n90\n100\nCutoff percentile\nHigh/low interaction coefficient plus 95% CI\nFigure 2. High/Low Interaction Coefficients for Gross Financial Openness and Private \nCredit to GDP at Different Sample Splits \nA. Fixed effects specification \n \nB. System GMM specification \n \n \n \nNotes: Specifications include base controls of Panel 3 of Table 2. Percentile cutoffs calculated for each period \non the basis of the distribution of private credit observations in that period. \n \n\n \n35\nFigure 3. Average Private Credit to GDP Relative to Estimated Thresholds: \nEmerging Market Economies, 1975-79 and 2000-04 \n \nTUR\nZAF\nJOR\nISR\nEGY\nTHA\nSGP\nPHL\nPAK\nMYS\nKOR\nIND\nCHN\nVEN\nPER\nMEX\nCOL\nCHL\nBRA\nARG\n0%\n25%\n50%\n75%\n100%\n125%\n150%\n0%\n25%\n50%\n75%\n100%\n125%\n150%\nPrivate credit to GDP, 2000-04\nPrivate credit to GDP, 1975-79\n45 degree \nline \nLower threshold (GMM)\nLower threshold (FE)\n \nNotes: Thresholds taken from quadratic interaction specification in Table 2, Panel 5. \n\n \n36\nFigure 4. Overall Financial Openness Coefficient Against Alternative Threshold \nVariables (based on GMM estimation) \n \n \na) Private credit to GDP as threshold variable \nb) Institutional quality index as threshold variable \n \n \nc) Trade openness to GDP as threshold variable \nd) Ln initial GDP per capita as threshold variable \n \n \nNotes: See Table 5 for estimation details. The lighter lines indicate 95 percent confidence intervals. \n-0.14\n-0.12\n-0.1\n-0.08\n-0.06\n-0.04\n-0.02\n0\n0.02\n0.04\n0.06\n0%\n20%\n40%\n60%\n80%\n100%\n120%\nOverall FO coefficient\nPrivate credit to GDP\n-0.25\n-0.2\n-0.15\n-0.1\n-0.05\n0\n0.05\n0.1\n-1.5 \n-1.0 \n-0.5 \n0.0\n0.5\n1.0\n1.5\n2.0\nOverall FO coefficient\nInstitutional quality index\n-0.12\n-0.1\n-0.08\n-0.06\n-0.04\n-0.02\n0\n0.02\n0.04\n0.06\n0%\n25%\n50%\n75%\n100%\n125%\n150%\nOverall FO coefficient\nTrade openness\n-2\n-1\n0\n1\n2\n3\n4\n6\n7\n8\n9\n10\n11\nOverall FO coefficient\nLn initial GDP per capita\n\n \n37\nFigure 5. Gross Financial Openness and Growth Residuals \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nNotes: The plots illustrate the relationship between five-year growth rates--once standard controls and dummy \nvariables have been controlled for (excluding the indirect effect of gross financial openness on these controls--\nand gross financial openness. A nonparametric relationship is then estimated and illustrated on the graph with \n95% confidence intervals indicated by vertical lines. Four alternative methods are illustrated. Three employ \nthe Robinson double residual estimator including local regression estimator (loess) using various spans of the \nobservations and a kernel smoother. The final one employs the differencing estimator described in the Semi-\nParametric Appendix (see working paper version of this paper). \n \n \n \n \n \n \n \n\n \n38\nFigure 6. Double Residual Nonparametric Interaction Effects \n(Credit to GDP as the threshold variable, interacted with gross financial openness to GDP) \n \n \n \n \nNotes: This plot illustrates the estimated nonparametric relationship between conditional growth once standard \ncontrols and dummy variables have been controlled for (excluding the indirect effect of gross financial \nopenness and credit-to-GDP on these controls) and gross financial openness and credit-to-GDP. The Robinson \ndouble residual estimator is employed using a local regression estimator (loess) with a span of 0.75. \n \n\n \n39\nFigure 7. Cross-Sections of Double Residual Nonparametric Interaction Effects \n(Credit to GDP as the threshold variable, interacted with gross financial openness to GDP) \n \nA. Sliced at different values of credit to GDP \n \nB. Sliced at different values of financial openness to GDP \n \n \nNotes: The six lower panels show the relationship between residual growth and financial openness in part (a) \nand credit to GDP in part (b) with 95% confidence intervals indicated by the vertical lines. The six plots are \ntaken at six equally spaced levels of credit-to-GDP and financial openness to GDP in parts A and B, \nrespectively. The lowest value of the given variable is represented in the bottom left-hand panel with the level \nrising in subsequent panels as one moves from left to right and then up and long the second panel. The \ncorresponding values of the given variable at which the slices are made are indicated by the dots in the \nuppermost plot across the width of the figure. \n\n \n40\nFigure 8. Double Residual Nonparametric Interaction Effects \n(Credit to GDP as the threshold variable, interacted with gross FDI and portfolio equity liabilities to \nGDP) \nNote: Similar to Figure 6 but with FDI and portfolio equity liabilities to GDP as the financial openness \nvariable rather than gross financial openness to GDP. \n \nFigure 9. Double Residual Nonparametric Interaction Effects \n(Credit to GDP as the threshold variable, interacted with gross external debt liabilities to GDP) \n \n \nNote: Similar to Figures 6 and 8 but with external debt liabilities to GDP used as the financial openness \nvariable. \n\n \n41\nTable 1. Long-term Growth in Emerging Markets and Other Developing Countries \n \n \n \nUnconditional growth (% per \nannum) \nConditional growth (% per \nannum) \n \n \nEM \nODCs \nEMs \nODCs \nOverall \n \n2.284 \n(1.937) \n0.820 \n(0.650) \n0.441 \n(0.533) \n-0.159 \n(-0.043) \nSplitting sub-samples: \n \n \n \n \n \nBy private credit to GDP \nHigh \n3.158 \n(3.113) \n0.656 \n(0.451) \n0.733 \n(0.673) \n-0.255 \n(-0.197) \n \nLow \n1.490 \n(1.410) \n0.983 \n(0.877) \n0.176 \n(0.503) \n-0.064 \n(0.139) \nDifference in means \n1.668* \n-0.327 \n0.557 \n-0.191 \nBy average WBGI \ninstitutional quality index \nHigh \n2.416 \n(1.878) \n1.217 \n(0.853) \n0.394 \n(0.418) \n0.369 \n(0.127) \n \nLow \n2.165 \n(1.937) \n0.422 \n(0.451) \n0.483 \n(0.633) \n-0.688 \n(-0.117) \nDifference in means \n0.251 \n0.795* \n-0.089 \n1.057** \nBy trade openness \nHigh \n2.923 \n(3.017) \n1.074 \n(0.710) \n0.644 \n(0.583) \n0.129 \n(0.127) \n \nLow \n1.704 \n(1.096) \n0.566 \n(0.493) \n0.256 \n(0.503) \n-0.448 \n(-0.094) \nDifference in means \n1.218 \n0.508 \n0.388 \n0.577 \nBy \nrigidity \nof \nemployment index \nLess rigid \n2.958 \n(2.440) \n0.787 \n(0.493) \n0.563 \n(0.533) \n-0.012 \n(-0.094) \n \nMore rigid \n1.544 \n(1.253) \n0.790 \n(0.927) \n0.306 \n(0.568) \n-0.344 \n(-0.168) \nDifference in means \n1.414 \n-0.003 \n0.257 \n0.333 \nBy st. dev of CPI inflation \nLow \n3.381 \n(3.365) \n1.509 \n(1.542) \n1.074 \n(0.968) \n0.398 \n(0.379) \n \nHigh \n1.078 \n(1.147) \n0.215 \n(0.346) \n-0.255 \n(-0.242) \n-0.841 \n(-0.810) \nDifference in means \n2.303*** \n1.294*** \n1.329*** \n1.239*** \nBy initial GDP per capita \nHigh \n1.105 \n(1.085) \n0.798 \n(1.034) \n-0.166 \n(-0.098) \n0.146 \n(0.276) \n \nLow \n3.357 \n(3.155) \n0.842 \n(0.493) \n0.993 \n(0.968) \n-0.464 \n(-0.506) \nDifference in means \n-2.253*** \n-0.044 \n-1.159** \n0.611 \nBy de jure financial \nopenness (IMF measure) \nHigh \n1.537 \n(1.211) \n0.730 \n(0.452) \n0.048 \n(-0.098) \n0.026 \n(-0.043) \n \nLow \n2.964 \n(2.431) \n0.901 \n(0.927) \n0.799 \n(0.813) \n-0.327 \n(-0.183) \nDifference in means \n-1.427 \n-0.171 \n-0.751 \n0.353 \nBy de facto gross financial \nopenness \nHigh \n1.502 \n(1.262) \n0.738 \n(0.853) \n0.036 \n(-0.248) \n-0.163 \n(0.009) \n \nLow \n2.995 \n(2.440) \n0.902 \n(0.493) \n0.810 \n(0.660) \n-0.155 \n(-0.094) \nDifference in means \n-1.493* \n-0.164 \n-0.774 \n-0.008 \n \nNotes: The numbers shown are average annual growth rates (medians are shown in parentheses below the \nmeans). The symbols *, ** and *** indicate statistical significance at the 10 percent, 5 percent and 1 percent \nlevels, respectively, of a t-test of mean equality across sub-samples. High/low sub-samples are defined relative \nto medians within groupings. See Appendix Table A.1 for definition of emerging market (EM) and other \ndeveloping country (ODC) sub-samples and Appendix Table A.2 for variable definitions. Conditional growth \nindicates residuals from a cross-section regression of growth on log initial GDP per capita, average investment \nto GDP, average years of schooling and average population growth rate. \n\n \n42\nTable 2. Interactions of Private Credit and Gross Financial Openness to GDP \n \n \n[1] Base \n \n[2] With FO \n \n[3] High/low interaction \n[4] Linear interaction \n[5] Quadratic interaction \n \nFE \nSys GMM \nFE \nSys GMM \nFE \nSys GMM \nFE \nSys GMM \nFE \nSys GMM \n-0.2769 \n-0.0505 \n-0.3028 \n-0.0529 \n-0.3122 \n-0.1028 \n-0.3096 \n-0.0762 \n-0.3196 \n-0.0847 \nLn initial income per capita \n \n[0.0560]*** \n[0.0657] \n[0.0460]*** \n[0.0533] \n[0.0473]*** \n[0.0483]** \n[0.0468]*** \n[0.0530] \n[0.0479]*** \n[0.0484]* \n0.8079 \n0.9852 \n0.8029 \n0.942 \n0.7534 \n0.8505 \n0.7521 \n0.9384 \n0.6835 \n0.9112 \nAv investment to GDP \n \n[0.3064]*** \n[0.2806]*** \n[0.3110]** \n[0.3097]*** \n[0.3126]** \n[0.2842]*** \n[0.3243]** \n[0.2862]*** \n[0.3025]** \n[0.2946]*** \nYears schooling \n0.0286 \n-0.0022 \n0.0305 \n0.0039 \n0.0301 \n0.0196 \n0.0301 \n0.0108 \n0.0252 \n0.011 \n \n[0.0140]** \n[0.0193] \n[0.0143]** \n[0.0168] \n[0.0145]** \n[0.0161] \n[0.0145]** \n[0.0161] \n[0.0148]* \n[0.0143] \n4.7321 \n-0.9328 \n4.8012 \n-0.1238 \n4.7648 \n-0.9325 \n4.7266 \n-0.8469 \n4.7277 \n-1.9786 \nPop growth \n \n[3.1908] \n[1.7681] \n[3.1706] \n[2.6259] \n[3.1514] \n[2.0722] \n[3.2036] \n[2.2271] \n[3.1587] \n[3.1068] \nGross FO to GDP \n \n \n-0.0008 \n-0.005 \n-0.0371 \n-0.0612 \n-0.0191 \n-0.0057 \n-0.0825 \n-0.0724 \n \n \n \n[0.0082] \n[0.0074] \n[0.0169]** \n[0.0221]*** \n[0.0187] \n[0.0228] \n[0.0277]*** \n[0.0325]** \nPrivate credit to GDP (PC) \n \n \n \n \n-0.0241 \n-0.0627 \n-0.0147 \n-0.0145 \n-0.1687 \n-0.2476 \n \n \n \n \n[0.0358] \n[0.0394] \n[0.0410] \n[0.0596] \n[0.0986]* \n[0.1535] \n \nGross FO*high PC \n \n \n \n \n0.0380 \n0.0628 \n \n \n \n \n \n \n \n \n \n[0.0160]** \n[0.0215]*** \n \n \n \n \nGross FO*PC \n \n \n \n \n \n \n0.0174 \n0.0018 \n0.1761 \n0.2024 \n \n \n \n \n \n \n \n[0.0152] \n[0.0195] \n[0.0518]*** \n[0.0814]** \nPC squared \n \n \n \n \n \n \n \n \n0.0798 \n0.157 \n \n \n \n \n \n \n \n \n[0.0436]* \n[0.0832]* \nFO* PC squared \n \n \n \n \n \n \n \n \n \n-0.0845 \n-0.115 \n \n \n \n \n \n \n \n \n \n[0.0242]*** \n[0.0464]** \nConstant \n2.1202 \n0.41 \n2.3375 \n0.3923 \n2.4632 \n0.8071 \n2.4252 \n0.5727 \n2.6252 \n0.7355 \n \n[0.4557]*** \n[0.4255] \n[0.3676]*** \n[0.3350] \n[0.3815]*** \n[0.3137]** \n[0.3802]*** \n[0.3393]* \n[0.3915]*** \n[0.3319]** \nObservations \n460 \n460 \n457 \n457 \n456 \n456 \n456 \n456 \n456 \n456 \nAdj R-squared \n0.2915 \n \n0.3131 \n \n0.3259 \n \n0.3149 \n \n0.338 \n \nAR2 test p-value \n \n0.3191 \n \n0.2498 \n \n0.2323 \n \n0.3333 \n \n0.4474 \nHansen p-value \n \n0.264 \n \n0.3873 \n \n0.4966 \n \n0.406 \n \n0.5246 \n \n \nNotes: All specifications include base controls in Table 2 and period effects, which are not reported. Standard errors in parentheses. The symbols *, **, *** indicate significance \n10%, 5% and 1% levels, respectively. FE: country fixed effects with robust standard errors clustered by country. GMM system (sys GMM) estimation: Two step using Windmeijer \nstandard errors with small sample correction and control variables treated as endogenous (instrumented using 2nd lag). \n \n\n \n43\nTable 3. Sub-sample Sensitivities: Private Credit and Gross Financial Openness to GDP Interaction Coefficients \n \n \n \n[1] Full sample \n[2] Ex OPEC \n[3] Ex OFCs \n[4] Ex Asian crisis countries \n \n \nFE \nSys GMM \nFE \nSys GMM \nFE \nSys GMM \nFE \nSys GMM \nGross FO \n-0.0008 \n-0.0050 \n-0.0007 \n-0.0041 \n-0.0235 \n-0.0267 \n0.0000 \n-0.0040 \nA. No \ninteraction \n \n[0.0082] \n[0.0074] \n[0.0080] \n[0.0071] \n[0.0080]*** \n[0.0138]* \n[0.0081] \n[0.0074] \nGross FO \n-0.0371 \n-0.0612 \n-0.0404 \n-0.0482 \n-0.0632 \n-0.0586 \n-0.045 \n-0.0499 \nB. High/low \ninteraction \n \n[0.0169]** \n[0.0221]*** \n[0.0169]** \n[0.0196]** \n[0.0163]*** \n[0.0231]** \n[0.0177]** \n[0.0200]** \n \nGross FO*high PC \n0.038 \n0.0628 \n0.042 \n0.0513 \n0.0415 \n0.0355 \n0.0472 \n0.0521 \n \n \n[0.0160]** \n[0.0215]*** \n[0.0164]** \n[0.0188]*** \n[0.0138]*** \n[0.0231] \n[0.0176]*** \n[0.0198]** \nGross FO \n-0.0191 \n-0.0057 \n-0.0157 \n-0.0026 \n-0.0526 \n-0.0479 \n-0.0204 \n-0.0027 \nC. Linear \ninteraction \n \n[0.0187] \n[0.0228] \n[0.0179] \n[0.0188] \n[0.0161]*** \n[0.0231]** \n[0.0187] \n[0.0222] \n \nGross FO*PC \n0.0174 \n0.0018 \n0.0147 \n-0.0002 \n0.0258 \n0.0188 \n0.0194 \n-0.0002 \n \n \n[0.0152] \n[0.0195] \n[0.0144] \n[0.0156] \n[0.0119]** \n[0.0155] \n[0.0159] \n[0.0198] \n \nPC cutoff for positive \noverall gross FO coeff. \n>1.10 \n>3.22 \n>1.07 \nn.a. \n>2.04 \n>2.55 \n>1.05 \nn.a. \nGross FO \n-0.0825 \n-0.0724 \n-0.0789 \n-0.0653 \n-0.0958 \n-0.0658 \n-0.0893 \n-0.0746 \nD. Quadratic \ninteraction \n \n[0.0277]*** \n[0.0325]** \n[0.0269]*** \n[0.0294]** \n[0.0267]*** \n[0.0488] \n[0.0281]*** \n[0.0365]** \n \nGross FO*PC \n0.1761 \n0.2024 \n0.1722 \n0.1844 \n0.1507 \n0.0673 \n0.1927 \n0.2195 \n \n \n[0.0518]*** \n[0.0814]** \n[0.0512]*** \n[0.0799]** \n[0.0542]*** \n[0.1002] \n[0.0532]*** \n[0.0957]** \n \nGross FO* PC squared \n-0.0845 \n-0.115 \n-0.0835 \n-0.1048 \n-0.0639 \n-0.0246 \n-0.0924 \n-0.1251 \n \n \n[0.0242]*** \n[0.0464]** \n[0.0241]*** \n[0.0457]** \n[0.0244]** \n[0.0494] \n[0.0247]*** \n[0.0551]** \n \n0.711 \n0.500 \n0.688 \n0.492 \nn.a. \nn.a. \n0.694 \n0.461 \n \nPC cutoffs at which overall \ngross FO coeff. is zero:a \n1.372 \n1.260 \n1.375 \n1.268 \nn.a. \nn.a. \n1.391 \n1.294 \n \n% observations above lower \ncutoff \n \n \n \n \n \n \n \n \n \nIndustrial countries \n60% \n80% \n62% \n80% \nn.a. \nn.a. \n62% \n81% \n \nEmerging economies \n21% \n42% \n25% \n46% \nn.a. \nn.a. \n20% \n43% \n \nOther developing countries \n1% \n10% \n1% \n8% \nn.a. \nn.a. \n1% \n12% \n \na Cutoff is not available if the overall FO coefficient estimated as a function of the threshold variable does not have a quadratic root. \n \nNotes: All specifications include base controls in Table 2 and period effects, which are not reported. Standard errors in parentheses. The symbols *, **, \n*** indicate significance at the 10%, 5% and 1% levels, respectively. FE: country fixed effects with robust standard errors clustered by country. GMM \nsystem estimation: Two step using Windmeijer standard errors with small sample correction and control variables treated as endogenous (instrumented \nusing 2nd lag). \n\n \n44\nTable 4A. Interaction Coefficients with Private Credit to GDP and Different Financial Openness Measures \n \nStock Measures (relative to GDP) \n \n \n \n[1] Gross measure \n \n[2] Total liabilities \n[3] FDI + portfolio equity \n[4] Debt liabilities \n \n \nFE \nSys GMM \nFE \nSys GMM \nFE \nSys GMM \nFE \nSys GMM \nFO \n-0.0008 \n-0.005 \n-0.0174 \n-0.0202 \n0.0352 \n0.0051 \n-0.0366 \n-0.031 \nA. No \ninteraction \n \n[0.0082] \n[0.0074] \n[0.0168] \n[0.0175] \n[0.0272] \n[0.0315] \n[0.0177]** \n[0.0237] \nFO \n-0.0371 \n-0.0612 \n-0.0722 \n-0.1025 \n-0.1764 \n-0.2233 \n-0.0839 \n-0.1231 \nB. High/low \ninteraction \n \n[0.0169]** \n[0.0221]*** \n[0.0205]*** \n[0.0286]*** \n[0.1135] \n[0.1756] \n[0.0213]*** \n[0.0332]*** \n \nFO*high PC \n0.038 \n0.0628 \n0.0619 \n0.1066 \n0.2205 \n0.2518 \n0.0574 \n0.1248 \n \n \n[0.0160]** \n[0.0215]*** \n[0.0189]*** \n[0.0303]*** \n[0.1119]* \n[0.1819] \n[0.0189]*** \n[0.0367]*** \nFO \n-0.0191 \n-0.0057 \n-0.0672 \n-0.0362 \n0.021 \n0.1107 \n-0.0792 \n-0.0581 \nC. Linear \ninteraction \n \n[0.0187] \n[0.0228] \n[0.0253]*** \n[0.0315] \n[0.1032] \n[0.1234] \n[0.0268]*** \n[0.0247]** \n \nFO*PC \n0.0174 \n0.0018 \n0.0591 \n0.024 \n0.0114 \n-0.084 \n0.0692 \n0.0477 \n \n \n[0.0152] \n[0.0195] \n[0.0256]** \n[0.0305] \n[0.0777] \n[0.0985] \n[0.0357]* \n[0.0502] \n \nPC cutoff for positive \noverall FO coefficient \n>1.10 \n>3.22 \n>1.14 \n>1.31 \nn.a. \n<1.97 \n>1.14 \n>-0.31 \nFO \n-0.0825 \n-0.0724 \n-0.1495 \n-0.1341 \n-0.3502 \n-0.1694 \n-0.1454 \n-0.159 \nD. Quadratic \ninteraction \n \n[0.0277]*** \n[0.0325]** \n[0.0330]*** \n[0.0402]*** \n[0.1622]** \n[0.2421] \n[0.0381]*** \n[0.0366]*** \n \nFO*PC \n0.1761 \n0.2024 \n0.3258 \n0.3715 \n0.8555 \n0.6364 \n0.3125 \n0.4258 \n \n \n[0.0518]*** \n[0.0814]** \n[0.0792]*** \n[0.1014]*** \n[0.2794]*** \n[0.4204] \n[0.1048]*** \n[0.1202]*** \n \nFO* PC squared \n-0.0845 \n-0.115 \n-0.1596 \n-0.2099 \n-0.4381 \n-0.3969 \n-0.1585 \n-0.249 \n \n \n[0.0242]*** \n[0.0464]** \n[0.0413]*** \n[0.0602]*** \n[0.1288]*** \n[0.1994]** \n[0.0548]*** \n[0.0715]*** \n \n0.711 \n0.500 \n0.697 \n0.505 \n0.584 \n0.337 \n0.752 \n0.551 \n \nPC cutoffs at which overall \nFO coefficient zero: \n1.372 \n1.260 \n1.345 \n1.264 \n1.369 \n1.266 \n1.220 \n1.159 \n \n% observations above lower \ncutoff \n \n \n \n \n \n \n \n \n \nIndustrial countries \n60% \n80% \n62% \n80% \n71% \n91% \n56% \n75% \n \nEmerging economies \n21% \n42% \n22% \n42% \n33% \n58% \n17% \n37% \n \nOther developing countries \n1% \n10% \n1% \n9% \n5% \n20% \n1% \n7% \n \nNotes: All specifications include the same base controls as in Table 2 and period effects, which are not reported. Standard errors in parentheses. The symbols *, **, *** \nindicate significance at the 10%, 5% and 1% levels, respectively. Also see notes to Table 3. \n\n \n45\nTable 4B. Interaction Coefficients with Private Credit to GDP and Different Financial Openness Measures \n \nFlow Measures (relative to GDP) \n \n \n \n \n[1] Gross flows \n \n[2] Total inflows \n[3] FDI + port. eq. inflows \n[4] Debt inflows \n \n \nFE \nSys GMM \nFE \nSys GMM \nFE \nSys GMM \nFE \nSys GMM \nFO \n0.0539 \n0.0277 \n0.1025 \n0.1343 \n0.3307 \n0.3683 \n0.0911 \n0.173 \nA. No \ninteraction \n \n[0.0368] \n[0.0440] \n[0.0808] \n[0.0616]** \n[0.0916]*** \n[0.1962]* \n[0.0919] \n[0.1203] \nFO \n0.3931 \n0.1229 \n0.8829 \n0.9599 \n-0.1891 \n0.1585 \n0.9959 \n1.4456 \nB. High/low \ninteraction \n \n[0.2046]* \n[0.6629] \n[0.2255]*** \n[0.5288]* \n[0.6742] \n[1.6524] \n[0.1858]*** \n[0.5775]** \n \nFO*high PC \n-0.3495 \n-0.0659 \n-0.8278 \n-0.8475 \n0.5464 \n0.2146 \n-0.9642 \n-1.3112 \n \n \n[0.2049]* \n[0.6572] \n[0.2305]*** \n[0.5197] \n[0.6829] \n[1.7114] \n[0.1901]*** \n[0.5558]** \nFO \n0.1447 \n0.189 \n0.3186 \n0.4339 \n0.9633 \n2.0915 \n0.456 \n0.6122 \nC. Linear \ninteraction \n \n[0.1168] \n[0.2165] \n[0.2057] \n[0.3120] \n[0.8197] \n[1.2152]* \n[0.2332]* \n[0.3992] \n \nFO*PC \n-0.1038 \n-0.1269 \n-0.2997 \n-0.3059 \n-0.6054 \n-1.5512 \n-0.582 \n-0.6076 \n \n \n[0.0929] \n[0.1724] \n[0.2214] \n[0.2691] \n[0.7299] \n[1.0439] \n[0.3666] \n[0.5376] \n \nPC cutoff for positive \noverall FO coefficient \n<1.39 \n<0.26 \n<1.06 \n<10.12 \n<1.59 \n<12.35 \n<0.78 \n<0.99 \nFO \n0.2085 \n-0.2087 \n0.9311 \n0.9015 \n-1.1963 \n-0.4571 \n1.1183 \n1.706 \nD. Quadratic \ninteraction \n \n[0.2317] \n[0.4957] \n[0.3238]*** \n[0.6844] \n[1.1833] \n[2.0809] \n[0.2762]*** \n[0.7045]** \n \nFO*PC \n-0.2824 \n0.6634 \n-2.0279 \n-1.7388 \n6.1905 \n6.5172 \n-2.5299 \n-3.8141 \n \n \n[0.5078] \n[0.9660] \n[0.8380]** \n[1.8620] \n[2.5132]** \n[5.4541] \n[0.6619]*** \n[1.8492]** \n \nFO* PC squared \n0.108 \n-0.3839 \n1.0608 \n0.8488 \n-4.2428 \n-5.0065 \n1.2849 \n1.9293 \n \n \n[0.2636] \n[0.4801] \n[0.4637]** \n[1.0722] \n[1.3188]*** \n[3.2263] \n[0.3488]*** \n[1.1016]* \n \nn.a. \n0.41 \n0.77 \nn.a. \n0.23 \n0.07 \n0.67 \n0.68 \n \nPC cutoffs at which overall \nFO coefficient zero: \nn.a. \n1.31 \n1.15 \nn.a. \n1.23 \n1.23 \n1.30 \n1.29 \n \n% observations above lower \ncutoff \n \n \n \n \n \n \n \n \n \nIndustrial countries \nn.a. \n87% \n50% \nn.a. \n98% \n100% \n64% \n63% \n \nEmerging economies \nn.a. \n50% \n16% \nn.a. \n78% \n98% \n25% \n24% \n \nOther developing countries \nn.a. \n14% \n1% \nn.a. \n48% \n88% \n2% \n2% \n \nNotes: Port. eq. denotes portfolio equity. All specifications include the same base controls as in Table 2 and period effects, which are not reported. Standard errors in \nparentheses. The symbols *, **, *** indicate significance at the 10%, 5% and 1% levels, respectively. Also see notes to Table 3. \n\n \n46\nTable 5. Alternative Threshold Variables: Interaction Coefficients with Gross Financial Openness to GDP \n \n \n \n[1] Private credit to GDP \nas threshold \n[2] Institutional quality index \nas threshold \n[3] Trade openness to GDP \nas threshold \n[4] Ln initial GDP per \ncapita as threshold \n \n \nFE \nSys GMM \nFE \nSys GMM \nFE \nSys GMM \nFE \nSys GMM \nGross FO \n-0.0371 \n-0.0612 \n-0.0657 \n-0.0773 \n-0.0155 \n-0.0618 \n-0.0383 \n-0.0774 \nA. High/low \ninteraction \n \n[0.0169]** \n[0.0221]*** \n[0.0196]*** \n[0.0396]* \n[0.0083]* \n[0.0254]** \n[0.0146]** \n[0.0322]** \n \nGross FO*high threshold \n0.038 \n0.0628 \n0.0721 \n0.0782 \n0.0143 \n0.0598 \n0.0419 \n0.0794 \n \n \n[0.0160]** \n[0.0215]*** \n[0.0208]*** \n[0.0396]* \n[0.0074]* \n[0.0247]** \n[0.0161]** \n[0.0327]** \nGross FO \n-0.0191 \n-0.0057 \n-0.0282 \n-0.0148 \n-0.0156 \n-0.0302 \n-0.1171 \n-0.266 \nB. Linear \ninteraction \n \n[0.0187] \n[0.0228] \n[0.0154]* \n[0.0197] \n[0.0102] \n[0.0189] \n[0.0785] \n[0.1185]** \n \nGross FO*threshold \nvariable \n0.0174 \n0.0018 \n0.0236 \n0.0113 \n0.0077 \n0.0154 \n0.0121 \n0.0263 \n \n \n[0.0152] \n[0.0195] \n[0.0099]** \n[0.0106] \n[0.0048] \n[0.0122] \n[0.0079] \n[0.0117]** \n \nThreshold cutoff for \npositive FO coefficient \n>1.10 \n>3.22 \n>1.19 \n>1.31 \n>2.02 \n>1.97 \n>9.64 \n>10.12 \nFO \n-0.0825 \n-0.0724 \n-0.0179 \n-0.0121 \n-0.0386 \n-0.0795 \n-1.3559 \n-1.7303 \nC. Quadratic \ninteraction \n \n[0.0277]*** \n[0.0325]** \n[0.0084]** \n[0.0108] \n[0.0137]*** \n[0.0262]*** \n[0.7836]* \n[1.2973] \n \nGross FO*threshold \nvariable \n0.1761 \n0.2024 \n0.0724 \n0.0779 \n0.0342 \n0.0733 \n0.289 \n0.3637 \n \n \n[0.0518]*** \n[0.0814]** \n[0.0256]*** \n[0.0262]*** \n[0.0161]** \n[0.0249]*** \n[0.1769] \n[0.3019] \n \n-0.0845 \n-0.115 \n-0.0339 \n-0.0421 \n-0.0056 \n-0.0147 \n-0.0153 \n-0.0191 \n \nGross FO* threshold \nvariable squared \n[0.0242]*** \n[0.0464]** \n[0.0152]** \n[0.0155]*** \n[0.0042] \n[0.0055]*** \n[0.0098] \n[0.0172] \n \n0.711 \n0.500 \n0.285 \n0.171 \n1.496 \n1.602 \n8.569 \n-6.129 \n \nThreshold cutoffs at which \noverall FO coeff. zero: \n1.372 \n1.260 \n1.848 \n1.681 \n4.633 \n3.368 \n10.368 \n3.827 \n \n% observations above lower \ncutoff \n \n \n \n \n \n \n \n \n \nIndustrial countries \n60% \n80% \n100% \n100% \n2% \n2% \n100% \n100% \n \nEmerging economies \n21% \n42% \n29% \n29% \n7% \n6% \n49% \n100% \n \nOther developing countries \n1% \n10% \n17% \n21% \n2% \n1% \n22% \n100% \n \nNotes: All specifications include the same base controls as in Table 2 and period effects, which are not reported. Standard errors in parentheses. The symbols *, **, *** \nindicate statistical significance at 10%, 5% and 1% levels, respectively. Also see notes to Table 3. \n\n \n47\nData Appendix \n \n \nTable A1. Country Sample \n \nIndustrial \nEmerging economies \n(EMs) \nOther developing \ncountries (ODCs) \n \nAustralia \nAustria \nBelgium \nCanada \nDenmark \nFinland \nFrance \nGermany \nGreece \nIreland \nItaly \nJapan \nNetherlands \nNew Zealand \nNorway \nPortugal \nSpain \nSweden \nSwitzerland \nUnited Kingdom \nUnited States \nArgentina \nBrazil \nChile \nChina \nColombia \nEgypt \nIndia \nIndonesia \nIsrael \nJordan \nKorea, Republic of \nMalaysia \nMexico \nPakistan \nPeru \nPhilippines \nSingapore \nSouth Africa \nThailand \nTurkey \nVenezuela \nAlgeria \nBangladesh \nBenin \nBolivia \nBotswana \nCameroon \nCongo, Republic of \nCosta Rica \nDominican Republic \nEcuador \nEl Salvador \nGhana \nGuatemala \nHaiti \nHonduras \nIran \nJamaica \nKenya \nKuwait \nMalawi \nMali \nMauritius \nMozambique \nNepal \nNicaragua \nNiger \nPanama \nPapua New Guinea \nParaguay \nRwanda \nSenegal \nSri Lanka \nSudan \nSyria \nTogo \nTrinidad &Tobago \nTunisia \nUganda \nUnited Arab Emirates \nUruguay \nZambia \nZimbabwe \n \nNotes: The sample comprises 84 countries—21 industrial and 63 developing (of which 21 are emerging \nmarket economies, EMs, and 42 are other developing countries, ODCs). \n \n\n \n48\nTable A2. Variable Definitions and Sources \n \nVariable \nSources \nGrowth rate of PPP per capita GDP (log \ndifference over period divided by length) \nPWT \nGDP per capita PPP, 1996 constant prices \nPWT \nAverage investment to GDP \nPWT \nAverage schooling years in population over 25 \nyears old \nUpdated Barro and Lee (2000) database. Data \navailable at \nhttp://www.cid.harvard.edu/ciddata/ciddata.html \nAverage annual population growth rate (log \ndifference over period divided by length) \nWDI \nGross de facto financial openness to GDP \nStock data from Lane and Milesi-Ferretti (2006). \nCurrent price US dollar GDP data from WDI. \nStock of external liabilities to GDP \nAs above \nStock of external FDI and portfolio equity \nliabilities to GDP \nAs above \nStock of external debt liabilities to GDP \nAs above \nGross flows to GDP defined as sum of \nabsolute inflows and absolute outflows \nFlow data from IMF IFS. Current price US dollar \nGDP data from WDI. \nTotal financial inflows to GDP \nAs above \nFDI plus portfolio equity inflows to GDP \nAs above \nDebt inflows to GDP \nAs above \nDomestic credit to private sector to GDP \nWDI \nCurrent price trade openness (exports plus \nimports) to GDP \nPWT \nAverage institutional quality index \nSimple average of six World Bank Governance \nIndicators (data available from 1996) \nRigidity of employment index for employing \nworkers \nWorld Bank / International Finance Corporate \nDoing Business Database (data available from \n2003) \nAnnual CPI inflation \nIFS \n \nNotes: PWT: Penn World Tables (version 6.2); IFS: International Financial Statistics; WDI: World \nDevelopment Indicators."},"Token_count":{"kind":"number","value":27763,"string":"27,763"}}},{"rowIdx":6696,"cells":{"Source":{"kind":"string","value":"NBER"},"Date":{"kind":"number","value":2009,"string":"2,009"},"Text":{"kind":"string","value":"NBER WORKING PAPER SERIES\nDISCLOSURE AND THE COST OF CAPITAL:\nEVIDENCE FROM FIRMS’ RESPONSES TO THE ENRON SHOCK\nChristian Leuz\nCatherine Schrand\nWorking Paper 14897\nhttp://www.nber.org/papers/w14897\nNATIONAL BUREAU OF ECONOMIC RESEARCH\n1050 Massachusetts Avenue\nCambridge, MA 02138\nApril 2009\n¸˛We appreciate helpful comments from Ray Ball, Christine Botosan, John Coates, Mihir Desai, Gene\nFama, Fabrizio Ferri, Luzi Hail, Anil Kashyap, Lubos Pastor, Raghu Rajan, Andrei Shleifer, Heather\nTookes, Annette Vissing-Jorgensen, Luigi Zingales, and workshop participants at Duke University,\nESMT Berlin, Harvard Business School, Harvard Law School, University of Chicago, University of\nIllinois at Chicago, Vienna University of Economics and Business, and Yale University. We thank\nShanshan Cao, Wannia Hu, Avery Michaelson and Eugene Soltes for their excellent research assistance.\nChristian Leuz gratefully acknowledges research funding provided by the Initiative on Global Markets\n(IGM) at the University of Chicago Booth School of Business. The views expressed herein are those\nof the author(s) and do not necessarily reflect the views of the National Bureau of Economic Research.\nNBER working papers are circulated for discussion and comment purposes. They have not been peer-\nreviewed or been subject to the review by the NBER Board of Directors that accompanies official\nNBER publications.\n© 2009 by Christian Leuz and Catherine Schrand. All rights reserved. Short sections of text, not to\nexceed two paragraphs, may be quoted without explicit permission provided that full credit, including\n© notice, is given to the source.\n\nDisclosure and the Cost of Capital: Evidence from Firms’ Responses to the Enron Shock\nChristian Leuz and Catherine Schrand\nNBER Working Paper No. 14897\nApril 2009\nJEL No. G01,G12,G14,G30,M41,M42\nABSTRACT\nThis paper examines the link between disclosure and the cost of capital. We exploit an exogenous\ncost of capital shock created by the Enron scandal in Fall 2001 and analyze firms’ disclosure responses\nto this shock. These tests are opposite to the typical research design that analyzes cost of capital responses\nto disclosure changes. In reversing the tests and using an exogenous shock, we mitigate concerns about\nomitted variables in traditional cross-sectional disclosure studies. We estimate shocks to firms’ betas\naround the Enron events and the ensuing transparency crisis. Our analysis shows that these beta shocks\nare associated with increased disclosure. Firms expand the number of pages of their annual 10-K filings,\nnotably the sections containing the financial statements and footnotes. The increase in disclosure is\nparticularly pronounced for firms that have positive cost of capital shocks and larger financing needs.\nWe also find that firms respond with additional interim disclosures (e.g., 8-K filings) and that these\ndisclosures are complementary to the 10-K disclosures. Finally, we show that firms’ disclosure responses\nreduce firms’ costs of capital and hence the impact of the transparency crisis.\nChristian Leuz\nGraduate School of Business\nUniversity of Chicago\n5807 S. Woodlawn Avenue\nChicago, IL 60637-1610\nand NBER\ncleuz@chicagobooth.edu\nCatherine Schrand\nUniversity of Pennsylvania\nschrand@wharton.upenn.edu\n\n-1- \n1. \nIntroduction \nThe link between corporate disclosure, investor information and the cost of capital is one \nof the most fundamental relations in finance and accounting. Understanding this link is of \nsubstantial interest to firms that provide information to capital markets as well as to financial \nmarket regulators who mandate disclosures. Various theoretical models predict that an increase \nin information quality is negatively related to the cost of capital (e.g., Diamond and Verrecchia, \n1991; Easley and O’Hara, 2004; Lambert et al., 2007). Similarly, the estimation risk literature \nsuggests that higher quality information should manifest in lower systematic risk and expected \nreturns (e.g., Barry and Brown, 1985; Coles and Loewenstein, 1988; Handa and Linn, 1993; \nColes et al., 1995; Kumar et al., 2008). \nDespite its importance, the empirical evidence on the link between disclosure, \ninformation quality, and the cost of capital is far from conclusive. In the estimation risk \nliterature, studies often rely on indirect proxies for information quality such as firm age or listing \nperiod and hence there are alternative interpretations of the results (e.g., Barry and Brown, 1984; \nClarkson and Thompson, 1990; Kumar et al., 2008). Studies that use more direct proxies for \ninformation quality suggest that firms with more extensive disclosures exhibit a lower cost of \ncapital, but the evidence is mostly cross-sectional in nature (see, e.g., survey by Leuz and \nWysocki, 2008). Thus, there are substantial concerns about whether these relations can be \ninterpreted in a causal way since there are reasons to believe that the same factors that drive \nfirms’ disclosure choices could also directly affect the cost of capital.1 \n \n1 For example, firms that need to raise external financing for new growth opportunities are likely to increase \ndisclosure. However, pursuing these opportunities may also change firms’ cost of capital. Separating these two \neffects in the cross-section is difficult. \n\n-2- \nTo mitigate these concerns, this paper relates an exogenous cost of capital shock – the \nEnron collapse – to firms’ disclosure responses, which is essentially the reverse experiment to \nthe prior literature. The events surrounding the collapse of Enron led to substantial concerns \nabout the transparency of other U.S. firms, in particular, the quality of their financial statements \nand disclosures. As such, we use the revelations about Enron as an information-related shock \nthat changed investors’ beliefs about the quality (or precision) of their information and hence \nfirms’ costs of capital. The cost of capital shock in turn should lead firms to re-evaluate their \ndisclosure policies and trigger changes in their disclosure behavior. By exploiting this natural \nexperiment, we provide new and complementary evidence that disclosure and the cost of capital \nare related as predicted by theory and that firms can influence their cost of capital through firm-\nspecific disclosures. \nOur analysis focuses on the period when the news about Enron’s losses and accounting \nirregularities hit the market in Fall 2001. The news quickly grew into widespread concerns about \nthe transparency of U.S. corporate reporting and, in particular, about the disclosures of special \npurpose vehicles, off-balance sheet financing, and related-party transactions (see Appendix for \ndetails). The concerns culminated in the passage of the Sarbanes-Oxley Act in August 2002. \nOur sample comprises 1,868 U.S. firms with December fiscal-year ends and required \nfinancial data from 1999 to 2001. We measure shocks to their costs of capital around the Enron \nevents using an econometric technique suggested by Lockwood and Kadiyala (1988) that allows \nfor a quadratic model to estimate beta during the event window as the transparency crisis was \nunfolding. We focus on the shock to systematic risk in an attempt to separate cost of capital \nchanges from updated assessments of firms’ future cash flows. The quadratic form is also \nimportant because it can account for the impact of a firm’s immediate responses during the event \n\n-3- \nperiod on its cost of capital.2 The first part of the analysis examines firms’ disclosure responses \nto the information-related cost of capital shock. We analyze responses in firms’ annual SEC \nfilings (Form 10-K), which occur after the event period, as well as potentially more immediate \nresponses in 8-K filings, conference calls and earnings announcements. \nWe document that the cost of capital shocks over the Enron event period are associated \nwith an increase in the firms’ disclosures in their subsequent annual 10-K filings. Firms extend \nthe number of pages in their 10-K filings, notably the sections containing narrative discussions \nlike Item 7, “Management’s Discussion and Analysis” (MD&A), and Item 13, “Certain \nRelationships and Related Transactions,” as well as the financial statements and footnotes. This \nevidence shows that firms respond to transparency concerns and cost of capital shocks with \nadditional disclosures. The documented link between cost of capital shocks and additional 10-K \ndisclosures is robust to a broad set of alternative specifications and sensitivity checks to address, \nin particular, concerns about the validity of the beta estimates, an undue influence of the events \nof September 11, 2001 and updated cash-flow estimates. \nFor firms that experience positive beta shocks, the increase in disclosure is particularly \npronounced for those that are likely to be more sensitive to their cost of capital because they \nhave larger external financing needs and more growth opportunities. This finding is consistent \nwith the idea that the firms most affected by the shock are most likely to respond. We also find \nthat Arthur Anderson clients respond more strongly to the beta shocks, increasing the 10-K pages \nand the section on related-party transactions compared to firms with other auditors. This finding \nis consistent with our interpretation that the disclosures are a response to the transparency \nconcerns created by the Enron scandal. \n \n2 However, we also show that our results are not driven by the choice of a quadratic model and that we obtain \nsimilar results from a standard market model. \n\n-4- \nWe also analyze disclosures made in the period immediately after the shock (hereafter \n“interim” disclosures). Although we do not find a significant relation between the beta shocks \nand the changes in the number of words in the firms’ annual earnings announcements, firms do \nrespond by increasing the number of 8-K filings. The additional 8-K disclosures mitigate the \neffects of the shock but they do not eliminate the relation between the beta shocks and disclosure \nin the 10-K. This finding is noteworthy because it suggests that firms view the annual 10-K \nfiling as a relevant disclosure mechanism to improve transparency and alleviate investor \nconcerns, despite its lack of timeliness. \nIt is also conceivable that firms respond to the Enron shock with additional disclosures to \navoid future litigation or to thwart a regulatory response, not to mitigate transparency concerns \nor to address changes in their cost of capital. This potential alternative explanation, however, \ndoes not predict the cross-sectional result, described previously, that firms with the largest \ndisclosure responses are those that have larger external financing needs and more growth \nopportunities. Moreover, the notion that firms provide more but not necessarily more \ninformative disclosures is difficult to reconcile with our final set of analyses showing that 10-K \ndisclosure increases are associated with abnormal volume reactions around the 10-K filing and \nwith a subsequent decline in beta. These findings suggest that the additional disclosures are \ninformative to investors and indeed lower the cost of capital. The latter result is important \nbecause it implies that even if the Enron scandal did not lead to cost of capital shocks per se but \nnonetheless triggered changes in other firms’ disclosures, these disclosures are related to \nsubsequent declines in the cost of capital. \nThis study contributes to the prior literature in several ways. First, by linking \ninformation-related beta shocks to firms’ disclosure responses, we provide evidence that is less \n\n-5- \nlikely to reflect an omitted variable that jointly affects disclosures and the cost of capital. Prior \nstudies generally provide evidence from cross-sectional analyses of disclosure or information \nlevels (e.g., Botosan, 1997; Botosan and Plumlee, 2002; Easley et al., 2002; Francis et al., 2004, \n2005). Similarly, the estimation risk literature generally analyzes cross-sectional differences in \nfirm characteristics (e.g., Barry and Brown, 1984; Clarkson and Thompson, 1990). There are a \nfew studies that examine changes in disclosure policies (e.g., Healy et al., 1999; Leuz and \nVerrecchia, 2000), use corporate events such as dividend changes or share repurchases (e.g., \nGrullon et al., 2002; Kumar et al., 2008) or exploit disclosure choices when firms access capital \nmarkets (e.g., Lang and Lundholm, 2000; Schrand and Verrecchia, 2005). But none of these \nstudies identifies an exogenous event that changes firms’ disclosure decisions (including those \nwith a design in changes). Second, our study provides novel evidence that firms respond to \ntransparency crises and credibility shocks in markets with increases in their disclosures. The \ndocumented disclosure increases occur shortly after the Enron scandal, but preceding the \npolitical response to the crisis, the Sarbanes-Oxley Act of 2002. Finally, we contribute to the \nestimation risk literature by showing that beta estimates respond to an information-related shock \nin predictable ways and that firms’ subsequent disclosure responses reduce the impact of the beta \nshocks, as predicted by theory. As Kumar et al. (2008) note, such evidence is also useful in the \nongoing debate about the conditional CAPM (e.g., Wang, 2003; Lettau and Ludvigson, 2001). \nThe paper is organized as follows. Section 2 develops the hypotheses and explains our \nresearch design exploiting the Enron events in 2001. In section 3, we define our disclosure and \ncost of capital proxies. Sections 4 and 5 present the results on the relation between the cost of \ncapital shocks and firms’ disclosure responses. Section 6 presents the analysis of subsequent \nmarket responses to firms’ disclosure responses. Section 7 concludes the paper. \n\n-6- \n2. \nHypothesis development and research design \nThere is a long-standing literature in financial economics providing theories on the \nrelation between investor information and the cost of capital (e.g., Diamond and Verrecchia, \n1991; Easley and O’Hara, 2004; Lambert et al., 2007). An implication of these theories is that a \nchange in investor information (e.g., its precision) leads to a change in the cost of capital. \nOptimal disclosure behavior in turn implies that firms respond to information-related shocks to \ntheir cost of capital by changing their disclosure policy. That is, assuming that a firm’s \ndisclosure policy is optimal at a given point in time, an information-related shock to the cost of \ncapital should lead to a reassessment of its disclosure policy. Our main prediction is that, ceteris \nparibus, firms with increases in their cost of capital are likely to increase their disclosures. \nFurthermore, firms’ disclosure increases should lead to subsequent reductions in the cost of \ncapital. Relating firms’ disclosure responses to an information-related shock as well as \nconsequent cost of capital changes to the new disclosures should provide evidence that investor \ninformation and the cost of capital are indeed related as predicted by theory. \nTo test these predictions, we exploit the events at Enron as an exogenous shock to other \nfirms’ costs of capital. On October 16, 2001, Enron announced a third quarter loss of $618 \nmillion and hinted at problems with its partnerships. On November 29, 2001, Enron announced \nit would restate its earnings back to 1997 and add $628 million of debt to its 2000 balance \nsheet. Between these events, the SEC launched an inquiry into Enron’s accounting practices \nand then initiated a formal investigation that included Enron’s auditors, Arthur Anderson. \nSubsequently, Enron was removed from the S&P 500, its debt was downgraded to junk and, \neventually, it filed for bankruptcy. Appendix A.1 provides more details on the sequence of \nevents. \n\n-7- \nWe believe that the nature of the Enron scandal is particularly suited to our analysis. \nThe announcements revealed that Enron had been able to hide its economic and financial \ncondition, despite compliance with SEC financial reporting requirements, which led to general \nconcerns about U.S. financial reporting quality. As such, the Enron events likely caused \ninvestors to revise their beliefs about the precision (or quality) of their information. \nFurthermore, even if the Enron scandal was a symptom of a larger transparency crisis, the \ntiming of this shock should be exogenous to other firms, which is all we need for our empirical \nstrategy. \nThe following timeline provides a stylized overview of our research design and how it \nrelates the events at Enron to our main hypothesis: \nPre-event-period \nEvent period \nPre-report period \nPost-report period \n \n↑↑↑↑ \n↑ \n \n \nEnron events unfold \nDisclosure Response \nIn the pre-event period, firms are in equilibrium with respect to their disclosure decisions. \nInvestors have rational expectations over firms’ future cash flows and the precision of their cash \nflow estimates. In the event period, the Enron scandal unfolds and leads to investor concerns \nabout a systematic lack of transparency in U.S. financial reporting.3 As a result, investors revise \ntheir beliefs about the precision of their estimates for other firms in the economy, which changes \nfirms’ cost of equity capital. This information-related shock to the cost of capital alters firms’ \noptimal disclosure levels and hence they adjust their disclosure policies. In response to firms’ \ndisclosures, investors revise and update their beliefs and the cost of capital adjusts again. \nBy relating investors’ uncertainty about their information set to expected returns and \nhence firms’ cost of capital, the estimation-risk literature provides the conceptual underpinnings \n \n3 See Appendix A.1 for citations to articles in the financial press that suggest that the market viewed the Enron \nscandal as indicative of widespread transparency problems in the U.S. \n\n-8- \nfor our analysis and, in particular, investors’ reactions to the Enron shock. This literature starts \nfrom the premise that the parameters of the return-generating process are generally not known \nand have to be estimated (e.g., Klein and Bawa, 1976, 1977). While early studies conclude that \nthe effects are diversifiable (e.g., Brown, 1979; Reinganum and Smith, 1983), Barry and Brown \n(1985) show that estimation risk is not diversifiable when information is differentially distributed \nacross securities. Extending this finding, Coles and Loewenstein (1988) and Coles et al. (1995) \nshow that when modeling estimation risk with respect to asset payoffs (rather than returns) \nestimation risk is not diversifiable and affects equilibrium returns and betas even when \ninformation is symmetrically distributed. Along similar lines, Lambert et al. (2007) analyze \ninformation structures that lend themselves more naturally to an interpretation as firm-specific \ndisclosures about future cash flows and show that more precise disclosures move firms’ cost of \ncapital closer to the risk-free rate. This effect is again not diversifiable and manifests itself in \nfirms’ beta factors as well as the market risk premium for the economy.4 Finally, Kumar et al. \n(2008) analyze the effects of estimation risk on the cross section of stock returns and firms’ cost \nof capital when investors learn from information of uncertain quality (or precision). They \ndemonstrate that, in this situation, asset prices, firms’ beta factors, and the market risk premium \nare estimation-risk dependent and respond to changes in the quality of information.5 \nBased on these theories, we expect that the Enron events led to an information-related \nshock to firms’ cost of capital. We focus on changes in firms’ beta factors for two reasons. \nFirst, the Enron events may have had an impact on investors’ expectations of future cash flows \n \n4 As Lambert et al. (2008) show, the link between information and the cost of capital in Easley and O’Hara (2004) is \nsolely driven by precision effects and very similar to the link described above, except that the analysis in Easley and \nO’Hara (2004) is limited to effects on firm-specific variances, which makes them diversifiable. \n5 Kumar et al. (2008) also provide empirical evidence that beta factors fall immediately after dividend initiations and \nshare repurchases, i.e., corporate events that can be interpreted as reducing information uncertainty. Similarly, Barry \nand Brown (1984) and Clarkson and Thompson (1990) find that systematic risk of low information firms decreases \nas time passes (e.g., the length of their listing period) and information increases. \n\n-9- \nfor some firms and not just on the precision of their cash flow estimates. By focusing on \nchanges in firms’ beta factors, rather than abnormal returns, we attempt to separate cost of \ncapital and cash flow effects. In addition, we include industry controls in all analyses and we \ncheck that our results are not driven by or sensitive to the inclusion of energy firms, for which \nthe Enron events likely had significant cash flow effects. We also conduct a robustness analysis \nthat uses changes in (analysts’) earnings expectations as a control for updated assessments of \nfuture cash flows. Second, it is likely that the Enron events increased the market risk premium. \nHowever, shocks to the risk premium apply to the entire economy and are constant across firms. \nThe beta factor in turn exacerbates or dampens any premium shocks for a given firm and hence \nappropriately captures cross-sectional differences in the cost of capital shock.6 \nIn addition, the Enron shock could have affected investors differentially and hence could \nhave altered information asymmetries among investors. Increased information asymmetry can \nraise the market risk premium demanded by investors as well as the firm-specific cost of capital \n(e.g., Amihud and Mendelson, 1986; Diamond and Verrecchia, 1991; Garleanu and Pedersen, \n2004; Lambert and Verrecchia, 2008). However, information asymmetry effects do not alter our \nprimary hypothesis that the Enron shock led to changes in firms’ cost of capital and that, ceteris \nparibus, firms with increases in their cost of capital respond with an increase in disclosure. \nWe also make several cross-sectional predictions. We predict that firms that are more \nsensitive to their cost of capital because, for example, they have greater external financing needs, \nincrease their disclosures more than less sensitive firms. We predict firms with larger credibility \nissues after the Enron events to respond more strongly, holding all else equal. Finally, it is \npossible that firms with a positive cost of capital shock respond by increasing their disclosure, \n \n6 We note that without estimating changes in the market risk premium, we cannot quantify the full magnitude of the \ncost of capital shock, but doing so is not necessary for a cross-sectional design. \n\n-10- \nwhile firms with a negative shock maintain their policies. Such a response is likely if the costs \nof retaining disclosures are small. \nTo test these hypotheses in the context of the Enron events, we estimate variations of the \nfollowing general specification of a cross-sectional model: \nChange in disclosure = α + beta shock + control variables + ε \n(1) \nand also separate models for firms that experience a positive shock and have a higher sensitivity \nto their cost of capital versus all other firms. We first analyze average changes in disclosure in \nfirms’ annual 10-K filings as the shock occurred during the fourth quarter of 2001. Thus, firms \ncould respond in their annual filings with the SEC, which are due 90 days after year end. In the \ntimeline above, we measure the beta shocks during the event period and then separately in the \npre-report period to account for the possibility that firms respond to the shock with interim \ndisclosures (or costly signals) during or after the event period but before filing the 10-K. \nAlthough the news about Enron’s problems was a surprise to the market and occurred during a \nrelatively short window, the event period is long enough that a firm may respond to the shock \nduring the event period or shortly thereafter. Responses could be interim disclosures, such as \nconference calls or press releases, or costly financial signals, such as changes in the dividend \npolicy or stock repurchases. \nWe explicitly account for such interim responses in our second set of analyses. We \nestimate equation (1) but include measures of changes in 8-K filings, conference call activity, \nand disclosures in the fourth-quarter earnings announcement as controls. We also estimate \nequation (1) using our various interim disclosure proxies as the dependent variable. These \nanalyses serve three purposes. First, evidence that firms respond immediately to the cost of \ncapital shock in predictable ways serves to increase the confidence in our shock proxies. \n\n-11- \nSecond, including these disclosures in the model of 10-K disclosures mitigates concerns about \nomitted variables. If an omitted variable drives firms to increase disclosure in the 10-K, it \nlikely also causes firms to increase other disclosure. In this case, including proxies for changes \nin the other disclosures in the model should attenuate the coefficient on the beta shocks, which \nwe examine. Third, the analysis provides evidence on the role of the 10-K filing as a disclosure \ninstrument. The 10-K must be filed within 90 days after the fiscal-year end and hence it may \nnot be a timely vehicle to reduce transparency concerns. At the same time, the 10-K may be \nparticularly appropriate to address such concerns because it contains audited financial \nstatements and it is subject to SEC review. Thus, if more timely but potentially less credible \ninterim disclosures are substitutes for disclosure responses in the 10-K, there may be no \nassociation between the beta shocks over the event period and firms’ 10-K responses when we \ncontrol for interim disclosures. If, however, interim and 10-K disclosures are complements, the \nrelation between the beta shock remaining after the interim response and the 10-K disclosures \nis attenuated unless we control for interim responses. \nIn our third and final set of analyses, we invert the model in equation (1) and examine \nsubsequent market reactions to firms’ disclosures responses after the Enron shock. If the \ndisclosures are useful to investors and increase the precision of their information, the market \nshould respond to the release of this information and the cost of capital should decline. In \nperforming these analyses, we also investigate the alternative hypothesis that firms increase their \ndisclosure following the crisis merely to address concerns about litigation or to thwart future \nregulation. If firms provide additional disclosures that are not informative to investors (but \nreduce litigation risk), we do not expect the disclosure changes to be related to subsequent \n\n-12- \ndeclines in the cost of capital.7 In this analysis, we measure the cost of capital response to a \ndisclosure change, which is closer in design to traditional disclosure studies, except that we use \ncross-sectional variation in disclosure that is generated by an exogenous shock. \nFinally, we note that there are several reasons the predicted relations may not prevail. \nFirst, the Enron crisis may not have resulted in an information-related shock for other firms, \ndespite widespread claims in the press at the time. Second, the link between financial disclosure \nand the cost of capital may not exist in the way predicted by theory. Third, it is conceivable that \nmanagers do not expect that they are able to address the transparency concerns via increased \ndisclosure, and thus they do not respond in this way. \n3. \nEmpirical proxies and descriptive statistics \n3.1 Sample \nThe sample includes all firms on Compustat that have fiscal year ends on December 31st \nand non-missing values for total assets and earnings announcement dates in each of the three \nyears from 1999 – 2001. We exclude firms with non-December fiscal-year ends to align firms \nwith respect to the timing of their disclosures. We exclude REITS, Limited Partnerships, Trusts \nand Funds from the sample. In addition, we eliminate 233 firms that likely were significantly \ninfluenced by the events of September 11, 2001. First, we exclude all airlines and insurance \ncarriers from our sample. Carter and Simkins (2004) and Doherty et al. (2003) show that the \nevents of September 11 had a significant effect on airlines (SIC 45) and insurance carriers, \nrespectively. Second, we eliminate firms that file an 8-K between September 11 and the peak of \nour event period (November 8) that mentions the terrorist attacks, suggesting that the events of \n \n7 Note that if firms provide additional disclosures in response to the Enron debacle but not necessarily to the cost of \ncapital shocks, our analysis of subsequent changes in the cost of capital is still useful and valid as long as the Enron \nshock produces exogenous variation in firms’ disclosure choices. \n\n-13- \nSeptember 11 had a material impact. While our event period does not start until mid October, \nexcluding these firms mitigates the concern that the returns and beta estimates for these firms \nreflect ongoing valuation adjustments associated with the terrorist attacks. \nWe require data for 1999 and 2000 to create control variables and also benchmarks to \nmeasure changes in disclosure. The final sample contains 1,868 firms for which we have return \ndata to compute the cost of capital shocks, Compustat data to compute the primary control \nvariables, and 10-K filings data to create the disclosure proxies. \n3.2 Event window \nWe consider seven different window specifications to define the pre-event period, the \nevent period, and the pre-report period (see Appendix A.2). In all cases, the pre-event window \nstarts on May 1, 2001, which for firms with December fiscal-year ends is after the year 2000 \nannual report season, and runs through August 31, 2001. This period precedes the market \nclosings caused by the terrorist attacks on September 11. These events are excluded from the \nevent period, which starts at the earliest on October 15, 2001. The pre-report period ends either \non January 15, 2002 or January 28, 2002, prior to the annual earnings announcement season. \nChoosing the later date increases the length of the pre-report period and hence the power to \nmeasure systematic risk, but it is more likely that the beta estimates include the effects of early \nearnings announcements. Similarly, the choice among the different windows involves trading \noff a longer event period, which allows us to more precisely measure the Enron shock, against a \nshorter pre-report period, which makes it more difficult to accurately measure the beta prior to \nthe 10-K filing. \nThroughout the paper, we present results for Window 1, which is centered on November \n8 and has an event-period length of 39 days. The pre-report period ends on January 15, 2002. \n\n-14- \nThe Appendix provides evidence to justify this choice. The results are robust to alternative \nwindows and generally stronger if we focus on those windows that maximize the length of the \nperiod over which the respective betas are estimated. The post-report period is consistently \ndefined across all analyses as May 1, 2002 to August 31, 2002. \n3.3 \nDisclosure proxies \nOur main proxy for a change in disclosure is the percent change in the page count of the \n10-K for 2001 from 2000 (% PAGES\nΔ\n). Page count data are from the Global Securities \nInformation (GSI) database, which covers all SEC filings that are available in the EDGAR \ndatabase. We eliminate all filings with less than 10 pages. Most of these observations are \nreferences to a fuller document and not the page count of the 10-K.8 The page counts are for the \nbody of the 10-K (i.e., Items 1 through 15), which includes the exhibits and financial statement \nschedules in Item 15. The page counts exclude separate exhibits and appendices to the 10-K \nfiling beyond Item 15. While those exhibits and appendices could include useful information \n(e.g., material contracts), they are not sufficiently standardized in format and content to provide a \nmeaningful proxy for variation in information content or disclosure. \nWe also measure page count changes for three key sections within the 10-K. The \nvariable \nMDA\n%Δ\n is the percent change in pages for Item 7, Management’s Discussion and \nAnalysis of Results of Operations and Financial Condition (MD&A). The variable \nFS\n%Δ\n is the \npercent change in the pages of 10-K sections that commonly include the financial statements and \nnotes. The sections used to create this proxy are Item 6 (Selected Financial Data), Item 8 \n(Financial Statements and Supplementary Data), and Item 15 (Exhibits, Financial Statement \n \n8 Fifty-six sample firms switch from (to) filing a 10-K to (from) a 10-KSB. Using a different form likely affects the \npage count. However, the results are virtually the same if we drop these firms. \n\n-15- \nSchedules, and Reports on Form 8-K). The variable \nDISCUSS\n%Δ\n is the percent change in the \npages of 10-K sections that commonly include important narrative discussions and information. \nThe sections used to create this proxy are Item 1 (Business), Item 2 (Properties), and Item 7 \n(MD&A). These three proxies for key subsections address concerns that the page changes for \nthe entire 10-K occur in less important parts. By combining multiple items, \nFS\n%Δ\n and \nDISCUSS\n%Δ\n further mitigate problems associated with small numbers of pages in the \nindividual sections and with rearranging the location of information within the 10-K, which are \nmore likely to arise for \nMDA\n%Δ\n.9 \nFor all of the percent change variables, the benchmark is the page count in the prior year. \nWe create an alternative proxy that uses the average of the 1999 and 2000 page counts. Results \nare very similar for this alternative variable and are not presented. We truncate % PAGES\nΔ\n \nabove the 98th percentile and below the 2nd percentile. For the other percentage change variables, \nwe have manually screened for a number of issues that could produce outliers (such as \nrearranging of information within the 10-K) and hence we truncate only above the 99th percentile \nand below the 1st percentile to conserve sample size.10 Winsorizing rather than truncating the \nvariables yields similar results. \nTable 1, Panel A reports descriptive statistics for the page count variables. The average \n10-K in 2001 has 64 pages (\n2001\nPAGES\n), which is almost a 17% increase over 2000. This \nincrease is statistically different from zero and far exceeds the average page change from 1999 to \n \n9 For all three variables, we attempt to exclude observations that appear to have rearranged the location of \ninformation within the 10-K. Observations that report an item in the current year that was incorporated by reference \nin the previous year (or vice versa) are eliminated under this criterion. We identify observations that potentially \nrearranged the 10-K using various data filters. For these observations, we examine the 10-K to verify that \ndisclosures within the 10-K were rearranged across various sections. \n10 Truncating the percentage change in total pages at the 99th and 1st percentile does not materially alter our results. \nHowever, the 43 additional observations that are gained due to the milder truncation at the 1% level are flagged by \ncommon outlier procedures. The inspection of these observations for the 10-K part analysis also confirms this \nassessment, which is why we choose to truncate at the 98th and 2nd percentile. \n\n-16- \n2000, which equals only 3.4% (untabulated). The difference between the increase from 1999 to \n2000 and the increase from 2000 to 2001 is statistically significant, suggesting that firms \nsignificantly expanded their 10-K filings in 2001 beyond the normal rate. \nWe investigate whether this increase in pages could be due to new disclosure \nrequirements or accounting standards. For the fiscal 2001 reporting year, a search of SEC \nactions affecting 10-K reporting yields two noteworthy changes. First, the SEC required firms to \ninclude an additional table in the 10-K related to components of its equity compensation plan. In \nterms of page increases, this requirement should have increased 10-K filings of all firms by one \npage. Second, the SEC issued new guidance for the MD&A suggesting that firms pay particular \nattention to discussions of off-balance-sheet arrangements, certain investment trading activities \nand related-party transactions. This guidance appears to be a direct response to the issues that \narose for Enron. In addition, the FASB issued three new accounting standards during the \nperiod.11 All of the new standards likely affected only the disclosures of firms that had an \nimpairment loss on their assets. Such losses are generally reflected as special items in the \nincome statement. We therefore check that our results are robust to a specification that controls \nfor special items in Section 4.2. In sum, it seems unlikely that the substantial increase in 10-K \ndisclosures from 2000 to 2001 is primarily the result of new accounting or disclosure \nrequirements. But even if new requirements were the primary impetus, our analysis that relates \ncross-sectional variation in the new disclosures to firms’ cost of capital shocks is still valid as \nlong as firms’ responses to the new standards are not also closely aligned with the beta shocks. \n \n11 Statement of Financial Accounting Standards (SFAS) No. 142, “Goodwill and other intangible assets;” SFAS No. \n143, “Accounting for asset retirement obligations;” and SFAS No. 144, “Accounting for the impairment or disposal \nof long-lived assets.” \n\n-17- \nTurning to the sub-sections of the 10-K filing, the MD&A section in 2001 has on average \n13.3 pages, which is a 39% increase over 2000. The section with narrative discussions (financial \ntables and footnotes) has on average 27 (34) pages, which represents an increase of 22% (18%) \nover year 2000. The MD&A, narrative, and financial sections are longer than in 2000 for over \n75% of the sample observations. \nBecause the Enron scandal involved special purpose vehicles and partnerships that were \nrun by corporate insiders, we also create a proxy for disclosure changes about related-party \ntransactions. In doing so, we face two challenges.12 First, while the 10-K requires disclosures on \nrelated-party transactions (Item 13), firms generally satisfy this requirement with a reference to \ntheir proxy statement. We therefore examine both the 10-K and the proxy statement and select \nthe page count from the appropriate document for each year before computing a change. \nSecond, disclosures on related-party transactions are often only one or two pages long, which can \ncreate a small-denominator problem when computing percentage changes. Moreover, it creates \nthe issue that changes of plus or minus one page can produce substantial percentage changes, \neven though they are essentially measurement error. For example, a one-page related-party \nsection that starts at the top of a page is counted as one page, but if it starts in the middle of the \npage next year, it may count as two pages. We therefore classify page changes of -1 and +1 as a \nzero change. \nAfter these adjustments, we compute the percentage change in the number of pages \nrelative to the fiscal 2000 related-party disclosure section (% REL_PAR\nΔ\n). The page counts are \nderived from the 2001 10-K or the proxy statement with a filing date between December 31, \n2001 and June 30, 2002. We delete extreme percentage changes that are in the 1st or the 99th \n \n12 In addition, there is the issue that a very meaningful (but short) disclosure may be to simply state that the firm has \nor had no related-party transactions. \n\n-18- \npercentile. Table 1, Panel A shows that the average related-party disclosure has 5 pages, but the \nmedian is equal to one page only. From 2000 to 2001, the related-party disclosures increase by \n19.4% which is statistically different from zero. \nOur next disclosure proxy is the change in the number of words of the fiscal 2001 \nearnings announcement relative to the fiscal 2000 earnings announcement (%ΔWORDS). The \nnumber of words in each announcement is determined based on a manual search of earnings \nannouncements for our sample firms on Dow Jones Interactive. We use only announcements on \nBusiness Wire or PR Newswire, as these two sources furnish unmodified press releases. We \ndelete extreme changes in the word count that are in the 1st and 99th percentile. Panel A of Table \n1 reports that the average earnings announcement has 2,003 words (WORDS2001), which is a \n15.5% increase over the number of words in the prior year fourth-quarter earnings \nannouncement. \nThe next disclosure proxy is constructed from 8-K filings. Firms are required to use the \nform 8-K to notify investors of any unscheduled material event that is important to shareholders \nor the SEC. The SEC defines required reportable events (e.g., shutting down a plant), and it also \nrequires more generally that firms use the 8-K form to disclose material information and to \nupdate any information provided in previous SEC filings. Short-window abnormal returns \naround 8-K filings suggest that they are informative (e.g., Beneish et al., 2005). In our context, \n8-K filings could be used to respond to the Enron shock and alleviate transparency concerns.13 \n \n13 For example, on December 5, 2001, American Express filed an 8-K, which simply reported: “In view of the \nsituation involving Enron, the company is analyzing its exposure, but preliminarily believes any impact will not be \nmaterial.” Another 8-K followed on February 6, 2002 and stated: “First, in terms of the financial impact of Enron's \nbankruptcy, back in early December we filed an 8-K indicating that we preliminarily believed the impact from \nEnron would not be material. Having now completed our review, this statement is still true. Second, we have also \nreviewed our business and accounting practices in light of Enron's recent issues. Based on this review, we can see \nno parallels between our businesses and what we understand the practices were at Enron. Here are a few specifics.” \n \n\n-19- \nWe count each firm’s 8-K filings during the period from October 15, 2001, which is the \nbeginning of our event period using Window 1, to March 31, 2002 and the same period one year \nearlier (8KCOUNT). In these counts, we exclude filings related to the terrorist attacks on \nSeptember 11, deleting those that refer to this date or include related words like “terrorist.” We \nthen compute the difference in the 8K counts from 2000 to 2001 (Δ8KCOUNT). We set all \ndifferences that are larger than three in absolute value equal to 3 and -3, respectively. Table 1, \nPanel A shows that sample firms file on average about one 8-K during the period, although the \nmedian number of filings is zero. Excluding 8-K filings related to the events of September 11, \nthe average change in the number of filings is close to zero. \nOur last disclosure proxy is based on changes in the number of conference calls. As for \nthe 8-K filings, we count the number of conference calls from October 15, 2001 to March 31, \n2002 (CONFCALL) and compute the difference relative to the same window the year earlier \n(ΔCALL). Increases (decreases) that are larger than three in absolute value are set equal to 3 (–\n3). Table 1, Panel A shows that, on average, firms hold one conference call over the \nmeasurement period and the number of calls decreases slightly relative to the prior year. \nTable 1, Panel B presents descriptive statistics for the control variables in the analyses. \nWe use proxies for firm characteristics that are commonly viewed as determinants of firms’ \ndisclosures, i.e., firm size, performance, financing needs and growth opportunities (e.g., Lang \nand Lundholm, 1993; Leuz and Wysocki, 2008). We truncate all control variables at the 1st \npercentile and the 99th percentile and measure them as of the beginning of the fiscal year, unless \nnoted otherwise. In untabulated regressions, we confirm that the levels of our disclosure proxies \nexhibit meaningful associations with the control variables. \n \nThe 8-K continues with a discussion of special purpose entities, off-balance-sheet financing, and employee conflict \nof interest, all three of which were problems at Enron. \n\n-20- \n3.4 \nCost of capital shocks \nTo construct measures of systematic risk in the pre-event period, the event period, and the \npre-report period, we employ a regime-switching model suggested by Lockwood and Kadiyala \n(1988). The method, which nests traditional methods, permits event-induced changes in \nsystematic risk and provides period-specific beta estimates, which we then use to construct cost \nof capital shocks. Separately, we estimate the post-report period beta after the annual report \nseason from May 1, 2002 to August 31, 2002 using a standard market model. \nFollowing Lockwood and Kadiyala (1988) and Cyree and DeGennaro (2002), the model \nspecification is: \n \nit\nmt\nit\ni\nit\nε\nR\nb\na\nR\n+\n+\n=\n \n(B1) \n \n]\n1\n)1\n(\n2\n)1\n2\n[(\n1\n)\n2\n)(\n1\n(\n3\n2\n1\nit\nit\ni\nit\ni\ni\nit\nD\nT\nt\nD\nT\nT\nb\nD\nT\nt\nt\nT\nb\nb\nb\n−\n+\n−\n+\n−\n−\n+\n=\n \n(B2) \nwhere \nit\nR is the daily holding period return for firm i from CRSP on day t (including dividends) \nand \nmt\nR is the value-weighted return on the market from CRSP for day t. Systematic risk varies \nas a function of the trading day t (equation B2). T1 and T2 define the event period. They are \nspecified as the number of trading days in the event period relative to day t. For window 1, our \nevent period extends from October 15 through December 5 (inclusive), which contains 39 CRSP \ntrading days, and it is centered on November 8 (t = 0), with T1 set at -19 and T2 set at +19. D1 \nand D2 are indicator variables that equal one if a trading day falls into the event period or the \npre-report period, respectively.14 \n \n14 The event window includes the third quarter earnings announcement period. We repeat the estimation excluding \nreturn observations on the earnings announcement date and the previous day. The estimates of \n1\nˆ\nib and \n3\nˆ\nib are \nvirtually identical. The exclusion of returns on these days has a bigger impact on \n2\nˆ\nib\nbut even then it is \neconomically small (at the 5th digit) and the estimates are not significantly different. The results presented in the \npaper are based on the estimates that include the returns on the third quarter earnings announcement date. \n\n-21- \nFrom the model parameters, we create separate cost of capital measures for the pre-event, \nevent, and pre-report periods. First, systematic risk for the pre-event period (BETA_PRE) is \nequal to \n1ˆ\nib . This parameter is a constant throughout the pre-event period. This estimate serves \nas a benchmark for the firm’s beta prior to the transparency shock. During the event period, \nwhen D1 = 1 and D2 = 0, systematic risk is determined by the parameter estimates for \n1\n2\n3\nˆ\nˆ\nˆ\n, \n, and \ni\ni\ni\nb\nb\nb . The quadratic specification allows for the initial shock to the cost of equity \ncapital to be either positive or negative. The parameter \n2ˆ\nib captures the direction and the \ncurvature of the initial shock. The quadratic specification also allows for (but does not require) a \nrecovery of the shock during the event period. Thus, the quadratic form of the regime-switching \nmodel can accommodate the impact of interim disclosure responses on firms’ costs of capital \nduring the event period. We confirm in Section 5 that the curvature and firms’ interim \ndisclosures are, in fact, related as hypothesized. \nWe compute the event beta (BETA_EVT) at day t = 0, which represents the peak given \nthe quadratic specification. This beta is computed from (B2) by setting t = 0 and recognizing \nthat our event window is symmetric and hence T2 = -1*T1: \n \n2\n2\n1\n2\n3\nˆ\nˆ\nˆ\nˆ\n( 2)\n2\ni\ni\ni\ni\nb\nb\nT\nb T\nβ\n=\n+\n+\n \n(B3) \nAfter the shock and during the pre-report period, systematic risk is given by: \n \n)1\n2\n(\nˆ\nˆ\nˆ\n3\n1\n3\nT\nT\nb\nb\ni\ni\ni\n−\n+\n=\nβ\n \n(B4) \nLike the pre-event beta, the pre-report or remaining beta (BETA_REM) is constrained to \nbe a constant across the period. \n\n-22- \nTo provide intuition for the parameters, Figure 1 illustrates time-series patterns of \nsystematic risk for the pre-event period, the event period, and the pre-report period under \nalternative estimates for \n2\n3\nˆ\nˆ\n and \ni\ni\nb\nb . Panels A through C illustrate the behavior of systematic risk \nin the cases of HIGH positive \n2ˆ\nib (equals 0.100), LOW positive \n2ˆ\nib (equals 0.002), and negative \n2ˆ\nib (equals -0.002), respectively. Within each panel, we present estimates of systematic risk in \nthe pre-report period for five levels of \n3ˆ\nib . In all three panels, the pre-event beta estimates are \nconstant over the period. The pre-report betas also are constant throughout the period, but they \ncan be higher or lower than the pre-event beta. When \n3ˆ\nib is positive, the pre-report beta is higher \nthan the pre-event beta. Furthermore, the pre-report beta can be higher than the event beta, even \nwhen \n2ˆ\nib is positive, if \n2\n3\n2\nˆ\nˆ\n2\n( 2)\ni\ni\nb T\nb\nT\n>\n. \nBefore computing the shocks, we truncate the beta estimates in each period at the 1st and \nthe 99th percentile. In addition, we drop observations for which the beta appears to unstable as \nindicated by a significantly positive estimate in the pre-event period but a significantly negative \nestimate in the post period (or vice versa). This criterion eliminates 20 observations but does not \nmaterially affect our results. We then use beta level estimates over the three periods to create \nproxies for changes in firms’ cost of capital (i.e., shocks) as a result of the Enron events. We \ndefine the initial shock (INIT_SHOCK) as the difference between the event beta in (B3) and the \npre-event beta: INIT_SHOCK = BETA_EVT - BETA_PRE. Recalling that BETA_EVT is the \npeak of the quadratic beta estimate, the initial shock metric captures the extent to which a firm’s \nsystematic risk increases during the Enron crisis. We define the remaining shock \n(REM_SHOCK) as the difference between the pre-report beta and the pre-event beta, i.e., \nREM_SHOCK = BETA_REM - BETA_PRE. The remaining shock captures the extent to which \n\n-23- \na firm’s systematic risk continues to be elevated after the Enron crisis and the firm’s immediate \nresponses, but before the 10-K is filed. \nTable 2, Panel A, provides descriptive statistics for the parameter estimates, systematic \nrisk and the beta shocks based on Window 1. The average pre-event beta for the sample firms is \n0.670 (and close to 1 computing a value-weighted average). The shock variables exhibit \nconsiderable cross-sectional variation and a reasonable parameter range. As the average (value-\nweighted) beta for the economy is by definition equal to 1 at any given point in time, it is \nexpected that the average shock is relatively close to zero. Nevertheless, firms with negative \nbeta shocks may experience an increase in the cost of capital if the equity risk premium \nincreases. Regardless, the beta shocks provide an appropriate cross-sectional ranking. \nPanel B reports the average initial shock (INIT_SHOCK) and the average remaining \nshock (REM_SHOCK) by 2-digit SIC industry code and the proportions of positive observations \nin each industry. The industries with the largest initial shocks are Furniture (SIC 25), Building \nproducts (SIC 15), and Auto dealers/gas stations (SIC 55). Industries that have a high proportion \n(i.e., greater than 80%) of firms with positive remaining shocks are Furniture (SIC 25), \nBuilding/Construction (15), Personal services (72), Apparel (SIC 23), Oil and gas extraction \n(SIC 13), Motion Pictures (78), and Water transportation (SIC 44). In the Appendix, we further \ncharacterize the shock variables and conduct various validity checks. \n4. \nAnalysis of disclosure responses to the cost of capital shocks \n4.1 \nMain results \nThe first analysis relates the percentage changes in the 10-K page count to the cost of \ncapital shocks. We implement the following specification of the general model in equation (1): \n\n-24- \nPGS\nE\nS\nPPE\nMB\nDERATIO\nROA\nSIZE\nPRE\nBETA\nSHOCK\nPAGES\nε\nδ\nδ\nδ\nδ\nδ\nβ\nβ\nα\n+\n+\n+\n+\n+\n+\n+\n+\n=\nΔ\n5\n4\n3\n2\n1\n \n \n \n_\n%\n \n(2) \nWe estimate the model including either the initial shock (INIT_SHOCK) or the \nremaining shock (REM_SHOCK). Based on Section 2, we predict a positive association \nbetween the initial shock or the remaining shock and the percent change in pages.15 \nWe include the pre-event beta (BETA_PRE) to account for the possibility that firms with \nhigher pre-period systematic risk are simply more responsive. In addition, we include control \nvariables for commonly cited determinants of disclosure such as firm size, performance, leverage \nand financing needs (Lang and Lundholm, 2000; Leuz and Wysocki, 2008). The proxies for \nthese constructs are the natural log of total assets at December 31, 2000 (LASSETS2000); return \nof operating income on average assets for the year ended December 31, 2000 (ROA2000); the \ndebt-equity ratio at December 31, 2000, which is the book value of long-term debt scaled by the \nmarket value of equity plus the book value of long-term debt and preferred shares \n(DERATIO2000); the market-book ratio at December 31, 2000 (MB2000) where observations with \na negative book value of equity are set to missing; and the ratio of property, plant and equipment, \nnet, to total assets (PPE/TA2000) as a proxy for capital intensity. An extended model also \nincludes changes in several of these variables from 2000 to 2001. All models include industry-\nfixed effects based on one-digit SIC codes and a separate indicator variable for firms in Enron-\nrelated industries (defined as SIC2 equal to 13, 29 or 49). \nTable 3, Panel A presents results for eight model specifications. As the Enron shock is \nlikely to affect firms in an industry similarly, our inferences are based on standard errors that are \n \n15 The relation between the initial shock and 10-K disclosure changes may be attenuated if firms address the shocks \nin the interim. This issue, which does not arise for the remaining shock, is discussed in more detail in Section 5. \n\n-25- \nclustered at the two-digit SIC level.16 The results for models (1) - (4), which include the initial \nshock variable and four variations of control variables, show a positive and significant relation \nbetween the initial shock (INIT_SHOCK) and the percentage change in the page count of the 10-\nK. The results for models (5) - (8), which include the remaining shock variable and four \nvariations of control variables, show that the relation between the remaining shock \n(REM_SHOCK) and page count changes also is significantly positive. These results are robust \nto alternative sets of control variables. In particular, the results are similar and the inferences the \nsame if we use the market value in computing firm size, employ contemporaneous, rather than \nlagged controls, and include only changes in the control variables in the model. \nThe positive associations for the initial and remaining shock with the disclosure changes \nin the 10-K are consistent with our hypotheses and the notion that firms’ disclosures reduce their \ncost of capital. Although we do not make ex-ante predictions about the control variables, the \nresults are consistent across all eight models and accord with our intuition. Firms with higher \npre-event period betas, deteriorating operating performance, and with greater and growing firm \nsize increase their 10-K disclosures more. \nStandard computations of the marginal effect of the cost of capital shock on 10-K page \nchanges from Models 2 and 4 suggest that an initial shock (or remaining shock) equal to the \ninter-quartile range of the pre-event beta is associated with an increase in 1.2 pages, which is 2% \nof the median number of 10-K pages. While this effect may appear small, we emphasize that \njudging the increase against the total number of pages may not be appropriate because the 10-K \ncontains many pages that are boilerplate and hence uninformative. Thus, a 2% increase in the \npage count could represent a substantial increase in relevant information to investors. In fact, as \n \n16 Alternatively, we cluster our standard errors by auditor and obtain very similar inferences. \n\n-26- \na percentage, the page increase is much larger if we gauge it against key (or relevant) subsections \nof the 10-K (e.g., DISCUSS2001, FS2001 in Table 1) or compute the marginal effect for key \nsubsections (e.g., \nFS\n%Δ\n in Table 3, Panel B). Moreover, the beta shocks are likely to be quite \nnoisy, which could substantially attenuate the coefficient magnitudes. To gauge the effect of \nmeasurement error on the coefficients, we follow common practice and run reverse regressions, \nessentially switching the independent and dependent variables. These regressions (not tabulated) \nsuggest that a 10% change in beta alone could explain the average page change in our sample. \nThus, there are several pieces of evidence suggesting that the disclosure changes we observe are \neconomically significant.17 \nWe next analyze the determinants of page count increases in specific sections of the 10-K \nthat are likely to contain important relevant disclosures: \nMDA\n%Δ\n, \nFS\n%Δ\n, \nDISCUSS\n%Δ\n and \n% REL_PAR\nΔ\n(see Section 3.3 for details). Table 3, Panel B, presents the results for models of \nthese four disclosure change variables on the initial shock (INIT_SHOCK) or the remaining \nshock (REM_SHOCK) and the levels of the same control variables and industry dummies as in \nPanel A (Model 2).18 We find that the initial shock is positively associated with the page count \nchanges for all four disclosure variables but the relation is statistically significantly for \nFS\n%Δ\n \nonly. However, the coefficients are still fairly close to the coefficient on the initial shock in \nPanel A (Model 2) and, for \nMDA\n%Δ\n and \nDISCUSS\n%Δ\n, the p-values of the coefficients are \nclose to conventional significance levels, suggesting that the reduced sample size and lack of \npower account for the drop in significance. The remaining shock is significantly associated with \nthree of the four disclosure changes, i.e., \nFS\n%Δ\n, \nDISCUSS\n%Δ\n and % REL_PAR\nΔ\n. \n \n17 In addition, we document in Section 6 that firms’ disclosure changes are associated with economically significant \nbeta declines (and also abnormal volume reactions around the 10-K filings). \n18 The results are similar but stronger when we use model 1 and weaker but consistent when we include changes in \nthe control variables (model 3). \n\n-27- \nNext, we estimate the same models as in Table 3 using the percentage change in the word \ncount of the earnings announcement in 2001 relative to 2000 (%ΔWORDS) as our disclosure \nresponse variable. In these regressions, the cost of capital shocks do not exhibit a positive \nrelation, are generally insignificant and the p-values are often far from conventional significance \nlevels. For brevity, we do not tabulate these results. We subject the word count proxy to \nnumerous refinements, e.g., we adjust the word counts for the occurrence of unusual corporate \nevents and by industry; we convert the percentage changes into a simple variable indicating \nincreases, decreases and approximately no change in the size of the earnings announcement; we \nfocus on large changes only, drop small earnings announcements (< 500 words), and use log \nchanges. These refinements do not change the conclusion that the shocks do not exhibit a \npositive association. Thus, it does not appear that noise in the earnings announcement proxy \ngenerates this (non-)result. While we are cautious about interpreting this finding, it is possible \nthat earnings announcements are simply not well suited to address the transparency concerns that \narose from the Enron collapse. They tend to be rather brief and they are not audited or subject to \nthe same enforcement as official SEC filings. Another (but more minor) issue is that firms’ \nearnings announcements and our pre-report beta estimation may overlap, which could result in \nconfounding effects. \n4.2 \nRobustness tests \nWe subject the results in the previous section to a large battery of robustness tests to \naddress a number of potential concerns. First, in an effort to rule out spurious correlations as an \nexplanation for the results, we examine whether the initial shock and remaining shock variables \nhave any explanatory power for the percent change in page counts from 1999 to 2000. They do \nnot; the coefficients of interest in these “placebo” regressions are close to zero, as they should be. \n\n-28- \nSecond, in Panel C of Table 3, we address a number of concerns about the use of betas to \nmeasure cost of capital shocks. For brevity, we report only the key coefficients of interest, but \nthe models include the full set of control variables from Models 2 and 6 in Panel A. In the first \nmodel, we use beta shocks derived from a linear market model. While quadratic beta estimates \nover the event period are conceptually preferred and better suited for our setting (e.g., to capture \ninterim responses), we obtain similar results using linear beta estimates, which is reassuring.19 \nAnother potential concern is that beta estimates for infrequently traded stocks are \ndownward biased (Scholes and Williams, 1977). As a result, the estimated beta shocks may in \npart reflect differences in liquidity, which matters because liquidity and corporate disclosure are \nknown to be related (e.g., Welker, 1995; Healy et al., 1999; Leuz and Verrecchia, 2000). To \naddress this potential issue, we examine the relation between %ΔPAGES and the two beta \nshocks for the subsample of stocks with a share price above $5 as they are less likely to have \nnon-synchronous returns. As reported in Panel C, the $5 restriction reduces the sample size to \n1,387 observations, but the results are very similar and the inferences are the same (see Model \n2). The results are also robust to controlling for the log of average daily volume as a proxy for \nthe downward bias in the beta estimates (not tabulated).20 \nNext, we address the concern that changes in the market’s expectations about future cash \nflows as a result of the Enron scandal or the September 11 terrorist attacks unduly affect our \nfindings. By using changes in beta, rather than abnormal returns, we should in principle separate \ncost of capital changes and updated assessments of future cash flows. Furthermore, our primary \n \n19 See Kumar et al. (2008) for a similar analysis of pre- and post-event betas using a linear market model. \n20 Another way to gauge the issue is to analyze the correlation between the residuals from the switching regime \nmodel and volume. This analysis (untabulated) reveals that the correlation is on average small, approximately 5-6%, \nwith approximately 20-30% of the correlations being significant and positive and 5-15% of the correlations being \nsignificant and negative. The correlations are similar across the pre-event, event, and pre-report periods. \n\n-29- \nsample excludes firms that potentially had severe cash flow shocks due to the September 11 \nterrorist attacks. Nevertheless, we include the revision in the analysts’ consensus forecast for \nnext year’s earnings, computed from August 2001 to December 2001, as a proxy for changes in \ncash flow expectations. The coefficient estimate on this variable is significantly negative, \nindicating that firms with larger downward revisions provide more disclosures. However, \ncontrolling for the forecast revisions does not alter our inferences for the beta shocks and, more \nimportantly, including this proxy for changes in cash flow expectations does not attenuate the \ncoefficients on INIT_SHOCK and REM_SHOCK as it would if our results were primarily driven \nby changes in cash flow expectations.21 \nModel 4 in Panel C includes a control variable for idiosyncratic return volatility to \naddress the concern that (a) the quadratic beta model insufficiently separates systematic and \nidiosyncratic components of risk and (b) that stocks with higher volatility are simply also more \nresponsive in their disclosures. We measure idiosyncratic volatility as the standard deviation of \nthe residual from a linear market model over the pre-event period. The results are similar and the \ninferences the same if we use raw volatility over the pre-event period or idiosyncratic volatility \nover the event period as controls. \nWe also perform several robustness checks related to specific variable definitions (results \nnot tabulated). First, we adjust the changes in page count and word count for the existence of \nevents that may reflect a fundamental change in the firm’s operations during the period that is \ncorrelated with both its beta and its required disclosures. For example, discontinued operations \nmay be associated with changes in systematic risk and it is likely to be associated with changes \nin page counts given the SEC disclosure requirements associated with discontinued operations. \n \n21 This result also holds if we use the two-year ahead consensus forecast revision, rather than next year’s revision. \nSimply controlling for analyst following does not alter our findings or inferences either. \n\n-30- \nOur reviews of financial statements as well as SEC disclosure requirements suggest that the \nexistence of the following irregular items (IRREGITEM) in the current year and two previous \nyears can affect the length of the financial statements: discontinued operations, new segments, \nextraordinary items, accounting changes, certain special items (specifically litigation reserves or \nrestructuring charges), and acquisition activity. We create seven indicator variables equal to one \nif the firm has evidence of these events based on Compustat data for the three years 1999, 2000, \nand 2001. We regress the changes in the page count on these indicator variables and use the \nresiduals from the following model (firm subscripts omitted) as dependent variable in our \nregressions in Table 3, Panel A: \n2001\n7\n1999\n1\n%\njy\njy\ny\nj\nPAGES\nd\nk IRREGITEM\n=\n=\nΔ\n=\n+ ∑∑\n \nWhile these adjustments provide a useful sensitivity check, it is a priori not obvious that \nthe adjusted proxies dominate the unadjusted page (or word) changes. The transparency \nconcerns after Enron and the cost of capital shocks may have induced firms to make decisions \nthat would result in such items. For instance, “coming clean” on certain transactions is likely to \nresult in extraordinary or special items. Nevertheless, our results are unaffected by these \nadjustments. That is, the positive association between the cost of capital shocks and the page \nchanges continues to be significant and, more importantly, the coefficients of key interest are not \nmaterially altered, indicating that unusual corporate events and fundamental changes in the firm \nare not responsible for our findings. \nFinally, we include controls for past performance, specifically for past losses as they may \ntrigger additional language in firms’ earnings announcements and 10-K filings (Li, 2008). \nModels (2) and (6) already include both ROA2000 and other models in Panel A of Table 3 include \n\n-31- \nthe change in ROA from 2000 to 2001. But given that operating performance generally lags \nstock returns, we expand these models to include the cumulative stock return over the pre-event \nperiod, the event or the pre-report period. In addition, we include an indicator variable for loss \nfirms (in 2001 and, alternatively, 2000). The results are robust to the inclusion of these variables \nand the inferences remain the same. \n4.3 \nCross-sectional analysis of the 10-K responses \nIn this section, we present a cross-sectional analysis of firms’ 10-K responses. As noted \npreviously, we predict cross-sectional variation in the expected benefits to expanding the 10-K \ndisclosures, which should lead to cross-sectional variation in firms’ disclosure responses. \nExploiting this cross-sectional heterogeneity should improve identification and increase the \nconfidence in our earlier findings. \nWe argue in Section 2 that we expect the strongest disclosure response for firms that have \nhigh external financing needs and larger growth opportunities. These firms are more likely to be \nsensitive to shocks to their cost of capital as they are expected to be more frequently accessing \ncapital markets. We also expect firms with positive shocks to respond more strongly, as the cost \nsavings to decreasing disclosure may be small, and disclosure decreases may in fact increase \nexpected litigation costs. \nTherefore, we estimate our main page change model separately for firms with positive \nshocks and a high sensitivity to cost of capital shocks vs. the (residual) sample of firms that have \neither a negative shock or low financing needs. This specification allows the control variables to \nvary across subsamples and hence is more general than an interaction term. We use two proxies \nto identify firms that are more likely to be sensitive to cost of capital shocks because they have \nhigh financing needs: 1) Firms with an above-median investment cash flow during fiscal 2001 \n\n-32- \nscaled by total assets, and 2) firms with an above-median difference between the average growth \nrate over two years and the maximum growth rate that can be financed by internal funds. \nFollowing Demirgüc-Kunt and Maksimovic (1998), we compute the maximum internally-\nfinanced growth as ROA/(1–ROA). We consider alternative partitioning variables for firms’ \nfinancing needs and growth opportunities, e.g., splitting by the market-to-book ratio, and obtain \nsimilar results. There are 946 (820) observations with a positive initial (remaining) shock and \n922 (1,048) observations with a negative one. We do not have data for the financing needs \nvariables for all firms and hence have fewer observations than in Table 3. All models include the \nsame control variables as specifications (2) and (6) in Table 3. The coefficients for the control \nvariables are not presented to conserve table space. \nTable 4, Panel A presents the cross-sectional results for %ΔPAGES. Columns (1) \nthrough (4) present results using above-median investment cash flow to identify firms that are \nmore sensitive to positive cost of capital shocks. Columns (5) through (8) present results using \nabove-median external financing needs to identify high-sensitivity firms. The coefficient \nestimates on INIT_SHOCK in models (1), (2) and (6) are significant at the 10% level or better. \nThe coefficient estimate on REM_SHOCK is significant in model (7). More importantly, the \ncoefficients on the shock variables, INIT_SHOCK and REM_SHOCK, for the group with \npositive shocks and high financing needs are much larger in magnitude than either the \ncoefficients for the residual group or the coefficients on the shocks in Panel A of Table 3. We \ntest whether the differences in the coefficients on the shocks are statistically significant across \nthe two subsamples and can reject the null hypothesis for models (7) and (8). For models (1) and \n(2), the p-value is close to conventional significance levels (0.14), especially when considering \nthat we cluster our standard errors at the industry level and hence draw conservative inferences. \n\n-33- \nIn Panel B, we report the same model specifications for the page changes in the MD&A. \nWe observe the same patterns for the coefficients on \nMDA\n%Δ\n as for the 10-K page changes. \nThat is, the estimated coefficient on \nMDA\n%Δ\n for firms with positive shocks and high financing \nneeds exceeds the respective coefficient for the residual group in all cases. With the exception of \nthe first model, the shock coefficients and their differences are not statistically significant but \ntheir magnitudes far exceed those in Panel B of Table 3, suggesting that the small sample size \nand hence lack of power are primarily responsible for the low t-statistics. \nOn November 29, 2001, the SEC expanded its investigation to include Enron’s auditor, \nArthur Andersen LLP (AA). The audit firm was indicted for obstruction of justice in 2002 and \neventually went out of business as a result of the Enron collapse. We predict that AA clients are \nmore likely to experience shocks to their cost of capital because of concerns over AA’s \ncredibility and hence are more likely to respond by increasing their disclosure.22 Consistent with \nthis expectation, Panel C shows that AA clients exhibit larger coefficients on the initial shock \nusing %ΔPAGES, %ΔΜDA and %ΔREL_PAR. The difference in the coefficients between AA \nclients and non-AA clients is statistically significant for the 10-K page changes and the related \nparty disclosure changes. \nIn a final set of cross-sectional tests (untabulated), we drop the beta shocks and the pre-\nevent beta from the models in Panel A of Table 4 and re-estimate the page change models \nsplitting the sample into firms with high and low financing needs. We then compare the \nconstants across the subsamples. The idea behind this analysis is to test whether firms that ex \nante are predicted to be more affected by the Enron shock do in fact increase their disclosures \n \n22 As discussed in Ball (2009), the Enron scandal was a more severe blow to the credibility of AA audit clients than \nprior financial reporting scandals had been for other audit firms because AA’s reputation was already tarnished by \nprior scandals including Waste Management and Sunbeam, and because there were allegations that the Enron audit \ndeficiencies involved top level management at AA and not just individuals. \n\n-34- \nmore than other firms. This analysis does not rely on the beta shocks and is merely built on the \npremise that the Enron scandal posed an exogenous transparency shock. We find that firms with \nlarger financing needs, as measured by the same proxies as in Panel A, exhibit significantly \nlarger disclosure increases, which supports our inferences. \n5. \nAnalysis of the role of interim disclosures \nIn this section, we analyze the relation between the cost of capital shocks and firms’ \ninterim disclosure responses. First, we include proxies for firms’ interim disclosures as controls \nin our 10-K page changes regressions. We use changes in 8-K filings, conference call activity, \nthe length of the earnings announcement as well as a summary measure of the firm’s response to \nthe Enron shock derived from the regime switching model, i.e., the parameter estimate for \n2\nˆ\nib . If \ninterim disclosures and 10-K filings are complements, we expect a positive relation between the \ninterim disclosure and the 10-K page changes. Moreover, introducing the interim disclosures \nshould strengthen the relation between the remaining shock and the 10-K page response if the \ntwo disclosures are complements because the decline in beta due to the interim disclosure occurs \nwith the expectation that firms still respond at the 10-K. Thus, controlling for the interim \ndisclosures should help the relation between REM_SHOCK and %ΔPAGES. \nWe expand Models (2) and (6) from Table 3 to include proxies for the four interim \ndisclosure proxies. Table 5 reports the results from these regressions. Models (1) and (2) \ninclude Δ8KCOUNT and Models (3) and (4) include ΔCALL. The coefficient estimates on \nINIT_SHOCK and REM_SHOCK are comparable to Table 3 and remain significant in both \ncases. The change in the number of 8-K filings is positively associated with the percentage \nchange in 10-K pages. The association between the change in the number of conference calls \nand the change in the 10-K page counts is negative but not significant. Models (5) and (6) \n\n-35- \ninclude the percent change in word count in the annual earnings announcement (%ΔWORDS). \nThe association between this change and the change in the 10-K page count is positive but again \nnot significant. This result is consistent with our earlier analyses relating changes in firms’ \nearnings announcements to the beta shocks and suggests that changes in earnings announcements \nare not suited to address investors’ transparency concerns as a result of the Enron collapse. \nModel (7) includes the \n2\nˆ\nib parameter from the regime-switching model as a proxy for \nfirms’ immediate responses to the initial shock and, hence, we estimate this model for the \nremaining shock only. As illustrated in Figure 1, \n2\nˆ\nib captures the direction of the beta shock and \nthe curvature of quadratic event beta. One explanation for this curvature is that a firm has \nresponded to the initial shock, for example, through interim disclosures or other actions that \nenhance a firm’s credibility, such as a change in dividend policy. In this sense, \n2\nˆ\nib is a summary \nmeasure of a firm’s responses to the initial shock and should capture the effects of all actions the \nfirm takes during the event period, regardless of whether we can separately identify or measure \nthem. The downside of this measure, however, is that \n2\nˆ\nib may also capture changes in the cost \nof capital unrelated to a firm’s actions, including recoveries in the cost of capital due to \ninformation transfers from other firms. However, as long as these other changes are not \nsystematically biased, they should introduce only noise into our analysis. \nThe percent change in 10-K pages has a positive and significant association with \n2\nˆ\nib , \nindicating that there are complementarities between a firm’s immediate responses to the shock \nand the disclosure response in the 10-K filing. Moreover, including \n2\nˆ\nib significantly increases \nthe coefficient on REM_SHOCK (using a Chow test), which is expected if interim responses and \nthe 10-K are complements. \n\n-36- \nNext, we directly examine the relation between firms’ interim disclosure responses and \nthe cost of capital shocks. Table 6 is the analog to the analysis of the 10-K responses in Table 3 \nusing the two immediate disclosure responses as dependent variables. Model (1) in Table 6 \nshows that the association between changes in firms’ 8-K filings and the initial shock is positive \nand significant, suggesting that firms respond to the shock with the release of 8-K filings. As the \nmeasurement of the change in 8-K filings and the event period overlap, we re-run the model \nmeasuring the change in 8-K filings starting in the middle of the event period (i.e., November 8), \nso that the shock is measured largely before the disclosure response. Again, the association \nbetween the shock and the 8-K response is positive and significant. Next, to see whether the \nadditional 8-K filings help mitigate the cost of capital shocks, we restrict the change in 8-K \nfilings to those that occur over the second-half of the event period, i.e., after the peak of the event \nperiod beta and up to the start of the pre-event period (i.e., December 6, 2001). Model (2) shows \nthat the coefficient on REM_SHOCK is negative and significant, consistent with the idea that \nimmediate 8-K responses mitigate the initial shock.23 \nModels (3) and (4) perform the same tests for the change in the number of conference \ncalls. We do not find a positive association with the initial shock, but the coefficient on the \nremaining shock is negative and close to conventional significance levels (p=0.185, two-sided), \nwhich is consistent with the results for the 8-K findings. Moreover, the latter association is \nsignificantly negative at the 1% level if we use changes in the conference calls over the entire \nevent window, i.e., from October 15 to December 6, 2001.24 \n \n23 We also confirm that the decline in the beta from the event period peak to the pre-report beta is significantly \nrelated to the change in 8-K filings. \n24 We also estimate seemingly unrelated regressions using both interim disclosure variables simultaneously to \naccount for dependencies among them and obtain the same inferences. \n\n-37- \nModels (5) and (6) estimate the relation between the interim disclosure responses and the \n2\nˆ\nib parameter from the regime switching model to see whether the curvature in the quadratic \nevent beta is indeed related to firms’ interim responses. Models (5) and (6) show a positive \nassociation for both interim disclosures, consistent with the idea that the curvature reflects \namong other things firms’ immediate responses, but the relation is significant only for the \nadditional 8-K filings. The weaker findings for the conference calls could stem from the fact that \nthey are purely voluntary and not subject to SEC review. \nTaken together, the results in Tables 5 and 6 suggest that some firms respond \nimmediately to the cost of capital shocks and that these interim responses are complements to \n10-K disclosures. The results also highlight that using a quadratic beta estimate during the event \nperiod is an important feature of our analysis. We intentionally define the event period to be \nlong enough to capture the series of Enron-related events that led to the transparency crisis. \nHowever, an extended event period increases the likelihood of confounding events, in particular, \ndue to firms’ responses to the initial shock. Our regime-switching model addresses this issue by \nbeing flexible enough to capture these interim responses in the beta estimation. \n6. \nAnalysis of market reactions to 10-K filings \nIn our final set of analyses, we examine whether firms’ disclosure changes have \nsubsequent market reactions and whether they address mitigate the cost of capital shocks. \nFinding such reactions would corroborate the results for firms’ interim disclosures in Table 6 and \naddress the concern that firms simply increase disclosures to reduce litigation risk or to thwart \nfuture regulation but do not provide information that is useful to investors. In relating changes in \ndisclosure to subsequent changes in the cost of capital, the tests in this section are similar to \n\n-38- \nthose in traditional disclosure studies, except that the disclosure changes in our study are induced \nby an exogenous information-related shock. \nWe analyze both post 10-K filing changes in systematic risk and short-window abnormal \nvolume responses to the 10-K filing. We begin with the beta response. We define the beta \nresponse (BRESPONSE) as the change in systematic risk from the event period to the post-filing \nperiod. We estimate the post-filing period beta (BETA_POST) after the annual report season \nfrom May 1, 2002 to August 31, 2002 using a standard market model. Thus, BRESPONSE = \nBETA_EVT - BETA_POST. A positive value of the beta response variable suggests a greater \nrecovery of the beta relative to the peak level of systematic risk during the event period. We are \nable to estimate BRESPONSE for 1,797 observations. \nThe measure of abnormal volume is a firm-specific measure (Asthana et al., 2004) that \nhas been applied specifically in the context of 10-K filings. We define abnormal volume \n(\ni\nt\nVOL\n_\nABN\n) for firm i on day t as: \n)\nVOL\n(\n)\nVOL\n(\nVOL\nVOL\n_\nABN\ni\nd\ni\ni\nd\ni\ni\nt\ni\nt\nσ\nμ\n−\n=\n \nwhere \ni\nt\nVOL represents daily trading volume for firm i on day t. Consistent with Asthana et al. \n(2004), we define the non-filing period as the 45-day window from 49 days before to five days \nbefore the 10-K filing date. We measure the mean (μ) and standard deviation (σ) of daily trading \nvolume for each firm i over this non-filing period and compute abnormal volume for a five-day \nevent window around the 10-K filing date (-1,+3). \nTable 7, Panel A presents the analysis for the beta response using various combinations \nof control variables. We use firm characteristics that are usually related to the cost of capital, \n\n-39- \nsuch as size, the market-to-book ratio and leverage as controls (in both levels and changes). We \nfind that the percentage change in the 10-K page is significantly related to the decline in beta for \nall three models. Gauging the magnitude of the effects suggests that the mean page increase is \nassociated with a 5% decline in beta. These findings are consistent with the interpretation that \nthe 10-K disclosure responses are useful to investors and alleviate the beta shocks. \nTable 7, Panel B presents the analysis of abnormal volume. The models include control \nvariables consistent with Asthani et al. (2004). In particular, we control for last year’s abnormal \nvolume response to the 10-K to ensure that any effect for %ΔPAGES stems indeed from the \nincrease in pages and not a cross-sectional difference in firms’ volume responses to the 10-K. \nFurthermore, we include a control variable capturing the delay in the filing of the 10-K, which \nAsthana et al. (2004) document as being related to the abnormal volume response. We find that \nthe %ΔPAGES is significantly related to the abnormal volume reaction in all three models. This \nfinding is consistent with the interpretation that the additional disclosures in the 10-K are \ninformative to investors and hence lead to greater turnover at the filing of the 10-K. \n7. \nConclusions \nIn this paper, we analyze the relation between disclosure and the cost of capital using an \nexogenous, information-related cost of capital shock created by the Enron scandal in Fall 2001. \nWe begin with tests that have the opposite structure of the typical research design. In reversing \nthe tests and using an exogenous shock, we mitigate endogeneity concerns in traditional cross-\nsectional disclosure studies, especially with respect to omitted factors that jointly determine \nfirms’ disclosure choices and their cost of capital. Our analysis is based on an event-study \ntechnique that allows us to estimate shocks to firms’ betas as well as accommodate subsequent \nchanges in beta due to firms’ interim responses to the transparency crisis. \n\n-40- \nOur analysis shows that beta shocks around the Enron scandal are associated with \nincreased disclosure. Firms expand the number of pages of their annual 10-K filings, notably the \nsections containing the narrative disclosures like the MD&A and related-party transactions as \nwell as the financial statements and footnotes. The increase in disclosure is particularly \npronounced for firms that have positive cost of capital shocks and larger financing needs and for \nfirms that have larger credibility issues. We also find that firms respond with additional interim \ndisclosures (e.g., 8-K filings), which have an immediate impact on the firm’s cost of capital. We \ndocument that the 10-K disclosures are complementary to the interim disclosures; firms provide \nthem even though the interim disclosures already reduce the beta shock. Finally, we show that \nfirms’ 10-K disclosure responses have significant subsequent market reactions (e.g., abnormal \nvolume) and are effective in reducing the cost of capital shocks. \nThese results complement prior disclosure studies by explicitly linking information-\nrelated cost of capital shocks to firms’ disclosure responses. In doing so, we increase the \nempirical support for the fundamental link between disclosure and firms’ cost of capital. Our \nstudy also provides novel evidence that firms respond to shocks to the credibility of their \nfinancial reports. This evidence is important as it shows that firms can mitigate the impact of a \nwider transparency crisis and that firm-specific disclosures can influence the cost of capital. We \nemphasize, however, that this result should not be used to justify additional disclosure \nrequirements as firms that voluntarily responded more to the cost of capital shock were those \nwith the greatest hypothesized benefits. \n\n-41- \nAppendix \nSection A.1 describes Enron-related events between October 16, 2001 and January 26, \n2002. From these, we identify seven potential sets of pre-event – event – pre-report windows \nthat are outlined in Section A.2. The final section of the Appendix reports the results of an ex \npost analysis of the beta estimates over the various windows to assess their validity. \nA.1 Event dates related to Enron \nWe identify potentially significant events from published timelines of the Enron scandal \n(e.g., Platts, 2002; Washington Post, 2002), and from independent searches of news reports. \nPotentially important event dates in the context of our study are those that changed investors’ \nprior beliefs about the likelihood or magnitude of private information about the sample firm. \nWe identify three types of event dates. First, we identify six Enron-specific events that \nare also indicators that financial reporting or corporate governance problems may be widespread. \nOct. 16: Enron first announces its huge losses and hints of conflict-of-interest problems with its \npartnerships. \nOct.22: Enron announces that the SEC has launched an inquiry. \nOct 29: The SEC moves its inquiry to DC signaling a higher likelihood of securities violations. \nOct 31: The SEC initiates a formal investigation. \nNov 8: \nEnron restates earnings back to 1997. \nNov 29: The SEC’s investigation is expanded to include Arthur Andersen LLP. \nThese events are specific to Enron, but they are not necessarily the most significant for \nEnron based on an analysis of its stock returns. This set does not include events such as a credit \nrating downgrade of Enron’s debt or a takeover offer of Enron which are significant for Enron \nbut unlikely (in our view) to have broad implications for information problems at other firms. \nSecond, we identify events that directly indicate that the information problems at Enron \nare expected to be systemic. \n\n-42- \nNov 19: Representative John Dingell requests that the Public Oversight Board (POB) review \nAndersen’s audits of Enron \nNov 29: SEC Chairman Harvey Pitt calls pro forma earnings \"unstructured and undisciplined\" \nand suggests that the SEC is focusing attention on misleading financial reporting. \nDec 4: \nThe Big-five firms pledge to work together to address financial reporting problems. \nJan. 7: \nIt is reported that the Big Five have petitioned the SEC to improve disclosure \nregulation. \nJan 9: \nThe Senate Banking Committee, headed by Sen. Paul Sarbanes, announces its plans to \nhold a hearing Feb. 12 to examine accounting and investor protections. The US Justice \nDepartment's fraud section also announces that it will form a special task force to \nexamine the collapse of Enron. \nThird, we identify financial press discussions that suggest that the Enron scandal was not \nan isolated problem but rather an indicator of corporate transparency and corporate governance \nproblems. \nNov 5: \n“What Enron’s Financial Reports Did – and Didn’t – Reveal --- Auditor Could Face \nScrutiny on Clarity of Financial Reports” (Wall Street Journal, p. C1) \nNov 8: \nIn a DJ Newswire column, the author speculates that Enron’s partnership accounting \nproblems, which resulted in significant earnings restatements, could push the FASB to \nprioritize its projects on SPEs from having standards of ownership including \nprovisions for minimum ownership by unaffiliated outsiders to control-based \nstandards. The article also questions Arthur Andersen’s responsibilities given the \nrestatements. \nNov 26: Business Week cover story is “CONFUSED ABOUT EARNINGS? You're not alone. \nHere's what companies should do--and what investors need to know.” \nJan 26: \n“Trying not to be the next Enron, companies scrutinize practices” appears on p.1 of \nthe New York Times Saturday business section. \nThese articles are not events per se. However, it is possible (or even likely) that the \narticles affected public opinion about the systemic nature of the transparency problem. Existing \nresearch has shown that articles in the financial press do affect investor opinions (e.g., Foster, \n1979; Foster, 1987; Chang and Suk, 1998; Pruitt et al. 2000; Huberman and Regev, 2001). It is \nalso possible that the articles reflect public opinion rather than create it, which is another reason \nto search for and consider these dates. We end our search for such articles in January 2001 \nbecause that is the start of the earnings announcement season. \n\n-43- \nA.2 Identification of event windows \nFigure A1 shows the value-weighted and equal-weighted index from August 1, 2001 \nthrough February 28, 2002 and provides some context for the window selection. The figure \nillustrates the high volatility period associated with the September 11th terrorist attacks. The \narrows point to the start dates and the end dates that we consider for the event period. \nFigure A1: Index returns \n \nWe identify seven combinations of windows to estimate systematic risk for the pre-event \nperiod (BETA_PRE), the event period (BETA_EVT), and the pre-report period (BETA_REM). \nIn the timelines below, T1 and T2 define the endpoints of the event period. The center date of \nthe event period is noted for each window. In all scenarios, the end of the pre-event period is \nAugust 31, 2001. There is a discontinuity between the end of the pre-event period and the \nbeginning of the event period ranging from 44 to 62 days. The discontinuity allows us to avoid \n\n-44- \nthe week-long trading halt and subsequent high volatility period associated with September 11. \nIn addition, the discontinuity is advantageous because the pre-event period is not contaminated \nwhen we do not properly identify the start of the event period. The discontinuity does not affect \nthe estimation procedure. \nAlternative Windows for the Beta Estimation \n \nT1 \nT2 \n \n−−−−−−−β1−−−−−− \n \n−−−−−−−−−β2−−−−−−−−−− \n−−−−−−−−β3−−−−−−−− \nWindow 1: \n \n \nNov 8 \n \n \n05/01/01–8/31/01 \n \n10/15/01–12/05/01 \n12/06/01 – 01/15/02 \n \nWindow 2: \n \n \nNov 12 \n \n \n05/01/01–8/31/01 \n \n10/19/01–12/05/01 \n12/06/01 – 01/15/02 \n \nWindow 3: \n \n \nNov 19 \n \n \n05/01/01–8/31/01 \n \n11/02/01–12/05/01 \n12/06/01 – 01/15/02 \n \nWindow 4: \n \n \nNov 27 \n \n \n05/01/01–8/31/01 \n \n10/15/01-01/10/02 \n1/11/02-01/28/02 \n \nWindow 5: \n \n \nNov 29 \n \n \n05/01/01–8/31/01 \n \n10/19/01–01/10/02 \n1/11/02-01/28/02 \n \nWindow 6: \n \n \nNov 20 \n \n \n05/01/01–8/31/01 \n \n10/15/01-12/31/01 \n1/01/02-01/28/02 \n \nWindow 7: \n \n \nNov 23 \n \n \n05/01/01–8/31/01 \n \n10/19/01–12/31/01 \n1/01/02-01/28/02 \n \nWe consider three start dates for the event window: October 15, October 19, and \nNovember 2, 2001. October 15 is day t-1 relative to Enron’s announcement that it lost $618 \nmillion, which provided the first indication of possible conflict-of-interest questions related to \nthe Fastow-run partnerships and the “quality” of Enron’s prior-period earnings. October 19 is \ntrading day t-1 relative to Enron’s announcement of an SEC investigation of the partnerships and \nof suspicions by analysts that Enron will release additional bad news (October 22). November 2 \nis the start date that makes November 19 the center of the event window. November 19 is \nsignificant because on that day it is first reported that Representative John Dingell requested that \n\n-45- \nthe Public Oversight Board (POB) review Andersen’s audits of Enron (and Waste Management). \nThe public oversight Board declines to review specific cases but indicates that it will consider a \nsystemic review. \nWe consider two end dates for the event window. The end date in windows 1-3 is \nDecember 5, which is day t+1 relative to the pledge by the big-five firms to work together to \naddress financial reporting problems, especially related to SPEs and market risks. The event \nperiods in these windows include all of the Enron-specific events and all but three of the \nsystemic and financial press events. The pre-report period begins December 6 and ends on \nJanuary 15, 2002, which is the first percentile of fourth quarter earnings announcement dates. \nIn windows 4 and 5, the event period extends to January 10, 2002, which is day t+1 \nrelative to the last systemic event and it includes all but the last financial press event. Because of \nthe longer event period, we are forced to extend the length of the pre-report period to January 28 \nin order to have a sufficiently long pre-report period to estimate the model. January 28 \nrepresents the first quartile of earnings announcement dates. In windows 6 and 7, we arbitrarily \nset the end of the event window at December 31, 2001. This specification provides more \nequally-sized windows for the event-period and the pre-report period. \nA.3 Analysis of beta estimates \nWe use beta estimates from window 1.25 In choosing among the windows, the primary \ntrade-off is that a longer event window, which is more inclusive of important events, \n \n25 There is no ex-post diagnostic tool that establishes the best window. Lockwood and Kadiyala (1988) estimate the \nmodel for every combination of event windows with a start date within 30 days prior to the one day event (T1) and \nan end date within 30 days after the event (T2). The window (T1, T2 combination) that maximizes the log \nlikelihood function is chosen as the best firm-specific model. However, they have firm-specific events whereas we \nhave a common event to all firms. \n\n-46- \nmechanically shortens the pre-report window given that the report date (either the earnings \nannouncement date or the 10-K date) is fixed. \nSeveral analyses favor the shorter event periods (windows 1-3). The means (medians) of \nthe adjusted R2s from the model estimation are higher for these windows than for the other \nwindows (although the differences are not significant). Windows 1-3 also produce the smallest \nnumber of negative \n1ˆ\nib estimates (regardless of the start date). Finally, the residuals for windows \n1, 2 and 3 are mean zero during the pre-event period and the event period. They are significantly \npositive during the pre-report period (at approximately the 10% level). In windows 4-7, the \nresiduals during the event (pre-report) period are positive (negative) and significant. None of the \nanalyses indicate a difference between windows 1-3. Thus, we report results for the window that \nallows the earliest possible start date, i.e., window 1. \nThe beta shocks estimated from window 1 have associations and exhibit cross-sectional \nvariation that support their validity. For example, the median initial shock for the 45 firms that \nhave Enron-related 8-Ks is 0.21, which is significantly greater than the median initial shock for \nthe remaining 2,056 firms of 0.006. The median remaining shocks also are greater (0.18 vs. \n0.10), but the difference is not significant. The median initial shocks for the Arthur Andersen \nclients are not significantly different (0.04 for AA clients vs. 0.005 for non-AA clients), but the \nremaining shocks are significantly higher by 0.043 at the 10% level. This pattern is consistent \nwith the fact that Arthur Andersen was not implicated in the scandal until later in the event \nwindow. 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Panel A provides descriptive statistics for the disclosure proxies in our analyses. It presents \ncounts of the number of pages for the entire 10-K in 2001 (PAGES2001); the management discussion and analysis \n(MDA2001) section (Item 7); the sum of Items 6 (Selected Financial Data), 8 (Financial Statements and \nSupplementary Data), and 15 (Exhibits, Financial Statement Schedules, and Reports on Form 8-K), which \ncommonly include financial tables (FS2001); and the sum of Items 1 (Business), 2 (Properties), and 7 (MD&A), \nwhich commonly include narrative discussions about the firm (DISCUSS2001). We also present the percent changes \n(%Δ) in these variables relative to the year 2000. We select the disclosures on related-party transactions from the \n10-K or the proxy statement. REL_PAR2001 is the number of pages of this section. %ΔREL_PAR is the percentage \nchange in these pages counts from 2000 to 2001, after setting changes of plus or minus one page to zero. \nWORDS2001 is the word count for the fourth quarter and annual 2001 earnings announcement and %ΔWORDS is the \npercent change in the word count relative to the prior year. The preceding change variables are truncated at the 1st \nand 99th percentile, except %ΔPAGES, which is truncated at 2nd and 98th percentile. 8ΚCOUNT is the number of 8-\nK filings between October 15, 2001 and March 31, 2002, but excluding filings related to the September 11th terrorist \nattacks. Δ8ΚCOUNT is the difference in the count over the same period in 2000, winsorized at +3 and –3. \nCONFCALL is the number of conference calls from October 15, 2001 to March 31, 2002. ΔCALL denotes the \ndifference in the number of calls relative to the same period one year earlier, winsorized at +3 and –3. \n*** (†††) and * (†) indicate that the mean (median) change in the respective variable is significantly different from \nzero at the 1% and 10% level, respectively. \nPanel A: Disclosure proxies \n \nN \nMean \nStd dev \np25 \nMedian \np75 \n10-K Page counts \n \n \n \n \n \n \n \nPAGES2001 \n1,868 \n63.619 \n29.720 \n44 \n62 \n79 \n \nMDA2001 \n1,116 \n13.320 \n8.574 \n7 \n11 \n18 \n \nDISCUSS2001 \n1,163 \n26.693 \n13.793 \n17 \n25 \n34 \n \nFS2001 \n1,163 \n34.013 \n16.995 \n25 \n32 \n40 \n \n%ΔPAGES \n1,868 \n16.6%*** \n31.5% \n0.0% \n9.0%††† \n23.6% \n \n%ΔMDA \n1,116 \n38.7%*** \n44.7% \n10.0% \n30.4%††† \n57.1% \n \n%ΔDISCUSS \n1,158 \n22.2%*** \n32.1% \n3.7% \n17.6%††† \n35.0% \n \n%ΔFS \n1,145 \n17.6%*** \n56.6% \n0.0% \n8.0%††† \n22.6% \n \n \n \n \n \n \n \nProxy statement counts \n \n \n \n \n \n \n \nREL_PAR2001 \n1441 \n4.822 \n9.003 \n0 \n1 \n4 \n \n%ΔREL_PAR \n1312 \n19.4%*** \n99.5% \n0 \n0 \n0 \n \n \n \n \n \n \n \nEA Word counts \n \n \n \n \n \n \n \nWORDS2001 \n1,788 \n2,003 \n1,262 \n1,197 \n1,706 \n2,446 \n \n%ΔWORDS \n1,715 \n15.5%*** \n32.2% \n-5.1% \n9.0%††† \n28.9% \n \n \n \n \n \n \n \n8-K counts \n \n \n \n \n \n \n \n8KCOUNT \n1,868 \n0.904 \n1.771 \n0 \n0 \n1 \n \n \n(without Sept 11) \n1,868 \n0.760 \n1.568 \n0 \n0 \n1 \n \n \n \n \n \n \n \n \nΔ8KCOUNT \n1,868 \n0.052* \n1.220 \n0 \n0 \n0 \n \n \n(without Sept 11) \n1,868 \n0.004 \n1.159 \n0 \n0 \n0 \n \n \n \n \n \n \n \nConference Call \n \n \n \n \n \n \n \nCONFCALL \n1,868 \n1.140 \n0.955 \n0 \n1 \n2 \n \nΔCALL \n1,868 \n-0.074*** \n0.642 \n0 \n0 \n0 \n \n\n \nTable 1 (continued) \nDescriptive statistics on disclosure proxies and firm characteristics \nPanel B provides descriptive statistics on firm characteristics that we use as control variables in subsequent models. \nTOTASS is the book value of total assets (in millions). SIZE is the market value of common equity plus the book \nvalue of preferred stock and long-term debt (in millions). ROA is return on assets using the operating income after \ndepreciation and amortization and scaling it by the average of total assets at the beginning and the end of the fiscal \nyear. CUMRET is the cumulative return from May 1, 2001 to August 31, 2001. DERATIO is long-term debt scaled \nby total assets at the end of the fiscal year. MB is the market-to-book ratio of common shareholders’ equity, setting \nthe ratio to zero for book value < 0. PPE/TA is the book value of net property, plant and equipment scaled by total \nassets. FCSTREV is the change in the mean (consensus) analyst forecast of next-fiscal-year annual earnings during \nthe period from August 2001 to December 2001 scaled by price. FIRMRETVOL is the firm’s idiosyncratic risk \nduring the pre-event period, where idiosyncratic risk is the standard deviation of the residual from a linear market \nmodel that includes the CRSP value-weighted return on the market. ABN_VOL is a firm-specific measure of \nabnormal volume computed as described in Section 6. All variables are reported for the fiscal year 2001 and \ntruncated at the 1th and 99th percentile of the respective distribution. \nPanel B: Firm characteristics \n \nN \nMean \nStd dev \np25 \nMedian \np75 \n \n \n \n \n \n \n \n \nTOTASS \n1,868 \n3,585.4 \n30,600.3 \n94.3 \n372.9 \n1,325.5 \n \nSIZE \n1,864 \n3,389.3 \n19,644.6 \n99.5 \n371.0 \n1,383.1 \n \nROA \n1,842 \n-0.002 \n0.189 \n-0.020 \n0.032 \n0.097 \n \nCUMRET \n1,837 \n0.032 \n0.300 \n-0.143 \n0.039 \n0.187 \n \nDERATIO \n1,849 \n0.270 \n0.257 \n0.023 \n0.201 \n0.466 \n \nΜΒ \n1,763 \n3.777 \n3.557 \n1.904 \n2.909 \n4.441 \n \nPPE_TA \n1,828 \n0.238 \n0.235 \n0.049 \n0.158 \n0.354 \n \nFCSTREV \n1,257 \n-0.011 \n0.045 \n-0.011 \n-0.001 \n0.000 \n \nFIRMRETVOL \n1,867 \n0.035 \n0.021 \n0.020 \n0.030 \n0.045 \n \nABN_VOL \n1,480 \n0.105 \n0.724 \n-0.377 \n-0.110 \n0.352 \n\nTable 2 \nDescriptive statistics on model parameter estimates, systematic risk and beta shocks \nPanel A reports descriptive statistics for the parameter estimates from the regime switching beta model in equations \n(B1) and (B2), measures of systematic risk over four time periods and the beta shocks. The pre-event period beta is \nequal to \n1\nˆ\nib (BETA_PRE). During the event period, \n)1\n(\nˆ\n)\n2\n)(\n1\n(\nˆ\nˆ\nˆ\n3\n2\n1\n2\nT\nt\nb\nT\nt\nt\nT\nb\nb\ni\ni\ni\ni\n−\n+\n−\n−\n+\n=\nβ\n. We compute the \nevent period beta (BETA_EVT) at day t = 0. The pre-report period beta (BETA_REM) is \n)1\n2\n(\nˆ\nˆ\nˆ\n3\n1\n3\nT\nT\nb\nb\ni\ni\ni\n−\n+\n=\nβ\n. We \nreport parameter and beta estimates choosing T1 and T2 according to Window 1 in Appendix A.2. During the post-\nreport period, systematic risk (BETA_POST) is estimated using a market model from May 1, 2002 to August 31, \n2002. The initial shock (INIT_SHOCK) is BETA_EVT - BETA_PRE. The remaining shock (REM_SHOCK) is \nBETA_REM - BETA_PRE. \nPanel A: \n \nN \nMean \nStd dev \np25 \nMedian \np75 \nModel parameters \n \n \n \n \n \n \n1\nˆ\nib \n1,868 \n0.66684 \n0.6898 \n0.18972 \n0.53558 \n0.95633 \n2\nˆ\nib\n \n1,868 \n-0.00029 \n0.00268 \n-0.00168 \n-0.00021 \n0.00107 \n3\nˆ\nib \n1,868 \n0.002 \n0.02398 \n-0.00936 \n0.00289 \n0.01457 \n \n \n \n \n \n \n \nEstimates of systematic risk \n \n \n \n \n \n \nBETA_PRE \n1,868 \n0.66684 \n0.6898 \n0.18972 \n0.53558 \n0.95633 \nBETA_EVT \n1,868 \n0.59982 \n0.97357 \n-0.00566 \n0.54162 \n1.12409 \nBETA_REM \n1,868 \n0.74266 \n1.01538 \n0.08332 \n0.68691 \n1.27465 \nBETA_POST \n1,797 \n0.67755 \n0.47763 \n0.30236 \n0.67339 \n0.97329 \n \n \n \n \n \n \n \nEstimates of shocks \n \n \n \n \n \n \nINIT_SHOCK \n1,868 \n-0.06702 \n0.94982 \n-0.56727 \n-0.01294 \n0.42015 \nREM_SHOCK \n1,868 \n0.07582 \n0.91106 \n-0.35574 \n0.10998 \n0.55355 \n \n \n \n \n \n \n \n \n\n \nTable 2 (continued) \nDescriptive statistics on model parameter estimates, systematic risk and beta shocks \nPanel B reports the means for the initial and the remaining shock by industry as well as the proportions of positive \ninitial and remaining shocks in each industry. We report only industries with more than 5 sample observations. \nPanel B: \nSIC \n \nN \nINIT_SHOCK \n% POS \nN \nREM_SHOCK\n% POS \n25 \nFurniture and Fixtures \n6 \n0.6402 \n0.8333 \n6 \n0.5301 \n0.8333 \n55 \nAuto Dealers, Gas Stations \n9 \n0.5488 \n0.7778 \n9 \n0.2675 \n0.7778 \n15 \nBldg Cnstr-Gen Contr, Op Bldr \n10 \n0.5215 \n0.8000 \n10 \n0.3981 \n0.9000 \n24 \nLumber and Wood Pds, Ex Furn \n7 \n0.4201 \n0.7143 \n7 \n0.1426 \n0.7143 \n23 \nApparel & Other Finished Pds \n6 \n0.3861 \n0.6667 \n6 \n0.2990 \n0.8333 \n72 \nPersonal Services \n7 \n0.3260 \n0.5714 \n7 \n0.7008 \n1.0000 \n57 \nHome Furniture & Equip Store \n6 \n0.2949 \n0.6667 \n6 \n-0.3993 \n0.3333 \n82 \nEducational Services \n6 \n0.2640 \n0.6667 \n6 \n0.1451 \n0.6667 \n29 \nPete Refining & Related Inds \n8 \n0.2596 \n0.3750 \n8 \n-0.1416 \n0.7500 \n34 \nFabr Metal, Ex Machy, Trans Eq \n29 \n0.2083 \n0.6207 \n29 \n0.2081 \n0.6552 \n31 \nLeather and Leather Products \n9 \n0.1957 \n0.6667 \n9 \n0.4491 \n0.7778 \n40 \nRailroad Transportation \n6 \n0.1714 \n0.6667 \n6 \n0.2418 \n0.6667 \n65 \nReal Estate \n20 \n0.1324 \n0.5500 \n20 \n0.1214 \n0.4500 \n37 \nTransportation Equipment \n35 \n0.1151 \n0.6571 \n35 \n0.1690 \n0.5714 \n49 \nElectric, Gas, Sanitary Serv \n59 \n0.0988 \n0.5593 \n59 \n0.0746 \n0.4746 \n20 \nFood and Kindred Products \n36 \n0.0848 \n0.5278 \n36 \n0.0638 \n0.4722 \n79 \nAmusements, Recreation \n16 \n0.0715 \n0.5000 \n16 \n0.1330 \n0.6250 \n30 \nRubber & Misc Plastics Prods \n22 \n0.0622 \n0.6364 \n22 \n-0.1174 \n0.5455 \n62 \nSecurity & Commodity Brokers \n21 \n0.0618 \n0.6667 \n21 \n0.3465 \n0.7143 \n61 \nNondepository Credit Instn \n17 \n0.0591 \n0.5294 \n17 \n0.5249 \n0.6471 \n17 \nConstruction-Special Trade \n6 \n0.0540 \n0.6667 \n6 \n-0.0408 \n0.3333 \n64 \nIns Agents, Brokers & Service \n8 \n0.0471 \n0.3750 \n8 \n-0.0381 \n0.3750 \n48 \nCommunications \n59 \n0.0398 \n0.5932 \n59 \n0.3605 \n0.6441 \n27 \nPrinting, Publishing & Allied \n24 \n0.0311 \n0.5417 \n24 \n0.2801 \n0.7083 \n58 \nEating and Drinking Places \n24 \n0.0205 \n0.4167 \n24 \n0.1284 \n0.7083 \n60 \nDepository Institutions \n252 \n0.0162 \n0.5278 \n252 \n0.1332 \n0.6071 \n67 \nHolding, Other Invest Offices \n29 \n-0.0213 \n0.5862 \n29 \n0.0816 \n0.5517 \n35 \nIndl,Comml Machy, Computer Eq \n93 \n-0.0294 \n0.5807 \n93 \n-0.0468 \n0.5484 \n80 \nHealth Services \n39 \n-0.0440 \n0.4359 \n39 \n-0.2189 \n0.3590 \n26 \nPaper and Allied Products \n17 \n-0.0767 \n0.5882 \n17 \n0.1145 \n0.5294 \n38 \nMeas Instr; PhotoGds; Watches \n96 \n-0.0816 \n0.4167 \n96 \n-0.0812 \n0.4896 \n32 \nStone, Clay, Glass, Concrete Pd \n15 \n-0.0823 \n0.5333 \n15 \n0.2790 \n0.7333 \n39 \nMisc Manufacturing Industries \n15 \n-0.0933 \n0.4667 \n15 \n-0.4003 \n0.4667 \n33 \nPrimary Metal Industries \n27 \n-0.1140 \n0.3704 \n27 \n0.3667 \n0.6296 \n36 \nElectr, Oth Elec Eq, Ex Cmp \n113 \n-0.1332 \n0.4336 \n113 \n0.0753 \n0.5487 \n13 \nOil and Gas Extraction \n71 \n-0.1358 \n0.4366 \n71 \n0.4118 \n0.8028 \n73 \nBusiness Services \n217 \n-0.1411 \n0.4839 \n217 \n0.0467 \n0.5438 \n50 \nDurable Goods-Wholesale \n49 \n-0.1631 \n0.4082 \n49 \n0.2468 \n0.5306 \n47 \nTransportation Services \n6 \n-0.2330 \n0.3333 \n6 \n0.0114 \n0.5000 \n51 \nNondurable Goods-Wholesale \n22 \n-0.2421 \n0.4091 \n22 \n0.2576 \n0.6818 \n28 \nChemicals & Allied Products \n163 \n-0.2812 \n0.4110 \n163 \n-0.2656 \n0.3742 \n10 \nMetal Mining \n10 \n-0.2854 \n0.5000 \n10 \n0.3835 \n0.7000 \n42 \nMotor Freight Trans, Warehouse \n21 \n-0.2957 \n0.3810 \n21 \n0.1334 \n0.6667 \n59 \nMiscellaneous Retail \n19 \n-0.3042 \n0.3684 \n19 \n0.0181 \n0.6842 \n70 \nHotels, Other Lodging Places \n17 \n-0.3445 \n0.2941 \n17 \n-0.0311 \n0.5294 \n87 \nEngr, Acc,Resh, Mgmt, Rel Svcs \n53 \n-0.3662 \n0.3396 \n53 \n-0.3568 \n0.3208 \n44 \nWater Transportation \n10 \n-0.4219 \n0.4000 \n10 \n0.3883 \n0.8000 \n78 \nMotion Pictures \n6 \n-0.5024 \n0.5000 \n6 \n0.7753 \n1.0000 \n16 \nHeavy Constr; not Bldg Cntractrs \n8 \n-0.5086 \n0.2500 \n8 \n0.4448 \n0.5000 \n22 \nTextile Mill Products \n8 \n-0.7936 \n0.5000 \n8 \n0.0101 \n0.5000 \n\n \nFigure 1 \nIllustrations of the time-series patterns in the systematic risk estimates for the pre-event period, the event period, and the pre-report period for alternative values \nof \n2\nˆ\nib\n. Panels A through C illustrate levels of systematic risk in the case of a HIGH positive \n2\nˆ\nib\n (equals 0.100), a LOW positive \n2\nˆ\nib\n (equals 0.002), and a \nnegative \n2\nˆ\nib\n (equals -0.002), respectively. Within each panel, we present estimates of systematic risk in the pre-report period for five levels of \n3\nˆ\nib . \n \n \n \nPanel A: HIGH Positive \n2\nˆ\nib \nPanel B: LOW Positive \n2\nˆ\nib \nPanel C: Negative \n2\nˆ\nib \n \n \n-1.5\n-1\n-0.5\n0\n0.5\n1\n1.5\n2\n20010501\n20010509\n20010517\n20010525\n20010605\n20010613\n20010621\n20010629\n20010710\n20010718\n20010726\n20010803\n20010813\n20010821\n20010829\n20010907\n20010921\n20011001\n20011009\n20011017\n20011025\n20011102\n20011112\n20011120\n20011129\n20011207\n20011217\n20011226\n20020104\n20020114\n-1\n-0.5\n0\n0.5\n1\n1.5\n2\n2.5\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n2E+07\n \n \n-5\n0 \n5 \n10 \n15 \n20 \n25 \n30 \n35 \n40 \n20010501 \n20010509 \n20010517 \n20010525\n20010605 \n20010613 \n20010621 \n20010629 \n20010710 \n20010718\n20010726 \n20010803 \n20010813 \n20010821 \n20010829 \n20010907\n20010921 \n20011001 \n20011009 \n20011017 \n20011025 \n20011102 \n20011112\n20011120 \n20011129\n20011207 \n20011217 \n20011226 \n20020104\n20020114\nb3 = -0.046\nb3 = -0.015\nb3 = -0.006\nb3 = 0.002\nb3 = 0.020\n\n \nTable 3 \nRelation between 10-K disclosure changes and cost of capital shocks \nThe sample comprises a maximum of 1,868 firms with a December fiscal year end and sufficient data to estimate \nsubsequent models. We eliminate airlines, insurance carriers, and firms that are majorly affected by the September 11th \nterrorist attacks as described in Section 3. Panel A reports results for eight models regressing the percent change in page \ncounts (%ΔPAGES) on combinations of initial shock (INIT_SHOCK), remaining shock (REM_SHOCK), pre-event beta \n(BETA_PRE), and control variables. LASSETS is natural log of total assets. ROA is the return on assets. DERATIO is \nlong-term debt scaled by total assets. MB is the market-to-book ratio. PPE/TA is the book value of net PPE scaled by total \nassets. The control variables are measured as of fiscal year 2000. The changes in the control variables are measured from \n2000 to 2001, except in for the market-to-book ratio which uses the market values of equity. The regression models \ninclude, wherever noted, industry-fixed effects based on one-digit SIC codes and a separate dummy for firms in Enron-\nrelated industries (defined as SIC2 equal to 13, 29 or 49). The t-statistics are reported in parentheses and based on standard \nerrors that are clustered at two-digit SIC level. *** {**} (*) indicate significance at the 1%, 5%, 10% level (two-sided). \nPanel A: \n \nModel specification \n \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n(8) \nConstant \n0.1072*** \n0.1015* \n0.0697 \n0.1274*** \n0.1057*** \n0.1067** \n0.0602 \n0.1173*** \n \n(4.17) \n(1.92) \n(1.27) \n(3.02) \n(4.02) \n(2.03) \n(1.11) \n(2.93) \nINIT_SHOCK \n0.0289*** \n0.0259*** \n0.0231*** \n0.0255*** \n \n \n \n \n \n(4.25) \n(3.88) \n(3.56) \n(3.68) \n \n \n \n \nREM_SHOCK \n \n \n \n \n0.0237*** \n0.0242*** \n0.0174* \n0.0207** \n \n \n \n \n \n(2.88) \n(3.15) \n(1.85) \n(2.23) \nBETA_PRE \n0.0501*** \n0.0434** \n0.0431** \n0.0463*** \n0.0429*** \n0.0368** \n0.0359* \n0.0401** \n \n(3.21) \n(2.36) \n(2.27) \n(2.81) \n(2.80) \n(2.03) \n(1.87) \n(2.38) \nLASSETS2000 \n0.0055 \n0.0078** \n0.0080** \n \n0.0062 \n0.0085** \n0.0092** \n \n \n(1.19) \n(2.10) \n(2.29) \n \n(1.32) \n(2.13) \n(2.30) \n \nROA2000 \n0.0089 \n0.0006 \n-0.0524 \n \n0.0074 \n0.0006 \n-0.0515 \n \n \n(0.40) \n(0.03) \n(1.48) \n \n(0.33) \n(0.03) \n(1.56) \n \nDERATIO2000 \n-0.0180 \n0.0017 \n0.0068 \n \n-0.0239 \n0.0014 \n0.0054 \n \n \n(0.54) \n(0.05) \n(0.19) \n \n(0.74) \n(0.04) \n(0.15) \n \nMB2000 \n \n-0.0004* \n-0.0006 \n \n \n-0.0003 \n-0.0006 \n \n \n \n(1.77) \n(0.76) \n \n \n(1.61) \n(0.75) \n \nPPE/TA2000 \n \n0.0432 \n0.0433 \n \n \n0.0316 \n0.0337 \n \n \n \n(0.68) \n(0.64) \n \n \n(0.50) \n(0.50) \n \nΔASSETS \n \n \n0.0573** \n0.0506** \n \n \n0.0607** \n0.0544** \n \n \n \n(2.15) \n(2.03) \n \n \n(2.35) \n(2.24) \nΔROA \n \n \n-0.1826** \n-0.1423* \n \n \n-0.1718** \n-0.1294 \n \n \n \n(2.40) \n(1.71) \n \n \n(2.29) \n(1.54) \nΔDERATIO \n \n \n0.0217 \n0.0372 \n \n \n0.0179 \n0.0324 \n \n \n \n(0.29) \n(0.52) \n \n \n(0.24) \n(0.46) \nΔMB \n \n \n0.0147 \n0.0177* \n \n \n0.0151 \n0.0175* \n \n \n \n(1.26) \n(1.71) \n \n \n(1.35) \n(1.77) \nIndustry controls \nNO \nYES \nYES \nYES \nNO \nYES \nYES \nYES \n \n \n \n \n \n \n \n \n \nObservations \n1,868 \n1,793 \n1,687 \n1,707 \n1,868 \n1,793 \n1,687 \n1,707 \nR-squared \n0.0169 \n0.0245 \n0.0253 \n0.0236 \n0.0147 \n0.0237 \n0.0236 \n0.0218 \n \n\n \nTable 3 (continued) \nRelation between 10-K disclosure changes and cost of capital shocks \nPanel B reports results for regression models of percent changes in page counts for three sections of the 10-K and one section from the proxy statement on \ncombinations of initial shock (INIT_SHOCK), remaining shock (REM_SHOCK), pre-event beta (BETA_PRE), and control variables (described in Panel A). \nThe percentage changes in the page counts refer to the management discussion & analysis (%ΔMDA), the sections that are generally narrative (%ΔDISCUSS) \nand the sections that include financial tables (%ΔFS). See Table 1, Panel A for more details. The regression models include industry-fixed effects based on one-\ndigit SIC codes and a separate dummy for firms in Enron-related industries (defined as SIC2 equal to 13, 29 or 49). The t-statistics are reported in parentheses \nand based on standard errors that are clustered at two-digit SIC level. *** {**} (*) indicate significance at the 1%, 5%, 10% level (two-sided). \nPanel B: \n \n \nDependent variable \n \n \n%ΔMDA \n%ΔDISCUSS \n%ΔFS \n%ΔREL_PAR \n%ΔMDA \n%ΔDISCUSS \n%ΔFS \n%ΔREL_PAR \nConstant \n0.3903*** \n0.1394*** \n0.0631 \n0.4637 \n0.3916*** \n0.1325*** \n0.0791 \n0.4513 \n \n(3.49) \n(2.85) \n(0.98) \n(1.56) \n(3.60) \n(2.72) \n(1.23) \n(1.50) \nINIT_SHOCK \n0.0166 \n0.0193 \n0.0343*** \n0.0343 \n \n \n \n \n \n(1.37) \n(1.63) \n(2.84) \n(0.98) \n \n \n \n \nREM_SHOCK \n \n \n \n \n0.0086 \n0.0181* \n0.0492* \n0.0473* \n \n \n \n \n \n(0.62) \n(1.88) \n(1.70) \n(1.72) \nBETA_PRE \n0.0352 \n0.0149 \n0.0640** \n0.0037 \n0.0278 \n0.0091 \n0.0594 \n0.0008 \n \n(1.37) \n(0.58) \n(2.06) \n(0.07) \n(1.08) \n(0.38) \n(1.62) \n(0.02) \nLASSETS2000 \n0.0110 \n0.0102 \n0.0108 \n-0.0065 \n0.0122 \n0.0108 \n0.0104 \n-0.0068 \n \n(1.23) \n(1.10) \n(1.19) \n(0.38) \n(1.39) \n(1.30) \n(0.98) \n(0.44) \nROA2000 \n-0.0160 \n0.0611** \n0.0172 \n-0.1202 \n-0.0184 \n0.0570* \n0.0080 \n-0.1212 \n \n(0.35) \n(2.04) \n(0.44) \n(1.15) \n(0.40) \n(1.97) \n(0.21) \n(1.13) \nDERATIO2000 \n-0.1149** \n-0.0627 \n-0.0060 \n0.0448 \n-0.1185** \n-0.0625 \n0.0038 \n0.0500 \n \n(2.03) \n(1.31) \n(0.08) \n(0.36) \n(2.01) \n(1.38) \n(0.05) \n(0.39) \nMB2000 \n0.0000 \n-0.0003 \n-0.0009** \n0.0008 \n0.0001 \n-0.0003 \n-0.0009** \n0.0007 \n \n(0.07) \n(1.00) \n(2.30) \n(0.21) \n(0.09) \n(1.06) \n(2.27) \n(0.20) \nPPE/TA2000 \n-0.0176 \n-0.0989** \n0.1003 \n-0.1234 \n-0.0224 \n-0.1074** \n0.0819 \n-0.1428 \n \n(0.28) \n(2.02) \n(1.03) \n(0.69) \n(0.36) \n(2.23) \n(0.90) \n(0.80) \nIndustry controls \nYES \nYES \nYES \nYES \nYES \nYES \nYES \nYES \n \n \n \n \n \n \n \n \n \nObservations \n1,069 \n1,109 \n1,096 \n1,257 \n1,069 \n1,109 \n1,096 \n1,257 \nR-squared \n0.0505 \n0.0411 \n0.0163 \n0.0107 \n0.0496 \n0.0405 \n0.0192 \n0.0114 \n \n\n \nTable 3 (continued) \nRobustness analysis with respect to the beta shocks \nRobustness analysis of the results presented in Table 3. All of the models regress the percent change in page counts (%ΔPAGES) on either the initial shock or \nremaining shock, the pre-event beta, and the control variables from Models (3) and (7) in Table 3. In Model 1, INIT_SHOCK, REM_SHOCK, and the pre-event \nbeta are estimated using coefficient estimates from a linear market model that includes the CRSP value-weighted return on the market, rather than from the \nquadratic model described in Table 2. In Model 2, the sample excludes firms with a stock price at December 31, 2001 of less than $5. In Model 3, we include an \nadditional control variable that measures the change in the mean (consensus) analyst forecast of next-fiscal-year annual earnings during the period from August \n2001 to December 2001 scaled by price (FCSTREV). In Model 4, we include an additional control variable that measures the firm’s idiosyncratic risk during the \npre-event period (FIRMRETVOL). Idiosyncratic risk is the standard deviation of the residual from the linear market model that includes the CRSP value-\nweighted return on the market. The t-statistics are reported in parentheses and are based on standard errors that are clustered at the two-digit SIC level. *** {**} \n(*) indicate significance at the 1%, 5%, 10% level (two-sided). \nPanel C: \n \n(1) \n(2) \n(3) \n(4) \n \nShocks computed using \nlinear market-model betas \nSample excludes stocks with \nprice < $5 \nModel includes control for \nanalyst forecast revision \nactivity \nModel includes control for \nidiosyncratic risk volatility \n \n \n \n \n \n \n \n \n \nConstant \n0.0593 \n0.0506 \n0.1054 \n0.0910 \n0.1624*** \n0.1694*** \n0.0438 \n0.0479 \n \n(1.12) \n(0.97) \n(1.62) \n(1.45) \n(2.85) \n(2.90) \n(0.71) \n(0.80) \nINIT_SHOCK \n0.0428*** \n \n0.0308*** \n \n0.0276*** \n \n0.0248*** \n \n \n(3.82) \n \n(3.34) \n \n(2.70) \n \n(3.54) \n \nREM_SHOCK \n \n0.0211*** \n \n0.0323*** \n \n0.0327*** \n \n0.0229*** \n \n \n(3.29) \n \n(2.74) \n \n(3.24) \n \n(3.00) \nAdditional control variables: \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFCSTREV \n \n \n \n \n-0.2652* \n-0.2935* \n \n \n \n \n \n \n \n(1.68) \n(1.89) \n \n \nFIRMRETVOL \n \n \n \n \n \n \n0.9826** \n0.9967** \n \n \n \n \n \n \n \n(2.10) \n(2.18) \n \n \n \n \n \n \n \n \n \nAll controls \nYES \nYES \nYES \nYES \nYES \nYES \nYES \nYES \n \n \n \n \n \n \n \n \n \nObservations \n1,784 \n1,784 \n1,387 \n1,387 \n1,220 \n1,220 \n1,792 \n1,792 \nR-squared \n0.0282 \n0.0245 \n0.0219 \n0.0224 \n0.0180 \n0.0194 \n0.0265 \n0.0257 \n \n\n \nTable 4 \nCross-sectional analysis of the relation between disclosure changes and cost of capital shocks \nPanels A and B present results on the relation between the beta shocks and firms’ disclosure choices for firms with a positive beta shock and high financing \nneeds. The first four columns present results partitioning the sample based on a positive initial or remaining shock and high financing needs (POS&HIGH=1), \nwhere the latter is defined as having a cash flow from investing activities scaled by total assets above the sample median. The last four columns present results \npartitioning the sample based on a positive initial or remaining shock and high financing needs (POS&HIGH=1), where the latter is defined as having external \nfinancing needs above the median for the sample. External financing needs are measured as the difference between a firm’s average asset growth rate over two \nyears and the maximum growth it can finance internally, which is ROA/(1-ROA), where ROA is income before extraordinary items over total assets. Panel A \npresents results for the percent change in the page counts of the 10-K (%ΔPAGES). Panel B presents results for the percent change in the page counts in the \nMD&A (%ΔMDA). The regression models include control variables as defined for Models (2) and (6) in Table 3 and industry-fixed effects based on one-digit \nSIC codes and a binary indicator for Enron-related industries. The coefficients of the control variables are not presented. The t-statistics are reported in \nparentheses and based on standard errors that are clustered at two-digit SIC level. *** {**} (*) indicate significance at the 1%, 5%, 10% level (two-sided). The \nlast row presents the p-value of a (Chow) test whether the coefficients on the beta shock are statistically different across the two subsamples. \n \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n(8) \nProxy for financing needs: \nCash flow from Investment \nExternal financing needs \n \nPOS&HIGH \nPOS&HIGH \nPOS&HIGH \nPOS&HIGH \n \n=1 \n=0 \n=1 \n=0 \n=1 \n=0 \n=1 \n=0 \nPanel A: % page changes in the 10-K \n \n \n \n \n \n \nConstant \n0.1563 \n0.1155** \n-0.0149 \n0.0946** \n0.0482 \n0.1097* \n-0.0224 \n0.0034 \n \n(1.12) \n(2.52) \n(-0.22) \n(2.49) \n(0.72) \n(1.83) \n(-0.32) \n(0.08) \nINIT_SHOCK \n0.0521** \n0.0167** \n \n \n0.0348 \n0.0202** \n \n \n \n(2.04) \n(2.25) \n \n \n(1.44) \n(2.34) \n \n \nREM_SHOCK \n \n \n0.0495 \n0.0108 \n \n \n0.0669** \n0.0101 \n \n \n \n(1.31) \n(1.27) \n \n \n(2.53) \n(1.02) \nAll controls \nYES \nYES \nYES \nYES \nYES \nYES \nYES \nYES \n \n \n \n \n \n \n \n \n \nObservations \n376 \n1,127 \n433 \n1,070 \n444 \n1,349 \n512 \n1,281 \nR-squared \n0.0285 \n0.0286 \n0.0498 \n0.0237 \n0.0347 \n0.0249 \n0.0466 \n0.0260 \np-value for difference \n0.1401 \n0.3075 \n0.5743 \n0.0389 \nPanel B: % page changes in MDA \n \n \n \n \n \n \nConstant \n0.6157 \n0.2729*** \n0.6980 \n0.2747*** \n0.2116* \n0.2994*** \n0.2564** \n0.3096*** \n \n(1.32) \n(3.44) \n(1.50) \n(3.52) \n(1.82) \n(4.08) \n(2.54) \n(4.25) \nINIT_SHOCK \n0.1121** \n-0.0049 \n \n \n0.0509 \n-0.0144 \n \n \n \n(2.03) \n(-0.35) \n \n \n(1.07) \n(-1.01) \n \n \nREM_SHOCK \n \n \n0.0447 \n-0.0117 \n \n \n0.0106 \n0.0031 \n \n \n \n(0.90) \n(-0.71) \n \n \n(0.28) \n(0.19) \nAll controls \nYES \nYES \nYES \nYES \nYES \nYES \nYES \nYES \n \n \n \n \n \n \n \n \n \nObservations \n232 \n695 \n232 \n695 \n280 \n789 \n280 \n789 \nR-squared \n0.1066 \n0.0249 \n0.0968 \n0.0255 \n0.0627 \n0.0589 \n0.0600 \n0.0581 \np-value for difference \n0.0478 \n0.2718 \n0.1515 \n0.8585 \n\n \nTable 4 (continued) \nCross-sectional analysis of the relation between disclosure changes and cost of capital shocks \nPanel C results on the relation between the beta shocks and firms’ disclosure choices for Arthur Andersen audit clients versus other firms. The first two columns \nuse %ΔPAGES as dependent variable. The next columns focus on page changes in the MD&A and the last two columns report results for regressions using \n%ΔREL_PAR. The regression models include control variables as defined for Models (2) and (6) in Table 3 and industry-fixed effects based on one-digit SIC \ncodes and a binary indicator for Enron-related industries. The t-statistics are reported in parentheses and based on standard errors that are clustered at two-digit \nSIC level. *** {**} (*) indicate significance at the 1%, 5%, 10% level (two-sided). The last row presents the p-value of a (Chow) test whether the coefficients on \nthe beta shock are statistically different across the two subsamples. \nPanel C: Split by auditor \n \n \n \n%ΔPAGES \n%ΔΜDA \n%ΔREL_PAR \n \nAA clients \nNon-AA clients \nAA clients \nNon-AA clients \nAA clients \nNon-AA clients \nConstant \n0.1307 \n0.0211 \n0.3094** \n0.3931*** \n0.1515 \n0.4364** \n \n(1.20) \n(0.38) \n(2.07) \n(5.01) \n(0.34) \n(2.00) \nINIT_SHOCK \n0.0612*** \n0.0193** \n0.0241 \n0.0149 \n0.2990*** \n-0.0222 \n \n(2.74) \n(2.11) \n(0.75) \n(0.91) \n(3.66) \n(0.66) \nBETA_PRE \n0.0519 \n0.0434*** \n0.0044 \n0.0319 \n0.2084* \n-0.0435 \n \n(1.54) \n(3.01) \n(0.09) \n(1.19) \n(1.65) \n(0.81) \nAll controls \nYES \nYES \nYES \nYES \nYES \nYES \n \n \n \n \n \n \n \nObservations \n333 \n1,460 \n206 \n863 \n249 \n1,008 \nR-squared \n0.0764 \n0.0205 \n0.0755 \n0.0644 \n0.1083 \n0.0104 \np-value for difference \n0.0439 \n0.6963 \n0.0016 \n \n\n \nTable 5 \nRelation between 10-K disclosure changes and cost of capital shocks controlling for interim disclosure changes \nResults from regressions of the percent change in pages (%ΔPAGES) on the cost of capital shocks and proxies for firms’ interim disclosures. The interim \ndisclosure proxies are: Δ8KCOUNT, which equals the change in the number of 8-K filings from 2000 to 2001; ΔCALL, which equals the change in the number \nof conference calls; %ΔWORDS, which is the percentage change in the number of words in the fourth quarter 2001 earnings announcement; and a market-based \nmeasure of interim disclosure, the parameter \n2\nibˆ\n from the regime-switching model. See Tables 1 and 2 for more details on these variables. All models include \ncontrol variables as defined for Models (2) and (6) in Table 3 and industry-fixed effects based on one-digit SIC codes and a binary indicator for Enron-related \nindustries. The t-statistics are reported in parentheses and based on standard errors that are clustered at two-digit SIC level. *** {**} (*) indicate significance at \nthe 1%, 5%, 10% level (two-sided), respectively. \n \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \nConstant \n0.0990* \n0.1040* \n0.1019* \n0.1070** \n0.0603 \n0.0486 \n0.1110** \n \n(1.88) \n(1.98) \n(1.93) \n(2.03) \n(1.07) \n(0.88) \n(2.12) \nINIT_SHOCK \n0.0254*** \n \n0.0258*** \n \n0.0259*** \n \n \n \n(3.78) \n \n(3.87) \n \n(3.84) \n \n \nREM_SHOCK \n \n0.0244*** \n \n0.0241*** \n \n0.0299*** \n0.0334*** \n \n \n(3.24) \n \n(3.15) \n \n(3.56) \n(3.78) \nBETA_PRE \n0.0431** \n0.0369** \n0.0434** \n0.0368** \n0.0477** \n0.0423** \n0.0495** \n \n(2.34) \n(2.04) \n(2.36) \n(2.03) \n(2.41) \n(2.14) \n(2.47) \nLASSETS2000 \n0.0084** \n0.0091** \n0.0078** \n0.0085** \n0.0092** \n0.0095** \n0.0061 \n \n(2.21) \n(2.23) \n(2.10) \n(2.13) \n(2.39) \n(2.30) \n(1.53) \nROA2000 \n-0.0013 \n-0.0016 \n0.0005 \n0.0005 \n-0.0133 \n-0.0155 \n0.0011 \n \n(0.06) \n(0.08) \n(0.02) \n(0.02) \n(0.58) \n(0.66) \n(0.05) \nDERATIO2000 \n0.0008 \n0.0006 \n0.0016 \n0.0012 \n0.0034 \n0.0037 \n0.0083 \n \n(0.02) \n(0.02) \n(0.05) \n(0.04) \n(0.09) \n(0.10) \n(0.24) \nMB2000 \n-0.0004* \n-0.0003 \n-0.0003* \n-0.0003 \n-0.0003 \n-0.0003 \n-0.0004* \n \n(1.78) \n(1.65) \n(1.69) \n(1.53) \n(1.45) \n(1.40) \n(1.89) \nPPE/TA2000 \n0.0439 \n0.0325 \n0.0427 \n0.0312 \n0.0446 \n0.0323 \n0.0362 \n \n(0.69) \n(0.52) \n(0.68) \n(0.50) \n(0.66) \n(0.48) \n(0.58) \nIndustry controls \nYES \nYES \nYES \nYES \nYES \nYES \nYES \nΔ8KCOUNT \n0.0080* \n0.0092* \n \n \n \n(1.77) \n(1.96) \n \n \nΔCALL \n \n \n-0.0032 \n-0.0027 \n \n \n \n(0.45) \n(0.39) \n%ΔWORDS \n \n \n \n \n0.0154\n0.0163\n \n \n \n \n \n(0.68)\n(0.72)\n2\nib\n \n \n8.4903***\n \n \n(3.63)\nObservations \n1,793 \n1,793 \n1,793 \n1,793 \n1,646 \n1,646 \n1,793 \nR-squared \n0.0253 \n0.0249 \n0.0245 \n0.0238 \n0.0275 \n0.0288 \n0.0279 \n\n \nTable 6 \nRelation between interim disclosures and cost of capital shocks \nThe first four columns of the table present results from four regression models of proxies for interim disclosures on \ncombinations of initial shock (INIT_SHOCK), remaining shock (REM_SHOCK), pre-event beta (BETA_PRE), and \ncontrol variables. Column 1 presents results for changes in the number of 8-K filings as defined in Table 1 \n(Δ8KCOUNT). In Column 2, we restrict the analysis to changes in the number of 8-K filings during the second-half of \nthe event period and hence before the pre-report period during which the remaining shock is measured. Column 3 \npresents results for changes in the number of conference calls as defined in Table 1 (ΔCALL). In Column 4, we restrict \nthe analysis to changes in the number of conference calls during the second-half of the event period. Columns 5 and 6 \npresent results from regressing \n2\nibˆ\n on Δ8KCOUNT and ΔCALL, respectively, relating the curvature of the quadratic \nbeta estimate to firms’ immediate disclosure responses. The control variables are measured as of fiscal year 2000. The \nmodels include industry-fixed effects based on one-digit SIC codes and a binary indicator for Enron-related industries. \nThe t-statistics are reported in parentheses and based on standard errors that are clustered at two-digit SIC level. \n*** {**} (*) indicates significance at the 1%, 5%, 10% level in a two-sided test, respectively. \n \nΔ8KCOUNT \nΔCALL \n2\nibˆ\n \n \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \nConstant \n0.3157*** \n0.1103*** \n0.1143 \n0.0333* \n-0.0049 \n-0.0009 \n \n(3.21) \n(3.56) \n(1.39) \n(1.69) \n(0.18) \n(0.03) \nINIT_SHOCK \n0.0637** \n \n-0.0136 \n \n \n \n \n(2.19) \n \n(0.71) \n \n \n \nREM_SHOCK \n \n-0.0220** \n \n-0.0085 \n \n \n \n \n(2.25) \n \n(1.34) \n \n \nBETA_PRE \n0.0412 \n-0.0154 \n-0.0157 \n0.0029 \n-0.1128*** \n-0.1127*** \n \n(0.76) \n(1.28) \n(0.44) \n(0.29) \n(16.15) \n(15.72) \nLASSETS2000 \n-0.0724*** \n0.0211*** \n-0.0114 \n0.0013 \n0.0177*** \n0.0186*** \n \n(3.98) \n(3.52) \n(0.82) \n(0.32) \n(3.95) \n(4.15) \nROA2000 \n0.2421 \n-0.1069*** \n-0.0434 \n-0.0081 \n0.0021 \n-0.0035 \n \n(1.36) \n(3.39) \n(0.79) \n(0.33) \n(0.07) \n(0.12) \nDERATIO2000 \n0.1102 \n-0.0581 \n-0.0437 \n0.0269 \n-0.0426 \n-0.0457 \n \n(0.81) \n(1.60) \n(0.66) \n(0.85) \n(1.47) \n(1.57) \nMB2000 \n-0.0001 \n0.0010 \n0.0029* \n0.0006 \n0.0004 \n0.0005 \n \n(0.03) \n(1.17) \n(1.95) \n(1.04) \n(1.26) \n(1.35) \nPPE/TA2000 \n-0.0967 \n0.0001 \n-0.1373 \n-0.0136 \n-0.0852** \n-0.0867** \n \n(0.67) \n(0.00) \n(1.38) \n(0.28) \n(2.19) \n(2.23) \nΔ8KCOUNT \n \n \n \n \n0.0247** \n \n \n \n \n \n \n(2.63) \n \nΔCALL \n \n \n \n \n \n0.0080 \n \n \n \n \n \n \n(0.57) \nIndustry controls \nYES \nYES \nYES \nYES \nYES \nYES \nObservations \n1,793 \n1,793 \n1,793 \n1,793 \n1,793 \n1,793 \nR-squared \n0.0204 \n0.0335 \n0.0122 \n0.0085 \n0.0770 \n0.0745 \n\n \nTable 7 \nAnalysis of market responses to the changes in the 10-K filings \nPanel A presents an analysis of beta response from the event period to the post-report period: BRESPONSE = \nBETA_EVT - BETA_POST. The models include page changes in the 10-K (%ΔPAGES) and control variables. \nLSIZE is the natural log of firm size as defined in Table 1. DERATIO is long-term debt scaled by total assets and \nMB is the market-to-book ratio as defined in Table 1. The control variables are measured as of fiscal year 2001. \nThe changes in the controls are measured from 2000 to 2001. The t-statistics are reported in parentheses and based \non standard errors that are clustered at the firm level. *** {**} (*) indicate significance at the 1%, 5%, 10% level \n(two-sided). \nPanel A: Post 10-K beta response \n \n \n \n(1) \n(2) \n(3) \nConstant \n-0.2543** \n-0.2957*** \n-0.2062* \n \n(2.32) \n(2.66) \n(1.77) \n%ΔPAGES \n0.1726*** \n0.1767*** \n0.1649*** \n \n(2.82) \n(2.81) \n(2.58) \nLSIZE2001 \n0.0106 \n0.0106 \n0.0166 \n \n(1.03) \n(0.93) \n(1.43) \nDERATIO2001 \n \n0.0353 \n-0.0133 \n \n \n(0.35) \n(0.13) \nMB2001 \n \n-0.0003 \n-0.0013 \n \n \n(0.04) \n(0.19) \nΔLSIZE \n \n \n-0.0438 \n \n \n \n(0.65) \nΔDERATIO \n \n \n0.3148 \n \n \n \n(1.36) \nΔMB \n \n \n0.0609 \n \n \n \n(0.88) \nIndustry controls \nYES \nYES \nYES \n \n \n \n \nObservations \n1,794 \n1,700 \n1,641 \nR-squared \n0.0168 \n0.0178 \n0.0201 \n \n\n \nTable 7 (continued) \nAnalysis of market responses to the changes in the 10-K filings \nPanel B presents models the short-window abnormal volume response around the 10-K filing on the percentage \nchange in 10-K pages and control variables. Abnormal volume is defined as stated in Section 6. We include the \nabnormal volume generated by last year’s 10-K filing as a control (ABN_VOL). FIL_DELAY measures the number \nof days the 10-K filing is delayed relative to the filing dates in 2000, adjusted for its delay relative to 1999. LSIZE is \nthe natural log of firm size as defined in Table 1, DERATIO is long-term debt scaled by total assets and MB is the \nmarket-to-book ratio as defined in Table 1. The control variables are measured as of fiscal year 2001. The \nregression models include industry fixed effects. The t-statistics are reported in parentheses and based on standard \nerrors that are clustered at the firm level. *** {**} (*) indicate significance at the 1%, 5%, 10% level (two-sided). \nPanel B: Short-window abnormal volume response to the 10-K \n \n(1) \n(2) \n(3) \nConstant \n1.0010 \n0.9116 \n0.8903 \n \n(1.62) \n(1.34) \n(1.47) \n%ΔPAGES \n0.1040* \n0.1127* \n0.1138* \n \n(1.66) \n(1.71) \n(1.83) \nABN_VOL2000 \n0.0465 \n0.0316 \n0.0323 \n \n(1.62) \n(1.10) \n(1.15) \nFIL_DELAY0199 \n-0.0015 \n-0.0013 \n-0.0022 \n \n(0.86) \n(0.66) \n(1.29) \nLSIZE2001 \n-0.0242** \n-0.0265** \n \n \n(2.46) \n(2.30) \n \nMB2001 \n \n0.0037 \n \n \n \n(0.39) \n \nDERATIO2001 \n \n \n0.1379 \n \n \n \n(1.47) \nLASSETS2001 \n \n \n-0.0357*** \n \n \n \n(2.89) \nROA2001 \n \n \n0.1681 \n \n \n \n(1.36) \nIndustry controls \nYES \nYES \nYES \n \n \n \n \nObservations \n1,389 \n1,316 \n1,361 \nR-squared \n0.0173 \n0.0174 \n0.0205"},"Token_count":{"kind":"number","value":38186,"string":"38,186"}}},{"rowIdx":6697,"cells":{"Source":{"kind":"string","value":"NBER"},"Date":{"kind":"number","value":2009,"string":"2,009"},"Text":{"kind":"string","value":"NBER WORKING PAPER SERIES\nINTERVIEWING IN TWO-SIDED MATCHING MARKETS\nRobin S. Lee\nMichael Schwarz\nWorking Paper 14922\nhttp://www.nber.org/papers/w14922\nNATIONAL BUREAU OF ECONOMIC RESEARCH\n1050 Massachusetts Avenue\nCambridge, MA 02138\nApril 2009, Revised March 2017\nWe thank Neil Thakral for exceptional research assistance, and Itay Fainmesser, Kyna Fong, \nRachel Kranton, David McAdams, Michael Ostrovsky, Al Roth, Lones Smith, two anonymous \nreferees and the editor for helpful comments. All errors are our own. The views expressed herein \nare those of the author(s) and do not necessarily reflect the views of the National Bureau of \nEconomic Research.\nNBER working papers are circulated for discussion and comment purposes. They have not been \npeer-reviewed or been subject to the review by the NBER Board of Directors that accompanies \nofficial NBER publications.\n© 2009 by Robin S. Lee and Michael Schwarz. All rights reserved. Short sections of text, not to \nexceed two paragraphs, may be quoted without explicit permission provided that full credit, \nincluding © notice, is given to the source.\n\nInterviewing in Two-Sided Matching Markets \nRobin S. Lee and Michael Schwarz\nNBER Working Paper No. 14922\nApril 2009, Revised March 2017\nJEL No. C78,D85,J01\nABSTRACT\nWe introduce the interview assignment problem, which generalizes the one-to-one matching \nmodel of Gale and Shapley (1962) by introducing a stage of costly information acquisition. Firms \nlearn preferences over workers via costly interviews. Even if all firms and workers conduct the \nsame number of interviews, realized unemployment depends also on the extent to which agents \nshare common interviewing partners. We introduce the concept of overlap that captures this \nnotion, and prove that unemployment is minimized with perfect overlap: i.e., if two firms \ninterview any common worker, they interview the exact same set of workers.\nRobin S. Lee\nDepartment of Economics\nHarvard University\nLittauer Center 120\nCambridge, MA 02138\nand NBER\nrobinlee@fas.harvard.edu\nMichael Schwarz\nGooge\nschwarz.m@gmail.com\n\n1\nIntroduction\nThe theory of two-sided matching generally assumes that agents know their true preferences over\npotential partners prior to engaging in a match.1 However, in many matching markets, information\nacquisition plays an important role: in labor markets, firms interview workers; in marriage markets,\nmen and women date; and in real estate markets, buyers attend open-houses. Such interviews,\ndates, and meetings are often costly and scarce—e.g., firms pay up to a third of an employee’s\nannual salary as commission to headhunters just to narrow down the potential field of candidates to\ninterview (cf. Coles et al., 2008)— and because these interviews affect the formation of preferences,\ntheir assignment can crucially affect the outcome of the eventual matching process.\nOur article generalizes the one-to-one matching model of Gale and Shapley (1962) to allow for a\nstage of costly information acquisition. To our knowledge, this article is one of the first to analyze\nthe interview assignment problem in the context of two-sided matching. Throughout this article\nwe will refer to agents as “firms” and “workers,” but this label can be changed to men and women,\ncolleges and students, hospitals and doctors, and so forth. Firms and workers do not ex ante know\ntheir idiosyncratic preferences over potential matching partners and must discover them through a\ncostly interviewing process.\nWe analyze a two-stage game: in the first stage, firms simultaneously choose a subset of work-\ners to interview, and in the second stage firms and workers are matched one-to-one using a firm-\nproposing deferred acceptance algorithm in which firms make “job offers” to workers. We primarily\nfocus on the first stage interviewing decisions, and use standard results from the one-to-one match-\ning literature to analyze the second stage assignment. Even so, the interview assignment problem\nis generally difficult and possibly intractable. To allow for analysis while still maintaining a model\nrich enough to yield meaningful results, we make the following assumptions: firms bear the full\ncost of interviewing; a firm and worker must interview in order to be matched; workers prefer being\nmatched to any firm than to being unemployed; firms may find some workers unacceptable and\nchoose to remain unmatched; and workers and firms are ex ante homogeneous, with preferences\nover partners independent and idiosyncratic to each agent.2\nEven if all firms and workers are ex ante identical (prior to the realization of their idiosyncratic\npreferences), our model emphasizes that agents are not indifferent over whom they interview with.\nAs interviews are costly, firms care about how many interviews a potential interviewee has: as\nthe number of interviews a worker has increases, the probability of a job offer being accepted\ndeclines as the worker might obtain and accept an offer from another firm. All else being equal,\nworkers who have few interviews are more attractive to interview because they are more likely\n1For a survey, see Roth and Sotomayor (1990).\n2Although homogeneity is a strong assumption to apply for an entire market, it is potentially plausible if a market\nis partitioned into subsets of workers and firms of similar quality. For example, although workers in general may\nhave a greater likelihood of preferring a “top-tier” firm in the entire market over a “bottom-tier” firm, within a tier\nworker preferences may be fairly evenly distributed; this also may be true with firm preferences over workers within\na tier. Thus, in the sense that workers and firms of similar quality are the only ones that interview and match with\neach other, and—conditional on all available information—worker and firm preferences are generally uniform and\ni.i.d. among a particular subset of agents, our model is less restrictive than it might appear.\n2\n\nto accept if an offer is made.\nAs a result, even though there are equilibria in which firms all\nconduct the same number of interviews, firms would prefer not to randomize over workers but\ninstead coordinate so that all workers receive the same number of interviews. We prove that for\nany number of interviews conducted by all firms, there are values of interviewing costs where firms\nrandomly choosing workers to interview comprise an equilibrium of our game; additionally, there\nare (potentially different) values of interviewing costs so that all firms and workers conducting the\nsame number of interviews is also an equilibrium.\nWe then investigate a more subtle form of coordination which also proves to have significant\neffects on equilibrium employment. For our analysis, we focus on symmetric interview assignments,\nwhich we define to be interview assignments in which each firm and worker has the same number\nof interviews, and even if two firms or two workers swap their interview assignments, the original\ninterview assignment can be obtained via a renaming of other agents. Although there may exist\nmany equilibria in which all agents conduct the same number of interviews, the number of expected\nhires in the match can be very different depending on whether or not the interview assignment\nexhibits greater overlap—a partial ordering that we define formally which captures the number of\ncommon interview partners among agents.\nTo illustrate why overlap matters, consider two firms, f and f′, who are the only firms that\ninterview workers w and w′: if f has an offer rejected by w, it must be that the worker accepted an\noffer from f′; consequently, f will then face no “competition” for w′ and obtain him for certain if\nit makes him an offer. If f and f′ did not interview the same set of workers, then f could possibly\nbe rejected by both w and w′ and not be matched despite making offers to both workers (because\nbeing rejected by w no longer implies obtaining w′ for certain). Thus, a firm’s expected payoff\ndepends not only on the number of interviews its workers receive, but also the identities of the\nfirms interviewing those workers.\nThe main result of this article is that if the ex ante probability that a worker is found to be\nunacceptable to a firm is sufficiently small, then among all symmetric interview assignments in\nwhich every firm and worker conducts the same number of interviews, any assignment with perfect\noverlap—i.e., if any two firms interview the same worker, they interview exactly the same set of\nworkers—minimizes unemployment for any firm or worker. We note that this result is also not\nfragile to the i.i.d. assumption over preferences: e.g., an interview assignment with perfect overlap\nalso minimizes unemployment when firm preferences over workers are perfectly correlated.\nWe also use simulation results to quantify the impact on unemployment from two forms of\ncoordination if all firms interview the same number of workers: (i) allowing firms to ensure that\nall workers receive the same number of interviews (as opposed to randomly assigning interviews),\nand (ii) conditional on all workers receiving the same number of interviews, increasing the amount\nof overlap in the interview assignment. We find that unemployment reductions from both forms of\ncoordination can be substantial.\nOur analysis suggests that institutions which, conditional on the total number of interviews that\nare conducted, limit the number of interviews that candidates receive or segment the market such\n3\n\nthat firms only interview workers within well defined categories can improve coordination, reduce\ninterviewing costs, and increase the number of successful matches. For example, with on-campus\nrecruiting at colleges, interviews are conducted on only a limited number of days, thereby making\ntime considerations a limiting factor on the number of interviews any given candidate can feasibly\nconduct; in academic job markets, placement officers aid in identifying candidates who have not\nreceived many interviews; and the “rush” system by which sororities on college campuses recruit\nnew members limits and equalizes interviews: “[a] rushee who receives more invitations than the\nnumber of parties permitted in a given round must decline, or ‘regret,’ the excess invitations”\n(Mongell and Roth, 1991). Our model’s reliance on a single simultaneous round of interviewing can\nalso be motivated by time constraints in these and other examples. With the possibility of delay,\nfirms may benefit from engaging in sequential rounds of interviews; this is outside of the scope of\nthe current analysis.\nExamples of improving overlap include segmentation via geography (e.g., firms interviewing\nonly local candidates) and other (potentially arbitrary) dimensions. Note that if the number of\ninterviews each firm conducts in equilibrium is close to the total number of workers or if the\npopulation could be partitioned such that agents in each group can only interview other agents\nin that group, interviews would be more evenly dispersed and the assignment would be closer to\nachieving perfect overlap. Thus, even if academic departments have no preference over the field in\nwhich to hire, each may still benefit from restricting its candidate search to a particular field if all\ndepartments did the same.\nFinally, we wish to emphasize that the use of the second stage “match” is only an approximation\nfor the dynamics of hiring processes in a variety of industries and settings. In some situations that\nutilize a centralized match such as the National Residency Matching Program, the relationship\nis quite exact; in others, a decentralized matching market may still be modeled as a centralized\nmechanism such as a deferred acceptance procedure given certain assumptions (Niederle and Yariv,\n2009; Schweinzer, 2008; Haeringer and Wooders, 2011). Whenever preferences are ex ante unknown\nand need be revealed through a costly interview, due diligence, or dating process, our analysis\nremains relevant.\nRelated Literature.\nThe interview assignment problem is a variant of the many-to-many match-\ning problem because firms may be assigned to many workers and workers to many firms in the\ninterview stage (Roth, 1984; Blair, 1988; Echenique and Oviedo, 2006; Konishi and ¨Unver, 2006).\nIn contrast to this literature, our model allows for externalities imposed on agents not directly\ninvolved in a particular pairwise match as firms care about the identities of other firms who in-\nterview its candidates.\nAdditionally, although we assume our second-stage assignment satisfies\nnon-equilibrium concepts such as pairwise stability, if we interpret this assignment as being gener-\nated from a centralized mechanism employing the deferred-acceptance algorithm, we are also able\nto utilize standard non-cooperative equilibrium conditions when analyzing interview assignments.\nIndeed, the special structure of the interview assignment problem considered herein enables us to\n4\n\nsolve for the equilibrium allocations.\nThe matching literature also generally assumes agents know their preferences before entering a\nmatch. There are notable exceptions: Das and Kamenica (2005) consider sequential learning in the\ncontext of dating markets where men and women repeatedly go on dates to learn their preferences;\nChade (2006) explores marriage markets where participants observe a noisy signal about the quality\nof their potential mates; Chakraborty, Citanna and Ostrovsky (2010) investigate the stability of\nmatching mechanisms with interdependent values over partners; Josephson and Shapiro (2016)\nstudy the role of adverse selection when firms sequentially interview workers in a decentralized\nmatching process; and Hoppe, Moldovanu and Sela (2009) examine the effectiveness of signaling\nin an assortative matching environment. Furthermore, the literature on college admissions such as\nNagypal (2004) and Chade, Lewis and Smith (2014) can be reinterpreted as matching with costly\ninformation acquisition, where the applicants bear the cost of providing colleges with a noisy signal\nof their own quality.\nA related issue not addressed in this article but examined in a companion piece (Lee and\nSchwarz, 2007) is the communication or signaling of preferences between agents once they are\nknown, but prior to making offers. Coles, Kushnir and Niederle (2013) consider settings where\nworkers initially know their preferences over firms, and examine mechanisms which allow them to\nsignal to firms prior to the assignment of interviews in decentralized labor markets; they find that\nallowing for signaling can improve employment and worker welfare.\nAlthough the literature on search seems related because it also explores the role of frictions in\nmatching markets, it is quite dissimilar: the sequential and directed search literature (e.g., Shimer\nand Smith (2000); Atakan (2006); Eeckhout and Kircher (2010)) primarily investigates frictions\ndue to delay and the impact on assortative matching; the model of simultaneous search in Chade\nand Smith (2006) focuses on the actions of a single decision maker who must choose a portfolio\nof ranked stochastic options, where the probability of obtaining a particular option is assumed to\nbe exogenous.3 In contrast, we abstract away from notions of assortative matching and explicitly\nexplore the miscoordination aspects of information acquisition by focusing on the externalities that\nfirms impose on other firms via their interview decisions, thereby endogenizing the probability\nthat a firm hires a worker. As a result, the interview assignment problem instead exhibits strong\nparallels to the literature on information acquisition in mechanism design: here firms (bidders)\nmust interview (invest) to learn their private values over workers (objects), and firms’ incentives\nto learn valuations over workers are reduced the more other firms choose to do so (cf. Bergemann\nand V¨alim¨aki (2005)).\nIn a sense, our interview stage is also similar to the bipartite network formation models of\nKranton and Minehart (2000, 2001), which explore the formation of trading links between buyers\nand sellers and where prices are influenced by these connections to potential trading partners.\nAlthough the economic environment appears to be very different from a matching market, there\nis a close mathematical connection. Indeed, their papers can be reinterpreted as an interviewing\n3Lien (2013) provides an example in which the assignment of interviews may be non-assortative.\n5\n\nproblem in a market with transferable utility: a seller (firm) may only trade with (hire) a buyer\n(worker) with whom it has formed a link (interviewed). A key mathematical difference, however, is\nthat Kranton and Minehart (2000, 2001) consider models with transferable utility (prices) whereas\nour article considers a traditional matching market model without transferable utility; furthermore,\nthey do not allow for buyers to vary in their valuations over sellers, whereas idiosyncratic preferences\nare central to our analysis.\n2\nModel\n2.1\nSetup and Definitions\nThere are N workers and N firms, represented by the sets W = {w1, . . . , wN} and F = {f1, . . . , fN}.\nA worker can only work for one firm and a firm can only hire one worker; we refer to this hiring\ndecision as a match between a firm and a worker.4 The main innovation of our model is that firm\npreferences for workers are unknown prior to a match, and can only be revealed through a costly\ninterview process. Firms and workers are allowed to conduct multiple interviews, but each interview\ncosts a fixed amount c ∈R+. Interviewing costs are borne by firms, and firms are assumed to be\nrisk-neutral profit maximizers.\nWhen a firm f interviews a worker w, it learns the value of δw,f ∈R which represents the\nfirm-specific profits it realizes if it hires w. If a firm does not hire a worker, it receives a reservation\nsurplus which we will assume to be 0. We assume {δw,f}w∈W,f∈F comprises private i.i.d. draws\nfrom the continuous distribution H, where H has density h, finite first moments so that all order\nstatistics have finite expectations (David and Nagaraja, 2003), and\nR\nxdH(x) < 0 which ensures\nthat a firm would not hire a worker it has not interviewed.5 We impose the following condition:\n(EH[δ|δ ≥y] −y) is weakly decreasing in y ∀y ≥0.\n(2.1)\nThis assumption is used in later proofs to guarantee that a firm’s expected gain from interviewing an\nadditional worker does not increase as it interviews more candidates.6 A sufficient condition for this\nis that the density h is weakly log-concave, which is satisfied by the normal, exponential, uniform,\nand Weibull (for γ > 1) densities (cf. Burdett (1996)). As H is continuous, the values of {δw,f}w∈W\nfor firm f can be used to generate strict preferences Pf over workers that are interviewed, as ties\noccur with zero probability.\nIf a worker w is hired by firm f, he realizes surplus ζw,f, where {ζw,f}w∈W,f∈F comprise i.i.d.\ndraws from the continuous distribution G with finite support [ζ, ζ]. If a worker is not matched to\n4We abstract away from wage negotiations and assume that there are no wages (as in dating markets), or that\nwages are fixed or already embedded in user preferences across firms (as may be the case in markets for some entry\nlevel jobs).\n5The National Residency Matching Program is a prominent example of a market between hospitals and medical\nschool graduates which utilizes a centralized match (Roth (1984)); hospitals rarely if ever rank students whom they\ndo not interview.\n6See Lemma A.2 in the appendix.\n6\n\na firm, he receives ζ0 < ζ, and thus a worker always prefers being employed to being unemployed.\nWorker preferences are bounded to ensure workers report preferences truthfully in the second stage\nmatch of our game. For each worker w, {ζw,f}f can be used to generate strict preferences Pw over\nfirms. In our model, workers cannot communicate preferences to firms, and thus whether they\nknow their preferences prior to being interviewed or learn them afterwards does not matter in this\nmodel.7 For expositional purposes, we assume workers know their preferences over all firms prior\nto interviewing; as a firm will never make a job offer to a worker whom it never interviewed, how\na worker ranks firms that do not interview him is irrelevant.\n2.2\nTiming and Description of Game\nThe timing of the interview and matching game is as follows:\n(1) In the first stage, each firm f chooses a set of workers Wf ⊂W to interview and bears an\ninterview cost c × |Wf|, where |Wf| is the number of interviews f conducts. We assume\nthat workers do not decline interviews.8 These choices define an interview assignment η, a\ncorrespondence from the set F ∪W into itself such that f ∈η(w) if and only if w ∈η(f).\nThus η(f) ≡Wf ⊂W represents the workers interviewed by firm f under η, and η(w) ⊂F\nrepresents the set of firms that interview w. Each firm privately realizes {δw,f}w∈Wf and\nforms ordinal preferences Pf over workers.9\n(2) In the second stage, we assume that firms and workers match according to the firm optimal\nstable matching (FOSM; cf. Gale and Shapley (1962); Roth and Sotomayor (1990)) given\npreferences {Pf, Pw}f∈F,w∈W learned in stage 1.10 The realized matching is represented by\nµ, where we say w is hired by f if µ(w) = f, and w is unemployed if µ(w) = w. Similarly, we\nsay f hires w if µ(f) = w and firm f does not hire anyone if µ(f) = f.\nWe note that the FOSM used in the second stage of our game is the equilibrium outcome when\nfirms and workers engage in a firm-proposing deferred acceptance algorithm (DAA) for employment\n(Gale and Shapley, 1962), and we utilize this particular procedure as an approximation for the\noutcome of the hiring process.\nIn Appendix A.1, we describe the algorithm (required for the\nconstruction of later proofs), and we also prove that—as long as workers sufficiently dislike being\n7Coles, Kushnir and Niederle (2013) and Lee and Schwarz (2007) consider settings where workers initially know\ntheir preferences over firms, and examine mechanisms which allow them to signal to firms prior to the assignment of\ninterviews.\n8Interviewing is costless from a worker’s perspective and a worker can elect to rank a firm as unacceptable in the\nsecond stage deferred acceptance algorithm. If we assume that workers’ dis-utility from unemployment is sufficiently\nlarge (which, as discussed below, is sufficient to ensure that workers report their true preferences in a firm-proposing\ndeferred acceptance algorithm), it can be shown that a worker will not wish to decline any interview.\n9Although a firm f only learns {δw,f} for those workers it interviews, as the expected value of hiring any w /∈\nη(f) < 0, how each firm ranks workers it does not interview is irrelevant.\n10A stable match is a matching in which there is no firm and worker pair who are not matched that would prefer\nto be matched to each other than to their existing partners. Firm optimal means that no firm can do better (match\nwith a more preferred worker) in another stable matching than in the FOSM, according to the preferences used.\n7\n\nf1\nf2\nf3\nf4\nw1\nw2\nw3\nw4\n(a)\nf1\nf2\nf3\nf4\nw1\nw2\nw3\nw4\n(b)\nf1\nf2\nf3\nf4\nf5\nw1\nw2\nw3\nw4\nw5\n(c)\nf1\nf2\nf3\nf4\nf5\nw1\nw2\nw3\nw4\nw5\n(d)\nFigure 1:\nExamples of Interview Assignments between firms {f1, f2, f3, . . .} and workers\n{w1, w2, w3, . . .}.\nunemployed—the only equilibrium of the DAA is for both firms and workers to utilize their true\npreferences learned from the first-stage interviewing process.\n3\nThe Interview Assignment Problem\nHaving specified the outcome of the second stage given any realization of preferences, we now focus\non the equilibrium first stage assignment of interviews.\n3.1\nInterview Assignments\nNote any interview assignment η can be represented by a bi-partite network or graph g(η) on nodes\n{F, W}, where we say fw ∈g(η) if f interviews w under interview assignment η (i.e., w ∈η(f)).\nIt will occasionally be convenient to represent interview assignments as networks; we provide some\nexamples of interview assignments in network form in Figure 1.\n3.2\nEquilibrium Analysis\nWe now turn to the equilibrium determination of an interview assignment. Formally, each firm’s\nstrategy during the interview assignment stage is a probability measure νf over the powerset of\nall workers P(W); i.e., for a set of workers Wf, νf assigns a probability that f interviews those\nand only those workers. We say that firm f (always) interviews K workers if for any Wf ⊂W,\nνf(Wf) > 0 implies |Wf| = K; and we say that firm f interviews K workers “(uniformly) at\nrandom” if νf(Wf) > 0 if and only if |Wf| = K, and |Wf| = |W ′\nf| implies νf(Wf) = νf(W ′\nf).\nA strategy profile ν ≡{νf}f∈F is a subgame perfect Nash Equilibrium of this game if and only\nif:\nZ\nνf\nZ\nν−f\nπf(Wf, W−f) ≥\nZ\nν′\nf\nZ\nν−f\nπf(Wf, W−f) ∀ν′\nf, f\n8\n\nwhere πf(Wf, W−f) is the expected profit for firm f given it interviews the set of workers Wf,\nand all other firms interview the workers W−f ≡{Wf′}∀f′∈F,f′̸=f. We will explicitly define each\nfirm’s expected profit in the next subsection, but for now it is sufficient to note that it includes the\nexpected value of δ for the worker that a firm expects to hire, minus the costs of interviewing |Wf|\nworkers. Imposing subgame perfection implies firms anticipate the outcome of the equilibrium play\nin the second stage match when conducting interviews in the first stage.\nWe are first interested in examining equilibria in which all firms interview the same number\nof workers. We show that even among this set of equilibria, the expected number of unemployed\nworkers or the costs expended on interviewing can vary significantly.\nFirst, consider an assignment where every firm randomly selects y workers to interview. For\ncertain values of c, this assignment is an equilibrium:\nProposition 3.1. For any x ∈{0, ..., N}, there exists c > 0 such that there is an equilibrium in\nwhich each firm interviews x workers at random.\n(All proofs are located in the appendix.) Such a mixed-strategy equilibrium may be a reasonable\noutcome if firms are unable to monitor how many interviews a worker receives, and if they are\nunable to coordinate with other firms on which workers to interview; however, this may lead some\nworkers to receive more interviews than others.\nNext, consider an assignment in which every firm selects a single subset of workers so that every\nfirm and every worker receives the same number of interviews. This is equivalent to firms employing\na strategy in which each firm f assigns probability 1 to one particular element Wf ∈P(W); if these\nstrategies constitute an equilibrium, we say the correspondence η : η(f) = Wf ∀f is an pure-\nstrategy equilibrium interview assignment. Such an equilibrium may also exist:\nProposition 3.2. For any x such that N is divisible by x, there exists c > 0 such that there exists\na pure-strategy equilibrium interview assignment η in which each worker and each firm conducts\nexactly x interviews.\nBoth existence proofs in this section rely on a result established in in the Appendix (Lemma A.2)\nthat conditional on other firms utilizing a particular strategy, a given firm’s utility from interviewing\nan additional worker is decreasing in the number of workers it is already interviewing. That is, a\nfirm gains more from the kth interview it conducts (holding everyone else’s actions fixed) than it\ngains from the k + 1th. Thus, if the cost of interviewing is less than the gain from interviewing the\nkth worker but greater than the gain from interviewing the k + 1th worker for a firm, every firm\ninterviewing k workers will be an equilibrium as it will not wish to add, remove, or replace any\nworkers in its set of interviewees.11\n11Due to integer constraints, there may exist values of c for which no equilibrium where all firms interview the same\nnumber of workers exists. To see why, consider the mixed-strategy case. Assume that no firm interviews any worker,\nand let G denote the gain from a firm deviating and randomly interviewing one worker. Let G′ represent the gain\nfrom interviewing one worker when every other firm also interviews one worker at random. Clearly G′ < G as the\ngain to interviewing a worker falls when other firms may also interview that worker. Thus as long as c ∈(G′, G), no\nsymmetric mixed-strategy equilibrium where all firms interview the same number of workers exists—neither everyone\n9\n\nUnlike in the mixed-strategy case, implicit in the construction of a pure-strategy equilibrium is\na means for firms to somehow distinguish subsets of workers when they are of the same size—i.e.,\na firm must be able to differentiate Wf from W ′\nf whenever |Wf| = |W ′\nf|. Furthermore, it also\nrequires a great deal of coordination among firms in terms of exactly how to partition the space of\nworkers or which particular equilibrium to play; there are a large number of different symmetric\npure-strategy equilibria in which k interviews are conducted by each firm and each worker. As a\nconsequence, firms need not only to be able to identify which workers to interview in a particular\npure-strategy equilibrium, but also need to coordinate with all other firms which particular pure-\nstrategy equilibrium to play.\nIf firms are able to coordinate, however, the following example shows they can achieve a better\noutcome in a pure-strategy equilibrium than mixed:\nExample 3.1. Consider N = 3 and the following symmetric interview assignment in which each\nfirm interviews two workers and each worker obtains two interviews:\nf1\nf2\nf3\nw1\nw2\nw3\nη(f1) = {w1, w2},\nη(f2) = {w2, w3},\nη(f3) = {w3, w1}\n(3.1)\nAssume δ = 1 + ϵ with probability .9 and δ = −10 + ϵ with probability 0.1, where ϵ is drawn iid\nfrom a continuous distribution with arbitrarily small bounded support [−¯ϵ, ¯ϵ].12 This corresponds\nto the case where a worker is likely to generate positive surplus to a firm, but may be very costly\nif the worker does not.\nIn the next subsection, we show how one can explicitly compute each firm’s expected profit from\nan interview assignment. For now, we note that each firm’s expected profit is π ≈0.88 −2c and\nthe probability of being unmatched is approximately 0.12. Such an assignment is an equilibrium\nfor c ∈(0.10, 0.24).\nHowever, consider the case where each firm now randomly selects 2 workers to interview. From\na given firm’s perspective, there are now several possible interview assignments—some workers may\nreceive 3 interviews and others 1 or 0; or, each worker gets exactly 2 interviews; and so on. In this\ncase it is also possible to compute the expected profits (π ≈0.85 −2c) and probabilities of being\nunmatched (0.15), indicating firms (and workers) are strictly worse offex ante by mixing. These\nare equilibrium strategies for c ∈(0.11, 0.27).\n■\ninterviewing no workers nor everyone interviewing one worker is an equilibrium (and as arguments in the proof of the\nprevious proposition can show, everyone interviewing more than one worker in not an equilibrium either). However,\nthere may exist equilibria in which firms interview different numbers of workers (as in a symmetric mixed strategy\nequilibrium where firms randomize over the number of interviews that they conduct).\n12The use of ϵ is to ensure the distribution of δ is continuous with no point masses.\n10\n\n3.3\nFirms’ Expected Profits\nWe now describe some of the details required to solve example 3.1 and provide insights into the\nresults of this article.\nIn section 2, we noted that the second stage FOSM between firms and workers can be generated\nfrom a firm-proposing DAA in which firms and workers use their true preferences. As a result, each\nfirm’s expected profit from interviewing any subset of workers Wf given the actions of other firms\nW−f can be computed using the logic of the DAA. For illustrative purposes, consider the expected\nprofit of a firm f from interviewing only w:\nπf({w}, W−f) = Pr(δw,f ≥0)\n|\n{z\n}\n(1)\nE [δw,f|δw,f ≥0]\n|\n{z\n}\n(2)\nPr(f ≻w f′ ∀f′ ⊂F w|f ∈F w)\n|\n{z\n}\n(3)\n−c\nwhere F w denotes the set of firms that make a job offer to w given all other firms interview the\nsubsets of workers W−f. When we say a firm f makes a job offer to w, we are referring to the event\nthat during any stage of the DAA, firm f finds itself proposing to w; this definition is independent\nof whether w rejects the offer, holds onto it, or ultimately accepts it. The expected profit can be\nseparated into three parts: (1) the probability that a job offer is made to the worker at some stage\nof the job matching process (which here, due to only interviewing one worker, is equivalent to the\nprobability that the firm receives a positive draw on δw,f), (2) the expected surplus this worker\nwill provide contingent on being hired, and (3) the probability the worker accepts this offer from\nthe firm given that the firm makes an offer (equivalent to the probability the worker prefers the\nfirm to all other firms who make him an offer). Notice conditional on receiving a job offer from\nfirm f, a worker’s δw,f is independent of his probability of actually accepting the offer—the latter\nis a function of his other δw· draws with other firms and his own preferences, both of which are\nindependent of δw,f. Thus, the expected value of δw,f conditional on being hired is simply the\nexpected value of δw,f conditional on being made an offer, which corresponds to (2).\nIf a firm decides to interview k > 0 workers, it is equivalent to taking k “draws” on δ. The\nrealization of the jth smallest δ draw is itself a random variable, known as the jth order statistic\nwhich we denote by δj:k.13 By the logic of the DAA, we can then construct the expected profit from\ninterviewing k workers as the expected surplus from hiring the top worker of k interviews times\nthe probability of hiring him, plus the expected profit from hiring the 2nd highest worker times the\nprobability of losing the highest worker times the probability of hiring the 2nd highest worker, and\nso forth. Formally then, a firm’s expected profit from interviewing the subset Wf (where |Wf| = k):\nπf(Wf, W−f)\n=\nΛk,kP\nη\n(k) + Λk−1,kP\nη\n(k−1)\n\u0010\n1 −P\nη\n(k)\n\u0011\n+\n(3.2)\n· · · + Λ2,kP\nη\n(2)\nk\nY\ni=3\n\u0010\n1 −P\nη\n(i)\n\u0011\n+ Λ1,kP\nη\n(1)\nk\nY\ni=2\n\u0010\n1 −P\nη\n(i)\n\u0011\n−ck\n13Thus δ1:k is the smallest and δk:k is the highest draw.\n11\n\nwhere Λj,k = Pr(δj:k ≥0)E[δj:k|δj:k ≥0] is the expected value of the jth lowest worker interviewed\nconditional on him being acceptable times the probability he is acceptable (equivalent to (1) and\n(2) in the single worker example), and P\nη\n(j) represents the probability that firm f “wins” its jth\nlowest worker conditional on making him an offer as a function of the entire interview assignment\nη—i.e., firm f was rejected by all workers which would yield higher surplus, and the worker prefers\nf over any other firm that makes him an offer (equivalent to (3) in the single worker example).\nFinally, the probability a firm eventually is matched to any worker is simply equation (3.2) with\nPr(δj:k ≥0) replacing Λj,k:\nPr(µ(f) ̸= f|Wf, W−f) =\nk\nX\nj=1\n\nPr(δj:k ≥0)P\nη\n(j)\nk\nY\ni=j+1\n(1 −P\nη\n(i))\n\n\n(3.3)\nAs the probabilities P\nη\n(j) are a function of the entire interview assignment, they may be difficult\nto compute. However, one observation that aids analysis is that from a firm’s perspective, any\nworker’s preferences are randomly generated uniformly over all the firms that interview him; conse-\nquently, if n firms make a job offer to a worker at any round of the DAA, each firm considers itself\nto have a 1/n probability of being the firm that the worker accepts (i.e., of being the highest ranked\nfirm for that worker). Thus, sufficient for determining P\nη\n(j) is simply the probability distribution\nover the number of firms that “compete” by making an offer to the jth ranked worker.\nLet P η\n(j),i indicate the probability, conditioned on the event that a firm makes an offer to its jth\nlowest worker, that i other firms also make that worker a job offer. Then it follows:\nP\nη\n(j) =\nN\nX\ni=0\n1\ni + 1P η\n(j),i\nThe following example illustrates how this symmetry can be used to compute expected utilities for\nfirms:\nExample 3.2. Let N = 4, and consider the interview assignment η depicted in Figure 1(a). As in\nexample 3.1, let δ = 1 + ϵ with probability .9 and δ = −10 + ϵ with probability 0.1.\nAs all firms have symmetric interview assignments, any firm’s profits can be expressed using\n(3.2) with the same values for each P η\n(j),i:\nπ = Λ2,2\n\u0012\nP η\n(2),0 + 1\n2P η\n(2),1\n\u0013\n|\n{z\n}\nP η\n(2)\n+Λ1,2\n\u0012\n1 −P η\n(2),0 −1\n2P η\n(2),1\n\u0013\n|\n{z\n}\n1−P η\n(2)\n\u0012\nP η\n(1),0 + 1\n2P η\n(1),1\n\u0013\n|\n{z\n}\nP η\n(1)\n−2c\n(3.4)\nwhere the first term is the expected gain times the probability of hiring the most preferred worker,\nand the second term is the expected gain times the probability of hiring the second most preferred\nworker (given it lost the first choice worker).\nConsider f1. Without loss of generality, assume f1 prefers w1 to w2. Assume now that f1\n12\n\nwas rejected by w1, and now makes an offer to w2. The probability f1 faces competition for w2\nconditional on having been rejected by w1 is given by:\nP η\n(1),1 = 1\n20.99\n| {z }\n(1)\n+ 1\n20.81\n\u00141\n2\n0.99\n2\n\u0015\n|\n{z\n}\n(2)\nwhere (1) represents the probability that f2’s most preferred acceptable worker is w2, and (2)\nrepresents the probability that w2 is f2’s second most preferred worker, is also acceptable, and f2\nis rejected by w3. As f1 could only have been rejected by w1 if f4 hired w1, f3 faces no competition\nfor w4, and thus f2 can only be rejected by w3 if w3 is f3’s top acceptable choice.\nNext, examine the probability f1 faces competition for w1 conditional on making him an offer;\nthis is given by:\nP η\n(2),1 = 1\n20.99\n| {z }\n(1)\n+ 1\n20.81\n\" ˜P η\n(2),1\n2\n#\n|\n{z\n}\n(2)\nwhere (1) is the probability that f4’s top acceptable worker is also w1, and (2) is the probability\nthat f4’s top acceptable worker is w4, is rejected by w4, and then subsequently makes an offer to\nw1. Firm f4 can only lose to w4 if f3 competes for the same worker, which occurs with probability\n˜P η\n(2),1. This will be different than the probability f1 faces competition for w1, because ˜P η\n(2),1 is\nconditional on f1 making an offer to w1.\nTo determine ˜P η\n(2),1, note if w4 is f4’s top acceptable choice, f4 will face competition only if f3’s\ntop acceptable choice is also w4, or if f3 is rejected by w3 first and then offers to w4. However,\nbecause w3 will only reject f3 if f2 makes w3 an offer, and f2 will do so only if w3 is its top acceptable\nchoice (otherwise, f2 would have hired w2 as f1 is not competing for w2 given it made an offer to\nw1 first), we see ˜P η\n(2),1 = P η\n(1),1.\nAs E[δ|δ ≥0] = 1, Λ2,2 = 0.99 and Λ1,2 = 0.81; also, noting P η\n(j),0 = 1−P η\n(j),1 for j = 1, 2 allows\nus to solve for a firm’s expected profit π ≈0.86 −2c from (3.4). Thus, if a worker can generate 100\nthousand surplus for a firm with 90% probability or lose 1 million with 10% probability, a firm will\nobtain in expectation approximately 86 thousand minus the cost of two interviews.\nFurthermore, the probability that a firm remains unmatched is\nPr(δ2:2 < 0) + Pr(δ1:2 > 0)\n\" P η\n(2),1\n2\n! P η\n(1),1\n2\n!#\n+ Pr (δ2:2 > 0, δ1:2 < 0)\n\"P η\n(2),1\n2\n#\n≈0.14\n■\n4\nOverlap\nIt is not surprising that coordinating on a pure-strategy equilibrium as opposed to playing a mixed-\nstrategy equilibrium can increase the probability of employment, as demonstrated by example 3.1\n13\n\nand supported by simulation results in the next section. However, this is not the only form of\ncoordination that can be achieved by firms in order to improve outcomes. It turns out that a\nfirm cares not only about the number of interviews its interviewees are already receiving, but the\nidentities of those firms that its interviewees are interviewing with.14\nWhy does the identity of other firms matter? Consider the decision of f choosing to interview\nan additional candidate when it is already interviewing w. Firm f can choose between w′ and w′′\nwho each already have the same number of interviews, except w′ also happens to be interviewing\nwith the same firms interviewing w, whereas w′′ is not. It turns out, the distinction between w′\nand w′′ is not trivial—f will strictly prefer to interview w′. This is due to the fact that if f loses\nits first choice worker (be it w or w′) to f′, then f will face less “competition” among firms for\nits second choice worker because f′ no longer needs to match. This generalizes naturally as well:\nif firm f’s candidates all overlap with the same other firms, then it means that for every worker\nwho rejects f’s job offer, effectively one fewer firm is then “competing” for its next highest ranked\nworker.\nWe discuss our main results by focusing on a a particular set of interview assignments that we\ncall symmetric:\nDefinition 4.1. An interview assignment η is symmetric if and only if for any f′, f′′ ∈F, there\nexist bijections γF : F →F and γW : W →W where γF (f′) = f′′ such that for all f ∈F, w ∈W,\nfw ∈g(η) if and only if γF (f)γW (w) ∈g(η).\nIn other words, in a symmetric interview assignment, any firm can switch its position in a\nnetwork with any other firm and an “identical” or isomorphic network can be obtained via a\nrelabeling of all other nodes. Not only does a symmetric network imply that each firm and worker\nreceives the same number of interviews (given there are the same number of firms and workers),\nbut also (because agents are ex ante symmetric) every agent will be indifferent between keeping\nand swapping all their interviews with any other agent.\nConsider the interview assignments depicted in Figure 1.\nBoth (a) and (b) are symmetric\nassignments where each firm and worker has exactly 2 interviews. However, although (c) and (d)\nare assignments where each firm and worker has exactly 3 interviews, (c) is symmetric and (d) is\nnot. To see why, note that in (d), f1 and f2 interview the exact same candidates, but, neither\nfirms f3, f4, or f5 share the exact same set of candidates with any other firm. Thus, if f1 switched\npositions with f3 in the network, there exists no relabeling of all other nodes such that f1 interviews\nthe same three candidates with any other firm.\nFinally, we note that a symmetric interview assignment implies not only that each component\nof the network contains the same number of firms and workers, but within each component the\ninterview assignment is also symmetric.15,16 In Figure 1, all assignments have 1 component except\n14In addition, a firm cares about the identities of the firms who interview the workers who are interviewed by the\nfirms who interview the same set of workers, and so on and so forth.\n15A component of a network is a distinct subgraph in which all nodes are connected via some path in the network\n(cf. Jackson, 2008).\n16It is straightforward to show that there must be the same number of firms as workers within any component\n14\n\nfor (b) which has 2.\nWe introduce a partial ordering over different symmetric interview assignments to capture this\nidea:\nDefinition 4.2. Consider two symmetric interview assignments η and η′ in which both firms and\nworkers receive k interviews. We say η exhibits greater overlap than η′ if and only if C(g(η)) <\nC(g(η′)), where C(g(·)) is the number of firms (or workers) in any component of g(·). We define\nthe degree of non-overlap of assignment η to be d(η, k) ≡(C(g(η)) −k). We say that η exhibits\nperfect overlap if d(η, k) = 0 and minimal overlap if d(η, k) = N −k.\nRecall for any symmetric interview assignment where each firm and worker receives k interviews,\nthe associated graph will have the same number of agents across each component. The smaller the\nsize of each component across different assignments, the greater the sense in which firms “share”\nthe same set of interviewees with other firms. Indeed, the smallest number of firms a component\ncan contain is k, which occurs under perfect overlap; this implies if any two firms interview the\nsame worker, those two firms interview the same exact set of workers: i.e., η(f) ∩η(f′) ̸= ∅implies\nη(f) = η(f′) ∀f, f′.\nAlthough perhaps subtle, the concept of greater overlap can have significant effects as demon-\nstrated by the following example.\nExample 4.1. Recall in example 3.2 that a firm’s expected profits was approximately 0.86 −2c,\nand the probability a firm is unmatched was approximately 0.14. Note this interview assignment,\ngiven by Figure 1(a), exhibited minimal overlap.\nNow take the same setup of example 3.2 except now consider the interview assignment η depicted\nin Figure 1(b) which exhibits perfect overlap. In this case, if both w1 and w2 are acceptable for f1,\nthen f1 is guaranteed to hire at least one of them with certainty: if f1 loses its top choice worker,\nit means f2 hired w1 and there no longer is competition for w2. It then follows that P η\n(1),0 = 1\n(the probability of facing no competition for the second best worker, given the first best worker\nrejected the firm). Additionally, the probability f1’s top worker receives another job offer is simply\nP η\n(2),1 = 1\n2 Pr(δ2:2 > 0), which is the probability that f2’s top choice worker coincides with f1’s top\nchoice. We thus find a firm’s expected profit is:\nπf = Λ2,2\n\u0012\nP η\n(2),0 + 1\n2P η\n(2),1\n\u0013\n+ Λ1,2\n\u0012\n1 −P η\n(2),0 −1\n2P η\n(2),1\n\u0013\n−2c ≈0.95 −2c\n,\nof a symmetric interview assignment (as there are N firms and N workers). To see why there must be the same\nnumber of firms (and workers) across components, consider an assignment η with two components, the first with\nexactly K firms and the other with exactly K′ firms, K > K′. Let f be in the larger component and f ′ in the\nother.\nIf the first component contains K firms, this means there exists an alternating sequence of unique firms\nand workers {f(1), w(1), f(2), w(2), . . . , w(K−1), f(K)} which contains f such that f(i)w(i) ∈g(η) ∀i ∈{1, . . . , K}\nand w(i)f(i+1) ∈g(η) ∀i ∈{1, . . . , K −1}.\nIf η is symmetric, then there must exist a bijection γF : F →F\nsuch that γF (f) = f ′ and a bijection γW : W →W such that γF (f(i))γW (w(i)) ∈g(η) ∀i ∈{1, . . . , K} and\nγW (w(i))γW (f(i+1)) ∈g(η) ∀i ∈{1, . . . , K −1}. But this implies f ′ is in a component with at least K firms, which\nis a contradiction.\n15\n\nand the probability of remaining unmatched is\nPr (δ2:2 < 0) + Pr (δ2:2 > 0&δ1:2 < 0)\n\u00121\n2P η\n(2),1\n\u0013\n≈0.05\n.\nHence, we see that with perfect overlap, the probability that any worker or firm is unmatched is\ndrastically reduced, and that a firm generates in expectation greater surplus from the same number\nof interviews—an increase of over 10%.\nIndeed, an interview assignment with minimal overlap given by Figure 1(a) is an equilibrium\nfor firms to follow for c ∈(0.09, 0.23). On the other hand, as long as c ∈(0.05, 0.26), the interview\nassignment depicted in 1(b) with perfect overlap is an equilibrium. Consequently, for any value of\nc ∈(0.09, 0.23), both interview assignments shown in Figure 1(a) and (b) are equilibria, but the\nlatter equilibrium dominates.\n■\nThus, there may be many different pure-strategy equilibria that still assign each firm and each\nworker the same number interviews, but exhibit different degrees to which workers coincide among\nfirms.\nWe prove that if the probability that a worker is found to be undesirable is sufficiently small,\nthen among the set of symmetric interview assignments in which firms and workers receive the\nsame number of interviews, the assignments that are characterized by perfect overlap minimize the\n(ex ante) probability that any firm or worker will remain unemployed:\nTheorem 4.1. For any x such that N is divisible by x, there exists ϵ > 0 such that if Pr(δ < 0) < ϵ,\nan interview assignment with perfect overlap with x interviews per firm and worker minimizes\nexpected unemployment over any other symmetric interview assignment in which each firm and\nworker receives x interviews.\nThe intuition for the proof is as follows. Note first that the probability that any firm is un-\nmatched (and hence a worker is unemployed) under perfect overlap is bounded above by the prob-\nability that a firm finds at least one worker to be undesirable. Second, note that without perfect\noverlap, the probability that any firm is unmatched is bounded below by the probability that a\nset of preferences are realized in which all of a firm’s interviewees are matched to other firms;\nby an application of Hall’s marriage theorem, we can show that a set of preferences resulting in\nsuch a match will always exist in a symmetric interview assignment without perfect overlap. If\nthe probability that a worker is found to be undesirable is sufficiently small, the upper bound on\nthe probability of unemployment under perfect overlap is smaller than the lower bound on the\nprobability of unemployment under “non-perfect” overlap, and the result follows.\nRemarks.\nIt is worth nothing that this result is not necessarily sensitive to the assumption that\nfirm preferences over workers are i.i.d.: e.g., consider the modification that firm preferences over\nworkers are perfectly correlated such that all firms share the same common value for each worker\n(but would have to still interview to learn these values, and could not share information). In this\n16\n\nTable 1: Comparison Between Pure and Mixed-Strategy Equilibria\nPure-Strategy (Perfect Overlap)\nMixed-Strategy (N=10)\nMixed-Strategy (N=100)\nProb.\nExp.\nEquil.\nProb.\nExp.\nEquil.\nProb.\nExp.\nEquil.\nUnemp.\nProfit\nCosts\nUnemp.\nProfit\nCosts\nUnemp.\nProfit\nCosts\n# Interviews Per Agent\n1\n0.10\n0.45\n(0.10,0.45)\n0.39\n0.30\n(0.18,0.30)\n0.40\n0.30\n(0.18,0.30)\n2\n0.05\n0.54\n(0.06,0.20)\n0.23\n0.43\n(0.11,0.17)\n0.25\n0.42\n(0.12,0.17)\n3\n0.03\n0.60\n(0.04,0.12)\n0.15\n0.50\n(0.09,0.12)\n0.18\n0.49\n(0.09,0.11)\n4\n0.02\n0.64\n(0.03,0.09)\n0.10\n0.56\n(0.06,0.09)\n0.13\n0.54\n(0.06,0.08)\n5\n0.02\n0.67\n(0.03,0.06)\n0.07\n0.60\n(0.05,0.06)\n0.10\n0.58\n(0.05,0.06)\n6\n0.01\n0.69\n(0.02,0.05)\n0.05\n0.64\n(0.04,0.05)\n0.08\n0.61\n(0.04,0.05)\n7\n0.01\n0.71\n(0.02,0.04)\n0.03\n0.68\n(0.03,0.04)\n0.06\n0.63\n(0.04,0.04)\n8\n0.01\n0.73\n(0.02,0.03)\n0.02\n0.71\n(0.03,0.03)\n0.05\n0.65\n(0.03,0.04)\n9\n0.01\n0.74\n(0.01,0.03)\n0.01\n0.73\n(0.02,0.03)\n0.05\n0.67\n(0.03,0.03)\n10\n0.00\n0.76\n(0.01,0.02)\n0.00\n0.76\n(0.00,0.02)\n0.04\n0.69\n(0.02,0.03)\nNotes: 10,000 simulation runs. δw,f is distributed iid uniformly on [0, 1] with 90% probability, and δw,f ≤−10 with\n10% probability. “Prob. Unemp.” is the probability that a firm or worker remains unemployed, “Exp. Profit”\ndoes not include interviewing costs, and “Equil. Costs” is the range of c such that the assignment in question is an\nequilibrium.\nsetting, an equilibrium with perfect overlap would also maximize expected employment among all\nother symmetric equilibria, including those in mixed-strategies. To see this, note that with perfect\noverlap, any worker that generates positive surplus will be employed as long as every firm conducts\nat least 1 interview; thus the expected probability of unemployment will simply be Pr(δ < 0). On\nthe other hand, either non-perfect overlap or random interviewing introduces matching frictions\nand strictly increases unemployment.\nAlso note that due to integer constraints, for a given c and N a symmetric pure-strategy\nequilibrium with perfect overlap may not exist: the construction of equilibria is sensitive to the\nrelationship of N versus x, where x is the number of interviews per firm in equilibrium. However,\nas we show in Appendix A.3, if there exists an equilibrium with perfect overlap for N = x for a\ngiven value of c, then as N grows large there exists a correlated equilibrium in which each firm\ninterviews x workers and achieves perfect overlap with probability close to 1.\n5\nSimulation Results\nThis article has examined two forms of coordination in the assignment of interviews: (i) pure vs.\nmixed-strategy (i.e., even though firms all conduct the same number of interviews, some workers\nmay receive different numbers); and (ii) the degree of non-overlap across assignments.\nIn this\nsection, we simulate interview markets to evaluate the impact of these forms of coordination on\nemployment.\n17\n\n5.1\nPure vs. Mixed-Strategy Equilibria\nTo compare outcomes under pure and mixed-strategy equilibria (i.e., when firms conduct the same\nnumber of interviews, but may not be able to coordinate on the set of workers that they interview),\nwe simulate markets where δw,f is distributed iid uniformly on [0, 1] with 90% probability, and\nδ ≤−10 with 10% probability; this also ensures the expected value of hiring a random worker\nwithout interviewing him is negative. We allow number of interviews each firm conducts to vary\nfrom 1 to 10, where firms coordinate on a perfect overlap equilibrium, or play a mixed-strategy\nequilibrium in which the number of total agents in the market varies from 10 to 100.17\nTable 1 reports the probability that any given firm is unemployed, its expected profit not in-\ncluding interviewing costs, and the range of interviewing costs such that the given assignment is an\nequilibrium. When firms can coordinate on a pure-strategy equilibrium, the probability of unem-\nployment is bounded above by 10% (the probability a worker is unacceptable); however, under a\nmixed-strategy equilibrium it is much higher, particularly when firms interview a small number of\nworkers. Furthermore, as the number of agents in the market grows larger, a mixed-strategy equi-\nlibrium performs worse; a pure-strategy equilibrium (because the size of each component remains\nfixed) is not affected.\nWhat is also evident from this exercise is that a pure-strategy equilibrium can result in either\nmore or fewer interviews being conducted than a mixed-strategy equilibrium.18 Although we do\nnot prove this result generally, in this example, for any c under which both a symmetric pure- and\nmixed-strategy equilibria exist, firms receive strictly higher expected profit (and higher probability\nof being matched) under the pure-strategy equilibrium.\n5.2\nComparing Overlap\nWe now focus on pure-strategy equilibria with symmetric interviewing assignments, and vary the\ndegree of non-overlap across assignments and the probability a worker is determined to be unac-\nceptable (but firms still do not hire workers they do not interview).19\nTable 2 reports the probability that a firm is unmatched across different specifications. Sim-\nulations suggest that as the degree of non-overlap increases for a given number of interviews or\nprobability that a worker is found undesirable, the probability of unemployment increases. Fur-\nthermore, the increase is the largest when moving from perfect overlap to an assignment with degree\nof non-overlap of 1; as the degree of non-overlap grows, the rate at which unemployment increases\n17For all simulations involving pure-strategy equilibria, we assume that N is equal to the number of firms in each\ncomponent (i.e., the number of interviews per worker plus the degree of non-overlap of the interview assignment);\nthe number of components in the market does not affect results.\n18For example, if c ∈(0.30, 0.45), a pure-strategy equilibrium exists where every firm exactly interviews 1 worker;\nhowever, no mixed-strategy equilibrium exists where all firms randomly interview at least 1 worker. However, if\nc = 0.06, a pure-strategy exists where firms interview 4 workers; a mixed-strategy equilibrium exists where firms\nrandomly select 5 workers.\n19There may be many symmetric interview assignments with the same degree of non-overlap that differ in expected\nprofits and unemployment. For our simulations, we construct the symmetric interview assignments according the\nprocedure used in the proof of Proposition 3.2.\n18\n\nTable 2: Probability of Being Unemployed\nDegree of non-overlap (|C(g(η))| −k)\n0\n1\n2\n3\n4\n5\n10\n25\n50\n100\n(Pr(δw,f) < 0) = 0%\n# Interviews Per Agent\n1\n0.0%\n0.0%\n0.0%\n0.0%\n0.0%\n0.0%\n0.0%\n0.0%\n0.0%\n0.0%\n2\n0.0%\n6.3%\n9.5%\n10.5%\n10.9%\n11.1%\n11.1%\n11.1%\n11.1%\n11.1%\n3\n0.0%\n2.2%\n3.9%\n4.8%\n5.3%\n5.5%\n5.7%\n5.7%\n5.7%\n5.7%\n4\n0.0%\n1.4%\n2.5%\n3.4%\n4.1%\n4.5%\n5.3%\n5.5%\n5.4%\n5.4%\n5\n0.0%\n0.9%\n1.6%\n2.2%\n2.7%\n3.1%\n3.9%\n4.2%\n4.2%\n4.1%\n6\n0.0%\n0.6%\n1.0%\n1.5%\n1.9%\n2.2%\n3.2%\n3.7%\n3.7%\n3.7%\n7\n0.0%\n0.4%\n0.7%\n1.0%\n1.3%\n1.6%\n2.5%\n3.1%\n3.1%\n3.1%\n8\n0.0%\n0.3%\n0.5%\n0.7%\n0.9%\n1.2%\n1.9%\n2.7%\n2.7%\n2.7%\n9\n0.0%\n0.2%\n0.4%\n0.5%\n0.7%\n0.9%\n1.5%\n2.3%\n2.4%\n2.4%\n10\n0.0%\n0.1%\n0.3%\n0.4%\n0.5%\n0.6%\n1.2%\n1.9%\n2.1%\n2.1%\n(Pr(δw,f) < 0) = 10%\n# Interviews Per Agent\n1\n10.0%\n10.0%\n10.0%\n10.0%\n10.0%\n10.0%\n10.0%\n10.0%\n10.0%\n10.0%\n2\n5.5%\n11.4%\n13.6%\n14.1%\n14.4%\n14.4%\n14.4%\n14.4%\n14.4%\n14.4%\n3\n3.4%\n5.7%\n7.3%\n8.0%\n8.3%\n8.4%\n8.5%\n8.5%\n8.5%\n8.5%\n4\n2.3%\n3.9%\n4.9%\n5.7%\n6.3%\n6.6%\n7.1%\n7.2%\n7.2%\n7.2%\n5\n1.6%\n2.6%\n3.4%\n4.0%\n4.5%\n4.8%\n5.4%\n5.6%\n5.6%\n5.6%\n6\n1.2%\n1.8%\n2.4%\n2.9%\n3.3%\n3.6%\n4.4%\n4.7%\n4.7%\n4.7%\n7\n0.9%\n1.4%\n1.8%\n2.1%\n2.4%\n2.7%\n3.5%\n3.9%\n4.0%\n3.9%\n8\n0.7%\n1.0%\n1.3%\n1.6%\n1.8%\n2.1%\n2.8%\n3.4%\n3.4%\n3.4%\n9\n0.5%\n0.8%\n1.0%\n1.2%\n1.4%\n1.6%\n2.3%\n2.9%\n3.0%\n3.0%\n10\n0.4%\n0.6%\n0.8%\n0.9%\n1.1%\n1.3%\n1.8%\n2.5%\n2.6%\n2.6%\n(Pr(δw,f) < 0) = 25%\n# Interviews Per Agent\n1\n25.0%\n25.0%\n25.0%\n25.0%\n25.0%\n25.0%\n25.0%\n25.0%\n25.0%\n25.0%\n2\n15.1%\n19.4%\n20.4%\n20.6%\n20.6%\n20.7%\n20.7%\n20.7%\n20.7%\n20.7%\n3\n10.1%\n12.0%\n13.2%\n13.6%\n13.7%\n13.8%\n13.9%\n13.9%\n13.9%\n13.9%\n4\n7.2%\n8.7%\n9.6%\n10.2%\n10.5%\n10.6%\n10.9%\n10.9%\n10.9%\n10.9%\n5\n5.4%\n6.4%\n7.2%\n7.6%\n8.0%\n8.2%\n8.5%\n8.6%\n8.6%\n8.6%\n6\n4.2%\n4.9%\n5.5%\n5.9%\n6.2%\n6.5%\n7.0%\n7.1%\n7.1%\n7.1%\n7\n3.3%\n3.8%\n4.3%\n4.6%\n4.9%\n5.1%\n5.7%\n5.9%\n5.9%\n5.9%\n8\n2.6%\n3.1%\n3.4%\n3.7%\n4.0%\n4.2%\n4.7%\n5.1%\n5.1%\n5.1%\n9\n2.1%\n2.5%\n2.8%\n3.0%\n3.2%\n3.4%\n4.0%\n4.4%\n4.4%\n4.4%\n10\n1.7%\n2.0%\n2.2%\n2.4%\n2.6%\n2.7%\n3.3%\n3.8%\n3.8%\n3.9%\nNotes: 10,000 simulation runs. Each cell indicates the probability a firm is unmatched or a worker is unemployed\ngiven every firm and worker receives 1-10 interviews, and the degree of non-overlap increases. Each panel of the table\nvaries the probability of any worker being unacceptable from 0% to 10% to 25%.\nfalls, and once the degree of non-overlap reaches 10, incremental reductions are marginal.\nSome intuition for this result can be gleaned from Figure 2: in panels (a)-(c), the first compo-\nnents of three different symmetric interview assignments {η0, η1, η2} are depicted in which each firm\nand worker receives 2 interviews; η0 has perfect overlap, whereas the other two have less overlap.\nIf all workers are acceptable to f1, then under η0 there is no possible stable match which results in\nfirm f1 being unmatched. However, there is one potential stable match under η1 and two under η2\nin which f1 is unmatched; as overlap decreases further, even more stable matches (supportable by\nsome set of preferences) can be constructed.\nOverall, the gains to overlap can be substantial: focusing on settings where firms interview more\nthan one worker and where the probability of a worker being unacceptable is 10%, the chance that a\nfirm or worker is unmatched is approximately 3-4 times smaller under a perfect overlap assignment\n19\n\nf1\nf2\nw1\nw2\n(a) η0\nf1\nf2\nf3\nw1\nw2\nw3\n(b) η1\nf1\nf2\nf3\nf4\nw1\nw2\nw3\nw4\n(c) η2\nf1\nf2\nf3\nw1\nw2\nw3\n(d) µ1\nf1\nf2\nf3\nf4\nw1\nw2\nw3\nw4\n(e) µ2\n1\nf1\nf2\nf3\nf4\nw1\nw2\nw3\nw4\n(f) µ2\n2\nFigure 2: (a)-(c) Symmetric interview assignments where each firm and worker receive 2 interviews,\neach with different levels of overlap. (d)-(f) Given interview assignments η1 and η2, only µ1, µ2\n1,\nand µ2\n2 are stable matches in which no worker is unacceptable and firm f1 is unmatched.\nthan under an assignment with a degree of non-overlap larger than 10.\n6\nConcluding Remarks\nThe cost of interviewing and the importance of how interviews are allocated has remained outside\nof the scope of the matching literature. This article takes a step towards incorporating interviewing\ninto matching models. We have studied the interview assignment problem and provided a model\nfor analysis when information acquisition is costly. We identified two distinct forms of potential\nmiscoordination in the assignment of interviews: (i) workers may receive varying numbers of inter-\nviews; (ii) more subtly, firms may not optimally overlap their interviews. Consequently, conditional\non a certain number of interviews being conducted, institutions that limit the number of interviews\nworkers can receive or artificially create labor market segmentation to encourage overlap may lead\nto better coordination in information acquisition activities.\nAs a first attempt to study interviewing in matching markets, our model is deliberately simple\nand hence relatively tractable. Within this model, there are several open questions, including: char-\nacterizing equilibrium existence when firms may interview different numbers of workers; whether\nfor fixed costs of interviewing, pure-strategy equilibria always outperform mixed-strategy equilib-\nria; and whether for certain types of symmetric interview assignments, greater overlap guarantees\nlower expected unemployment.\nIt also is worth mentioning several aspects of interviewing markets that are beyond the scope\n20\n\nof this article but are interesting for future work. For example, our article assumes that firms bear\nthe full cost of interviewing; this may be a reasonable approximation in some markets, but less so\nin others (e.g., economic PhDs do not pay for travel and accommodations for academic interviews\nwhereas medical residents do). Can this division—or even the interviewing costs themselves—be\nendogenized? There is also a normative question of what division of interviewing costs reduces\nunemployment and how it differs from market equilibrium.\nOther potentially important extensions include allowing for (ex-ante) heterogeneous agents or\ncorrelated preferences, and determining the socially efficient allocation of interviews in this envi-\nronment; this analysis becomes even more complex when interviews can be assigned dynamically\n(i.e., sequentially) based on information revealed in earlier interviews. Finally, although the present\nmodel considered a matching market without transfers (or rigid wages that cannot be tailored to an\nindividual candidate), generalizing the model to environments with endogenous transfers is another\nchallenging and important problem.\n21\n\nA\nAppendix\nA.1\nSecond Stage Matching\nThe firm optimal stable matching (FOSM) used in the second stage of our game is the outcome of a firm-proposing\ndeferred acceptance algorithm (DAA) for employment when firms and workers utilize their true preferences (Gale\nand Shapley, 1962). We first describe the algorithm (whose features are leveraged in proving this article’s main\nresults), and then prove that—as long as workers place sufficient disutility on being unemployed—firms and workers\nwill utilize their true preferences in equilibrium when the firm-proposing DAA is used.\nBefore proceeding, note that in the main text, we have assumed that workers have only ordinal preferences over\nfirms; this was sufficient as long as we assumed the outcome of the second stage was the FOSM. However, if we\nwish to interpret the FOSM as arising from a firm-proposing DAA, we will need to make assumptions on the relative\nutility of being employed versus unemployed (see Lemma A.1 to follow); this is the only instance in which we assume\nworkers have cardinal preferences.\nThe Deferred Acceptance Algorithm (DAA).\nIn the firm-proposing DAA, each firm f utilizes pref-\nerences Pf over acceptable workers and each worker w utilizes preferences Pw. The algorithm proceeds as follows:\n• Step 1: Each firm makes a job offer to its highest-ranked worker (or, if all workers are unacceptable, does not\nmake any offers). Each worker who receives an offer “holds” onto its most preferred offer and rejects the rest.\n• In general, at step t: Each firm who was rejected in step t −1 makes a job offer to the most preferred and\nacceptable worker who has not yet rejected it. Each worker who receives an offer compares all offers received\n(including an offer he may be holding from a previous round), holds onto his most preferred offer, and rejects\nthe rest.\nThe algorithm stops after any step in which no firm’s offer is rejected; at this point all firms have either a worker\nholding its job offer, or has no workers it wishes to make an offer to that has not already rejected it. At this point, any\nworker who is holding a job offer from a firm is hired by that firm (an event we also refer to as the worker accepting\nan offer), and any worker who does not have a job offer remains unemployed.\nTruthful Revelation of Preferences.\nA matching mechanism such as the DAA can be susceptible to\n“gaming” in that participants may find it preferable to misrepresent their true preferences; indeed, there are several\nnegative results in the two-sided matching literature on the existence of mechanisms that elicit truth-telling from\nparticipants. Nonetheless in this section, we prove that as long as workers place a high enough disutility of being\nunemployed, firms and workers will utilize their true preferences realized during the interview stage under the firm-\nproposing DAA. This is formally stated and proven in the following lemma:\nLemma A.1. There exists a finite β > 0 such that if ζ −ζ0 > β(ζ −ζ), the only equilibrium of the firm-proposing\ndeferred acceptance algorithm is for both workers and firms to report their true preferences.\nProof of Lemma A.1. As this is equivalent to the marriage problem that yields the M-optimal stable matching (with\nfirms as men), firms have a dominant strategy to report their preferences truthfully (Dubins and Freedman (1981),\nRoth (1982)). For workers, it is sufficient to rule out two types of deviations: (i) a worker may rank some firm as\n“unacceptable” and reject any offer from that firm; (ii) a worker may rank firm j′ higher than j in his reported\npreferences despite preferring j to j′ in his true preferences.\nTo see why deviation (i) may be effective, note that declaring a firm as unacceptable can lead to the following\n“chain” of events: some worker w rejects some firm f’s offer (instead of holding onto it or accepting it); f then makes\nan offer to another worker w′ who accepts and rejects some other firm f ′; this f ′ in turn proposes to another worker,\nwho then may reject another firm; and so on, until a firm f ′′, who was rejected by another worker, makes an offer\nto the original worker w, and w prefers f ′′ to f. As long as the gain to such a deviation is never greater than the\npotential loss from employing it, a worker will never choose to reject any firm.\nLet L ≤N be the maximum number of interviews any worker receives in any equilibrium. Assume w is considering\nrejecting a firm’s offer that it finds preferable to the one w is already holding. In order for this to be profitable, f\nupon being rejected (conditional on making w an offer) must propose to another worker that already has an existing\noffer from another firm f ′, and that worker must prefer f to f ′. The probability that f is preferred to any f ′ by\nanother worker is at most 1/2, which is an upper bound on the probability that rejecting a firm leads to a profitable\nmanipulation. Thus the gain to rejecting a firm’s offer is bounded by (ζ −ζ)/2, where the term in parenthesis is\nthe maximum gain possible to w by obtaining a more preferred firm. However, if a worker receives L interviews and\nrejects an offer, the probability that he receives no other offer is at least (λ)L−1, where λ = Pr(δw,f < 0) > 0 is\nthe probability that a worker is unacceptable to a firm. Consequently, by rejecting firm j, he risks losing at least\n22\n\n(λ)L−1(ζ −ζ0). Clearly as long as β > (λ)L−1/2, the inequality holds and no worker will find it profitable to reject\nany firm’s offer (unless it is holding an offer from a more preferred firm).\nTo rule out deviation (ii), we first establish the following claim: prior to engaging in the match, the expected\nprobability of being hired by a firm is strictly decreasing in the rank a worker orders that firm in his reported\npreferences. First recall preferences are independently drawn and privately realized for all agents, and workers do not\nobserve the complete interview assignment. Thus, a worker perceives the probability of receiving a job offer as the\nsame for any firm. If this probability is denoted by p, then the expected probability of being hired by a firm ranked\nin the nth position is (1 −p)n−1 × p (as in order to be hired by the nth ranked firm, all firms that were ranked higher\nmust not have made a job offer). This expression is decreasing in n.\nHaving established the claim, it is straightforward to show that if any worker ranked f ′ higher than f despite\npreferring f to f ′, he would be better offreporting truthfully.\nA.2\nProofs\nProof of Proposition 3.1. Assume each firm selects x ≤N workers to interview uniformly at random: i.e., each firm\nplays a strategy νf which assigns equal positive probability to all distinct subsets of workers of size x. We show that\nthere exists a c such that this is an equilibrium.\nConsider firm f. If every other firm is randomizing uniformly, firm f is indifferent over interviewing any particular\nworker, so for any given number of interviews k that f conducts, any set of workers is optimal. Let\ngf(k, ν−f; x) = Eν−f [πf(Wf, W−f) −πf(W ′\nf, W−f) | |Wf| = k, |W ′\nf| = k −1, |W−f| = x]\n(A.1)\ndenote f’s expected gain to interviewing an additional kth worker (not including costs) given all other firms are\ninterviewing x workers uniformly at random, and πf(·) is defined in (3.2).\nWe first prove the following lemma:\nLemma A.2. gf(k, ν−f; x) is strictly decreasing in k.\nProof. Let w′ = µk−1(f) be the worker f would have matched to if it only conducted k −1 interviews. Note f only\nbenefits from interviewing the kth worker (denoted wk) if µk(f) = wk, as otherwise f would still be matched to w′.\nWe decompose gf(k, ν−f) as follows:\ngf(k, ν−f; x) = Pr(µk(f) = wk)\n|\n{z\n}\n(1)\n× (E[δwk,f −δw′,f|µk(f) = wk & µk−1(f) = w′])\n|\n{z\n}\n(2)\n(A.2)\n(1) Probability that interviewing wk results in being matched to wk: As f is indifferent between interviewing any two\nworkers, it must be that Pr(µk(f) = wk) = Pr(µk(f) = w) for any w that f is already interviewing. Intuitively,\nit is clear that interviewing an additional worker strictly reduces the probability that f is eventually matched\nto a given worker it already is interviewing. Formally, consider any w, and assume that µk−1(f) = w. The\nprobability that f is still matched to w after k interviews is:\nPr(µk(f) = w|µk−1(f) = w) = Pr(δw,f ≥δwk,f) + Pr(δw,f < δwk,f) × Pr(µk−1(wk)Pwkf)\nwhere the first term is the probability that f does not make an offer to wk after interviewing him, and the second\nterm is that f makes an offer to wk but is rejected. This is strictly less than 1 because Pr(µk−1(wk)Pwkf) < 1.\nThus (1) is strictly decreasing in k.\n(2) Expected gain from being matched to wk given f was matched to w′: Note (2) is equivalent to E[δwk,f −\nδw′,f|δwk,f > δw′,f]: i.e., f would only make an offer to wk after interviewing him if δwk,f > δw′,f. As δw′,f\nis weakly increasing in k (i.e., the value of f’s existing match cannot fall by conducting more interviews), and\nbecause Assumption 2.1 implies that [E[δwk,f −y|δwk,f ≥y] weakly falls as y increases, if follows that (2) is\nat least weakly decreasing in k.\nAs (1) is strictly and (2) is at least weakly decreasing in k, the lemma is proved.\nBy the previous lemma, we can find c ∈(gf(x, ν−f; x), gf(x −1, ν−f; x)). For such c, given every other firm\nis interviewing a subset of x workers uniformly at random, no individual firm will wish to interview more than x\ncandidates (because doing so earns an expected gain of less than c per additional candidate) or fewer than x candidates\n(because doing so gives up an expected gain greater than c per candidate). Furthermore, every firm is indifferent\nover all subsets of x workers, so the set of mixed-strategies {νf}∀f is an equilibrium.\n23\n\nProof of Proposition 3.2. Assume N is divisible by x.\nConstruct a symmetric perfect-overlap interview assign-\nment η as follows:\ndivide firms and workers into M ≡N/x components, where component m ∈{1, . . . , M}\ncontains firm and worker indexed by {(m −1)x + 1, . . . , (m −1)x + x}.\nIn the first component (m = 1), let\nη(fi) = {wi, w[i+1]x, . . . , w[i+x−1]x} for each i ∈{1, . . . , x}, where we define [a]b ≡mod(a −1, b) + 1;20 repeat the\nsame assignment within each component for each m ≥2.\nW.l.o.g, consider f ≡f1, and let\ngf(k, η−f) = [πf({w1, . . . , wk}, W−f) −πf({w1, . . . , wk−1}, W−f) | W−f = {η(f ′)}∀f′∈F,f′̸=f]\nbe f’s expected gain to interviewing wk when it is already interviewing {w1, . . . , wk−1} and all other firms are\ninterviewing according to η.\nLemma A.3. gf(k, η−f) is strictly decreasing in k.\nProof. The proof is similar to Lemma A.2: decompose gf(k, η−f) as follows:\ngf(k, η−f) = Pr(µk(f) = wk)\n|\n{z\n}\n(1)\n× (E[δwk,f −δw′,f|µk(f) = wk & µk−1(f) = w′])\n|\n{z\n}\n(2)\n(A.3)\nFor k ∈{1, . . . , x}: Within the first component, (1) is strictly decreasing in k by the same logic as in Lemma A.2:\nas {w1, . . . , wx} are identical from the perspective of f (they are all receiving x −1 interviews from {f2, . . . , fx}),\nthe probability of being matched to the kth worker (given the first k workers are interviewed) is the same as being\nmatched to any w ∈{wj}j<k. As before, because interviewing an additional worker must reduce the probability f is\nmatched to a given worker, (1) is strictly decreasing in k. Also, as before, (2) is weakly decreasing in k.\nFor k = x + 1: For the k + 1 interview, f must interview a worker who already has x interviews, and is in a\ndifferent component of the graph associated with η. To see that the gain to interviewing wx+1 is less than the gain\nto interviewing the wx, note that (1) can be further decomposed as:\nPr(µk(f) = wk) = Pr(δwk,f > δw′,f)\n|\n{z\n}\n(1a)\n× Pr(fPwkµk−1(wk)|δwk,f > δw′,f)\n|\n{z\n}\n(1b)\n(A.4)\nwhere (1a) is the probability that f makes an offer to the kth worker, and (1b) is the probability that wk accepts an\noffer from f given f proposes to wk (where µk−1(wk) is the firm that wk would match to if f had conducted k −1\ninterviews). (1a) is decreasing with k because the the value of δw′,f is weakly increasing as more interviews are made.\nNote that (1b) is strictly increasing for k ∈{1, . . . , x} (i.e., within a component). To see why, consider the probability\nw1 accepts f’s offer conditional on f having made an offer, and examine the difference if f interviews the first k −1\nversus k workers within the first component. If f interviews k candidates and wkPfw1, then f will propose to w1 if\nand only if wk rejected f; however, if f only interviewed k −1 candidates, it would have proposed to w1 regardless of\nwhere it ranked wk (because it didn’t interview wk). As wk rejecting f raises the probability w1 accepts f’s offer (as\nthere are now fewer firms that w1 can match to), the probability of w1 accepting an offer conditional on one being\nmade strictly increases in k. Furthermore, as (1b) is higher for k = 1 than k = x + 1 (as w1 receives fewer interviews\nthan wk+1), we see that (1b) is higher for k = x compared to k = x + 1. Thus, given (2) is still decreasing, the gain\nto interviewing wk+1 is less than the gain to interviewing wk.\nFor k = {x + 2, . . . , N}: As f is indifferent between interviewing any set of workers within the same component,\nthe same logic as with k ∈{1, . . . , x} can be used to show that gf(k, η−f) is strictly decreasing in k for k ∈\n{mx + 2, . . . , mx + x}, m = {1, . . . , M −1}. Finally, the same logic for k = x + 1 can be applied to show that\ngf(k, η−f) is strictly decreasing between components (i.e., k = (m + 1)x + 1, m ∈{1, . . . , M −2}).\nNote that Lemma A.3 implies that if f were to interview k workers where k ≤x, it is optimal for f to interview\nonly workers contained within {w1, . . . , wx}; otherwise, f would have a profitable deviation by swapping any worker\nthat it is interviewing not in that set with one within it (as doing so would strictly increase its profits). Furthermore,\nby Lemma A.3, we can find c ∈(gf(x, η−f), gf(x −1, η−f)) such that η is an equilibrium interview assignment: i.e.,\nno individual firm will wish to interview more or fewer than x candidates or change the set of workers that it is\ninterviewing.\nProof of Theorem 4.1. Consider a perfect overlap interview assignment η where every firm and worker receives k = x\ninterviews. Under this assignment, the ex ante probability that any firm is unmatched is bounded above by Pr(δ1:k <\n0): i.e., the probability that the first order statistic out of k draws is negative. To see this, note that if every worker\nthat a firm interviews yields non-negative surplus (which occurs with probability Pr(δ1:k ≥0)), then that firm is\n20mod(a, b) = a −⌊a\nb ⌋b, where ⌊x⌋represents the greatest integer less than or equal to x.\n24\n\nguaranteed to be matched during the second stage deferred acceptance algorithm: if a firm is not matched by the\nkth round, it must have been because the previous k −1 workers have rejected that firm and accepted an offer from\nanother firm; thus, by the kth round, no other firms are “competing” for the kth worker and the firm will be able\nto hire him. Consequently, a firm can only remain unmatched under a perfect overlap interview assignment if at\nleast one worker is found by that firm to be unacceptable; the probability of this occurrence can be no greater than\n1 −Pr(δ1:k ≥0).\nConsider now any symmetric interview assignment η′ where every firm and worker receives k interviews and\nd(η′, k) > 0 (i.e., η′ does not have perfect overlap). As all firms and workers are ex ante identical, consider any\nparticular firm f. Let the set of workers interviewed by f be given by Wf ≡η′(f), where |Wf| = k (by construction).\nIf η′(Wf) \\ f represents the set of all firms excluding f that interview workers in Wf, note that |η′(Wf) \\ f| ≥k:\ni.e., there must be at least k other firms that share at least one worker with f under the interviewing assignment η′\n(otherwise, if |η′(Wf) \\ f| = k −1, η′ would have perfect overlap). Furthermore, note that for any subset of workers\nW ′ ⊂Wf, |η′(W ′) \\ f| ≥k −1 ≥|W ′|: i.e., there are at least k −1 other firms than f that interview the workers in\nW ′. Thus, by Hall’s marriage theorem, there exists a one-to-one matching µ such that every worker in Wf is matched\nto a firm that interviews that worker under η′ and is not f. In such a matching, f1 is unmatched (as all of the workers\nit interviews are matched with another firm). Consequently, the probability that f is unmatched given η′ can be no\nless than the probability that preferences for all agents are such that each worker in Wf prefers its matched partner\nunder µ to f, and each firm other than f matched to a worker in Wf finds that worker to be desirable and its most\npreferable.\nThe upper bound on the probability that any firm is unmatched under perfect overlap assignment η is Pr(δ1:k < 0),\nwhich is decreasing as Pr(δ < 0) decreases, and limPr(δ<0)→0 Pr(δ1:k < 0) = 0. The lower bound on the probability\nthat any firm is unmatched under η′ is strictly positive (the realization of a non-empty set of preferences). Thus,\nthere exists an ϵ > 0 such that if Pr(δ < 0) < ϵ, this upper bound is below the lower bound, and η has a lower\nprobability of unemployment than η′.\nA.3\nExistence of Correlated Equilibrium with Almost Perfect Overlap\nIn small markets, a given c might require an x such that the integer constraints N precludes perfect overlap. Nonethe-\nless, with a large enough market, this issue is not a problem: with a correlated device or an intermediary, there still\nwill exist an equilibrium where each firm and worker receives x interviews, and the probability each firm is located\nin a component with exactly x workers and x firms approaches 1; i.e., even though N may not be a multiple of x,\nas long as N is sufficiently large, the perfect overlap quantity of interviews can still be achieved for almost every\nfirm. Note the induced interview assignment may not be symmetric, as exactly one component may have more than\nx firms or workers.\nProposition A.4. If there exist N, c such that a symmetric pure-strategy equilibrium where each firm and each\nworker conducts x interviews with perfect overlap exists, then for any ϵ > 0, there exists an N such that ∀˜\nN > N\na correlated equilibrium exists in which each firm interviews x workers, each worker receives x interviews, and with\nprobability 1 −ϵ each firm obtains the same expected profit as under a perfect overlap equilibrium.\nProof of Proposition A.4. For any ˜\nN, we can partition the population into ⌊\n˜\nN\nx ⌋−1 groups of exactly x workers and x\nfirms, and 1 group of x + ( ˜\nN −⌊\n˜\nN\nx ⌋x) workers and firms. For any such partition π, associate an interview assignment\nη(π) whereby in each of the groups with exactly x workers and firms, every firm interviews every worker in that\ngroup, and in the group with slightly more than x workers and firms, the interview assignment among workers and\nfirms assigns each agent x interviews as in the proof of proposition 3.2. Thus, η(π) gives each firm and each worker\nx interviews, and for but only x + ( ˜\nN −⌊\n˜\nN\nx ⌋x) workers and firms, firms and workers are located components with\nonly 2x total agents (i.e., perfect overlap).\nConsider the space of all possible π and associated η(π). For any ε > 0, there exists an N such that for any\n˜\nN > N, if a π is chosen at random, the probability that a given firm achieves perfect overlap in the interview\nassignment η(π) is at least 1−ε. 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Cambridge, MA:Econometric Society Monographs.\nSchweinzer, Paul. 2008. “Labour Market Recruiting with Intermediaries.” Review of Economic\nDesign, 12(2): 119–127.\nShimer, Robert, and Lones Smith. 2000. “Assortative Matching and Search.” Econometrica,\n68: 343–369.\n27"},"Token_count":{"kind":"number","value":24385,"string":"24,385"}}},{"rowIdx":6698,"cells":{"Source":{"kind":"string","value":"NBER"},"Date":{"kind":"number","value":2009,"string":"2,009"},"Text":{"kind":"string","value":"NBER WORKING PAPER SERIES\nADVERSE SELECTION IN COMPETITIVE SEARCH EQUILIBRIUM\nVeronica Guerrieri\nRobert Shimer\nRandall Wright\nWorking Paper 14915\nhttp://www.nber.org/papers/w14915\nNATIONAL BUREAU OF ECONOMIC RESEARCH\n1050 Massachusetts Avenue\nCambridge, MA 02138\nApril 2009\nWe are grateful for comments from Daron Acemoglu, Hector Chade, Guido Lorenzoni, Giuseppe\nMoscarini, Iván Werning, and numerous seminar participants. Shimer and Wright thank the National\nScience Foundation for research support. The views expressed herein are those of the author(s) and\ndo not necessarily reflect the views of the National Bureau of Economic Research.\nNBER working papers are circulated for discussion and comment purposes. They have not been peer-\nreviewed or been subject to the review by the NBER Board of Directors that accompanies official\nNBER publications.\n© 2009 by Veronica Guerrieri, Robert Shimer, and Randall Wright. All rights reserved. Short sections\nof text, not to exceed two paragraphs, may be quoted without explicit permission provided that full\ncredit, including © notice, is given to the source.\n\nAdverse Selection in Competitive Search Equilibrium\nVeronica Guerrieri, Robert Shimer, and Randall Wright\nNBER Working Paper No. 14915\nApril 2009\nJEL No. D82,E24,J6\nABSTRACT\nWe extend the concept of competitive search equilibrium to environments with private information,\nand in particular adverse selection. Principals (e.g. employers or agents who want to buy assets) post\ncontracts, which we model as revelation mechanisms. Agents (e.g. workers, or asset holders) have\nprivate information about the potential gains from trade. Agents observe the posted contracts and decide\nwhere to apply, trading off the contracts' terms of trade against the probability of matching, which\ndepends in general on the principals' capacity constraints and market search frictions. We characterize\nequilibrium as the solution to a constrained optimization problem, and prove that principals offer separating\ncontracts to attract different types of agents. We then present a series of applications, including models\nof signaling, insurance, and lemons. These illustrate the usefulness and generality of the approach,\nand serve to contrast our findings with standard results in both the contract and search literatures.\nVeronica Guerrieri\nUniversity of Chicago\nBooth School of Business\n5807 South Woodlawn Avenue\nChicago, IL 60637\nand NBER\nvguerrie@chicagobooth.edu\nRobert Shimer\nDepartment of Economics\nUniversity of Chicago\n1126 East 59th Street\nChicago, IL 60637\nand NBER\nshimer@uchicago.edu\nRandall Wright\nDepartment of Economics\nUniversity of Pennsylvania\n3718 Locust Walk\nPhiladelphia, PA 19104\nand NBER\nrwright@econ.upenn.edu\n\n1\nIntroduction\nThis paper studies equilibrium and efficiency in economies with adverse selection as well\nas the frictions modeled in competitive search theory. In our framework, a large number of\nuninformed principals compete to attract agents. For example, firms may compete to attract\nworkers who have private information about their productivity or preferences, or buyers of\nassets may compete to attract sellers who have private information concerning the quality\nof their holdings. Principals post incentive-compatible contracts that specify actions if they\nmatch with particular agent types. Agents observe the posted contracts and direct their\nsearch, or apply, to the most attractive ones. Matching is limited by capacity constraints\n(each principal can match with at most one agent) and possibly by search frictions (it may\nbe more difficult to contact a principal when more agents apply). Principals and agents form\nrational expectations about the market tightness for each contract—the ratio of the measure\nof principals posting that contract to the measure of agents who apply—as well as the types\nof agents who apply.\nPart of our contribution is technical: we extend the standard competitive search model\n(Montgomery, 1991; Peters, 1991; Moen, 1997; Shimer, 1996; Acemoglu and Shimer, 1999;\nBurdett, Shi and Wright, 2001; Mortensen and Wright, 2002) to allow ex-ante heterogeneous\nagents with private information about their type. We prove that, under mild assumptions,\nincluding a single-crossing condition, equilibrium exists where principals offer separating\ncontracts: each contract posted attracts only one type of agent, and different types direct\ntheir search to different contracts. The expected utility of each type of agent is uniquely\ndetermined in equilibrium. Moreover, equilibrium is easily found by sequentially solving a\nconstrained optimization problem for each type. We also present a series of examples and\napplications to illustrate the generality and usefulness of our approach. These examples also\nshow that some well-known results in contract theory and search theory change when we\ncombine elements of both in the same model, and they allow us to explore the role of several\nkey assumptions.\nThe first application is a classic signaling problem (Spence, 1973; Akerlof, 1976). Suppose\nthat workers are heterogeneous with respect to both their productivity and their cost of\nworking longer hours, where more productive workers find long hours less costly, and firms\ncare about productivity but not the length of the workday. In equilibrium, firms require\nthat more productive workers accept longer hours than they would under full information—\na version of the rat race. In this example, although hours are distorted, the probability that\nany worker gets a job is not. The equilibrium is not generally efficient: a Pareto improvement\nis possible if there are sufficiently few low-productivity agents or the difference in the cost of\n1\n\nworking is small relative to the difference in productivity. We also consider the case where\nlong workdays are less costly for low-productivity workers; in this case there is an equilibrium\nwith pooling and no distortion in the workday (although other equilibria exist).\nOur second example is a version of the Rothschild and Stiglitz (1976) insurance model.\nConsider a labor market interpretation, rather than pure insurance, as in the original model.1\nRisk-averse workers and risk-neutral firms match in pairs to produce output, but only some\npairs are productive. Workers differ in the probability that they can form a productive match,\nand if a match is unproductive the worker is let go. In equilibrium, firms separate workers\nby only partially insuring them against the probability the match will be unproductive. This\nmeans that workers can be worse offif they lose their job than if they had never found a one.\nWe interpret this as an explanation for the observation that firms do not fully insure workers\nagainst layoffrisk: if they did, they would attract low-productivity applicants.\nIn this\nexample, we show that a partial-pooling allocation—pooling only some types of workers—\nmay Pareto dominate the equilibrium.\nCompetitive search also offers a novel resolution to the famous nonexistence problem\nin Rothschild and Stiglitz (1976). When there are relatively few low productivity workers,\nequilibrium may not exist in the original model, for the following reason: given any separating\ncontract, profit for an individual firm can be increased by a deviation to a pooling contract\nthat cross subsidizes low-productivity workers. Here, such a pooling contract will not increase\nprofit. The key difference is that, in our model, because firms are capacity constrained, a\ndeviation cannot serve the entire population. Suppose a firm posts a contract designed to\nattract a representative cross section of agents. Because of capacity constraints, the more\nworkers that apply for this contract the less likely it is that any one will match.\nThis\ndiscourages some workers from applying. Critically, it is the most productive workers who\nare the first to withdraw their applications, because their outside option—trying to obtain\na separating contract—is more attractive. Hence, only undesirable workers are attracted by\nsuch a deviation, making it unprofitable.\nIn the above applications, asymmetric information affects the contracts offered in equi-\nlibrium, but not market tightness. Our third example reverses this: adverse selection makes\nit harder for agents to find a partner, but it does not affect the terms of trade conditional on\na match. Consider a market where agents want to sell a heterogeneous object with uncertain\nquality (Akerlof, 1970). Imagine these objects are apples, meant to represent assets, that\ncould be high or low quality.2 Principals want to buy apples from agents, but some apples\n1We present the results in a labor market context because the assumption that a firm can only hire\na fraction of the available workforce may seem more reasonable than the assumption that an insurance\ncompany can only serve a fraction of the potential customers.\n2In our static environment, apples stand in for the claims to trees bearing fruit as dividends in the\n2\n\nare bad—they are lemons. To make the case stark, assume there are no fundamental search\nfrictions, so that everyone on the short side of the market can match. But since the short\nside of the market is endogenous, in equilibrium, agents with good apples only match proba-\nbilistically. This serves to screen out low quality. This is different from related results in the\nliterature (e.g. Nosal and Wallace 2007), where lotteries are used to similar effect. Interest-\ningly, screening through matching saves resources, since posting contracts and trading with\nlotteries is wasteful. Still, equilibrium here can be Pareto dominated by a pooling allocation\nif there are few bad apples. We also find that in some cases adverse selection completely\nshuts down the market—an extreme lemons problem that may have something to do with\nthe recent collapse in credit markets.\nOur paper is related to a growing literature exploring competitive search theory with\ninformational frictions, including Faig and Jerez (2005), Guerrieri (2008), and Moen and\nRos´en (2006), who propose different extensions of the standard model to allow one-sided\nprivate information. However, in all those papers, agents are ex ante homogeneous, and\nheterogeneity is match-specific. Inderst and M¨uller (1999) provide a model that extends\ncompetitive search to an environment with ex ante heterogeneous agents; their model is a\nspecial case of our first example, where the single-crossing condition is satisfied. Inderst and\nWambach (2001) explore a version of the Rothschild and Stiglitz (1976) model with a finite\nnumber of principals and agents, and capacity constraints; this model is related to our second\nexample. Our paper is the first to develop a general framework for analyzing competitive\nsearch with adverse selection and present a variety of applications.\nMany papers study adverse selection without search, of course. Driven by the nonex-\nistence issue in Rothschild and Stiglitz (1976), Miyazaki (1977), Wilson (1977), and Riley\n(1979) propose alternative notions of equilibrium. In contrast to those papers, we do not\nget cross subsidization in equilibrium. As we have mentioned above, the key difference with\nRothschild and Stiglitz (1976) in our model is this: since no principal can serve all the agents,\neach must deduce which agents are most attracted to pooling contracts, and this eliminates\nthe incentive to pool. Prescott and Townsend (1984) study adverse selection in competitive\neconomies, concluding that “there do seem to be fundamental problems for the operation of\ncompetitive markets for economies or situations which suffer from adverse selection” (p. 44).\nThis is consistent with our substantive findings, even though the models are very different.\nMore recently, Bisin and Gottardi (2006) study adverse selection with competitive markets,\nwhere agents are restricted to trade incentive-compatible contracts, and also show that there\nstandard asset-pricing model of Lucas (1978). Lester, Postlewaite and Wright (2007) review some related\nasset models with search and private information, but that literature does not analyze adverse selection and\ncompetitive search.\n3\n\nexists a separating equilibrium. Although our equilibrium concept is more strategic, some\nfeatures are similar, and the incentive-compatibility condition they impose is analogous to\none that we generate endogenously.\nThe rest of the paper is organized as follows.\nIn Section 2, we develop the general\nenvironment, define equilibrium, and discuss some critical assumptions. In Section 3, we\nshow how to find equilibrium by solving a constrained optimization problem, prove that\na separating equilibrium always exists, and show that equilibrium payoffs are unique. In\nSection 4, we define a class of incentive-feasible allocations in order to discuss whether\nequilibrium outcomes are efficient within this class. In Sections 5–7, we present the series of\napplications discussed above, in each case characterizing equilibria and discussing efficiency.\nSection 8 concludes. All proofs are relegated to the Appendix.\n2\nThe Model\nThere is a measure 1 of agents, a fraction πi > 0 of whom are of type i ∈I ≡{1, 2, . . ., I}.\nType is an agent’s private information. There is a large set of ex ante homogeneous principals.\nA principal may post a contract, at cost k > 0, that provides an opportunity to match with\nan agent (we discuss the nature of a contract below).\nTo keep the analysis focused, we\nconsider a static environment, where principals and agents have a single opportunity to\nmatch.\nThere is a compact and nonempty set of feasible actions for principals and agents who\nare matched, Y, contained in a metric space with metric d(y, y′) for y, y′ ∈Y. A typical\nelement y ∈Y may specify actions by the principal, actions by the agent, transfers between\nthem, and other possibilities; as we make explicit in the applications, it can include lotteries.\nA principal who matches with a type i agent gets a payoffvi(y) −k if they undertake action\ny. A principal who does not post a contract gets a payoffnormalized to 0, while one who\nposts a contract but fails to match gets −k. A type i agent matched with a principal gets a\npayoffui(y) if they undertake y, while an unmatched agent gets a payoffalso normalized to\n0. For all i, ui : Y 7→R and vi : Y 7→R are continuous.\nBy the revelation principle, without loss of generality we assume that contracts are rev-\nelation mechanisms. Precisely, a contract is a vector of actions, C ≡{y1, . . . , yI} ∈YI,\nspecifying that if a principal and agent match, the latter (truthfully) announces a type i and\nthey implement yi. Contract C is incentive compatible if ui(yi) ≥ui(yj) for all i, j.3 Let\nC ⊂YI denote the set of incentive compatible contracts. Principals only post contracts in\n3Since we are not concerned with moral hazard, here, we assume that y can be implemented by any\nprincipal and agent.\n4\n\nC.\nWe turn now to the matching process. All agents observe the set of posted contracts and\napply, or direct their search to the most attractive ones. Let Θ(C) denote the principal-agent\nratio associated with contract C, defined as the measure of principals offering C over the\nmeasure of agents who apply to that contract, Θ : C 7→[0, ∞]. Let γi(C) denote the share\nof agents that apply to C whose type is i, with Γ(C) ≡{γ1(C), . . . , γi(C), . . . , γI(C)} ∈∆I,\nthe I-dimensional unit simplex. That is, Γ(C) satisfies γi(C) ≥0 for all i and P\ni γi(C) = 1,\nand Γ : C 7→∆I.\nThe functions Θ and Γ are determined endogenously in equilibrium,\nand are defined for all incentive compatible contracts, not only the ones that are posted in\nequilibrium.\nAn agent who applies to contract C matches with a principal with probability µ(Θ(C)),\nindependent of type, where µ : [0, ∞] 7→[0, 1] is nondecreasing.\nA principal offering C\nmatches with a type i agent with probability η(Θ(C))γi(C), where η : [0, ∞] 7→[0, 1] is\nnonincreasing. We impose µ(θ) = θη(θ) for all θ, since the left hand side is the matching\nprobability of an agent and the right hand side is the matching probability of a principal\ntimes the principal-agent ratio. Together with the monotonicity of µ and η, this implies both\nfunctions are continuous. It is convenient to let ¯η ≡η(0) > 0 denote the highest probability\nthat a principal can match with an agent, obtained when the principal-agent ratio is 0.\nSimilarly let ¯µ ≡µ(∞) > 0 denote the highest probability that an agent can match with a\nprincipal. Conversely, µ(θ) = θη(θ) ensures that η(∞) = µ(0) = 0.\nThe expected utility of principals who post C = {y1, . . ., yI} is\nη(Θ(C))\nI\nX\ni=1\nγi(C)vi(yi) −k.\nThe expected utility of type i agents who apply to C and report type j if they match is\nµ(Θ(C))ui(yj).\nWe now have the following definition of equilibrium.\nDefinition 1 A competitive search equilibrium is a vector ¯U = { ¯Ui}i∈I ∈RI\n+, a measure\nλ on C with support ¯C, a function Θ(C) : C 7→[0, ∞], and a function Γ(C) : C 7→∆I\nsatisfying:\n(i) principals’ profit maximization and free-entry: for any C = {y1, . . . , yI} ∈C,\nη(Θ(C))\nI\nX\ni=1\nγi(C)vi(yi) ≤k,\n5\n\nwith equality if C ∈¯C;\n(ii) agents’ optimal search: let\n¯Ui = max\n\u001a\n0 ,\nmax\nC′={y′\n1,...,y′\nI}∈¯C µ(Θ(C′))ui(y′\ni)\n\u001b\nand ¯Ui = 0 if ¯C = ∅; then for any C = {y1, . . ., yI} ∈C and i,\n¯Ui ≥µ(Θ(C))ui(yi),\nwith equality if Θ(C) < ∞and γi(C) > 0; moreover, if ui(yi) < 0, either Θ(C) = ∞\nor γi(C) = 0;\n(iii) market clearing:\nZ\n¯C\nγi(C)\nΘ(C)dλ({C}) ≤πi for any i,\nwith equality if ¯Ui > 0.\nTo understand our notion of equilibrium, first consider contracts that are actually posted\nin equilibrium, C ∈¯C. Part (i) of the definition implies that principals earn zero profits\nfrom any such contract. Since η(∞) = 0 < k, it must be that Θ(C) < ∞. Part (ii) then\nimplies that if type i agents apply for a contract, γi(C) > 0, they cannot earn a higher level\nof utility from any other posted contract. And part (iii) guarantees that all type i agents\napply to some contract, unless they are indifferent about participating in the market, which\ngives them the outside option ¯Ui = 0.\nEquilibrium also imposes restrictions on contracts that are not posted in equilibrium.\nEach principal anticipates that it cannot affect the utility of any type of agent, and so takes\n¯U = { ¯Ui}i∈I as given. Intuitively, when a principal considers posting a new contract C ∈C,\nhe initially imagines an infinite principal-agent ratio. Some contracts will not be able to\nattract agents even at that ratio, ¯Ui ≥¯µui(yi) for all i, in which case Θ(C) = ∞and the\nchoice of Γ(C) is arbitrary and immaterial. Otherwise, agents would be attracted to the\ncontract, pulling down the principal-agent ratio. This process would stop at the value of\nΘ(C) such that ¯Ui = µ(Θ(C))ui(yi) for some i and ¯Uj ≥µ(Θ(C))uj(yj) for all j. Moreover,\nonly type i agents with ¯Ui = µ(Θ(C))ui(yi) would be attracted to the contract, pinning\ndown Γ(C). This is all captured by equilibrium condition (ii).4 Equilibrium condition (i)\n4The requirement that if ui(yi) < 0 then either Θ(C) = ∞or γi(C) = 0 rules out the possibility that type\ni agents earn zero utility, but apply for contract C = {y1, . . . , yI} with the expectation they will not be able\nto get it, Θ(C) = 0. Other agents with ¯Uj > 0 would then not apply for the contract. Such a belief might\n6\n\nthen imposes that, for principals not to post C, in equilibrium, they must not earn positive\nprofit from it, given Θ(C) and Γ(C).\nLet\n¯Yi ≡\n\b\ny ∈Y | ¯ηvi(y) ≥k and ui(y) ≥0\n\t\nbe the set of actions for a type i agent that deliver nonnegative utility while permitting the\nprincipal to make nonnegative profits if the principal-agent ratio is 0, and\n¯Y ≡\n[\ni\n¯Yi.\nIn equilibrium, actions that are not in ¯Y are not implemented.\nFor the benchmark analysis, we make three assumptions on preferences over y ∈¯Y, as\nwe now discuss.\nAssumption A1 Monotonicity: for all y ∈¯Y,\nv1(y) ≤v2(y) ≤. . . ≤vI(y).\nThis mild assumption says that for any fixed action, principals weakly prefer higher types.\nFor the next assumption, let Bε(y) ≡{y′ ∈Y|d(y, y′) < ε} be a ball of radius ε.\nAssumption A2 Local non-satiation: for all i ∈I, j < i, y ∈¯Yi, and ε > 0, there exists a\ny′ ∈Bε(y) such that vi(y′) > vi(y) and uj(y′) ≤uj(y).\nAnother mild assumption, A2 is satisfied in any application where actions allow transfers.5\nThe next assumption guarantees that it is possible to design contracts that attract some\nagents without attracting less desirable agents.\nAssumption A3 Sorting: for all i ∈I, y ∈¯Yi, and ε > 0, there exists a y′ ∈Bε(y) such\nthat\nuj(y′) > uj(y) for all j ≥i and uj(y′) < uj(y) for all j < i.\nOne should interpret A3 as a generalized version of a standard single-crossing condition.\nAssume for illustration that: y ≡(y1, y2) ∈R2; ¯Y is open; and the functions ui(y1, y2) are\nmake a deviation unprofitable, if ¯ηvi(yi) < k, but we find it implausible. In particular, it is inconsistent with\nthe adjustment process described in the text.\n5Moreover, we use A2 only in the proof of Proposition 2 below, and nowhere else. We use it to establish\nthat it is possible to make a principal better offwhile not improving the well-being of agents. If η is strictly\ndecreasing, we could do this by adjusting market tightness, but since for some examples it is interesting to\nhave η only weakly decreasing, we introduce A2.\n7\n\ndifferentiable. Then A3 holds if the marginal rate of substitution between y1 and y2 is higher\nfor higher types—i.e., if\n∂ui(y1, y2)/∂y1\n∂ui(y1, y2)/∂y2\nis monotone in i.6 Assumption A3 is important for the results: in Section 5.4 we show how\nthe nature of equilibrium changes when A3 fails.\n3\nEquilibrium Characterization\nWe now show that equilibrium solves a set of optimization problems. For any type i, consider\nthe following problem:\nmax\nθ∈[0,∞],y∈Y µ(θ)ui(y)\n(P-i)\ns.t. η(θ)vi(y) ≥k,\nand µ(θ)uj(y) ≤¯Uj for all j < i.\nIn terms of economics, (P-i) chooses market tightness θ and an action y to maximize the\nexpected utility of type i subject to a principal at least breaking even when only type i\nagents apply, and subject to types lower than i not wanting to apply.\nNow consider the larger problem (P) of solving (P-i) for all i. More precisely, we say that\na set I∗⊂I and three vectors { ¯Ui}i∈I, {θi}i∈I∗, and {yi}i∈I∗solve (P) if:\n1. I∗denotes the set of i such that the constraint set of (P-i) is non empty and the\nmaximized value is strictly positive, given ( ¯U1, . . ., ¯Ui−1);\n2. for any i ∈I∗, the pair (θi, yi) solves problem (P-i) given ( ¯U1, . . . , ¯Ui−1), and ¯Ui =\nµ(θi)ui(yi)\n3. for any i /∈I∗, ¯Ui = 0.\n6Actually, A3 is more general. For example, suppose\nui(y1, y2) =\n\u0000 1\n2yρi\n1 + 1\n2yρi\n2\n\u00011/ρi −y2\n1 + y2\n2\n2\n−1\n8.\nfor ρi < 1, and ρi is higher for higher types. Then the elasticity of substitution between y1 and y2, 1/(1−ρi),\nis increasing in i. Let Y = [0, 1]2, a superset of the points where ui(y1, y2) > 0. Any point (y1, y2) on the\nboundary of Y is not in ¯Y since ui(y1, y2) < 0. This example fails the single-crossing condition when y1 = y2,\nbut satisfies A3, since it is possible to increase the spread between y1 and y2 and attract higher types while\nrepelling lower ones.\n8\n\nProposition 1 below says that we can find any equilibrium by solving (P); conversely,\nProposition 2 says that any solution to (P) generates an equilibrium. As a preliminary step,\nwe prove that (P) has a solution and provide a partial characterization, by showing that the\nzero profit condition binds and that higher types are not attracted by (θ, y). This implies\nthat only downward incentive constraints are relevant in equilibrium (while, in fact, in the\nprimitive definition incentive compatibility requires no j > i or j < i would want to apply\nand pretend to be type i).\nLemma 1 Assume A1-A3. There exists I∗, { ¯Ui}i∈I, {θi}i∈I∗, and {yi}i∈I∗that solve (P). At\nany solution,\nη(θi)vi(yi) = k for all i ∈I∗,\nµ(θi)uj(yi) ≤¯Uj for all j ∈I and i ∈I∗.\nAll formal proofs are in the Appendix, but the idea is to notice that (P) has a recursive\nstructure. As a first step, (P-1) depends only on exogenous variables and thus determines\n¯U1. In general, at step i, (P-i) depends on the previously determined values of ¯Uj for j < i\nand determines ¯Ui. Thus, we can solve (P) in I iterative steps.\nWe now show that a solution to (P) can be used to construct an equilibrium in which\nsome principals offer a contract to attract type i ∈I∗, while keeping out other types. The\nrelevant contract is suggested by the solution to (P), but we must also show that no other\ncontract gives positive profit.\nProposition 1 Assume A1-A3. Suppose I∗, { ¯Ui}i∈I, {θi}i∈I∗, and {yi}i∈I∗solve (P). Then\nthere exists a competitive search equilibrium { ¯U, λ, ¯C, Θ, Γ} with ¯U = { ¯Ui}i∈I, ¯C = {Ci}i∈I∗\nwhere Ci = (yi, . . . , yi), Θ(Ci) = θi, and γi(Ci) = 1.\nThe next result establishes that any equilibrium can be characterized using (P). The\nproof is based on a variational argument, showing that if (θi, yi) does not solve (P), it cannot\nbe part of an equilibrium.\nProposition 2 Assume A1-A3. Let { ¯U, λ, ¯C, Θ, Γ} be a competitive search equilibrium. Let\n{ ¯Ui}i∈I = ¯U and I∗= {i ∈I| ¯Ui > 0}. For each i ∈I∗, there exists a contract C ∈¯C\nwith Θ(C) < ∞and γi(C) > 0. Moreover, take any {θi}i∈I∗and {yi}i∈I∗such that for each\ni ∈I∗, there exists a contract Ci = {y1, . . . , yi, . . ., yI} ∈¯C (so the ith element of Ci is yi)\nwith θi = Θ(Ci) < ∞and γi(Ci) > 0. Then I∗, { ¯Ui}i∈I, {θi}i∈I∗, and {yi}i∈I∗solve (P).\nThe above results imply that in equilibrium any contract C = {y1, . . . , yi, . . . , yI} that\nattracts type i solves (P-i), in the sense that the solution to the problem has θ = Θ(C) and\n9\n\ny = yi. This does not necessarily mean that each contract attracts only one type of agent.\nIn general, a contract could attract more than one type, or the same type could apply to\nmore than one type of contract. But if a contract C = {y1, . . . , yI} attracts two types, say i\nand j, then Θ(C) and yi solve (P-i) and Θ(C) and yj also solve (P-j).\nThe existence of equilibrium and uniqueness of equilibrium payoffs follow immediately\nfrom the above results.\nProposition 3 Assume A1-A3. Then competitive search equilibrium exists, and the equi-\nlibrium ¯U is unique.\nThe last result of this section shows that, when there are strict gains from trade for all\ntypes, all agents get strictly positive utility.\nProposition 4 Assume A1-A3, and that for all i there exists y ∈Y with ¯ηvi(y) > k and\nui(y) > 0. Then in any competitive search equilibrium, ¯Ui > 0 for all i, and in particular\nthere exists a contract C ∈C with Θ(C) < ∞and γi(C) > 0.\nThe proof follows by showing that the maximized value of any (P-i) is positive as long as\n¯Uj > 0 for j < i. One might imagine a stronger claim, that if there are strict gains from\ntrade for any type i then ¯Ui > 0, but Section 7 shows that this may not be the case. In\nparticular, if there are no gains from trade for some type j < i and ¯Uj = 0, it may be that\n¯Ui = 0 even though there would be gains from trade for type i with full information.\n4\nIncentive Feasible Allocations\nTo set the stage for studying efficiency, we define an incentive feasible allocation. We begin\nby defining an allocation, by which we basically mean a description of the posted contracts\ntogether with the implied search behavior and payoffs of agents.\nDefinition 2 An allocation is a vector ¯U of expected utilities for the agents, a measure λ\nover the set of incentive-compatible contracts C with support ¯C, a function ˜Θ : ¯C 7→[0, ∞],\nand a function ˜Γ : ¯C 7→∆I.\nNote that ˜Θ and ˜Γ are different from the Θ and Γ in the definition of equilibrium, because\nthe former are defined only over the set of posted contracts, while the latter are defined for\nall incentive compatible contracts.\nAn allocation is incentive feasible whenever: (1) each posted contract offers the maximal\nexpected utility to agents who direct their search for that contract and no more to those\n10\n\nwho do not; (2) the economy’s resource constraint is satisfied; and (3) markets clear.7 More\nformally, we have:\nDefinition 3 An allocation { ¯U, λ, ¯C, ˜Θ, ˜Γ} is incentive feasible if\n1. for any C ∈¯C and i ∈{1, . . . , I} such that ˜γi(C) > 0 and ˜Θ(C) < ∞,\n¯Ui = µ(˜Θ(C))ui(yi),\nand\n¯Ui ≡max\nC′∈¯C µ(˜Θ(C′))ui(y′\ni)\nwhere C′ = {y′\n1, . . . , y′\nI};\n2.\nZ \nη(˜Θ(C))\nI\nX\ni=1\n˜γivi(yi) −k\n!\ndλ(C) = 0;\n3. for all i ∈{1, . . ., I},\nZ ˜γi(C)\n˜Θ(C)\ndλ(C) ≤πi,\nwith equality if ¯Ui > 0.\nThe set of incentive feasible allocations provides a benchmark for what the economy could\nconceivably achieve, perhaps through legal or other restrictions on the contracts that can be\noffered.\n5\nApplication I: The Rat Race\nWe now proceed with the first of our three main applications, a version of a classic signaling\nmodel (Akerlof, 1976). For concreteness, think of agents here as workers who are heteroge-\nneous in terms of both their productivity and their cost of working long hours, and principals\nas firms that are willing to pay more for high-productivity workers and can observe hours\nbut not productivity. The main substantive result in this example is this: if the cost of a long\nworkday is lower for more productive workers, equilibrium will separate types by distorting\nhours relative to the first best, but will not distort market tightness.8\n7The resource constraint in the formal definition to follow can be read as saying that principals’ profits\nare 0 on average, but this is not the only interpretation, since there may be different ways to implement the\nsame outcome—e.g. we can imagine a planner posting the contracts directly.\n8This example is essentially a static version of Inderst and M¨uller (1999), although we also extend it to\nconsider cases where the sorting assumption A3 fails.\n11\n\n5.1\nSetup\nAn action here is y = {t, x}, where t is a transfer from the firm to the worker and x ≥0 is\nthe length of the workday. The payoffof a type i worker who undertakes {t, x} is\nui(t, x) = t −x\nai\n,\nwhere higher values of ai imply that x is less costly. The payoffof a firm matched with type\ni who undertakes {t, x} is\nvi(t, x) = bi −t,\nwhere bi is the productivity the worker. We assume I = 2 and b2 > b1. We restrict the set of\nfeasible actions to Y = {(t, x)|t ∈[−ε, b2] and x ∈[0, b2 max{a1, a2}]} for some ε > 0. This\nensures Y is compact but is not otherwise important (although it is convenient to allow for\nthe possibility of a small negative transfer).9\nThe space of actions that provide nonnegative utility to type i and nonnegative profit\nwhen the firm-worker ratio is 0 is\n¯Yi = {(t, x) ∈Y|x/ai ≤t ≤bi −k/¯η}.\nThe fact that b2 > b1 implies A1, and A2 holds because (t, x) ∈¯Yi implies t ≥0, so t\ncan be reduced to raise vi(t, x) and lower uj(t, x) (this is where it is convenient to allow\nnegative transfers). In terms of A3, consider (t, 0) ∈¯Y. There are nearby points (t′, x′) with\nu1(t′, x′) < u1(t, 0) and u2(t′, x′) > u2(t, 0) if and only if a2 > a1, so more productive workers\nfind it less costly to have higher x. At any other {t, x} ∈¯Y, A3 holds for all a1 and a2. We\nimpose a2 > a1, for now, and discuss later what happens if this is violated. Finally, assume\n¯ηb1 > k, so that there are gains from trade for both types, and hence Proposition 4 implies\n¯U1 > 0 and ¯U2 > 0.\n9A firm would never offer t > b2 and the worker would never accept t < 0. Also, a worker would never\nprovide x > b2 max{a1, a2}, given that b2 ≤t.\n12\n\n5.2\nEquilibrium\nBased on Section 3, we characterize equilibrium by solving (P). In this example, (P-i) is\n¯Ui =\nmax\nθ∈[0,∞],(t,x)∈Y µ(θ)\n\u0012\nt −x\nai\n\u0013\ns.t. η(θ)(bi −t) ≥k,\nand µ(θ)\n\u0012\nt −x\naj\n\u0013\n≤¯Uj for j ≤i.\nLet us assume µ is strictly concave and continuously differentiable.\nThen we have the\nfollowing (again, all proofs are in the Appendix):\nResult 1 There exists a unique competitive search equilibrium with µ′(θi)bi = k, for i = 1, 2,\nand so θ1 < θ2. Moreover, we have\nx1 = 0; x2 =\na1\nµ(θ2)\n\u0014\u0012 µ(θ2)\nµ′(θ2) −θ2\n\u0013\n−\n\u0012 µ(θ1)\nµ′(θ1) −θ1\n\u0013\u0015\nk > 0;\nti =\n\u0012\n1\nµ′(θi) −\nθi\nµ(θi)\n\u0013\nk, for i = 1, 2;\n¯U1 =\n\u0012 µ(θ1)\nµ′(θ1) −θ1\n\u0013\nk and ¯U2 =\n\u0014a1\na2\n\u0012 µ(θ1)\nµ′(θ1) −θ1\n\u0013\n+\n\u0012\n1 −a1\na2\n\u0013 \u0012 µ(θ2)\nµ′(θ2) −θ2\n\u0013\u0015\nk.\nIn equilibrium, some firms post contracts to attract only type 1 workers, while others\npost contracts to attract only type 2 workers. To achieve separation, the contracts of type 1\nare undistorted, while those of type 2 are distorted by having them work (unproductively)\nlong hours. Notice that market tightness is equal to the first-best or full-information level\nfor both types, so matching and hence employment are not distorted in this example. Also,\nnote that without additional assumptions we cannot rule out the possibility that t2 < t1, so\ntype 2 workers may actually receive less compensation in addition to working longer hours;\nthey are compensated for this by a higher probability of being hired, θ2 > θ1.\n5.3\nEfficiency\nEquilibrium is not necessarily efficient. Consider an allocation that treats the two types\nidentically, ¯C = {C}, where C = {(t, 0), (t, 0)}.\nThen let ˜Θ(C) = θ∗, with θ∗solving\nµ′(θ∗)(π1b1 + π2b2) = k, ˜γi(C) = πi, and λ({C}) = 1/θ∗. Note that θ1 < θ∗< θ2, so market\ntightness now is in between the levels for the different types in equilibrium. Then choose t\n13\n\nso profits are 0:\nt =\n\u0012\n1\nµ′(θ∗) −\nθ∗\nµ(θ∗)\n\u0013\nk\nThis contract is obviously incentive compatible, since types are treated identically.\nAll\nworkers want to apply, and get the same expected utility, so condition (1) of feasibility is\nsatisfied. And the resource and market clearing conditions are satisfied by the choice of t\nand λ. Hence, this constitutes an incentive feasible allocation.\nThe expected utility for all workers is\n¯U =\n\u0012 µ(θ∗)\nµ′(θ∗) −θ∗\n\u0013\nk.\nComparing this with equilibrium, ¯U > ¯U1 since θ∗> θ1, and ¯U ≥¯U2 if and only if\nµ(θ∗)\nµ′(θ∗) −θ∗≥a1\na2\n\u0012 µ(θ1)\nµ′(θ1) −θ1\n\u0013\n+\n\u0012\n1 −a1\na2\n\u0013 \u0012 µ(θ2)\nµ′(θ2) −θ2\n\u0013\n.\nThis inequality holds if a1/a2 is close to 1 (screening is very costly) or if π1 is close to zero\n(there are very few type 1 agents). The reason is that, in equilibrium, firms who want to\nattract type 2 need to screen out type 1, or they would be swamped by applications from\nlow-productivity workers. Screening may not be socially optimal, however, if there are few\ntype 1 workers or screening is costly. Thus it may be socially preferable to subsidize type 1\nworkers and eliminate screening, but this is not consistent with equilibrium, any individual\nfirm has incentive to deviate and try to screen.\n5.4\nPooling\nThe sorting condition A3 plays a critical role. To understand this, consider a variant where\nsignaling is cheaper for less productive workers, a1 ≥a2 while b1 < b2.10 Now firms would like\nto screen out low-productivity workers, but cannot: if they try to attract high-productivity\ntypes, the low-productive types would apply and claim to be high-productivity. In this case,\nwe now show that there is a class of equilibria in which firms attract both types, and ask\nthe low-productivity agents to work x ≥0 unproductive hours.\nWhen A3 is violated, we cannot use the analysis in Section 3 to characterize the outcome,\nand we must go back to the primitive definition of equilibrium. Given this, we describe a\n10We claim this violates A3. Fix t and set x = 0. For any nearby contract (t′, x′),\nu1(t′, x′) −u1(t, 0) = t′ −t −x′/a1 ≥t′ −t −x′/a2 = u2(t′, x′) −u2(t, 0),\nsince x′ ≥0. Hence, there is no (t′, x′) with u1(t′, x′) < u1(t, 0) and u2(t′, x′) > u2(t, 0).\n14\n\nclass of equilibria indexed by x1 ∈[0, a1(b2 −b1)(1 −π1)/π1].\nAll firms post the same\ncontract C = {(t + x1/a1, x1), (t, 0)}, where t is chosen to make profit 0, all workers apply,\nand γi(C) = πi. Given this, suppose a firm considers offering a contract that attracts only\none type. If it tries to attract type 1, it loses the benefit of cross-subsidization, and is unable\nto attract them while earning positive profits. If it tries to attract type 2 workers, it is unable\nto devise a contract that will exclude type 1, again making the deviation unprofitable.\nResult 2 Suppose a1 ≥a2. For any x1 ∈[0, a1(b2−b1)(1−π1)/π1], there exists a competitive\nsearch equilibrium where ¯C = {C} with C = {(t + x1/a1, x1), (t, 0)} and\nt = π1\n\u0012\nb1 −x1\na1\n\u0013\n+ π2b2 −\nθ\nµ(θ)k,\nwhere θ solves\nµ′(θ)\n\u0012\nπ1\n\u0012\nb1 −x1\na1\n\u0013\n+ π2b2\n\u0013\n= k.\nThe expected utility for all workers is ¯U = µ(θ)t.\nThere are equilibria where x > 0, because if firms did not set x > 0 they would be stuck\nexclusively with type 1 workers. Clearly, x > 0 is socially wasteful, and these equilibria can\nbe Pareto ranked, with x1 = 0 being the best. Moreover, if a1 > a2, then not only do there\nexist the equilibria in Result 2, we can show that any equilibrium where all firms post the\nsame contract must be in that class.\nResult 3 Suppose a1 > a2. If there exists a competitive search equilibrium with ¯C = {C}\nwhere C = {(t1, x1), (t2, x2)}, γi(C) = πi, and Θ(C) < ∞, then x2 = 0, and t1 = t2 + x1/a1.\n6\nApplication II: Insurance\nOur next application is based on Rothschild and Stiglitz (1976), where risk neutral prin-\ncipals offer insurance to risk averse agents who are heterogeneous in their probability of a\nloss. This illustrates several features. First, although we had linear utility in the previous\nsection, this is not necessary in the general framework. Second, we show here that even if\na pooling allocation does not Pareto dominate the equilibrium, a partial-pooling allocation\nmay. Finally, to illustrate that traditional search frictions are not necessary, in this example\nwe allow the short side of the market to match for sure: µ(θ) = min{θ, 1}.\n15\n\n6.1\nSetup\nWe again specify the model in terms of worker-firm matching.11 Now the productivity of a\nmatch is initially unknown by both the worker and firm. Some workers are more likely than\nothers to generate productive matches, but firms cannot observe this: type i produce 1 unit\nof output with probability pi and 0 otherwise, and pi is the agent’s private information. A\ncontract specifies a transfer to the worker conditional on realized productivity and reported\ntype. Workers are risk averse and firms are risk neutral. In the absence of adverse selection,\nfull insurance equates the marginal utility of agents across states. We show that firms here\ndo not provide full insurance, because incomplete insurance helps keep undesirable workers\nfrom applying.\nAn action consists of a pair of consumption levels, conditional on employment or unem-\nployment after match productivity has been realized, y = {ce, cu}. The payoffof a matched\ntype i worker given y is\nui(ce, cu) = piU(ce) + (1 −pi)U(cu),\nwhere p1 < p2 < · · · < pI < 1 and U : [c, ∞) →R is increasing and strictly concave with\nlimc→c U(c) = −∞for some c < 0 and U(0) = 0. The payoffof a firm matched with type i\ngiven y is\nvi(ce, cu) = pi(1 −ce) −(1 −pi)cu.\nTo ensure A1 is satisfied, we restrict the set of feasible actions to Y = {(ce, cu)|cu + 1 ≥\nce ≥c and cu ≥c}. The assumption limc→c U(c) = −∞ensures that actions of the form\n{ce, c} yield negative utility for all types and so are not in ¯Y. Then, since a reduction in\ncu raises vi(y) and lowers ui(y), and is feasible for all y ∈¯Y, A2 is satisfied.\nTo verify\nA3, consider an incremental increase in ce to ce + dce and an incremental reduction in cu\nto cu −dcu for some dce > 0 and dcu > 0.\nFor a type i worker, this raises utility by\napproximately piu′(ce)dce −(1 −pi)u′(cu)dcu, which is positive if and only if\ndce\ndcu\n> 1 −pi\npi\nu′(cu)\nu′(ce).\nSince (1 −pi)/pi is decreasing in i, an appropriate choice of dce/dcu yields an increase in\nutility if and only if j ≥i, which verifies A3.\nFinally, assume p1 ≤k < pI, which ensures that there are no gains from employing the\n11Again, we frame the discussion in terms of labor markets, rather than general insurance, because it\nseems more reasonable to assume an employer can only hire a fraction of the available workforce than to\nassume an insurance company can only serve a fraction of its potential customers.\n16\n\nlowest type, even in the absence of asymmetric information, but there may be gains from\ntrade for higher types, say by setting ce = cu = pI −k > 0. Let i∗denote the lowest type\nwithout gains from trade, so pi∗≤k < pi∗+1 (the results extend to the case k < p1 by\ndefining i∗= 0).\n6.2\nEquilibrium\nWe can again characterize equilibrium using (P), leading to:\nResult 4 There exists a competitive search equilibrium where for all i ≤i∗, ¯Ui = 0; and for\nall i > i∗, θi = 1, ¯Ui > 0, and ce,i > ce,i−1 and cu,i < cu,i−1 are the unique solution to\npi(1 −ce,i) −(1 −pi)cu,i = k\nand pi−1U(ce,i) + (1 −pi−1)U(cu,i) = pi−1U(ce,i−1) + (1 −pi−1)U(cu,i−1),\nwhere ce,i∗= cu,i∗= 0.\nOne interesting feature of equilibrium here is that cu,i < 0 for all i > i∗, and therefore a\nworker is worse offwhen he matches and turns out to be unproductive than when he does\nnot match in the first place. If one interprets a bad match as a layoff, contracts give laid-off\nworkers less utility than those who never match, because this keeps inferior workers from\napplying for the job.12\n6.3\nEfficiency\nAgain we show equilibrium may not be efficient. First note that a worker with pi close to 1\nsuffers little from the distortions introduced by the information problem. At the extreme, if\npI = 1, cu,I = c excludes other workers without distorting the type I contract at all. Gener-\nally, adverse selection has the biggest impact on the utility of workers with an intermediate\nvalue of pi. We show that a Pareto improvement may result from partial pooling. Consider\np1 = 1/4, p2 = 1/2, and p3 = 3/4 and suppose there are equal numbers of type 1 and 3,\nso that half of all matches are productive. Set U(c) = log(1 + c) and k = 3/8. Then in\nequilibrium, ¯U1 = 0; ce,2 = 0.344, cu,2 = −0.094, and ¯U2 = U(0.104); and ce,3 = 0.576,\ncu,3 = −0.227, and ¯U3 = U(0.319).\nPooling all three types, the best incentive-feasible allocation involves ce = cu = 1/8 and\n¯Ui = U(1/8). Compared to the equilibrium, this raises the utility of type 1 and type 2\n12To interpret a bad match as a layoff, it might help to imagine that an agent actually must work for some\ntime before productivity is realized, as in Jovanovic (1979).\n17\n\nworkers, but reduces the utility of type 3 workers. Now consider an allocation that pools\ntypes 1 and 2. If there are sufficiently few type 1 workers, it is feasible to set ce = cu > 0.104,\ndelivering utility greater to types 1 and 2.\nFor example, suppose π1 = π3 = 0.01 and\nπ2 = 0.98. Then the utility of types 1 and 2 rises to U(0.122). By raising the utility of type\n2, it is easier to exclude them from type 3 contracts, reducing the requisite inefficiency of\nthose contracts. This raises the utility of type 3, in this case, to U(0.325).\n6.4\nRelationship to Rothschild-Stiglitz\nRothschild and Stiglitz (1976, p. 630) “consider an individual who will have income of size\nW if he is lucky enough to avoid accident. In the event an accident occurs, his income will\nbe only W −d. The individual can insure against this accident by paying to an insurance\ncompany a premium α1 in return for which he will be paid ˆα2 if an accident occurs. Without\ninsurance his income in the two states, ‘accident,’ ‘no accident,’ was (W, W −d); with\ninsurance it is now (W −α1, W −d + α2) where α2 = ˆα2 −α1.”\nWe can normalize the utility of an uninsured individual to zero and express the utility of\none who anticipates an accident with probability pi as\nui(α1, α2) = piU(W −α1) + (1 −pi)U(W −d + α2) −κi,\nwhere κi ≡piU(W) + (1 −pi)U(W −d). Setting W = d = 1 and defining ce = 1 −α1\nand cu = α2, this is equivalent to our example. Our results apply to their setup, with one\nwrinkle: our fixed cost of posting contracts.\nRothschild and Stiglitz (1976) show that in any equilibrium, principals who attract type\ni agents, i > 1, offer incomplete insurance to deter type i −1 agents, which is of course\nvery similar to our finding. Under some conditions, however, their equilibrium does not\nexist. Starting from a configuration of separating contracts, suppose one principal deviates\nby offering a pooling contract to attract multiple types. In their setup, this is profitable if the\nleast-cost separating contract is Pareto inefficient. But such a deviation is never profitable\nin our environment.\nThe key difference is that in the original model a deviating principal can capture all\nthe agents in the economy, or at least a representative cross-section, while in our model a\nprincipal cannot serve all the agents who are potentially attracted to a contract. Instead,\nagents are rationed thorough the endogenous movement in market tightness θ. Whether\nsuch a deviation is profitable depends on which agents are most willing to accept a decline\nin θ.\nWhat we find is that high types are the first to give up on the pooling contract\nwhen applications get crowded by low types. A lower type, with an inferior outside option\n18\n\n¯Ui−1 < ¯Ui, is more inclined to stick it out. Hence, a principal who tries to offer a pooling\ncontract will end up with a long queue of type 1 agents—the worst possible outcome. For this\nreason, a deviation to a pooling contract is not profitable, and equilibrium with separating\ncontracts always exists.\n7\nApplication III: Asset Markets\nA feature of the previous two examples is that market tightness is not distorted: θi is at\nits full-information level for all i. We now present a model where tightness can be used to\nscreen bad types. Although the results hold more generally, to stress the point, we again\nabstract from traditional search frictions and assume µ(θ) = min{θ, 1}, so that matching is\ndetermined by the short side of the market and ¯η = 1.\n7.1\nSetup\nConsider an asset market with lemons, in the sense of Akerlof (1970). Buyers (principals)\nalways value an asset more than sellers (agents) value it, but some assets are better than\nothers and their values are private information to the seller. Market tightness, or probabilistic\ntrading, seems in principle a good way to screen out low quality asset holders, since sellers\nwith more valuable assets are more willing to accept a low probability of trade at any given\nprice. This model shows how an illiquid asset market may have a useful role as a screening\ndevice.\nEach type i seller is endowed with one indivisible asset, which call an apple, of type\ni, with value aS\ni > 0 to the seller and aB\ni > 0 to the buyer, both expressed in units of a\nnumeraire good. An action for type i sellers is a pair {αi, ti}, where αi is the probability\nthat the seller gives the buyer the apple and ti is the transfer of the numeraire to the seller.13\nThe payoffof a matched type i seller who reports type j is\nui(αj, tj) = tj −αjaS\ni ,\n13Given that apples are indivisible, it may be efficient to use lotteries, with α the probability apples change\nhands, as e.g. in Prescott and Townsend (1984) and Rogerson (1988). Nosal and Wallace (2007) provide a\nrelated model of an asset (money) market where probabilistic trade is useful due to private information, but\nthey use random and not directed search, leading to quite different results. It would actually be equivalent\nhere to assume apples are perfectly divisible and preferences are linear, with α reinterpreted as the fraction\ntraded, but we like the indivisibility since it allows us to contrast our results with models that use lotteries.\n19\n\nwhile the payoffof a buyer matched with a type i seller who reports truthfully is\nvi(αi, ti) = αiaB\ni −ti.\nNote that we have normalized the no-trade payoffto 0.\nWe set I = 2 and impose a number of restrictions on payoffs. First, both buyers and\nsellers prefer type 2 apples and both types of agents like apples:\naS\n2 > aS\n1 > 0 and aB\n2 > aB\n1 > 0.\nSecond, there would gains from trade, including the cost k of posting, if the buyer were sure\nto trade:\naS\ni + k < aB\ni for i = 1, 2.\nThe available actions are Y = [0, 1] × [0, 1], with ¯Yi = {(α, t) ∈Y|αaS\ni ≤t ≤αaB\ni −k}.\nUsing these restrictions, we verify our three assumptions. As a preliminary step, note that\n(α, t) ∈¯Yi implies α ≥k/(aB\ni −aS\ni ) > 0 and t ≥kaS\ni /(aB\ni −aS\ni ) > 0, so in any equilibrium\ncontract, trades are bounded away from zero.\nSince α > 0 whenever (α, t) ∈¯Yi, the restriction aB\n1 < aB\n2 implies A1. Also, A2 holds\nbecause for any (α, t) ∈¯Yi, a movement to (α, t −ε) with ε > 0 is feasible and raises buyer\nutility. The important assumption is again A3, which is here guaranteed by aS\n1 < aS\n2 . Fix\n(α, t) ∈¯Y and ¯a ∈(aS\n1 , aS\n2 ). For arbitrary δ > 0, consider (α′, t′) = (α −δ, t −¯aδ). This is\nfeasible for small δ because (α, t) ∈¯Y guarantees that α > 0 and t > 0. Then\nu2(α′, t′) −u2(α, t) = δ(aS\n2 −¯a) > 0\nu1(α′, t′) −u1(α, t) = δ(aS\n1 −¯a) < 0.\nNow for fixed ε > 0, choose δ ≤ε/\n√\n1 + ¯a2. This ensures (α′, t′) ∈Bε(α, t), so A3 holds.\n7.2\nEquilibrium\nWe again use problem (P) to characterize the equilibrium.\nResult 5 There exists a unique competitive search equilibrium with αi = 1, ti = aB\ni −k,\nθ1 = 1, ¯U1 = aB\n1 −aS\n1 −k, θ2 = aB\n1 −aS\n1 −k\naB\n2 −aS\n1 −k < 1, and ¯U2 = θ2(aB\n2 −aS\n2 −k).\nWith full information, θ2 = 1 and ¯U2 = aB\n2 −aS\n2 −k. Relative to this benchmark, buyers\npost too few contracts designed to attract type 2 sellers, and hence too many of them fail to\n20\n\ntrade. Since type 2 sellers have better apples than type 1, they are more willing to accept this\nin return for a better price when they do trade. Agents with inferior assets are less willing to\naccept a low probability of trade because they do not want to be stuck with their own apple,\nwhich is in fact a lemon. Note that the alternative of setting θ2 = 1 but rationing though the\nprobability of trade in a match, α2 < 1, wastes resources, because it involves posting more\ncontracts at cost k. In other words, reducing the matching rate is a cost-effective screening\ndevice compared to lotteries.\n7.3\nEfficiency\nConsider a pooling contract, with α1 = α2 = 1 and t1 = t2 = t.\nThat is, ¯Y = {C},\nwhere C = {(1, t), (1, t)}. Moreover, ˜Θ(C) = 1, ˜γi(C) = πi, and λ({C}) = 1. Finally, set\nt = π1aB\n1 + π2aB\n2 −k. The choice of t ensures that the resource constraint holds and the\nchoice of λ ensures that markets clear. All sellers apply and the contract is trivially incentive\ncompatible. The expected payofffor type i sellers is\n¯Ui = π1aB\n1 + π2aB\n2 −aS\ni −k.\nWith this pooling allocation, type 1 sellers are always better offthan they were in equi-\nlibrium since aB\n1 < aB\n2 , and type 2 sellers are better offif and only if\nπ1aB\n1 + π2aB\n2 −aS\n2 −k > (aB\n2 −aS\n2 −k)(aB\n1 −aS\n1 −k)\naB\n2 −aS\n1 −k\n.\nSince π2 = 1 −π1, this reduces to\nπ1 < aB\n2 −aS\n2 −k\naB\n2 −aS\n1 −k =\n¯U2\n¯U1\n.\nBoth the numerator and denominator are positive, but the numerator is smaller (gains from\ntrade are smaller for type 2 sellers) because aS\n2 > aS\n1 . Thus type 2 sellers prefer the pooling\nallocation when there is not too much cross subsidization, or π1 is small, so that the cost of\nsubsidizing type 1 sellers is worth the increased efficiency of trade.\n7.4\nNo Trade\nSo far, we have assumed there are gains from trade for both types of sellers. Now suppose\nthere are no gains from trade for type 1 apples, aB\n1 ≤aS\n1 + k. Then not only will type 1\nseller fail to trade, in equilibrium, the entire market will shut down:\n21\n\nResult 6 If aB\n1 ≤aS\n1 + k then, in any equilibrium, ¯U1 = ¯U2 = 0.\nNotice that the market shuts down here even if there are still gains from trade in good\napples, aB\n2 > aS\n2 + k. Intuitively, it is only possible to keep bad apples out of the market\nby reducing the probability of trade in good apples. If there is no market in bad apples,\nhowever, agents holding them would accept any probability of trade. Hence we cannot screen\nout bad apples, and this renders the good apple market inoperative. Whether this is related\nto the recent collapse in asset-backed securities markets seems worth further exploration.\n8\nConclusion\nWe have developed a tractable general framework to analyze adverse selection in competitive\nsearch markets. Under our assumptions, there is a unique equilibrium, where principals post\nseparating contracts. We characterized the equilibrium as the solution to a set of constrained\noptimization problems, and illustrated the use of the model through a series of examples.\nWe expect that one could extend the framework to dynamic situations, with repeated\nrounds of posting and search, as seems relevant in many applications—including labor and\nasset markets. It may also be interesting to study the case opposite to the one analyzed\nhere, where the informed instead of the uninformed parties post contracts.\nIn standard\ncompetitive search theory the outcome does not depend on who posts. With asymmetric\ninformation, contract posting by informed parties may introduce multiplicity of equilibrium\nthrough the usual signaling mechanism (see Delacroix and Shi (2007)). All of this is left for\nfuture work.\nAPPENDIX\nProof of Lemma 1.\nIn the first step, we prove that there exists a solution to (P). The\nsecond and third steps establish the stated properties of the solution.\nStep 1: Consider i = 1. If the constraint set in (P-1) is empty, set ¯U1 = 0. Otherwise,\n(P-1) is well-behaved, as the objective function is continuous and the constraint set compact.\nHence, (P-1) has a solution and a unique maximum m1. If m1 ≤0 set ¯U1 = 0; otherwise set\n¯U1 = m1 and let (θ1, y1) be one of the maximizers.\nWe now proceed by induction. Fix i > 1 and assume that we have found ¯Uj for all j < i\nand (θj, yj) for all j ∈I∗, j < i. Consider (P-i). If the constraint set is empty, set ¯Ui = 0.\nOtherwise, (P-i) again has a solution and a unique maximum mi. If mi ≤0, set ¯Ui = 0;\notherwise let ¯Ui = mi and let (θi, yi) be one of the maximizers.\n22\n\nStep 2: Suppose by way of contradiction that there exists i ∈I∗such that (θi, yi) solves\n(P-i) but η(θi)vi(yi) > k. This together with ¯Ui = µ(θi)ui(yi) > 0 implies that yi ∈¯Yi and\nµ(θi) > 0. Fix ε > 0 such that η(θi)vi(y) ≥k for all y ∈Bε(yi). Then A3 ensures there\nexists a y′ ∈Bε(yi) such that\nuj(y′) > uj(yi) for all j ≥i,\nuj(y′) < uj(yi) for all j < i.\nThen the pair (θi, y′) satisfies all the constraints of problem (P-i):\n1. η(θi)vi(y′) ≥k from the choice of ε;\n2. µ(θi)uj(y′) < µ(θi)uj(yi) ≤¯Uj for all j < i, where the first inequality is by construction\nof y′ and µ(θi) > 0, while the second holds since (θi, yi) solves (P-i).\nNow (θi, y′) achieves a higher value than (θi, yi) for the objective function in (P-i), given\nµ(θi)ui(y′) > µ(θi)ui(yi). Hence, (θi, yi) does not solve (P-i), a contradiction.\nStep 3: Fix i ∈I∗and suppose by way of contradiction that there exists j > i such that\nµ(θi)uj(yi) > ¯Uj. Let h be the smallest such j. Since i ∈I∗, µ(θi)ui(yi) = ¯Ui > 0, which\nimplies µ(θi) > 0 and ui(yi) > 0. Also, from the previous step (θi, yi) satisfies η(θi)vi(yi) = k,\nwhich ensures η(θi) > 0 and vi(yi) > 0. In particular, this implies that yi ∈¯Yi.\nThe pair (θi, yi) satisfies the constraints of (P-h) since\n1. η(θi)vh(yi) ≥η(θi)vi(yi) = k, where the first inequality holds by A1 given h > i and\nyi ∈¯Yi ⊂¯Y, and the second comes from the previous step;\n2. µ(θi)ul(yi) ≤¯Ul for all l < h, which holds for\n(a) l < i because (θi, yi) satisfy the constraints of (P-i),\n(b) l = i because ¯Ui = µ(θi)ui(yi) since i ∈I∗,\n(c) i < l < h by the choice of h as the smallest violation of µ(θi)uj(yi) > ¯Uj.\nSince µ(θi)uh(yi) > ¯Uh ≥0, (θi, yi) is in the constraint set of (P-h) and delivers a strictly\npositive value for the objective function; hence h ∈I∗. But then the fact that ¯Uh is not the\nmaximized value of (P-h) is a contradiction.\nProof of Proposition 1.\nWe proceed by construction.\n• The vector of expected utilities is ¯U = { ¯Ui}i∈I.\n23\n\n• The set of posted contracts is ¯C = {Ci}i∈I∗where Ci ≡(yi, . . . , yi).\n• λ is such that λ({Ci}) = πiΘ(Ci) for any i ∈I∗.\n• For i ∈I∗and Ci = (yi, . . . , yi), Θ(Ci) = θi. Otherwise, for any incentive compatible\nC′ = {y′\n1, . . . , y′\nI} ∈C, let J(C′) = {j|uj(y′\nj) > 0} denote the types that attain positive\nutility from C′. If J(C′) ̸= ∅and minj∈J(C′){ ¯Uj/uj(y′\nj)} < ¯µ then\nµ(Θ(C′)) = min\nj∈J(C′)\n¯Uj\nuj(y′\nj).\nIf this equation is consistent with multiple values of Θ(C′), pick the largest one. Oth-\nerwise, if J(C′) = ∅or minj∈J(C′){ ¯Uj/uj(y′\nj)} ≥¯µ, then Θ(C′) = ∞.\n• For i ∈I∗and Ci = (yi, . . ., yi), let γi(Ci) = 1 and so γj(Ci) = 0 for j ̸= i. For any\nother C′, define Γ(C′) such that γh(C′) > 0 only if h ∈arg minj∈J(C′){ ¯Uj/uj(y′\nj)}. If\nthere are multiple minimizers, let γh(C′) = 1 for the smallest such h. If J(C′) = ∅,\nagain choose Γ(C′) arbitrarily, e.g. γ1(C′) = 1.\nWe now verify that all of the equilibrium conditions hold.\nCondition (i): For any i ∈I∗, (θi, yi) solves (P-i), and Lemma 1 implies η(θi)vi(yi) = k.\nThus, profit maximization and free entry hold for any {Ci}i∈I∗. Now consider an arbitrary\nincentive compatible contract; we claim that principals’ profit maximization and free-entry\ncondition is satisfied.\nSuppose, to the contrary, that there exists C′ = (y′\n1, . . ., y′\nI) ∈C\nwith η(Θ(C′)) P\ni γi(C′)vi(y′\ni) > k. This implies η(Θ(C′)) > 0, so Θ(C′) < ∞, and there\nexists some type j with γj(C′) > 0 and η(Θ(C′))vj(y′\nj) > k.\nSince γj(C′) > 0 and\nΘ(C′) < ∞, our construction of Θ(C′) and Γ(C′) implies that j is the smallest solution\nto minh∈J(C′){ ¯Uh/uh(y′\nh)} and hence uj(y′\nj) > 0 and ¯Uj = µ(Θ(C′))uj(y′\nj). So for all h < j\nwith uh(y′\nh) > 0,\n¯Uh > µ(Θ(C′))uh(y′\nh) ≥µ(Θ(C′))uh(y′\nj),\nwhere the first inequality is by construction and the second follows because C′ is incentive\ncompatible.\nMoreover, if uh(y′\nh) ≤0, ¯Uh ≥µ(Θ(C′))uh(y′\nh) since ¯Uh ≥0.\nThis proves\nthat (Θ(C′), y′\nj) satisfies the constraints of (P-j).\nThen, since η(Θ(C′))vj(y′\nj) > k and\n¯Uh ≥µ(Θ(C′))uh(y′\nj) for all h < j, with strict inequality when uh(y′\nh) > 0, there exist a\nθ > Θ(C′) such that (θ, y′\nj) satisfies all the constraints of (P-j) and achieves a higher value\nthan (Θ(C′), y′\nj). This implies ¯Uj is not the maximized value of (P-j), a contradiction.\nCondition (ii): By construction, Θ and Γ ensure that ¯Ui ≥µ(Θ(C′))ui(y′\ni) for all contracts\nC′ = {y′\n1, . . . , y′\nI}, with equality if Θ(C′) < ∞and γi(C′) > 0. Moreover, for any i ∈I∗,\n24\n\n¯Ui = µ(θi)ui(yi) > 0 where θi = Θ(Ci) and Ci = {yi, . . . , yi} is the equilibrium contract\noffered to i. Finally, if ui(yi) < 0 so i /∈J(C′), either J(C′) = ∅, in which case Θ(C′) = ∞,\nor J(C′) ̸= ∅, in which case γh(C′) = 1 for some h ∈J(C′) and so γi(C′) = 0.\nCondition (iii): Market clearing obviously holds given the way we construct λ. Since all\nthe equilibrium conditions are satisfied the proof is complete.\nProof of Proposition 2.\nFrom equilibrium condition (i), any C ∈¯C has η(Θ(C)) > 0,\nhence Θ(C) < ∞. From condition (iii), ¯Ui > 0 implies γi(C) > 0 for some C ∈¯C. This\nproves that for each i ∈I∗, there exists a contract C ∈¯C with Θ(C) < ∞and γi(C) > 0.\nThe remainder of the proof proceeds in five steps. The first four steps show that for any\ni ∈I∗and Ci ∈¯C with θi = Θ(Ci) < ∞and γi(Ci) > 0, (θi, yi) solves (P-i). First, we\nprove that the constraint η(θi)vi(yi) ≥k is satisfied. Second, we prove that the constraint\nµ(θi)uj(yi) ≤¯Uj is satisfied for all j. Third, we prove that the pair (θi, yi) delivers ¯Ui to\ntype i. Fourth, we prove that (θi, yi) solves (P-i). The fifth and final step shows that for any\ni /∈I∗, either the constraint set of (P-i) is empty or the maximized value is nonpositive.\nStep 1: Take i ∈I∗and Ci ∈¯C with θi = Θ(Ci) < ∞and γi(Ci) > 0. We claim\nthe constraint η(θi)vi(yi) ≥k is satisfied in (P-i). Note first that i ∈I∗implies ¯Ui > 0. By\nequilibrium condition (ii), ¯Ui = µ(θi)ui(yi), and µ(θi) > 0. To derive a contradiction, assume\nη(θi)vi(yi) < k. Equilibrium condition (i) implies η(θi) P\nj γj(C)vj(yj) = k, so there is an\nh with γh(C) > 0 and η(θi)vh(yh) > k. Since η(θi) ≤¯η, ¯ηvh(yh) > k. Moreover, because\nθi = Θ(C) < ∞and γh(C) > 0, optimal search implies uh(yh) > 0. This proves yh ∈¯Yh.\nNext, fix ε > 0 such that η(θi)vh(y) > k for all y ∈Bε(yh). Then A3 together with\nyh ∈¯Y guarantees that there exists y′ ∈Bε(yh) such that\nuj(y′) > uj(yh) for all j ≥h,\nuj(y′) < uj(yh) for all j < h.\nNotice that y′ ∈¯Yh as well, given that uh(y′) > uh(yh) > 0 and ¯ηvh(y′) ≥η(θi)vh(y′) > k.\nNow consider C′ = {y′, . . . , y′} and θ′ ≡Θ(C′). Note that\nµ(θ′)uh(y′) ≤¯Uh = µ(θi)uh(yh) < µ(θi)uh(y′),\nwhere the weak inequality follows from optimal search, the equality holds because θi < ∞\nand γh(C) > 0, and the strict inequality holds by the construction of y′, since µ(θi) > 0.\nThis implies µ(θ′) < µ(θi), and θ′ < θi. Next observe that for all j < h, either uj(y′) < 0, in\n25\n\nwhich case γj(C′) = 0 by equilibrium condition (ii), or\nµ(θ′)uj(y′) < µ(θi)uj(yh) ≤µ(θi)uj(uj) ≤¯Uj,\nwhere the first inequality uses µ(θ′) < µ(θi), the second uses incentive compatibility, and the\nthird follows from optimal search. Hence, γj(C′) = 0 for all j < h.\nFinally, profits from posting C′ are\nη(θ′)\nI\nX\nj=1\nγj(C′)vj(y′) ≥η(θ′)vh(y′) ≥η(θi)vh(y′) > k.\nThe first inequality follows because γj(C′) = 0 if j < h and vh(y′) is nondecreasing in h by\nA1 together with y′ ∈¯Yh ⊂¯Y, the second follows because θ′ < θi, and the last inequality\nuses the construction of ε. Posting C′ is therefore strictly profitable, which is a contradiction,\nand this completes Step 1.\nStep 2: Again take i ∈I∗and Ci ∈¯C with θi = Θ(Ci) < ∞and γi(Ci) > 0. Equilibrium\ncondition (ii) implies µ(θi)uj(yj) ≤¯Uj for all j while incentive compatibility implies uj(yi) ≤\nuj(yj). Hence the constraint µ(θi)uj(yi) ≤¯Uj in (P-i) is satisfied for all j.\nStep 3: Again take i ∈I∗and Ci ∈¯C with θi = Θ(Ci) < ∞and γi(Ci) > 0. Equilibrium\ncondition (ii) implies ¯Ui = µ(θi)ui(yi), since θi < ∞and γi(Ci) > 0. Hence(θi, yi) delivers ¯Ui\nto type i.\nStep 4: Again take i ∈I∗and Ci =∈¯C with θi = Θ(Ci) < ∞and γi(Ci) > 0. To find a\ncontradiction, suppose there exists (θ′, y′) that satisfies the constraints of (P-i) but delivers\nhigher utility. That is, η(θ′)vi(y′) ≥k, µ(θ′)uj(y′) ≤¯Uj for all j < i, and µ(θ′)ui(y′) > ¯Ui.\nWe now use A2. Note that µ(θ′)ui(y′) > ¯Ui > 0 implies µ(θ′) > 0 and ui(y′) > 0, while\nη(θ′)vi(y′) ≥k implies vi(y′) > 0 and so ¯ηvi(y′) ≥k. In particular, y′ ∈¯Yi. We can therefore\nfix ε′ > 0 such that for all y ∈Bε′(y′), µ(θ′)ui(y) > ¯Ui, and then choose y′′ ∈Bε′(y′)\nsuch that vi(y′′) > vi(y′) and uj(y′′) ≤uj(y′) for all j < i. This ensures η(θ′)vi(y′′) > k,\nµ(θ′)uj(y′′) ≤¯Uj for all j < i, and µ(θ′)ui(y′′) > ¯Ui. Note that we still have y′′ ∈¯Yi.\nWe now use A3. Fix ε′′ > 0 such that for all y ∈Bε′′(y′′), η(θ′)vi(y) > k and µ(θ′)ui(y) >\n¯Ui. Choose y′′′ ∈Bε′′(y′′) such that\nuj(y′′′) > uj(y′′) for all j ≥i,\nuj(y′′′) < uj(y′′) for all j < i.\nThis ensures η(θ′)vi(y′′′) > k, µ(θ′)uj(y′′′) < ¯Uj for all j < i, and µ(θ′)ui(y′′′) > ¯Ui. Note\nthat we still have y′′′ ∈¯Yi.\n26\n\nNow consider C′′′ ≡{y′′′, . . . , y′′′}. From equilibrium condition (ii), µ(θ′)ui(y′′′) > ¯Ui\nimplies that µ(θ′) > µ(Θ(C′′′)), which guarantees η(Θ(C′′′))vi(y′′′) > k. This also implies\nΘ(C′′′) < ∞.\nWe next claim γj(y′′′) = 0 for all j < i. Suppose γj(y′′′) > 0 for j < i. Since Θ(C′′′) <\n∞, equilibrium condition (ii) implies uj(y′′′) ≥0. We have already shown that µ(θ′) >\nµ(Θ(C′′′)), and uj(y′′) > uj(y′′′) by construction. Thus µ(θ′)uj(y′′) > µ(Θ(C′′′))uj(y′′′). The\nresult follows from equilibrium condition (ii) and ¯Uj ≥µ(θ′)uj(y′′); i.e., ¯Uj > µ(Θ(C′′′))uj(y′′′)\nand Θ(C′′′) < ∞implies γj(C′′′) = 0, a contradiction.\nThe profit from offering this contract is\nη(Θ(C′′′))\nI\nX\nj=1\nγj(C′′′)vj(y′′′) ≥η(Θ(C′′′))vi(y′′′) > k,\nwhere the first inequality uses γj(C′′′) = 0 for j < i and A1. This contradicts the first\ncondition in the definition of equilibrium, and proves (θi, yi) solves (P-i).\nStep 5: Suppose there is an i /∈I∗for which the constraint set of (P-i) is nonempty and the\nmaximized value is positive. That is, suppose there exists (θ′, y′) such that η(θ′)vi(y′) ≥k,\nµ(θ′)uj(y′) ≤¯Uj for all j < i, and µ(θ′)ui(y′) > ¯Ui = 0. Replicating step 4, we can first find\ny′′ such that η(θ′)vi(y′′) > k, µ(θ′)uj(y′′) ≤¯Uj for all j < i, and µ(θ′)ui(y′′) > 0. Then we\nfind y′′′ such that η(θ′)vi(y′′′) > k, uj(y′′) > uj(y′′′) for j < i, and uj(y′′) < uj(y′′′) for j ≥i.\nFinally, C′′′ = {y′′′, . . ., y′′′} only attracts type i or higher and hence must be profitable and\ndeliver positive utility, a contradiction.\nProof of Proposition 3.\nBy Lemma 1 there is a solution to (P). Proposition 1 shows\nthat if I∗, { ¯Ui}i∈I, {θi}i∈I∗, and {yi}i∈I∗solve (P), there is an equilibrium { ¯U, λ, ¯C, Θ, Γ} with\nthe same ¯U, ¯C = {Ci}i∈I∗, where Ci = (yi, . . . , yi), Θ(Ci) = θi, and γi(Ci) = 1. This proves\nexistence. Proposition 2 shows that in any equilibrium { ¯U, λ, ¯C, Θ, Γ}, ¯Ui is the maximum\nvalue of (P-i) for all i ∈I∗, and ¯Ui = 0 otherwise. Lemma 1 shows there is a unique maximum\nvalue ¯Ui for (P-i) for all i ∈I∗. This proves that the equilibrium ¯U is unique.\nProof of Proposition 4.\nConsider i = 1. Fix y satisfying ¯ηv1(y) > k and u1(y) > 0.\nThen fix θ > 0 satisfying η(θ)v1(y) = k. These points satisfy the constraints of (P-1) and\ndeliver utility µ(θ)u1(y) > 0. This proves ¯U1 > 0. Now suppose ¯Uj > 0 for all j < i. We\nclaim ¯Ui > 0. Again fix y satisfying ¯ηvi(y) > k and ui(y) > 0. Then fix θ > 0 satisfying\nη(θ)v1(y) ≥k and µ(θ)uj(y) ≤¯Uj for all j < i; this is feasible since ¯Uj > 0 and µ(0) = 0.\nThese points satisfy the constraints of (P-i) and deliver utility µ(θ)ui(y) > 0, which proves\n27\n\n¯Ui > 0. By induction, the proof is complete.\nProof of Result 1.\nUsing η(θ) = µ(θ)/θ, write (P-1) as\n¯U1 =\nmax\nθ∈[0,∞],(t,x)∈Y µ(θ)\n\u0012\nt −x\na1\n\u0013\ns.t. µ(θ)\nθ\n(b1 −t) ≥k.\nBy Lemma 1 the constraint is binding, so we can eliminate t and reduce the problem to\n¯U1 =\nmax\nθ∈[0,∞],x∈[0,b2a2] µ(θ)\n\u0012\nb1 −x\na1\n\u0013\n−θk.\nAt the solution, x = 0 and θ = θ1 solves µ′(θ1)b1 = k. Using this to eliminate b1 from the\nobjective function delivers ¯U1, and the constraint delivers t1.\nNext, solve (P-2) using the ¯U1 derived in the previous step:\n¯U2 =\nmax\nθ∈[0,∞],(t,x)∈Y µ(θ)\n\u0012\nt −x\na2\n\u0013\ns.t. µ(θ)\nθ (b2 −t) ≥k,\nand µ(θ)\n\u0012\nt −x\na1\n\u0013\n≤\n\u0012 µ(θ1)\nµ′(θ1) −θ1\n\u0013\nk.\nAgain we can eliminate t. It is easy to see that the second constraint binds, so we can\neliminate x and then check that at the solution x ≥0. Thus, the problem reduces to\n¯U2 = max\nθ∈[0,∞]\n\u0014a1\na2\n\u0012 µ(θ1)\nµ′(θ1) −θ1\n\u0013\nk +\n\u0012\n1 −a1\na2\n\u0013\n(µ(θ)b2 −θk)\n\u0015\n.\nNow θ = θ2 solves µ′(θ2)b2 = k, where the concavity of µ implies θ1 < θ2. Substituting into\nthe objective function gives ¯U2. Then the constraints give t2 and x2. Concavity of µ ensures\nthat µ(θ)/µ′(θ) −θ is increasing in θ, which implies x2 > 0.\nProof of Result 2.\nFix x1 ≤a1(b2 −b1)(1 −π1)/π1. We now construct an equilibrium.\nAssume ¯C = {C} where C = {(t + x1/a1, x1), (t, 0)}, and ¯U1 = ¯U2 = ¯U = µ(θ)t, where t\nand θ are defined above. Moreover, Θ(C) = θ, γi(C) = πi for i = 1, 2, and λ({C}) = θ. For\nany other incentive compatible contract C′ = {(t′\n1, x′\n1), (t′\n2, x′\n2)} ∈C, C′ ̸= C, suppose Θ(C′)\nsolves\n¯U = µ(Θ(C′))\n\u0012\nt′\n1 −x′\n1\na1\n\u0013\nif this defines Θ(C′) < ∞; if ¯U ≥¯µ(t′\n1 −x′\n1/a1), Θ(C′) = ∞. And suppose γ1(C′) = 1 and\nγ2(C′) = 0 for all such contracts. By construction, profit maximization and free entry hold\n28\n\nfor C, with t chosen so that firms break even.\nFor any other incentive-compatible C′ ̸= C, profit maximization and free entry reduce to\nη(Θ(C′))(b1 −t′\n1) ≤k.\nSince η(∞) = 0, this holds if Θ(C′) = ∞. Otherwise, use Θ(C′) to eliminate t′\n1. We need to\nshow that\nµ(Θ(C′))\n\u0012\nb1 −x′\n1\na1\n\u0013\n−Θ(C′)k ≤¯U.\nAn upper bound on the left hand side is obtained setting x′\n1 = 0 and choosing Θ(C′)\nto maximize µ′(Θ(C′))b1 = k.\nThe restriction x1 ≤a1(b2 −b1)(1 −π1)/π1 implies that\nb1 ≤π1\n\u0000b1 −x1\na1\n\u0001\n+ π2b2, from which it follows that Θ(C′) ≤θ. That is,\nµ(Θ(C′))\n\u0012\nb1 −x′\n1\na1\n\u0013\n−Θ(C′)k ≤\n\u0012 µ(Θ(C′))\nµ′(Θ(C′)) −Θ(C′)\n\u0013\nk ≤\n\u0012 µ(θ)\nµ′(θ) −θ\n\u0013\nk = ¯U,\nwhere the first inequality uses the preceding discussion, the second inequality holds because\nΘ(C′) ≤θ, and the third holds from the construction of ¯U.\nNext, optimal search holds by construction for type 1. For type 2, we need to verify\n¯U ≥µ(Θ(C′))\n\u0012\nt′\n2 −x′\n2\na2\n\u0013\n.\nTo this end, note that t′\n1 −x′\n1/a1 ≥t′\n2 −x′\n2/a1 ≥t′\n2 −x′\n2/a2, where the first inequality comes\nfrom incentive compatibility of C′ and the second from a1 ≥a2 and x′\n2 ≥0. This implies\nthe desired inequality:\nµ(Θ(C′))\n\u0012\nt′\n2 −x′\n2\na2\n\u0013\n≤µ(Θ(C′))\n\u0012\nt′\n1 −x′\n1\na1\n\u0013\n= ¯U,\nFinally, market clearing holds by construction.\nProof of Result 3.\nThroughout this proof, we suppose there is an equilibrium character-\nized by the single incentive compatible contract C = {(t1, x1), (t2, x2)}. In the first step we\nprove that x2 = 0 and in the second that t1 = t2 + x1/a1.\nStep 1: Suppose x2 > 0. Given that γi(C) = πi > 0 for i = 1, 2 and Θ(C) < ∞, optimal\nsearch requires\n¯Ui = µ(Θ(C))\n\u0012\nti −xi\nai\n\u0013\n,\nfor i = 1, 2. Now consider C′ = {(t2 −x2/a2, 0), (t2 −x2/a2, 0)}. Optimal search by type 2\n29\n\nrequires\nµ(Θ(C′))\n\u0012\nt2 −x2\na2\n\u0013\n≤¯U2 = µ(Θ(C))\n\u0012\nt2 −x2\na2\n\u0013\n,\nwhich implies that Θ(C′) ≤Θ(C). Moreover, notice that\nt1 −x1\na1\n≥t2 −x2\na1\n> t2 −x2\na2\n,\nwhere the first inequality follows from incentive compatibility and the second from a1 > a2\nand x2 > 0.\nTogether with Θ(C′) ≤Θ(C), this implies\n¯U1 = µ(Θ(C))\n\u0012\nt1 −x1\na1\n\u0013\n> µ(Θ(C′))\n\u0012\nt2 −x2\na2\n\u0013\n.\nThus, γ1(C′) = 0. Hence, given that Θ(C′) ≤Θ(C) < ∞, γ2(C′) = 1 and Θ(C′) = Θ(C).\nThen expected profit for a firm posting C′ is\nη(Θ(C′))\n\u0012\nb2 −t2 + x2\na2\n\u0013\n> η(Θ(C))(π1(b1 −t1) + π2(b2 −t2)) = k.\nThe first inequality follows from b1 < b2, t1 > t2 −x2/a2 + x1/a1 ≥t2 −x2/a2, and Θ(C′) =\nΘ(C), while the second follows from the fact that in the proposed equilibrium firms post C\nand break even. Hence, C′ represents a profitable deviation, a contradiction.\nStep 2: Incentive compatibility and x2 = 0 imply that t1 −x1/a1 ≥t2.\nBy way of\ncontradiction, suppose t1 −x1/a1 > t2. Consider C′ = {(t2 + x1/a1, x1), (t2, 0)}. Then\nµ(Θ(C′))t2 ≤¯U2 = µ(Θ(C))t2,\nwhere the first inequality follows from optimal search by type 2 for C′ and the second from\noptimal search for C together with γ2(C) = π2 > 0. This implies that Θ(C′) ≤Θ(C). Hence,\n¯U1 = µ(Θ(C))\n\u0012\nt1 −x1\na1\n\u0013\n> µ(Θ(C′))t2,\nwhere the first equality follows from optimal search by type 1 for C and γ1(C) = π1 > 0,\nwhile the second comes from Θ(C′) ≤Θ(C) and t1 −x1/a1 > t2. Hence, γ1(C′) = 0, and,\ngiven that Θ(C′) ≤Θ(C) < ∞, γ2(C′) = 1 and Θ(C′) = Θ(C). So the expected profit from\nC′ is\nη(Θ(C′)))(b2 −t2) > η(Θ(C))(π1(b1 −t1) + π2(b2 −t2)) = k,\n30\n\ngiven that Θ(C′) = Θ(C) and b2 −t2 > b1 −(t1 −x1/a1) > b1 −t1. Hence C′ represents a\nprofitable deviation, a contradiction.\nProof of Result 4.\nFor i ≤i∗, consider (P-i) without the constraint of keeping out lower\ntypes. This relaxed problem should yield a higher payoff\n¯Ui ≤\nmax\nθ∈[0,∞],(ce,cu)∈Y min{θ, 1}\n\u0000piU(ce) + (1 −pi)U(cu)\n\u0001\ns.t. min{1, θ−1}(pi(1 −ce) −(1 −pi)cu) ≥k.\nAt the solution, cu,i = ce,i = ci, so this reduces to\n¯Ui ≤\nmax\nθ∈[0,∞],c≥cmin{θ, 1}U(c)\ns.t. min{1, θ−1}(pi −c) ≥k.\nEither the constraint set is empty (if pi < k + c) or there are no points in the constraint set\nthat give positive utility (given that pi −k ≤0). In any case, this gives ¯Ui = 0.\nTurn next to a typical problem (P-i), i > i∗:\n¯Ui =\nmax\nθ∈[0,∞],(ce,cu)∈Y min{θ, 1}\n\u0000piU(ce) + (1 −pi)U(cu)\n\u0001\ns.t. min{1, θ−1}(pi(1 −ce) −(1 −pi)cu) ≥k\nand min{θ, 1}\n\u0000pjU(ce) + (1 −pj)U(cu)\n\u0001\n≤¯Uj for all j < i.\nWe claim the solution sets θi ≤1 (if θi > 1, reducing θi to 1 relaxes the first constraint\nwithout otherwise affecting the problem). Hence, we can rewrite the problem as\n¯Ui =\nmax\nθ≤1,(ce,cu)∈Y θ\n\u0000piU(ce) + (1 −pi)U(cu)\n\u0001\ns.t. pi(1 −ce) −(1 −pi)cu ≥k\nand θ\n\u0000pjU(ce) + (1 −pj)U(cu)\n\u0001\n≤¯Uj for all j < i.\nLemma 1 ensures the first constraint is binding, which proves pi(1 −ce,i) −(1 −pi)cu,i = k.\nIt remains to prove that θi = 1, ce,i > ce,i−1, cu,i < cu,i−1, and the constraint for j = i −1\nbinds.\nWe start by establishing these claims for i = i∗+ 1. In this case, ce,i = cu,i = 0 satisfies\nthe constraints but leaves the first one slack. Lemma 1 implies that it is possible to do better,\nwhich proves ¯Ui > 0. On the other hand, consider any (ce,i, cu,i) that delivers positive utility\n31\n\nand satisfies the last constraint, so\npiU(ce,i) + (1 −pi)U(cu,i) > 0 and pjU(ce,i) + (1 −pj)U(cu,i) ≤0\nfor all j < i. Subtracting inequalities gives\n(pi −pj)(U(ce,i) −U(cu,i)) > 0,\nwhich proves ce,i > cu,i. Now if ce,i > cu,i ≥0, pjU(ce,i) + (1 −pj)U(cu,i) > 0, so this is\ninfeasible. If 0 ≥ce,i > cu,i, piU(ce,i) + (1 −pi)U(cu,i) < 0, so this is suboptimal. This proves\nce,i > 0 > cu,i when i = i∗+ 1. Finally, since pjU(ce,i) + (1 −pj)U(cu,i) ≤0 for all j < i,\nsetting θi = 1 raises the value of the objective function without affecting the constraints and\nso is optimal.\nWe now proceed by induction. Fix i > i∗+1 and assume that for all j ∈{i∗+1, . . ., i−1},\nwe have ce,j > ce,j−1, cu,j < cu,j−1, θj = 1, and\npj−1U(ce,j) + (1 −pj−1)U(cu,j) = pj−1U(ce,j−1) + (1 −pj−1)U(cu,j−1) = ¯Uj−1.\nWe establish the result for i. Setting ce,i = ce,i−1, cu,i = cu,i−1, and θi = 1 satisfies the\nconstraints in (P-i). Since it leaves the first constraint slack, Lemma 1 implies it is possible\nto do better. Thus\nθi\n\u0000piU(ce,i) + (1 −pi)U(cu,i)\n\u0001\n> piU(ce,i−1) + (1 −pi)U(cu,i−1).\nOn the other hand, incentive compatibility implies,\nθi\n\u0000pi−1U(ce,i) + (1 −pi−1)U(cu,i)\n\u0001\n≤¯Ui−1 = pi−1U(ce,i−1) + (1 −pi−1)U(cu,i−1).\nSubtracting inequalities gives (pi −pi−1)\n\u0000θiU(ce,i) −U(ce,i−1) −θiU(cu,i) + U(cu,i−1)\n\u0001\n>\n0. Using pi > pi−1, this implies θiU(ce,i) −U(ce,i−1) > θiU(cu,i) −U(cu,i−1). As before,\nwe can rule out the possibility that θiU(cu,i) ≥U(cu,i−1), because this is infeasible. We\ncan rule out the possibility that θiU(ce,i) ≤U(ce,i−1), because this is suboptimal. Hence\nθiU(ce,i) −U(ce,i−1) > 0 > θiU(cu,i) −U(cu,i−1). Now since U(ce,i−1) > 0 and θi ∈[0, 1],\nthe first inequality implies ce,i > ce,i−1. Since U(cu,i−1) < 0, the second inequality implies\ncu,i < cu,i−1.\nNext suppose θi < 1 and consider the following variation: raise θi to 1 and increase ce\nand reduce cu while keeping both θ\n\u0000piU(ce)+(1−pi)U(cu)\n\u0001\nand pice +(1−pi)cu unchanged;\n32\n\ni.e. set ce > ce,i and cu < cu,i so that\nθi\n\u0000piU(ce,i) + (1 −pi)U(cu,i)\n\u0001\n= piU(ce) + (1 −pi)U(cu)\nand pice,i + (1 −pi)cu,i = pice + (1 −pi)cu.\nFor all j < i, (pj −pi)\n\u0000θiU(ce,i) −U(ce) + U(cu) −θiU(cu,i)\n\u0001\n> 0 since pj < pi, 0 < ce,i < ce,\ncu < cu,i < 0, and θi < 1. Add this to θi\n\u0000piU(ce,i)+(1−pi)U(cu,i)\n\u0001\n= piU(ce)+(1−pi)U(cu) to\nobtain θi(pjU(ce,i)+(1−pj)U(cu,i)) > pjU(cu)+(1−pj)U(ce). This implies the perturbation\nrelaxes the remaining constraints and so is feasible. This proves θi = 1.\nFinally, we prove that the constraints for j < i −1 are slack. If not then\npjU(ce,i) + (1 −pj)U(cu,i) = ¯Uj ≥pjU(ce,i−1) + (1 −pj)U(cu,i−1),\nwhere the inequality uses the constraints in problem (P-(i −1)). On the other hand,\npi−1U(ce,i) + (1 −pi−1)U(cu,i) ≤¯Ui−1 = pi−1U(ce,i−1) + (1 −pi−1)U(cu,i−1),\nwhere the inequality is a constraint in (P-i) and the equality is the definition of ¯Ui−1. Sub-\ntracting these equations, (pi−1 −pj)\n\u0000U(ce,i−1) −U(ce,i) + U(cu,i) −U(cu,i−1)\n\u0001\n≥0. Since\npi−1 > pj, ce,i > ce,i−1, and cu,i < cu,i−1, we have a contradiction. The constraints for all\nj < i−1 are slack, while the constraint for i−1 binds, otherwise the solution to (P-i) would\nhave ce,i = cu,i = pi −k.\nProof of Result 5.\nWrite (P-1) as\n¯U1 =\nmax\nθ∈[0,∞],(α,t)∈Y min{θ, 1}\n\u0000t −αaS\n1\n\u0001\ns.t. min{1, θ−1}\n\u0000αaB\n1 −t\n\u0001\n≥k.\nBy Lemma 1 we can rewrite the problem as\n¯U1 =\nmax\nθ∈[0,∞],α∈[0,1] min{θ, 1}α\n\u0000aB\n1 −aS\n1\n\u0001\n−θk.\nSince aB\n1 > aS\n1 + k, it is optimal to set α = θ = 1. It follows that ¯U1 = aB\n1 −aS\n1 −k.\n33\n\nNow consider (P-2)\n¯U2 =\nmax\nθ∈[0,∞],(α,t)∈Y min{θ, 1}\n\u0000t −αaS\n2\n\u0001\ns.t. min{1, θ−1}\n\u0000αaB\n2 −t\n\u0001\n≥k\nmin{θ, 1}\n\u0000t −αaS\n1\n\u0001\n≤aB\n1 −aS\n1 −k.\nAgain, we can eliminate t using the first constraint to write\n¯U2 =\nmax\nθ∈[0,∞],α∈[0,1]min{θ, 1}α\n\u0000aB\n2 −aS\n2\n\u0001\n−θk\ns.t. min{θ, 1}α\n\u0000aB\n2 −aS\n1\n\u0001\n−θk = aB\n1 −aS\n1 −k.\nThen use the last constraint to eliminate α and write\n¯U2 = max\nθ∈[0,∞]\naB\n1 −aS\n1 −(1 −θ)k\naB\n2 −aS\n1\n\u0000aB\n2 −aS\n2\n\u0001\n−θk\ns.t. aB\n1 −aS\n1 −(1 −θ)k\nmin{θ, 1}(aB\n2 −aS\n1 ) ∈[0, 1],\nwhere the constraint here ensures that α is a probability. Since aS\n1 < aS\n2 < aB\n2 , the objective\nfunction is decreasing in θ, and we set it to the smallest value consistent with the constraints:\nθ2 = aB\n1 −aS\n1 −k\naB\n2 −aS\n1 −k < 1.\nThis implies α2 = 1, so the constraint binds. Then ¯U2 is easy to compute.\nProof of Result 6.\nWrite problem (P-1) as\n¯U1 =\nmax\nθ∈[0,∞],(α,t)∈Y min{θ, 1}\n\u0000t −αaS\n1\n\u0001\ns.t. min{1, θ−1}\n\u0000αaB\n1 −t\n\u0001\n≥k.\nAgain we can eliminate t and rewrite this as\n¯U1 =\nmax\nθ∈[0,∞],α∈[0,1] min{θ, 1}α\n\u0000aB\n1 −aS\n1\n\u0001\n−θk.\nSince aB\n1 ≤aS\n1 + k, ¯U1 = 0, which is attained by θ = 0.\n34\n\nNow consider (P-2):\n¯U2 =\nmax\nθ∈[0,∞],(α,t)∈Y min{θ, 1}\n\u0000t −αaS\n2\n\u0001\ns.t. min{1, θ−1}\n\u0000αaB\n2 −t\n\u0001\n≥k\nmin{θ, 1}\n\u0000t −αaS\n1\n\u0001\n≤0.\nEliminating t using the first constraint gives\n¯U2 =\nmax\nθ∈[0,∞],α∈[0,1] min{θ, 1}α\n\u0000aB\n2 −aS\n2\n\u0001\n−θk\ns.t. min{θ, 1}α\n\u0000aB\n2 −aS\n1\n\u0001\n−θk = 0.\nEliminating α using the last constraint gives\n¯U2 = max\nθ∈[0,∞]\naS\n1 −aS\n2\naB\n2 −aS\n1\nθk\ns.t.\nθk\nmin{θ, 1}(aB\n2 −aS\n1 ) ∈[0, 1].\nSince aB\n2 > aS\n2 > aS\n1 , the fraction in the objective function is negative. Hence θ = 0 and\n¯U2 = 0.\n35\n\nReferences\nAcemoglu, Daron and Robert Shimer, “Efficient Unemployment Insurance,” Journal\nof Political Economy, 1999, 107 (5), 893–928.\nAkerlof, George A., “The Market for “Lemons”: Quality Uncertainty and the Market\nMechanism,” The Quarterly Journal of Economics, 1970, 84 (3), 488–500.\n, “The Economics of Caste and of the Rat Race and Other Woeful Tales,” Quarterly\nJournal of Economics, 1976, 90 (4), 599–617.\nBisin, Alberto and Piero Gottardi, “Efficient Competitive Equilibria with Adverse Se-\nlection,” Journal of Political Economy, 2006, 114 (3), 485–516.\nBurdett, Kenneth, Shouyong Shi, and Randall Wright, “Pricing and Matching with\nFrictions,” Journal of Political Economy, 2001, 109 (5), 1060–1085.\nDelacroix, A. and S. Shi, “Pricing and Signaling with Frictions,” 2007. 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Townsend, “Pareto Optima and Competitive\nEquilibria with Adverse Selection and Moral Hazard,” Econometrica, 1984, 52 (1), 21–45.\nRiley, John G., “Informational Equilibrium,” Econometrica, 1979, 47 (2), 331–360.\nRogerson, Richard, “Indivisible Labor, Lotteries and Equilibrium,” Journal of Monetary\nEconomics, 1988, 21 (1), 3–16.\nRothschild, Michael and Joseph Stiglitz, “Equilibrium in Competitive Insurance Mar-\nkets: An Essay on the Economics of Imperfect Information,” Quarterly Journal of Eco-\nnomics, 1976, 90 (4), 629–649.\nShimer, Robert, “Contracts in a Frictional Labor Market,” 1996. MIT mimeo.\nSpence, Michael, “Job Market Signaling,” Quarterly Journal of Economics, 1973, 87 (3),\n355–374.\nWilson, Charles, “A Model of Insurance Markets with Incomplete Information,” Journal\nof Economic Theory, 1977, 16 (2), 167–207.\n37"},"Token_count":{"kind":"number","value":27355,"string":"27,355"}}},{"rowIdx":6699,"cells":{"Source":{"kind":"string","value":"NBER"},"Date":{"kind":"number","value":2009,"string":"2,009"},"Text":{"kind":"string","value":"NBER WORKING PAPER SERIES\nEXPERTS AND THEIR RECORDS\nAlexander Frankel\nMichael Schwarz\nWorking Paper 14921\nhttp://www.nber.org/papers/w14921\nNATIONAL BUREAU OF ECONOMIC RESEARCH\n1050 Massachusetts Avenue\nCambridge, MA 02138\nApril 2009\nWe would like to thank Heski Bar-Isaac, Martin Cripps, Drew Fudenberg, Andrzej Skrzypacz, and\nSteve Tadelis for their helpful comments, as well as seminar audiences at Stanford and Stanford GSB.\n Alexander Frankel primarily worked on this during an internship at Yahoo! Research. The views\nexpressed herein are those of the author(s) and do not necessarily reflect the views of the National\nBureau of Economic Research.\nNBER working papers are circulated for discussion and comment purposes. They have not been peer-\nreviewed or been subject to the review by the NBER Board of Directors that accompanies official\nNBER publications.\n© 2009 by Alexander Frankel and Michael Schwarz. All rights reserved. Short sections of text, not\nto exceed two paragraphs, may be quoted without explicit permission provided that full credit, including\n© notice, is given to the source.\n\nExperts and Their Records\nAlexander Frankel and Michael Schwarz\nNBER Working Paper No. 14921\nApril 2009\nJEL No. C7,C73,D82,J01\nABSTRACT\nConsider an environment where long-lived experts repeatedly interact with short-lived customers.\nIn periods when an expert is hired, she chooses between providing a profitable major treatment or\na less profitable minor treatment. The expert has private information about which treatment best serves\nthe customer, but has no direct incentive to act in the customer's interest. Customers can observe the\npast record of each expert's actions, but never learn which actions would have been appropriate. We\nfind that there exists an equilibrium in which experts always play truthfully and choose the customer's\npreferred treatment. The expert is rewarded for choosing the less profitable action with future business:\ncustomers return to an expert with high probability if the previous treatment was minor, and low probability\nif it was major. \nIf experts have private information regarding their own payoffs as well as what treatments are appropriate,\nthen there is no equilibrium with truthful play in every period. But we construct equilibria where\nexperts are truthful arbitrarily often as their discount factor converges to one.\nAlexander Frankel\nStanford Graduate School of Business\n518 Memorial Way\nStanford, CA 94305\nafrankel@stanford.edu\nMichael Schwarz\nYahoo! Research\n2397 Shattuck Ave\nBerkeley, CA 94704\nand NBER\nmschwarz@yahoo-inc.com\n\n1\nIntroduction\nIn many economic environments, uninformed customers must rely on experts to both diag-\nnose and treat their problems. Doctors, dentists, mechanics, and management consultants\nall help to determine what services their clients need in addition to providing those services.\nThere is a misalignment of incentives when experts earn higher profits on certain treatments\nthan on others.\nThis paper considers a repeated environment in which experts are long-lived and cus-\ntomers can use experts’ records of past actions to determine whom to hire. We take treat-\nments to be pure credence goods: customers observe past treatments, but they never receive\nsignals about what treatments would have been appropriate. This makes it difficult for cus-\ntomers to punish experts for past dishonesty, as dishonesty is never revealed. Moreover, we\nassume that an expert’s payoffs from each action are completely independent of the cus-\ntomer’s underlying need, and that customers are short-lived players for whom long-term\ncontracts are impossible.\nSince experts’ payoffs do not depend on the problem a customer faces, an expert will\nplay truthfully in a period only if her expected discounted profit is equal across actions. An\nhonest expert is likely to have a balanced record over time, with the proportion of major\nand minor treatments close to the probability that each is needed, but it will not be an\nequilibrium for customers to choose the expert whose record is most balanced. If customers\nchose in this way, then experts would just take actions to keep their records balanced.\nExperts may have the proper incentives to be truthful if customers give more business to\nexperts who have chosen the less profitable treatments in the past. The logic is illustrated by\nan anecdote about McKinsey & Co.’s former managing director Marvin Bower, relayed in a\nBusiness Week obituary. “In the 1950s, Bower was summoned to Los Angeles by billionaire\nHoward Hughes, who wanted him to study Paramount Pictures.... But Bower sensed that\nnothing good could come of working for Hughes. He found the entrepreneur’s approach\nto business ‘so unorthodox and so unusual’ that he felt he would never be able to help\nParamount. Instead of taking the assignment and reaping a big fee, he walked away. The\nmove was classic Bower. He built McKinsey into a global consulting powerhouse by insisting\nthat values mattered more than money” (Byrne (2003)). In other words, by publicly rejecting\na profitable action, McKinsey increased its future business.\nWe study a repeated game between customers and experts modeled after the above\ninteractions and look for conditions under which the experts may be truthful in equilibrium.\nIn each period, a new customer arrives on the market and chooses an expert. There are\n1\n\ntwo possible states of the world: the customer might need a major treatment, or a minor one.\nThe customer prefers that the appropriate action be taken but has no information about\nthe state of the world. His only action is to choose an expert. Once a choice is made, the\ncustomer must defer to the expert’s judgment.\nThe chosen expert observes the state and then decides whether to provide a major or\nminor treatment, and the expert’s payoffs depend on what she does but not on what the\ncustomer needs. In other words, her payoffis a function of the action but not the state.\nThis can be thought of as an environment where prices are exogenously fixed at industry\nstandard levels. Experts always prefer some work to no work, and when chosen they will\nearn a higher profit on major treatments than on minor ones.\nWhile true states are never revealed to future customers, we assume the full history of\nexperts chosen and actions taken to be observable. Experts are infinitely long lived, but\ncustomers disappear from the market after they receive a treatment.\nIn Section 3 we show that if experts are homogeneous then a truthful equilibrium of\nthe repeated game can be found; the promise of future business removes the incentive to\nplay major treatments over minor ones. Customers only need to look at the most recent\naction taken.\nIf it was a minor treatment, they return to the last period’s expert with\nhigh probability.\nIf it was a major treatment, they return with a low probability.\nBy\nsetting appropriate probabilities, they can make experts exactly indifferent between major\ntreatments with a high short term payoffand minor ones with a high continuation value.\nThis explains the intuition behind the McKinsey story: it can be an equilibrium for experts\nto report their private information truthfully when their likelihood of future work rises with\nless profitable actions and falls with more profitable ones.\nIn Section 4 we consider a more general model with heterogeneous experts who have\nprivate information along two dimensions. In addition to observing each period’s hidden\nstate, experts privately observe their own relative payoffs from providing a minor versus\nmajor treatment. Now customers will not know what probability would make an expert\nindifferent across actions, and so we cannot enforce the truthful equilibria above.\nCustomers are short-lived and so long-term contracts are impossible. If experts could\ncommit to long-term contracts over their actions, however, and if they did not discount\nfuture payoffs, then a quota system could allow for truthfulness in some periods. Say that\ncontracts were written over two-period blocks, and an expert was required to play one major\nand one minor treatment in each block. Whatever she played in the first period, she would\nplay the opposite in the second period and get one major payoffand one minor payoffover\n2\n\nthe block. Regardless of her relative payoffs across actions, then, the expert would agree\nto be truthful in the first period of any block. In a similar contract with blocks of length\nK > 2, the expert would be truthful until one of the actions reached its quota and would\nthen play the other action deterministically to the end of the block. If the quota for action\na were set to be close to K times the probability of a being appropriate, then the share of\ntruthful actions would approach 100% as K grew large.\nEven if long-term contracts were possible, the discounting of payoffs would prevent us\nfrom exploiting this idea directly. Under a quota like the one described above, experts would\nshift all of the profitable major treatments into the early periods. One way to return experts\nto truthful play in early periods would be to allow their payoffs to depend slightly on the\nstate of the world, and in particular to suppose that experts receive a utility benefit from\ntaking the appropriate action in a period. Consider an expert who observes that a minor\ntreatment is appropriate in an early period of a block. Playing the minor action over the\nmajor one would impose a cost from delaying the major payoff, but would yield a benefit\nfrom aligning the action with the state in this period. If she were sufficiently patient then\nthe cost would be small, so she would prefer to be truthful and would play the minor action.\nIn Section 4 we show that the logic of a quota can be recovered to induce heterogenous\nexperts to act truthfully in almost every period, even in an environment in which there is no\ncommitment over time and experts have no preference for aligning actions with states. In a\nstandard quota, the number of plays on an action is constant over prospective equilibrium\npaths; once one action hits its quota, the underplayed action must be played until the end of\nthe block. Instead of a literal quota where the number of times an action is played is constant\nover prospective equilibrium paths, we use a “discounted quota” in which the number of\nexpected discounted plays is constant over paths. In each block, experts are truthful in early\nperiods, and in later periods they deterministically choose the underplayed action until a\nnew block begins.\nHere is a basic example to illustrate the idea of such a discounted quota (see Example\n1).\nStrategies repeat every three periods.\nIn the first period the expert acts truthfully,\nperforming a major or minor treatment. In the second and third periods any expert who is\nchosen plays the opposite action of the first period, regardless of the state. After the first\n(truthful) period, the new customer keeps the old expert with some probability q and moves\nto a new one with probability 1−q. After the second and third periods, the customer retains\nthe expert if the suggested action was played and otherwise goes to a new one.\nSay that experts have discount factor β in this example. Over the course of the three\n3\n\nperiods, an expert chosen in the first period gets an expected discounted weight of 1 play\ntowards whichever action is taken first, and a weight of q(β+β2) towards the opposite action.\nIf the retention probability q is\n1\nβ+β2 then the weight is 1 on both actions. So over the three\nperiods an expert gets one major payoffand one minor payoffin expectation along either\npath, and is willing to condition her first period action on the state of the world.\nThe customer facing a truthful expert is happy, and has no incentive to deviate. But\ncustomers facing deterministic experts are stuck – whenever a customer switches experts,\nthe new expert plays exactly like the old one would have.\nThe example allows for truthfulness in every third period. Taking the experts’ discount\nfactor to 1 and enforcing the discounted quotas over longer and longer blocks, we can con-\nstruct equilibria with an arbitrarily high share of truthful periods.\nIn this paper, we examine whether experts can be induced to act truthfully under very\nstrict assumptions: experts have no intrinsic motivation to aid customers, and customers\nnever learn whether past experts had been truthful. Even after stacking the deck in this\nmanner, we find that truthful play is still possible so long as records are available.\nIn\nmore realistic settings experts may be somewhat altruistic, or customers (one-time or repeat\nvisitors) may observe signals about the quality of past play. In either of these cases, our\nresults provide a lower bound on what is achievable; the equilibria we construct continue to\nhold.\n2\nLiterature Review\nDarby and Karni (1973) introduced the concept of credence goods, goods whose value is\nknown by a seller but never fully revealed to the consumer. Dulleck and Kerschbamer (2006)\nprovide a recent review of the literature on when and how credence goods can be provided\nefficiently. In the credence goods literature, most work focusses on inducing truthfulness in\none-shot settings in which expert payofflevels are common knowledge.\nIn the terminology of Dulleck and Kerschbamer (2006), we impose the Verifiability rather\nthan Liability assumption: customers can confirm that the announced treatment has been\nperformed, but the success or failure of the treatment is not publicly observed and is non-\ncontractible. When the Liability assumption holds instead, experts will correctly treat the\nproblems but may attempt to overcharge customers, performing a cheap treatment but re-\nporting an expensive one.\nWe also impose what that paper calls the Commitment assumption, that a customer who\n4\n\ngoes to an expert must be treated by that expert. When this is relaxed, truthfulness can\nbe induced by having one expert diagnose the problem and another perform the treatment.\nThis is explored in Wolinsky (1993), Dulleck and Kerschbamer (2006, 2008), and Alger and\nSalanie (2006). In these models, agents may incur inefficient search and diagnosis costs. An\nalternative way of relaxing the Commitment assumption is to prevent the customer from\nseeking other experts, but allowing him to refuse treatment after observing a diagnosis.\nPitchik and Schotter (1987) take this approach and find a mixed strategy equilibrium in\nwhich customers sometime reject expensive treatments and experts are sometimes truthful.\nIf the prices for treatments are set so that profits are equal across all actions, then experts\nwill play truthfully. Emons (1997) and Dulleck and Kerschbamer (2006) build models of\ncredence goods which exploit this solution. In this sense, when the parties can bargain over\nprices they may achieve truthful play without resorting to a repeated game. But such an\napproach only works when, as in our model of homogeneous experts in Section 3, experts\nhave no private information about their costs for each action. If costs are privately observed,\nas in Section 4, then customers have no way to know what prices would induce truthful play,\nand experts will have no incentive to report their costs honestly. In the Web Appendix,\nwe elaborate on the impossibility of endogenous prices to induce truthful play in a one-shot\nsetting.\nThere is a large body of work on repeated games outside of the context of credence goods.\nIt is common for players to have private information on their own payoffs, as experts do in\nSection 4. Our model diverges from a standard set-up in that some players – the customers\n– do not know their own payoffs over others’ actions.\nBar-Isaac and Tadelis (2008) provide a recent survey of results on repeated games with\n“reputations” in which some players have hidden types and other players have beliefs about\nthese types. The uninformed agents observe past actions (or signals thereof) to generate\ninferences about others’ types, and therefore about the future actions they may play. When\nwe consider the case where experts have private information on their own payoffs in Section 4,\nwe will construct equilibria where reputational dynamics are trivial: all experts play identical\nstrategies, and customers make no inferences about types or strategies based on histories.\nBar-Isaac and Tadelis (2008) also discuss the problem of firms’ trying to develop a rep-\nutation for having expertise, which – although orthogonal to our analysis – is more in the\nspirit of credence goods. To demonstrate expertise, firms may may have incentives to skew\ntheir reports to match or go against what customers expect to hear.\nThere is a small literature involving repeated markets for credence goods.\n5\n\nFudenberg and Levine (1994) present a number of general results about payofffrontiers\nin repeated games with long-run and short-run players who do not have private payofftypes.\nTheir Investment Game example shares a number of features with our model of homoge-\nneous experts in Section 3. As in our paper, short-lived players offer business to long-lived\nplayers who may secretly take advantage of them.\nIn equilibrium, the long-run players’\ntemptation to cheat can largely be overcome by the threat that future short-run players will\nwithdraw their business after suspect outcomes. Our model allows us to explicitly construct\nefficient equilibria; Fudenberg and Levine (1994) focus on conditions under which there exist\nequilibria approaching efficiency.\nSchneider (2007) studies repeated interactions in the market for car repair, an example\nof a credence good. He considers a 2-period model with multiple experts, and runs a field\nexperiment to test the predictions. As in our paper, Schneider (2007) takes prices to be fixed\nexogenously and shows that there is an equilibrium in which customers return to an expert\nwith lower probability after an expensive repair. In this equilibrium, profit-maximizing me-\nchanics are honest in the first period and do unnecessary major repairs in the second period.\nOur equilibrium in Section 3 demonstrates a similar intuition for inducing truthfulness in\nevery period in a fully repeated setting.\nWolinsky (1993) also considers a setting where customers return to a market twice and\nthe choice of expert in the second period depends on the expert’s first period action. In\nthis model experts may reject customers with expensive problems, and customers return to\nexperts who had been willing to treat them in the past.\nPark (2005) studies an infinitely repeated game in which there are finitely many expert\nmechanics, and a diagnosis reveals to the mechanics which of them can best perform the\nrepair. After a diagnosis, mechanics report their diagnoses (possibly falsely) and the customer\nchooses an expert based on the reports. Experts have no opportunities to lie about the actual\nrepair they perform; once an expert is chosen, she fixes the car and receives a deterministic\npayoff. Park (2005) finds that equilibria with many or even 100% truthful reports are possible\nwhen payoffs do not vary too much across periods. One crucial feature of the model is that\ncustomers learn the true state at the end of a period, which lets them punish liars. This\nmeans that experts do not have to be made precisely indifferent over reports in order to be\ntruthful, and so the equilibria are robust against some uncertainty in the experts’ payoffs.\nEly and Valimaki (2003) study a model where short-lived customers play a repeated\ngame with long-lived mechanics who privately observe the state and determine the proper\nrepair.\nIn their model altruistic mechanics strictly prefer to act truthfully and perform\n6\n\nrepairs which are appropriate to the state, while bad mechanics always want to do engine\nrepairs. But instead of being truthful at first, the good mechanics will do a tune-up in\norder to separate themselves from bad mechanics and prove their goodness to all future\ncustomers. No consumer wants to be the first to go to an expert, so the market breaks\ndown. Ely, Fudenberg, and Levine (2008) extend this work and find sufficient conditions for\nwhen observable histories lead to market collapse.\nWhile we set up the problem similarly to Ely and Valimaki (2003), two key differences\nmake our strategic environment vastly different. First, there is no altruism in our model\n– the utility an expert receives from each repair is independent of the state of the world.\nSecond, in their model customers will exercise an outside option rather than receive a repair\nthat is independent of the state; we do not allow customers to opt out of the market. This\nprevents the market from breaking down.\nEly and Valimaki (2003) and Schneider (2007) consider the possibility that some propor-\ntion of experts are altruistic. One of the key innovations of our paper is to treat a different\nform of unobserved heterogeneity, in which experts are profit maximizers but are privately\ninformed about their payoffs across actions. Park (2005) does allow for private information\non instantaneous profits, but in that paper customers can discover and punish deviations.\nThe punishments give experts a strict incentive to prefer truthful reports over a range of\nprofit levels.\nFinally, there is another set of related papers which bears mentioning. Recall that in our\nmodel an expert can never be given a strict incentive to act truthfully. We can achieve truth-\nful play via indifference by enforcing “discounted quotas” which fix the number of expected\ndiscounted times that each action can be played by an expert. Past work has explored the\nuse of standard quotas, which fix the absolute number of times that an action is played, to\ninduce truthful revelation of private information. Townsend (1982) shows how quotas can\nbe applied to the context of repeated bilateral trade, and Jackson and Sonnenschein (2007)\nextends this to a general environment with independent and ex ante identical allocation de-\ncisions. Agents are asked to report types jointly over many decisions, and the distribution of\nreported types is restricted to match the theoretical distribution. Jackson and Sonnenschein\n(2007) call this the “linking” of separate decisions through “budgets” or “rations.” As more\ndecisions are linked, the mechanisms approach efficiency.\nAs mentioned in the introduction, when agents discount future payoffs a standard quota\nwill only work if agents receive some benefit from telling the truth. In these papers there\nis such a benefit because the agents’ private information regards their own preferences, and\n7\n\nconditional on their reported types efficient outcomes are realized. If a trade is more likely\nwhen a buyer reports a high value, and if a buyer can only report that he has a high value\na limited number of times, he prefers to report this when his value truly is high. In both\npapers, the mechanisms fall apart if agents cannot commit in advance to participate over\nlong time horizons.\nAlthough less directly related to our own work, Pesendorfer (2000) employs a similar\nintuition. He studies a bidding cartel for procurement contracts in which firms link separate\nauctions by reporting to each other a desirability ranking of the available contracts, and\ndetermine bidders from the cartel based on these rankings. As the number of linked auctions\nincreases, the mechanism gives rise to nearly optimal bids. A quota is analogous to a ranking\nwhen information is binary rather than continuous.\n3\nHomogeneous Experts\nThere is a set of experts E = {e1, e2, ...} and a set of customers C = {c1, c2, ...}. Customers\nare short-term players, while experts are infinitely long-lived and have discount factor β ∈\n(0, 1).1\nIn period t ∈{1, 2, ...}, customer ct arrives on the market and observes the past history\nof experts chosen and actions taken. The customer then chooses a single expert et from E.\n(Superscripts denote elements of the set E, while subscripts represent time periods). The\nexpert observes the state θt and then chooses an action at.\nIn each period the set of possible states is Θ = {θm, θM}, and the set of actions for the\nchosen expert is A = {m, M}, where m refers to a “minor” treatment and M a “major”\none. The customers always want the expert to be “truthful” and choose a when the state is\nθa. But in the short term, every expert prefers action M. Formally, write stage payoffs in\nperiod t as\nCustomer ct :\nU t(at|θt)\nExpert ei :\n\n\n\n\n\n\n\n\n\n0\nif et ̸= ei\nR(m) = r\nif et = ei & at = m\nR(M) = 1\nif et = ei & at = M\n1The set of experts is modeled as countably finite, but our arguments will not hinge on this assumption;\nwe discuss the case of finitely many experts in the Web Appendix.\n8\n\nwhere these stage payoffs satisfy\nU t(a|θa) > U t(a′|θa) for a′ ̸= a\n0 < r < 1.\nFor each expert only the relative payoffs of the different actions matter, so we have normalized\nR(M) to 1 and the payoffwhen not selected to 0. R(m) = r is in between these two. Expert\nei’s lifetime utility is P\n{t|et=ei} βt−1R(at).\nA customer only receives a payoffin the period in which she chooses an expert, and this\npayoffis a function of the treatment received along with the current state of the world. The\npayoffdoes not depend on the identity of the expert.\nAlthough we do not analyze efficiency explicitly, efficiency is synonymous with truthful-\nness if the benefit to customers from an appropriate treatment always outweighs the costs\nto the experts.\nIn each period, the state is θm with probability 0 < p < 1 and θM with probability 1 −p.\nThe identities of experts chosen in past periods and the actions taken by these experts\nare publicly observable to all players. We write this list of experts and actions observed prior\nto period t as a “public history” Ht = (e1, a1, ...et−1, at−1), with H1 ≡(∅). Let Ht be the set\nof all possible public histories at time t, and let H be the set of possible public histories at\nany time: H ≡∪tHt .\nLet ⊕be the concatenation operator, so that for histories H and H′, the notation H ⊕H′\nmeans history H followed by history H′. Say that a history G begins with history H if\nG = H ⊕H′ for some H′.\nCustomers observe the list of past experts and actions, but they have no way of discerning\nwhether past actions were appropriate. A customer’s only decision is to use the observable\npublic history to choose an expert to treat his problem. The customer may play a mixed\nstrategy and choose experts probabilistically. We write customer ct’s strategy in period t as\nρt : Ht →∆(E)\nwhere, for any countable set S, ∆(S) denotes the space of probability distributions over S.\nIn order to avoid awkward descriptions of pure strategies, we will slightly abuse notation\nand use s to denote the element of ∆(S) which places probability 1 on s ∈S. Define the\ncollective strategies of all customers as ρ ≡(ρ1, ρ2, ...).\nEach expert also sees the public history, and once chosen, she also observes the current\n9\n\nstate. The expert then chooses a treatment based on all of this information. We write expert\nei’s strategy conditional on being chosen as\nσi : H × Θ →∆(A).\nMore generally, an expert could also condition her strategy on privately observed values of\nthe state in previous periods in which she was chosen. Past states are payoffirrelevant to all\nplayers, so allowing this would complicate notation without affecting our results.2\nIn this model, the expert’s utility is independent of the true state, and the customers\ncannot confirm whether an expert has acted honestly or dishonestly in the past. Moreover,\neach customer is a short term player who is unable to reward or punish an expert after\nchoosing her.\nThere exists an inefficient equilibrium in which every expert always performs the more\nprofitable major treatment M. We will show that when experts are patient enough a truthful\nequilibrium will also exist, in which experts always play the action corresponding to the\ntrue state. Here and in the rest of the paper, the term equilibrium refers to a sequential\nequilibrium.\nDefinition. The expert ei with strategy σi is said to be truthful at history Ht if σi(Ht, θa) = a\nfor each a ∈A. A truthful equilibrium is an equilibrium in which, at every equilibrium history,\nevery expert who may be chosen with positive probability is truthful.\nProposition 1. A truthful equilibrium exists if and only if β ≥1 −r.\nBefore proving the proposition, it is useful to state a lemma. Recall that an expert’s\npayoffs are independent of the state, so an expert is only willing to play truthfully if she is\nindifferent across possible actions. The following lemma states that if truthful play on some\nset of histories is a best response, then the strategy remains a best response if the expert\nswitches to arbirtrary play at those histories.\nLemma 1. Fix the customers’ strategy ρ and the strategies σ−i of experts aside from ei, and\nsuppose that expert ei has a sequentially rational best response σi. Take ˆH ⊆H to be a set of\nhistories for which σi is truthful. Now construct a new strategy ˆσi, where ˆσi : H×θ →∆(A)\n2Any strategy that is optimal in the class of those which do not depend on past states is also optimal in the\nlarger class of strategies which do. So the equilibria we construct will remain equilibria in the more general\nstrategy space. The results that no truthful equilibria can exist under various conditions (see Proposition 1\nand Remark 1) also hold in the general strategy space.\n10\n\nis identical to σi on all H\\ ˆH, and is arbitrary on ˆH. The strategy ˆσi gives the same expected\nutility as σi starting at every history and is also a sequentially rational best response.\nProof. See Appendix.\nProof of Proposition 1. First, we will show the “If” direction. Suppose that β ≥1 −r; we\nwill now construct a truthful equilibrium.\nExperts play truthfully: for each a ∈A, for each ei ∈E, and for each Ht ∈H, let\nσi(Ht, θa) = a.\nThe customer’s strategy is the following. c1 chooses ρ1(H1) = e1. For t > 1, ct chooses\nρt\u0000Ht−1 ⊕(et−1 = ei, at−1)\n\u0001\n=\n\n\n\nei with Prob q(at−1)\nei+1 with Prob 1 −q(at−1)\nwith q(m) = 1, q(M) = r+β−1\nrβ\n. Notice that 0 ≤q(M) < q(m) = 1.\nBecause all experts play truthfully at every history, customers are indifferent across\nexperts and any strategy is a best response.\nTo check that truthful play is a best response for the experts, consider the expected\npayoffV for an expert who follows the strategy, conditional on being chosen in a given\nperiod but unconditional on the realization of θ. By the one-shot deviation principle, the\nexpert’s strategy is optimal if she always plays a maximizer of R(a) + βq(a)V , and if V =\nmaxa∈A{R(a) + βq(a)V }. This holds if we can find a V such that\nV = 1 + β r + β −1\nrβ\nV = r + βV\n⇐⇒V\n\u0012\n1 −r + β −1\nr\n\u0013\n= 1 & V (1 −β) = r\n⇐⇒V =\nr\n1 −β .\nSo the continuation value V is\nr\n1−β, and there are no profitable deviations.\nThis shows that the above strategies are an equilibrium when β ≥1 −r – the experts\nand customers are indifferent with respect to all actions at all histories.\nNow, in the “Only If” direction, suppose that a truthful equilibrium exists. By Lemma\n1, if a given expert is willing to play truthfully at every period in which she is chosen then\nshe must be indifferent to switching to the strategy of playing m at every period, or to\nthe strategy of playing M in every period. An expert selected in the current period whose\n11\n\nstrategy is to always play m will get stage payoffs of at most r in each period (0 in any\nperiod in which she is not chosen) and so her present value of future payoffs is at most\nr\n1−β.\nAn expert who plans to play M in every period gets a stage payoffof 1 today, and some\nnonnegative payoffin the future. The expert can only be indifferent over these two strategies\nif\nr\n1−β ≥1, or rather β ≥1 −r.\nTo implement this equilibrium customers only need to observe the previous period’s\naction, and experts can ignore the histories entirely. The customer utility functions never\nappear in the construction. Moreover, the equilibrium is completely independent of p, the\nparameter controlling the probabilities of the various states of the world.\nThe expert’s\ncontinuation probabilities from different actions are set such that every expert is exactly\nindifferent between the higher payofftoday versus the lower payoffin the future from playing\nM, and p plays no role in this.\nIn the above equilibrium, customers return to an expert with probability 1 if the minor\naction m was played last period, and probability less than 1 if the major action M was\nplayed. In the Web Appendix we show that similar truthful equilibria can be constructed if\nexpert utility functions differ but are observed by customers, or if experts have more than\ntwo possible actions.\n4\nHeterogeneous Experts\nIn Section 3 customers always return to an expert who has just performed a minor treatment,\nand rehire an expert who has just performed a major treatment with a probability less than\n1. This probability is chosen so that each expert will be exactly indifferent between a major\nand minor treatment in every period. But this equilibrium falls apart if customers are no\nlonger certain about an expert’s relative payofffrom the two treatments. An expert with\na slightly lower payofffrom the major treatment will find it profitable to play the minor\ntreatment in each period she is chosen, and vice versa.\nWe will proceed to consider whether any truthful play is possible when an expert’s payoffs\nare private information. We find that experts can often be incentivized to play truthfully\nwhen strategies depend on more than just the last period’s play.\n12\n\nIn this section, we take expert stage payoffs to be\nExpert ei :\n\n\n\n\n\n\n\n\n\n0\nif et ̸= ei\nRi(m) = ri\nif et = ei & at = m\nRi(M) = 1\nif et = ei & at = M\n.\nWe maintain the normalization of the instantaneous payoffof the major treatment to 1,\nand we now let the relative payoffof the minor to major treatment be an expert’s private\ninformation.\nEach expert ei realizes a relative payoffri drawn from a distribution over\nR+, the set of nonnegative real numbers; some (or all) experts may prefer m to M. These\ndistributions need not be identical or independent. The realization of ri is privately observed\nby expert ei at the start of the game, and is fixed over time.\nExperts have discount factor β as before, and β is common knowledge.\nCustomer payoffs are as in Section 3. Customers prefer to receive appropriate treatments,\nbut conditional on the action and state have no preferences over the identity of the expert.\nEach customer still observes only the public history of past experts and actions when choosing\na new expert, so customer strategies are also as above; the collective strategy of the customers\nis ρ : H →∆(E).\nWe now have to generalize the strategy space of each expert to depend not only on the\nhistory and state but also her realized type ri. In this context, an expert ei’s strategy is a\nfunction φi : H × Θ × R+ →∆(A). For ease of notation, we will use the term conditional\nstrategy to refer to maps σ : H × Θ →∆(A). An expert’s strategy can be thought of as a\nmap from types to conditional strategies. The conditional strategy for an expert ei of type\nr ∈R+ will be denoted by σi\nr(Hi, θ) ≡φi(Ht, θ, r).\nAt any history Ht with state θt at which the expert et = ei is chosen, a conditional\nstrategy σ for the expert induces weights on the number of expected discounted times that\nm and M will be played in the current and future periods. For a ∈A, we can define the\nweight on action a by\nW i\na(Ht, σ|θt, φ−i, ρ) ≡\n∞\nX\nτ=t\nβτ−tProb[eτ = ei and aτ = a|θt, σ, Ht, φ−i, ρ; et = ei]\nwhere the probability is taken with respect to the (Bayesian) beliefs of ei . The expected\npresent value of conditional strategy σ at some history Ht (conditional on φ−i, ρ, θt, and\nconditional on et = ei, but suppressing these from the notation) in the current and future\n13\n\nperiods is\nri · W i\nm(Ht, σ) + W i\nM(Ht, σ).\nSequential rationality requires that σi\nri be a maximizer of this expression for each ri ∈R+.\nNotice that this value depends on θτ in periods where eτ = ei only through the effect of θτ\non σ.\nIt can be natural to think of an expert ei as choosing a bundle (W i\nm, W i\nM) at each action\nnode rather than a strategy. Let the set of available bundles be denoted\nWi(Ht) ≡{(W i\nm(Ht, σ), W i\nM(Ht, σ))|σ : H × Θ →∆(A)},\nsuppressing the dependence on φ−i and ρ. Wi(Ht) is independent of θt because no matter\nwhat the state is, a strategy exists that plays “as if” the state were the opposite.\nPoints in Wi(Ht) fully determine an expert’s utility going forward, so we can consider\nindirect preferences over this set. Expert ei’s indifference curves over Wi(Ht) are straight\nlines with slope −ri.\nThe equilibrium in Proposition 1 holds up because the customers\nconstruct Wi(Ht) such that all points lie on a straight line with slope −r; every expert is\nindifferent between every action at every history.\nNo experts with different values of ri can be indifferent over distinct pairs of (W i\nm, W i\nM)\nbecause the indifference curves have a unique intersection point. However, if two distinct\nstrategies yield the same pair of weights, then experts of any ri ∈R+ will be indifferent. We\nformalize this in the following lemma.\nLemma 2. Take two conditional strategies σ′, σ′′ : H × Θ →∆(A), and take r′ ̸= r′′ ∈R+.\nAn expert ei chosen at history Ht is indifferent between σ′ and σ′′ for both possible types r′\nand r′′ if and only if\nW i\nm(Ht, σ′) = W i\nm(Ht, σ′′) & W i\nM(Ht, σ′) = W i\nM(Ht, σ′′).\nProof. See Appendix.\nLemma 1 of Section 3 continues to hold: an expert whose best response includes truthful\nactions is indifferent to switching her strategy arbitrarily at truthful histories.\nRemark 1. If each expert can realize at least two possible types then no truthful equilibrium\nexists. More formally, there is no truthful equilibrium if for no expert ei does there exist\nr ∈R+ such that ri = r with probability 1.\n14\n\nProof. Suppose there is a truthful equilibrium. Take ei to be some expert who is selected\nwith positive probability at H1. In a truthful equilibrium, there is a probability one that ri\nis such that σi\nri is truthful at every equilibrium history in which ei is chosen with positive\nprobability. Take some such ri. (An equilibrium history is an element of H which occurs\nwith positive probability in equilibrium).\nFor each a ∈A, consider the conditional strategy σa of playing action a at every history.\nThe payoffto the expert of playing σa is exactly the same as the payoffof a strategy which\nplays a at all equilibrium histories in which she is selected with positive probability, and\nmimics σi\nri at other histories. And by Lemma 1, such a strategy is optimal; the expert is\ntruthful at all equilibrium histories in which she may be chosen, and so is indifferent to\nmodifying her strategy arbitrarily at these periods. So the expert ei of type ri is indifferent\nbetween σm and σM at H1.\nBut the two conditional strategies σm and σM induce different points in Wi(Ht): if expert\nei is chosen at H1, then W i\nM(H1, σM) ≥1 while W i\nM(H1, σm) = 0. So by Lemma 2, there is\nat most a single type in R+ for which ei would be indifferent between σm and σM at H1. This\ntype must be realized almost surely in a truthful equilibrium, and so there is no truthful\nequilibrium if no single type is realized with probability 1.\nWhile an equilibrium with truthful play at every history cannot exist, we can still find\nequilibria in which experts play truthfully at some histories. Here is an example with truthful\nplay once every third period.\nExample 1. We will show that for any β ≳.80, the following strategy profile constitutes\nan equilibrium in which experts are truthful at periods t = 1, 4, 7, 10, and so on.\nFirst we define T0 be the set of time periods of the form 3n + 1, and t0(t) ∈T0 to be the\nmost recent period in T0 up to period t:\nT0 ≡{τ|τ = 3n + 1 for some n ≥0}\nt0(t) ≡max{τ ∈T0|τ ≤t}.\nSo t0(t) is 1 for t = 1, 2, 3, and t0(t) = 4 for t = 4, 5, 6. It will be convenient to use t0 for\nboth the function t0(t) and as a representative element of the set T0.\nThe experts’ strategies are as follows. All experts of all types play identical conditional\nstrategies. They will be truthful at the T0 periods, experts will play the action opposite of\nthat played at t0 ∈T0 in the periods t0 + 1 and t0 + 2. Strategies repeat every three periods.\n15\n\nSo for all ei ∈E, for all r ∈R+, and for each a ∈A,\nσi\nr(Ht, θa) =\n\n\n\n\n\n\n\n\n\na\nif t0(t) = t\nM\nif t0(t) > t & at0(t) = m\nm\nif t0(t) > t & at0(t) = M\n.\nSee Figure 1 for an illustration of the experts’ strategies.\nWe move now to the customers’ strategies. After each period, the customer either returns\nto the previous expert or “fires” her and moves to an entirely new one. After the first period\nin each repeating block, the truthful period t0, the customer arriving at t0 +1 has some fixed\npositive probability of firing the old expert. After the two deterministic periods, the next\ncustomer returns to the expert if she played the suggested deterministic action and fires her\notherwise. Formally, at period 1 c1 chooses ρ1(H1) = e1. At period t > 1, if the previous\nexpert chosen was ei (so that Ht = Ht−1 ⊕(ei, at−1)), the customer chooses\nρt(Ht) =\n\n\n\nei\nwith Prob q(Ht)\nei+1\nwith Prob 1 −q(Ht)\nwhere the function q : H \\ H1 →[0, 1] determines the probability of continuing with an\nexpert rather than moving to the next one. q(Ht) satisfies\nq(Ht) =\n\n\n\n\n\n\n\n\n\n1\nβ+β2\nif t0(t) = t −1\n1\nif t0(t) ̸= t −1 & at0(t−1) ̸= at−1\n0\nif t0(t) ̸= t −1 & at0(t−1) = at−1\n.\nSuch a strategy profile can be constructed so long as\n1\nβ+β2 ≤1, that is, β ≥\n√\n5−1\n2\n∼= .62.\nThe customers are necessarily best responding because at every history, each expert plays\nidentically to every other expert. So the customer is indifferent over who he chooses. We will\nshow that the experts’ strategies also constitute a best response when their discount factor\nis large enough, and therefore that this strategy profile is indeed an equilibrium.\nConsider the contribution to W i\nM(Ht0, σi\nr) and W i\nm(Ht0, σi\nr) from the three periods t0, t0+\n1, t0 + 2, for t0 ∈T0. If θt0 = m then the expert is instructed to play m in t0, and M in\nt0 + 1 and t0 + 2 if chosen. This gives a weight of 1 towards the minor action m and a\nweight of\n1\nβ+β2 (β + β2) = 1 towards the major action M over the block. The same holds\n16\n\nm\nM\nt0\nFigure 1: Expert Strategies in Example 1.\nThis picture illustrates the equilibrium paths of play in a single 3-period block of the strategies in Example\n1. At the end of the block, the strategy repeats. The open circle represents a truthful period; the closed\ncircles represent deterministic periods. The style of the lines is varied in order to show which histories lead\nto which actions at deterministic periods.\nif θt0 = M and the expert is instructed to play M, m, m; the expert gets one minor payoff\nand one major payoffin expected discounted terms over the three periods. She is therefore\nindifferent between m and M at the t0 periods and is willing to follow the suggested strategy\nof truthful play.\nWe still have to show that there are no profitable deviations at the t0 +1 or t0 +2 stages.\nAt a period with t equal to t0 + 1 or t0 + 2, suppose that a is the suggested action and the\nselected expert considers a deviation to a ̸= a. Deviating gives a weight of 1 towards a and\n0 towards a. So a sufficient condition for deviations to be unprofitable is that the weight on\na from following the equilibrium is at least 1.\nAn expert chosen at t0 receives a weight of 1 towards both m and M over the three periods\nof the block. Iterating this out, the weight over all periods from following the strategy is\nW i\na(Ht0, σi\nr) =\n1\n1−\n1\nβ+β2 β3 =\n1+β\n1+β−β2 for a = m, M. So following the equilibrium starting at\nt0+1 gives a weight of β2\n1+β\n1+β−β2 towards a; starting at t0+2, the weight is β\n1+β\n1+β−β2. Deviating\nis unprofitable as long as β2\n1+β\n1+β−β2 ≥1, or rather β3 + 2β2 −β −1 ≥0.3\n3This is a sufficient but not necessary condition. We may have an equilibrium for values of β which don’t\nsatisfy this, if the support of possible values of ri is limited to some subset of R+ bounded away from 0 and\ninfinity. In particular, for any β ≥\n√\n5−1\n2\nthere are no profitable deviations if all values of ri are known to be\nin the interval [1 −β,\n1\n1−β ].\n17\n\nThis gives us a condition under which the proposed strategy will be a best response at all\nhistories for all experts, and we already determined that the customers are best responding.\nThe proposed strategy is an equilibrium as long as β3 + 2β2 −β −1 ≥0, which holds for\nβ ≳.80.\n■\nIn the above example, we have blocks of length three in which there is a truthful period\nfollowed by two deterministic periods in each block. The weight on each action is constant\nacross all prospective equilibrium action paths. At truthful periods different equilibrium\npaths allow for different actions in the current period, and so the expert is willing to condition\nher choice of path on the payoff-irrelevant state of the world. At deterministic periods, the\ncontinuation payofffrom following the suggested strategy and receiving future work is greater\nthan the benefit from deviating and never being chosen again.\nAll action paths consistent with equilibrium play give an expert the same weights towards\neach action; for each action, she faces a “quota” or a “budget” on the number of expected\ndiscounted plays. When the discount factor is large enough any off-equilibrium strategy\nprovides a weakly lower weight on both actions, and hence has a weakly lower payofffor an\nexpert of any type.\nAs the discount factor increases, we can find similar equilibria in which truthful periods\noccur more frequently. We take strategies that repeat in blocks of longer than three periods,\nand have experts play truthfully until some number of either m or M actions are played\nwithin the current block.\nOnce one of these actions reaches enough plays, the opposite\naction is played until the end of the block. Taking the length of blocks to be large and\ntaking the discount factor to 1, we can get the long-term proportion of truthful periods to\napproach 1.\nProposition 2. Take ϵ > 0.\nFor β large enough, there is an equilibrium in which the\nlong-run proportion of truthful periods is greater than 1 −ϵ with probability 1.\nProof. We will use the notation ⌊x⌋to denote the “floor” of a number x, the greatest integer\nless than or equal to x. First, a technical lemma:\nLemma 3. For any k ∈N, there exists K ≥k such that\ni. 1 < pK < K −1,\nii. pK is not an integer,\niii. 2p −1 + ζ < pK −⌊pK⌋< 2p −ζ for ζ ≡min{p/2, (1 −p)/2} > 0, and\n18\n\niv. min\nn\n(1−p)⌊pK⌋\np⌊(1−p)K⌋, p⌊(1−p)K⌋\n(1−p)⌊pK⌋\no\n> 1 −1\n4 min\nn\np\n1−p, 1−p\np\no\n.\nProof. See Appendix.\nTake some K satisfying conditions (i)-(iv) of the above lemma; we will construct a strat-\negy profile for which strategies repeat every K periods. Conditions (i) and (ii) are necessary\nfor constructing the strategies of the experts. Conditions (iii) and (iv) will guarantee that\nthe proposed probabilities chosen by the customers are valid for a large enough discount\nfactor, and also that the experts’ responses are optimal. We will show that the strategy\nis well-defined and is an equilibrium for β large enough, and that as K is taken to ∞the\nlong-term proportion of truthful periods will converge to 1. Because we can find K arbitrar-\nily large that satisfies the above conditions, this means that we can find equilibria in which\ntruthful periods occur arbitrarily often.\nStrategies will be defined on blocks of K periods, and will reset at periods of the form\nnK + 1. T0 will denote the set of these periods at which new blocks begin, and t0(t) will be\nthe most recent period in T0 starting at period t:\nT0 ≡{τ|τ = nK + 1 for some n ≥0}\nt0(t) ≡max{τ ∈T0|τ ≤t}.\nThe term t0 will express this function as well as a representative element of the set T0.\nEach block begins with a segment of truthful periods. Once m or M is played a certain\nnumber of times within the block, the players move into a segment where experts take\ndeterministic actions. After K periods in the block, t0(t) increments up by K and strategies\nrepeat. We now construct these strategies.\nPartition H into “deterministic histories” H(D) and “truthful histories” H(T). A history\nis truthful if m has been played less than ⌊pK⌋times in the current block and M has been\nplayed less than ⌊(1 −p)K⌋times in the current block. Once either action has been played\nthis many times, histories are deterministic for the rest of the block:\nHt ∈\n\n\n\n\n\n\n\n\n\nH(T)\nif #{τ|t0(t) ≤τ ≤t −1, aτ = m} < ⌊pK⌋\nand #{τ|t0(t) ≤τ ≤t −1, aτ = M} < ⌊(1 −p)K⌋\nH(D)\notherwise\n.\nFor Ht ∈H(D), let X(Ht) be the first deterministic period in the block containing period\nt, and let N a be the number of times that action a had been played over the truthful periods\n19\n\nin the block:\nX(Ht) ≡max{τ ≤t|Hτ−1 ∈H(T)}\nN a(Ht) ≡#{τ|t0(t) ≤τ < X(Ht), aτ = a}.\nOn H(D), if N M = ⌊(1 −p)K⌋then let a(Ht) = m and a(Ht) = M; otherwise, if\nN m = ⌊pK⌋, then let a(Ht) = M and a(Ht) = m. Either N M = ⌊(1 −p)K⌋or N m = ⌊pK⌋\nat a deterministic period, because periods only become deterministic once one of these holds.\nIn words, a is the action that has been played “enough” over the truthful periods while a is\nthe action which “needs more plays”.\nNow, let all experts of all types share the following conditional strategy. At any his-\ntory in H(T) the expert plays truthfully, and at any history in H(D) the expert plays the\nunderplayed action a(Ht): for all ei, in state is θa,\nσi\nr(Ht, θa) =\n\n\n\na\nif Ht ∈H(T)\na(Ht)\nif Ht ∈H(D)\nSee Figure 2 for an illustration of the experts’ strategy.\nNow we construct the customers’ strategy ρ. At period 1, c1 chooses ρ1(H1) = e1. At\nperiod t > 1, if the previous expert chosen was ei (so that Ht = Ht−1 ⊕(ei, at−1)), the\ncustomer chooses\nρt(Ht) =\n\n\n\nei\nwith Prob q(Ht)\nei+1\nwith Prob 1 −q(Ht)\nwhere the function q : H \\ H1 →[0, 1] determines the probability of continuing with an\nexpert rather than firing her and moving to the next one. q(Ht) satisfies\nq(Ht) =\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nqStartT(Ht)\nif t = t0(t) & at−1 = a(Ht−1)\n0\nif t = t0(t) & at−1 ̸= a(Ht−1)\n1\nif Ht ∈H(T) & t > t0(t)\nqStartD(Ht)\nif Ht ∈H(D) & t = X(Ht)\n1\nif Ht ∈H(D) & t > X(Ht) & at−1 = a(Ht)\n0\nif Ht ∈H(D) & t > X(Ht) & at−1 ̸= a(Ht)\nwith qStartD and qStartT defined below. At t0, the start of a new block – and therefore the end\n20\n\nt\nm\nM\nt + (1-p)K\nt + pK\n0\n0\n0\nFigure 2: Expert Strategies in Proposition 2, for K = 10 and p = 2/3.\nThis picture illustrates the strategy of the experts in Proposition 2 in a single K-period block, for K = 10\nand p = 2/3. At the end of the block, the strategy repeats. The open circles represent truthful periods; the\nclosed circles represent deterministic periods. The style of the lines is varied in order to show which histories\nlead to which actions at deterministic periods.\nOn the equilibrium path, the customer chooses the continuation probability as q = 1 at every history except\nfor the first deterministic history in a block, at which qStartD is chosen; and the truthful history at t0, at\nwhich qStartT is chosen. These q’s depend on the full history of actions at all truthful periods in the most\nrecent block.\nIn this example, ⌊pK⌋= 6 and ⌊(1 −p)K⌋= 3. The expected number of truthful periods in a block is about\n6.60, so the long-term proportion of truthful actions is 66%.\nof an old block – there is probability qStartT of keeping the previous expert if she played the\nsuggested action in the previous period, and probability 0 otherwise. At every other truthful\nperiod, the customer returns to the previous expert with probability 1 no matter what.\n21\n\nAt the start of the first deterministic period, customers again move to a new expert with\nprobability 1 −qStartD. For the remaining deterministic periods in the block, the customer\nkeeps the previous expert with probability 1 if the expert plays the suggested action a and\nfires her otherwise.\nIt will be useful to define a few other terms on the way to constructing qStartD and qStartT.\nFor a deterministic history Ht, let Za(Ht) be the weight that would accumulate towards\naction a (relative to t0(t)) for an expert chosen at t0 intending to play actions consistent\nwith Ht over the truthful periods in the block, t0 through X(Ht) −1. Let W a(Ht) be the\nweight that would accumulate over all periods in the block, from t0 through t0 + K −1, for\nan expert chosen at t0 intending to play actions consistent with Ht over the truthful periods\nand the equilibrium action a(Ht) for the rest of the block. That is, on Ht ∈H(D),\nZa(Ht) ≡\nX\nτ s.t. aτ=a\n& t0(t)≤τ<X(Ht)\nβτ−t0(t)\nW a(Ht) ≡\n\n\n\nZa(Ht)\nif a = a(Ht)\nZa(Ht) + qStartD(Ht) Pt0(t)+K−1\nτ=X(Ht) βτ−t0(t)\nif a = a(Ht)\n.\nAdjusting qStartD lets us adjust the weight W a that accumulates towards a within a single\nblock without affecting the weight W a that accumulates towards a. We want to choose qStartD\nso that the ratio of weights W m\nW M is equal to the respective ratio of the probabilities of the\nactions being appropriate,\np\n1−p, across all equilibrium paths. For Ht ∈H(D), let qStartD(Ht)\nbe defined by\nqStartD(Ht) ≡\n\n\n\n\n\np\n1−pZM(Ht)−Zm(Ht)\nPt0(t)+K−1\nτ=X(Ht)\nβτ−t0(t)\nif a(Ht) = m\n1−p\np Zm(Ht)−ZM(Ht)\nPt0(t)+K−1\nτ=X(Ht)\nβτ−t0(t)\nif a(Ht) = M\n.\nRearranging, we see that qStartD has been chosen so that\npW M(Ht) = (1 −p)W m(Ht) if a = m\n(1 −p)W m(Ht) = pW M(Ht) if a = M\nand so in either case, W m\nW M =\np\n1−p as desired.\nNow, for Ht ∈H(D), define Y a(Ht) as\nY a(Ht) ≡\nW a(Ht)\n1 −βKqStartD(Ht)\n22\n\nand define Y a as\nY a ≡\nmin\nHt∈H(D) Y a(Ht).\nThe minimum is well-defined because blocks are identical, and there are only finitely many\naction paths along the truthful periods of a block. If two deterministic histories share the\nsame action path over the truthful periods in their respective blocks then the histories have\nidentical Y a values. Y m is equal to\np\n1−pY M and so Y m =\np\n1−pY M.\nY a(Ht) would be the lifetime weight on a, relative to t0, that an expert chosen at t0 would\nreceive if she planned to repeat the actions played in the truthful periods of the current block\nof Ht in the truthful periods of every future block and to play the suggested actions in all\ndeterministic periods, if the continuation probability across blocks were qStartD. But in fact\nthe continuation probability is not qStartD but qStartD · qStartT, because the expert may be\nfired at both the first deterministic period in a block and also at the start of the next block.\nJust as adjusting qStartD allowed us to manipulate the relative weights on m and M along a\npath, adjusting qStartT will let us affect the level of the weights along a repeating path while\nholding the relative weights fixed. We want to set qStartT(Ht) so that the lifetime weight on\na is equal to Y a along any repeating path.\nFor t0 ∈T0 with t0 > 1, let\nqStartT(Ht0) ≡\n1 −\nW m(Ht0−1)\nY m\nβKqStartD(Ht0−1) =\n1 −\nW M(Ht0−1)\nY M\nβKqStartD(Ht0−1).\nIt holds that\nY a =\nW a(Ht0−1)\n1 −βKqStartD(Ht0−1)qStartT(Ht0)\nwhere the right-hand side is the actual lifetime weight on a for an expert picked at t0 who\nplans to repeat the path consistent with Ht. This is constant across all Ht. It will turn out\nthat if these levels are constant across repeating paths, they will also be constant across all\nprospective equilibrium paths.\nThis completes the descriptions of the strategies. Before we check whether these strategies\nimply an equilibrium for high discount factors, we need to show that qStartD and qStartT are\nvalid probabilities for β large enough, i.e., that they are numbers in [0,1].\n• limβ→1 qStartD(Ht) ∈(1\n4 min{\np\n1−p, 1−p\np }, 1):\n23\n\nAs β →1,\nZm →N m =\n\n\n\n(X −t0) −⌊(1 −p)K⌋\nif a = m\n⌊pK⌋\nif a = M\nZM →N M =\n\n\n\n⌊(1 −p)K⌋\nif a = m\n(X −t0) −⌊pK⌋\nif a = M\n.\nSo\nqStartD =\n\n\n\n\n\np\n1−pZM−Zm\nPt0(t)+K−1\nτ=X(Ht)\nβτ−t0(t)\nβ→1\n−→\np\n1−p⌊(1−p)K⌋−(X−t0)+⌊(1−p)K⌋\nK−(X−t0)\n=\n1\n1−p⌊(1−p)K⌋−(X−t0)\nK−(X−t0)\nif a = m\n1−p\np Zm−ZM\nPt0(t)+K−1\nτ=X(Ht)\nβτ−t0(t)\nβ→1\n−→\n1−p\np ⌊pK⌋−(X−t0)+⌊pK⌋\nK−(X−t0)\n=\n1\np⌊pK⌋−(X−t0)\nK−(X−t0)\nif a = M\n.\nBecause pK is not an integer, in either case the numerator is strictly smaller than the\n(positive) denominator; the limit of qStartD is strictly less than 1.\nNow we wish to show that the minimum value of lim qStartD over all deterministic\nhistories is greater than 1\n4 min{\np\n1−p, 1−p\np } .\nThe maximum value that (X −t0) can take is ⌊pK⌋+ ⌊(1 −p)K⌋−1 = K −2. And\nfor any fixed a, the above expression for lim qStartD is decreasing in (X −t0) as long as\n(X −t0) < K. This implies that for any Ht ∈H(D),\nlim qStartD(Ht) ≥min\n(\n1\n1−p⌊(1 −p)K⌋−K + 2\n2\n,\n1\np⌊pK⌋−K + 2\n2\n)\n.\nNoting that ⌊(1 −p)K⌋= K −⌊pK⌋−1, the first fraction can be reduced to\n⌊(1 −p)K⌋−(1 −p)K + 2(1 −p)\n2(1 −p)\n= pK −⌊pK⌋−2p + 1\n2(1 −p)\n>\nζ\n2(1 −p)\nand the second can be reduced to\n⌊pK⌋−pK + 2p\n2p\n> ζ\n2p\nwhere the inequalities come from condition (iii) of Lemma 3, with ζ = min{p/2, (1 −\np)/2}.\nTherefore lim qStartD(Ht) is greater than min{\nζ\n2(1−p), ζ\n2p} = 1\n4 min{\np\n1−p, 1−p\np }.\n24\n\n• limβ→1 qStartT(Ht) ∈(0, 1]:\nFor t = t0 −1,\nqStartT(Ht+1) =\n1 −W m(Ht)\nY m\nβKqStartD(Ht) ≤\n1 −W m(Ht)\nY m\nβKqStartD(Ht) =\n1 −\nW m(Ht)\nW m(Ht)\n1−βK qStartD(Ht)\nβKqStartD(Ht)\n= 1\nand so qStartT ≤1. Now we will show that the limit of qStartT(Ht+1) as β goes to 1 is\nstrictly positive.\nlim qStartT(Ht+1) =\n1 −lim W m(Ht)\nlim Y m\nlim qStartD(Ht) =\n1 −lim W m(Ht)\nˆ\nW m\n1−ˆqStartD\nlim qStartD(Ht)\nwhere ˆqStartD = lim qStartD( ˆH) and ˆW m = lim W m( ˆH) for some ˆH ∈H(D) with\nlim Y m( ˆH) = lim Y m. This is positive if and only if\nˆW m\nlim W m(Ht) > 1 −ˆqStartD.\nWe know that ˆqStartD > 1\n4 min{\np\n1−p, 1−p\np } so it suffices to show that\nˆW m\nlim W m(Ht) > 1 −1\n4 min{\np\n1 −p, 1 −p\np\n}.\n(1)\nNotice that lim W m(Ht) can be expressed as\nlim W m(Ht) =\n\n\n\nN m(Ht) = ⌊pK⌋\nif a = M\np\n1−pN M(Ht) =\np\n1−p⌊(1 −p)K⌋\nif a = m\nand the same holds for ˆW m. Therefore (1) follows from condition (iv) of Lemma 3.\nSo for β sufficiently large, we have defined a valid strategy profile. It remains to be shown\nthat this strategy profile is an equilibrium when β is close to 1, and that the proportion of\ntruthfulness in this equilibrium goes to 1 as K increases.\nBecause all experts act identically at every history, on or offthe equilibrium path, any\ncustomer strategy will be a best response. So to show that the strategy is an equilibrium,\nit will suffice to show that the experts play best responses at every history when β is large\nenough.\n25\n\nBy the following lemma, to show that the strategy is a best response for experts at\ntruthful periods, we only need to look at how deviations at later periods would affect the\nweights relative to t0. If an expert chosen at t0 would get the same set of t0-weights from\nplanning to deviate to m as she would deviating to M at any later truthful period in the\nblock, then there will be no profitable deviation once any such period is reached.\nLemma 4. Let σHτ,a : H × Θ →∆(A) for a ∈A, Hτ ∈H be identical to the conditional\nstrategy for σi\nr, defined above, except at the history Hτ. At Hτ, σHτ,a plays a for either θ.\nTake some t0 ∈T0 to be the start of a block, and suppose that for all Hτ ∈H(T) satisfying\nt0(τ) = t0 (that is, for all truthful histories in that block) it holds that W i\na(Ht0, σHτ,m) =\nW i\na(Ht0, σHτ,M) for a = m, M. Then ei has no profitable deviation if selected at any Hτ ∈\nH(T) satisfying t0(τ) = t0.\nProof. See Appendix.\nWe will show that the conditions of Lemma 4 hold, implying that our strategy is in fact\na best response at truthful periods.\nIf expert ei is picked at time t0 ∈T0, she receives weights W i\nm(Ht0, σi), W i\nM(Ht0, σi) from\nfollowing the proposed strategy. Consider a path of actions from t0 to t0 + K −1, going\nfrom the first period in the block to the last period before the block repeats. Relative to t0,\na weight of W a(Ht0+K−1) accumulates towards W i\na(Ht0, σi). Because strategies repeat anew\nevery K periods, W i\na(Ht0, σi) = W i\na(Ht0+K, σi), and so W i\na(Ht0, σi) satisfies the recursive\nformula\nW i\na(Ht0, σi) = E\nh\nW a(Ht0+K−1) + βKqStartD(Ht0+K−1)qStartT(Ht0+K)W i\na(Ht0, σi)\n\f\fHt0\ni\nwhere the expectation is taken over Ht0+K, given Ht0. On each path, qStartT =\n1−W a\nY a\nqStartDβK .\nPlugging this in to the recursive formula gives\nW i\na(Ht0, σi)\n= E\nh\nW a(Ht0+K−1) +\n\u0010\n1 −\nW a(Ht0+K−1)\nY a\n\u0011\nW i\na(Ht0, σi)\n\f\fHt0, σii\n=⇒\n0\n=\n\u0010\n1 −\nW i\na(Ht0,σi)\nY a\n\u0011\nE\nh\nW a(Ht0+K−1)|Ht0, σii\n=⇒\nW i\na(Ht0, σi)\n= Y a.\nNow consider a deviation of the form discussed in Lemma 4 to the strategy σi\nHτ,a. Following\n26\n\nthe same substitutions, with W i\na(Ht0, σi) = Y a, this gives weights\nW i\na(Ht0,σi\nHτ,a)\n= E\nh\nW a(Ht0+K−1) + βKqStartD(Ht0+K−1)qStartT(Ht0+K)W i\na(Ht0, σi)\n\f\fHt0, σi\nHτ,a\ni\n= W i\na(Ht0, σi) +\n\u0012\n1 −W i\na(Ht0, σi)\nY a\n\u0013\nE\nh\nW a(Ht0+K−1)|Ht0, σi\nHτ,a\ni\n= W i\na(Ht0, σi)\n(In equilibrium, if expert ei is selected at t0 then the same expert will be selected at every\ntruthful period in a block, and W a, qStartD, and qStartT are determined only by the play at\ntruthful periods. So when the expectation is taken, only ei’s strategy need be considered.).\nSo all such deviations yield the same point in (Wm, WM)-space, and Lemma 4 applies; there\nare no profitable deviations by experts at any H(T) periods.\nTo finish the proof that the strategies are an equilibrium for β and K large enough, we\nnow only need to show that the expert prefers not to deviate at H(D) periods. Starting at\nany H(D) period Ht, the expert is at most K periods away from reaching the next block\nat t0(t + K) with probability qStartT. So from following the equilibrium path, W i\na(Ht, σi) ≥\nβKqStartTW i\na(Ht0(t+K), σi) = βKqStartTY a. Fixing K and taking β →1, βK goes to 1 and\nqStartT approaches some value at least equal to min{\nζ\n2(1−p), ζ\n2p}. The limit of Y m is bounded\nbelow by an expression which goes to infinity:\nlim\nβ→1 Y m ≥\nmin\nHτ∈H(D) lim\nβ→1 W m(Hτ) ≥min\n\u001a\npK −1,\np\n1 −p((1 −p)K −1)\n\u001b\n= pK−max\n\u001a\n1,\np\n1 −p\n\u001b\nand a similar argument shows that Y M also diverges. Therefore, for β and K large enough,\nW i\na(Ht0, σi) becomes arbitrarily large and in particular is greater than 1. Deviating from a\nto a at Ht ∈H(D) gives a weight of 1 on a and 0 on a, and so for K and β large enough\nthis is strictly dominated by not deviating because ri ≥0.\nSo the strategies we have constructed do form an equilibrium. Now we show that as K\nincreases, these strategies give an arbitrarily high long-term proportion of truthful play.\nConsider the probability of having less than or equal to n truthful periods in a block of\nlength K. Writing pa as the probability of state θa in a period, so pm = p and pM = 1 −p,\n27\n\nand letting xn\na denote a binomial random variable of n draws from probability pa,\nProb\n\u0002\nAt most n truthful periods in a block\n\u0003\n= Prob\n\u0002\nAt least ⌊paK⌋θa’s after n periods, for some a\n\u0003\n≤\nX\na\nProb\n\u0002\nAt least ⌊paK⌋θa’s after n periods\n\u0003\n=\nX\na\nProb\n\u0002\nxn\na ≥⌊paK⌋\n\u0003\n≤\nX\na\nProb\n\u0002xn\na\nn ≥pa + pa\nK −n −1\nn\n\u0003\n.\n(2)\nThe random variable xn\na/n has mean pa and standard deviation\nq\npa(1−pa)\nn\n. So if we take\nn ≃K −Ks for some s ∈(1/2, 1) then pa\nK−n−1\nn\ndivided by the standard deviation goes to\ninfinity as K goes to infinity:\npa\nK−n−1\nn\nq\npa(1−pa)\nn\n≃\npa\nKs−1\nK−Ks\nq\npa(1−pa)\nK−Ks\n=\nr\npa\n1 −pa\nKs −1\n√\nK −Ks ≥\nr\npa\n1 −pa\nKs −1\n√\nK\n−→\nK→∞∞.\nTherefore each of the probability terms in (2) goes to 0 as K increases (for instance, by\nChebyshev’s theorem), and the probability of more than n ≃K −Ks out of K truthful\nperiods goes to 1. Moreover, since\nn\nK ≃K−Ks\nK\ngoes to 1 for large enough K, the expected\nproportion of truthful periods in a block must approach 1 for large enough K.4\nBecause blocks are independent, the Law of Large Numbers tells us that the long-term\nproportion of truthful periods approaches the expected proportion in a given block, and we\ncan get this arbitrarily close to 1.\n5\nExtensions\nWe consider an extreme environment in which customers are short term players who never\nreceive signals about the true state in past periods, and in which expert preferences are\ncompletely independent of the state of the world. Any combination of these assumptions\ncan be relaxed without fundamentally altering our conclusions.\nThe stream of customers can be thought of as a single long-term player, or some combina-\ntion of short- and long-term players, without affecting any of the equilibria. The customers’\n4This implies that the number of deterministic periods in a block is of order at most\n√\nK; taking s < 1/2,\nthe difference divided by the standard deviation goes to 0, which can be used to show that the order is at\nleast\n√\nK.\n28\n\nactions have no effect on the current or future play of experts, and so any strategy is both\nmyopically and dynamically optimal. We focussed on short-lived customers to highlight the\nfact that long-term relationships between individuals are unnecessary so long as histories are\nobservable.\nIf experts receive a small amount of disutility from mismatching the action and the state\n– due to guilt from lying, a fear of God or audits, or because their underlying cost structure\ndepends on the state – then all equilibrium strategy profiles in the paper remain equilibria.\nIndeed, a slight preference for truth makes truthfulness a strict rather than a weak best\nresponse at the appropriate periods.\nMoreover, because all of the equilibria we construct are fully pooling, signals about the\ntrue state in past periods reveal no new information about an expert’s type or about how\nan expert will play in the future. Signals may alter the set of equilibria, but do not disrupt\nthe ones we lay out. (It is easy to imagine that long-lived customers in particular might\nobserve signals of an expert’s past truthfulness – for instance, a customer brought his car\ninto a mechanic to have it repaired, and the car still had problems when he got it back).\nThe assumption that the outside option of an expert is 0 may not make sense if we\ninterpret experts as having no business outside of this market.\nHow could we possibly\nsupport a large (infinite) number of experts, when almost all of them get no business? But\nif we think that the experts have nonbinding capacity constraints and otherwise linear costs,\nthen this constructed market can be thought of as being on top of whatever other business\nthey get – possibly from identical markets running in parallel.\nIn this model, the infinity of available experts stands in for the ability of a customer to\ngo to a new expert in each period. We can implement all of the equilibria considered so\nlong as the previously chosen expert and a single new one are always available. In the Web\nAppendix we show how such an equilibrium can work in an environment akin to Section 3 in\nwhich customers must always return to a fixed, finite set of experts. We also extend Section\n3 to a setting with a larger action space in which there may be observable heterogeneity.\nFinally, in the Web Appendix we discuss the possibility of truthful play in a one-shot\nsetting where prices may not be exogenously fixed. When experts are homogeneous or when\ntheir costs are commonly known, prices for each treatment can be found at which experts\nearn the same profit for each action. At these prices, experts will be truthful even when\nno future business depends on the action chosen today. But it is impossible to find such\nprices when an expert’s costs are drawn from a nondegenerate distribution and are privately\nobserved. There is no way to condition prices on reported costs appropriately without giving\n29\n\nexperts an incentive to lie about these costs.\n6\nConclusion\nWe looked at a model where short-lived customers successively choose long-lived experts\nwho decide on and then perform actions for the customers. Customer utilities depend on\nthe state of the world along with the chosen action, but customers cannot observe the state.\nCustomers only observe the history of past actions taken by experts. The experts see the\nstate, but it doesn’t factor into their utility function. We considered how experts could be\ninduced to take the state-dependent action preferred by the customers.\nIn the game where experts do not have private information about their own preferences,\na truthful equilibrium can be implemented in the following manner. If the previous period’s\nexpert just performed a minor treatment, then the next customer returns to that expert; if\nthe expert just performed a major one then the customer moves to a new expert with some\nprobability.\nThe expert is indifferent between the actions because she gets more money\ntoday but less future business from a major treatment, and the customer is indifferent across\nexperts because all would be truthful.\nWhen experts do have private information about their own preferences, fully truthful\nequilibria are no longer possible. But customers can play a strategy in which all types of\nexperts will be indifferent over actions, and will therefore play truthfully, in certain periods.\nIn other periods, the experts are told to ignore the state and perform some predetermined\naction. At the truthful periods, experts are indifferent because either action will lead to the\nexpert’s performing the same number of expected, discounted lifetime minor treatments and\nthe same number of expected, discounted lifetime major treatments. As the discount factor\napproaches 1, we can achieve truthful actions in nearly all periods.\nAppendix: Proofs of Lemmas\nProof of Lemma 1. The strategy σi is sequentially rational and gives an optimal payoffstart-\ning from every history, so it is sufficient to show that ˆσi gives an optimal payoffstarting from\nany history in ˆH.\nWe divide this into three cases.\n• Case 1: There is a single history ˆHt ∈ˆH. The strategy of playing m at ˆHt and\ncontinuing with σi in the future gives the same payoffas the strategy of playing M\n30\n\nat ˆHt followed by σi; otherwise, it would not be optimal for ei to be truthful. So for\nany realization of θt, any mixture of these two strategies also gives this same optimal\npayoff.\n• Case 2: ˆH is finite. We can apply the argument of Case 1 inductively, changing the\nstrategies at each element of ˆH in any order. After each of these changes, the strategy\nremains sequentially rational and payoffs remain the same.\n• Case 3: ˆH is countably infinite. Suppose that ˆσi gives δ > 0 less utility than σi to ei if\nshe is chosen at some history Ht. For any positive N, we can change the strategy from\nσi to ˆσi at the finitely many histories ˆHτ ∈ˆH which satisfy t ≤τ ≤t+N, and utilities\nat all periods will remain constant as in Case 2. Call this intermediate strategy σi\nN.\nThe highest stage payoffthat the player can receive is 1 and the lowest is 0, and so\nstarting at Ht the utilities from strategies ˆσi and σi\nN can differ by at most P∞\nτ=N+1 βτ =\nβN+1\n1−β . For N large enough, this difference must be less than any fixed δ > 0. Contra-\ndiction.5\nProof of Lemma 2. The “If” part is immediate from the fact that, given on an expert’s type,\nthe utility of a conditional strategy is determined entirely by the weights it induces on M\nand m.\nTo show the “Only If” part, let ei be indifferent between σ′ and σ′′ at history Ht for both\nr′ and r′′. Then\n(\nr′W i\nm(Ht, σ′) + W i\nM(Ht, σ′) = r′W i\nm(Ht, σ′′) + W i\nM(Ht, σ′′)\nr′′W i\nm(Ht, σ′) + W i\nM(Ht, σ′) = r′′W i\nm(Ht, σ′′) + W i\nM(Ht, σ′′)\n=⇒\n(\nr′\u0000W i\nm(Ht, σ′) −W i\nm(Ht, σ′′)\n\u0001\n= W i\nM(Ht, σ′′) −W i\nM(Ht, σ′)\nr′′\u0000W i\nm(Ht, σ′) −W i\nm(Ht, σ′′)\n\u0001\n= W i\nM(Ht, σ′′) −W i\nM(Ht, σ′)\n=⇒(r′ −r′′)\n\u0000W i\nm(Ht, σ′) −W i\nm(Ht, σ′′)\n\u0001\n= 0.\nBecause r′ ̸= r′′, it must hold that W i\nm(Ht, σ′) = W i\nm(Ht, σ′′). Plugging this back into the\noriginal indifference r′W i\nm(Ht, σ′) + W i\nM(Ht, σ′) = r′W i\nm(Ht, σ′′) + W i\nM(Ht, σ′′) implies that\nW i\nM(Ht, σ′) = W i\nM(Ht, σ′′) as well.\nProof of Lemma 3. Conditions (i) and (iv) hold for any K large enough. Condition (ii) holds\nif pK −⌊pK⌋̸= 0.\nCondition (iii) is equivalent to pK −⌊pK⌋∈(5p−2\n2 , 3p\n2 ) for p ≤\n1\n2, or pK −⌊pK⌋∈\n(3p−1\n2 , 5p−1\n2 ) for p ≥\n1\n2.\nIn either case, both conditions (ii) and (iii) will be satisfied if\npK −⌊pK⌋is in some small neighborhood Np ⊆(0, 1) about p. (In fact, even for K small,\ncondition (iv) is satisfied if pK −⌊pK⌋is close to p).\n5When this lemma is applied in Section 4, the highest stage payoffis max{ri, 1} rather than 1, and so the\nmaximum payoffdifference is βN+1\n1−β · max{ri, 1} rather than βN+1\n1−β . The argument is otherwise unchanged.\n31\n\nFor p ∈(0, 1) irrational, {pn−⌊pn⌋|n ∈N} is dense on (0, 1). For p rational with reduced\ndenominator d, any K of the form K = nd + 1 will have pK −⌊pK⌋= p ∈Np. In each case\nan arbitrarily large K can be found with pK −⌊pK⌋in Np.\nProof of Lemma 4. Let Gτ = {Hτ ∈H(T)} be the set of possible truthful histories at time τ.\nSay that H′\nτ and H′′\nτ in Gτ are equivalent if the actions from periods t0(τ) through τ −1 are\nthe same in both histories. For τ = t0(τ), all Hτ are equivalent. The experts’ and customers’\nstrategies are such that picking an expert ei at two equivalent histories yields identical play\nby all agents going forward.\nTake t0 ∈T0, and fix some τ such that t0(τ) = t0 and Gτ is nonempty. Suppose that ei is\nselected at Ht. Define Gi\nτ(Hτ|Ht0) to be the single history following Ht0 that is equivalent\nto Hτ, in which ei is selected at period t0 and is selected with positive probability at period\nτ according to the customers’ equilibrium strategy. It is the element of the equivalence class\nfollowing Ht0 in which ei is chosen in every period from t through τ −1 and the actions\ncorresponding to Hτ are played. For τ = t0, define Gi\nτ(Hτ|Ht0) to be Hτ even if ei is not\nselected by the customers with positive probability at Ht0.\nConditional on customer ct0 selecting expert ei at history Ht0, given some arbitrary\nconditional strategy σ′ of ei, denote the probability of a history Hτ occurring and ei being\nselected at Hτ by πi(Hτ, Ht0|σ′). For any history Hτ ̸= Gi\nτ(Hτ|Ht0), the customers’ strategy\nis such that either Hτ occurs with probability 0 or ei is selected with probability 0 at Hτ;\nin either case, πi(Hτ, Ht0|σ′) = 0 for any σ′. Under the equilibrium conditional strategy σi\nr,\nthe probability πi(Gi\nτ(Hτ|Ht0), Ht0|σi\nr) is in fact positive. An expert chosen at t0 is never\nfired before the start of deterministic periods in a block, so πi(Gi\nτ(Hτ|Ht0), Ht0|σi\nr) is just\nthe probability that the states θt0 through θτ−1 are such that truthful play gives the correct\nsequence of actions.\nNow, consider some conditional strategy σ′ which differs from the equilibrium strategy\nσi\nr only at Hτ. We can see that W i\na(Ht0, σ′) is equal to some constant (the weight added\nalong all histories which do not follow Hτ) plus βτ−t0π(Gi\nτ(Hτ|Ht0), σi\nr) times a convex com-\nbination of W i\na(Hτ, σHτ,m) and W i\na(Hτ, σHτ,M). The convex combination places a weight on\nW i\na(Hτ, σHτ,a′) equal to the probability (unconditional on θτ) of ei playing a′ at Hτ if chosen.\nIn particular, if σ′ = σHτ,a′ then the convex combination places a weight of 1 on W i\nHτ(a, σHτ,a′)\nand 0 on W i\na(Hτ, σHτ,a′′), for a′′ ̸= a′.\nSuppose that the condition of the lemma holds: W i\na(Ht0, σHτ,m) = W i\na(Ht0, σHτ,M). Then\nit must be the case that either π(Hτ, Ht0|σi\nr) = 0, or W i\na(Hτ, σHτ,m) = W i\na(Hτ, σHτ,M). These\nweights are equal for equivalent histories, and each history is equivalent to one for which π\nis positive. So in fact W i\na(Hτ, σHτ,m) = W i\na(Hτ, σHτ,M) for all Hτ ∈H(T).\nTherefore at the history Hτ, if ei is selected, she has no profitable deviations. She can\neither deviate to m, M, or some mixture of the two; and any such deviation yields the same\nweights on each action, that is, the same number of expected discounted lifetime plays. Any\nsuch mixture of actions is optimal at a truthful period Hτ.\n32\n\nReferences\nIngela Alger and Francois Salanie. A theory of fraud and overtreatment in experts markets.\nJournal of Economics & Management Strategy, 15(4):853–881, Winter 2006.\nHeski Bar-Isaac and Steve Tadelis. Seller reputation. Foundations and Trends in Microeco-\nnomics, 4(4):273–351, 2008.\nJohn A. Byrne. Goodbye to an ethicist. Business Week, page 38, February 10, 2003.\nMichael R. Darby and Edi Karni. Free competition and the optimal amount of fraud. Journal\nof Law and Economics, 16(1), 1973.\nUwe Dulleck and Rudolf Kerschbamer. On doctors, mechanics and computer specialists:\nThe economics of credence goods. Journal of Economic Literature, 44:5–42, 2006.\nUwe Dulleck and Rudolf Kerschbamer. Experts vs. discounters: Consumer free-riding and\nexperts withholding advice in markets for credence goods. International Journal of Indus-\ntrial Organization, 2008.\nJeffey C. Ely and Juuso Valimaki. Bad reputation. Quarterly Journal of Economics, 118(3):\n785–814, 2003.\nJeffrey C. Ely, Drew Fudenberg, and David K. Levine. When is reputation bad. Games and\nEconomic Behavior, 63:498–526, 2008.\nWinand Emons. Credence goods and fraudulent experts. RAND Journal of Economics, 28\n(1):107–119, Spring 1997.\nDrew Fudenberg and David K. Levine. Efficiency and observability in games with long-run\nand short-run players. Journal of Economic Theory, 62:103–135, 1994.\nMatthew O. Jackson and Hugo Sonnenschein. Overcoming incentive constraints. Economet-\nrica, 75(1):241–257, January 2007.\nIn-Uck Park. Cheap talk referrals of differentiated experts in repeated relationships. RAND\nJournal of Economics, 36(2), Summer 2005.\nMartin Pesendorfer. A study of collusion in first price auctions. Review of Economic Studies,\n67:381–411, 2000.\nCarolyn Pitchik and Andrew Schotter. Honesty in a model of strategic information trans-\nmission. American Economic Review, 77(5):1032–1036, December 1987.\nHenry Schneider. Agency problems and reputation in expert services: evidence from auto\nrepair. Johnson School Research Paper Series #15-07, October 2007.\n33\n\nRobert M. Townsend.\nOptimal multiperiod contracts and the gain from enduring rela-\ntionships under private information. The Journal of Political Economy, 90(6):1166–1186,\nDecember 1982.\nAsher Wolinsky. Competition in a market for informed experts’ services. RAND Journal of\nEconomics, 24:380–398, 1993.\n34"},"Token_count":{"kind":"number","value":22589,"string":"22,589"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":66,"numItemsPerPage":100,"numTotalItems":6965,"offset":6600,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4Njg5MjAzMywianRpIjoiYmIyYWNlMmEtNzU1Ni00MDM2LWI3OWEtYmU5NTczMzY2ODIzIiwic3ViIjoiL2RhdGFzZXRzL1RoZUZpbkFJL05CRVIiLCJleHAiOjE3ODY4OTU2MzMsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.kEsi_LncbFW4NceD9pg9dcnYtYHnEQY3FH1FxoJ1gfEMQtjL30s68zhwBorgiafrl7OGmtUBfP99tmG8RUHpAg","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":6965,"numBytesParquet":319955443}]},"discussionsStats":{"closed":0,"open":0,"total":0},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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NBER | 2,009 | NBER WORKING PAPER SERIES
SHOW ME THE MONEY:
DOES SHARED CAPITALISM SHARE THE WEALTH?
Robert Buchele
Douglas Kruse
Loren Rodgers
Adria Scharf
Working Paper 14830
http://www.nber.org/papers/w14830
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Earlier drafts of this paper w... | 18,515 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE RISE OF THE SERVICE ECONOMY
Francisco J. Buera
Joseph P. Kaboski
Working Paper 14822
http://www.nber.org/papers/w14822
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
March 2009
We have benefited from helpful comments of the editor, three anonymous referr... | 36,360 |
NBER | 2,009 | NBER WORKING PAPER SERIES
SELECTIVE SWAP ARRANGEMENTS AND THE GLOBAL FINANCIAL CRISIS:
ANALYSIS AND INTERPRETATION
Joshua Aizenman
Gurnain Kaur Pasricha
Working Paper 14821
http://www.nber.org/papers/w14821
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
March 2009
We are grateful to ... | 13,145 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE CONSEQUENCES OF RADICAL REFORM:
THE FRENCH REVOLUTION
Daron Acemoglu
Davide Cantoni
Simon Johnson
James A. Robinson
Working Paper 14831
http://www.nber.org/papers/w14831
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank Jared Diamond, Ro... | 38,837 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE CAUSES AND EFFECTS OF INTERNATIONAL MIGRATIONS:
EVIDENCE FROM OECD COUNTRIES 1980-2005
Francesc Ortega
Giovanni Peri
Working Paper 14833
http://www.nber.org/papers/w14833
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are thankful to Greg W... | 30,357 |
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FINANCIAL INSTABILITY, RESERVES, AND CENTRAL BANK SWAP LINES
IN THE PANIC OF 2008
Maurice Obstfeld
Jay C. Shambaugh
Alan M. Taylor
Working Paper 14826
http://www.nber.org/papers/w14826
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
March 2009
We thank the Fo... | 10,684 |
NBER | 2,009 | NBER WORKING PAPER SERIES
MARRIAGE MEETS THE JONESES:
RELATIVE INCOME, IDENTITY, AND MARITAL STATUS
Tara Watson
Sara McLanahan
Working Paper 14773
http://www.nber.org/papers/w14773
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
March 2009
The authors thank Martha Bailey, Betsy Braine... | 31,988 |
NBER | 2,009 | NBER WORKING PAPER SERIES
CURRENCY MISALIGNMENTS AND OPTIMAL MONETARY POLICY:
A REEXAMINATION
Charles Engel
Working Paper 14829
http://www.nber.org/papers/w14829
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
I am indebted very much to Gianluca Benigno, Mick Devereux, Jon ... | 44,183 |
NBER | 2,009 | NBER WORKING PAPER SERIES
A MODEL OF INTERNATIONAL CITIES:
IMPLICATIONS FOR REAL EXCHANGE RATES
Mario J. Crucini
Hakan Yilmazkuday
Working Paper 14834
http://www.nber.org/papers/w14834
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We especially thank Mary Amiti, Eric Bond... | 22,817 |
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PREFERENCES FOR REDISTRIBUTION
Alberto F. Alesina
Paola Giuliano
Working Paper 14825
http://www.nber.org/papers/w14825
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
March 2009
¸ The views expressed herein are those of the author(s) and do not necessarily re... | 27,162 |
NBER | 2,009 | NBER WORKING PAPER SERIES
TEACHING THE TAX CODE:
EARNINGS RESPONSES TO AN EXPERIMENT WITH EITC RECIPIENTS
Raj Chetty
Emmanuel Saez
Working Paper 14836
http://www.nber.org/papers/w14836
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are extremely grateful to Joe Cresta, ... | 34,298 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE LAW OF ONE PRICE WITHOUT THE BORDER:
THE ROLE OF DISTANCE VERSUS STICKY PRICES
Mario J. Crucini
Mototsugu Shintani
Takayuki Tsuruga
Working Paper 14835
http://www.nber.org/papers/w14835
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are ind... | 23,129 |
NBER | 2,009 | NBER WORKING PAPER SERIES
CAN A RATIONAL CHOICE FRAMEWORK MAKE SENSE OF ANOREXIA NERVOSA?
Robert S. Goldfarb
Thomas C. Leonard
Sara Markowitz
Steven Suranovic
Working Paper 14838
http://www.nber.org/papers/w14838
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank Bern... | 21,832 |
NBER | 2,009 | NBER WORKING PAPER SERIES
CONCORDING U.S. HARMONIZED SYSTEM CATEGORIES OVER TIME
Justin R. Pierce
Peter K. Schott
Working Paper 14837
http://www.nber.org/papers/w14837
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Schott thanks the National Science Foundation (grants SES-... | 11,993 |
NBER | 2,009 | NBER WORKING PAPER SERIES
DETERMINANTS OF TECHNOLOGY ADOPTION:
PRIVATE VALUE AND PEER EFFECTS IN MENSTRUAL CUP TAKE-UP
Emily Oster
Rebecca Thornton
Working Paper 14828
http://www.nber.org/papers/w14828
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
March 2009
The Menstruation and Edu... | 24,537 |
NBER | 2,009 | NBER WORKING PAPER SERIES
EMPLOYER-SPONSORED HEALTH INSURANCE AND THE PROMISE OF HEALTH
INSURANCE REFORM
Thomas C. Buchmueller
Alan C. Monheit
Working Paper 14839
http://www.nber.org/papers/w14839
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
This research was funded by t... | 13,762 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE ANALYTICS OF THE WAGE EFFECT OF IMMIGRATION
George J. Borjas
Working Paper 14796
http://www.nber.org/papers/w14796
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
March 2009
I am grateful to Edward Glaeser, Jeffrey Grogger, Daniel Hamermesh, Gordon Hanson... | 64,521 |
NBER | 2,009 | NBER WORKING PAPER SERIES
ENERGY, THE ENVIRONMENT, AND TECHNOLOGICAL CHANGE
David Popp
Richard G. Newell
Adam B. Jaffe
Working Paper 14832
http://www.nber.org/papers/w14832
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
This paper was prepared for the forthcoming Handbook ... | 53,666 |
NBER | 2,009 | NBER WORKING PAPER SERIES
ESTIMATION OF CAUSAL EFFECTS IN EXPERIMENTS WITH MULTIPLE SOURCES
OF NONCOMPLIANCE
John Engberg
Dennis Epple
Jason Imbrogno
Holger Sieg
Ron Zimmer
Working Paper 14842
http://www.nber.org/papers/w14842
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009... | 16,175 |
NBER | 2,009 | NBER WORKING PAPER SERIES
FINANCIAL OPENNESS AND PRODUCTIVITY
Geert Bekaert
Campbell R. Harvey
Christian Lundblad
Working Paper 14843
http://www.nber.org/papers/w14843
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We appreciate the helpful comments of the participants at ... | 25,879 |
NBER | 2,009 | NBER WORKING PAPER SERIES
DECENTRALIZED MATCHING WITH ALIGNED PREFERENCES
Muriel Niederle
Leeat Yariv
Working Paper 14840
http://www.nber.org/papers/w14840
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful to Attila Ambrus, Federico Echenique, Guillaume Haerin... | 24,996 |
NBER | 2,009 | NBER WORKING PAPER SERIES
FACE VALUE: INFORMATION AND SIGNALING IN AN ILLEGAL MARKET
Trevon Logan
Manisha Shah
Working Paper 14841
http://www.nber.org/papers/w14841
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
George Akerlof, Rodney J. Andrews, Raj Arunachalam, Andrea A.... | 33,455 |
NBER | 2,009 | NBER WORKING PAPER SERIES
A UNIFIED THEORY OF TOBIN'S Q, CORPORATE INVESTMENT, FINANCING,
AND RISK MANAGEMENT
Patrick Bolton
Hui Chen
Neng Wang
Working Paper 14845
http://www.nber.org/papers/w14845
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful to Andrew Ab... | 30,748 |
NBER | 2,009 | NBER WORKING PAPER SERIES
EARLY ADMISSIONS AT SELECTIVE COLLEGES
Christopher Avery
Jonathan D. Levin
Working Paper 14844
http://www.nber.org/papers/w14844
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
This paper developed out of work done independently by the authors, mos... | 22,574 |
NBER | 2,009 | NBER WORKING PAPER SERIES
SURPRISING COMPARATIVE PROPERTIES OF MONETARY MODELS:
RESULTS FROM A NEW DATA BASE
John B. Taylor
Volker Wieland
Working Paper 14849
http://www.nber.org/papers/w14849
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful for excellent res... | 17,016 |
NBER | 2,009 | NBER WORKING PAPER SERIES
ENTREPRENEURIAL FINANCE AND NON-DIVERSIFIABLE RISK
Hui Chen
Jianjun Miao
Neng Wang
Working Paper 14848
http://www.nber.org/papers/w14848
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank Yakov Amihud, Patrick Bolton, John Cox, Bob Hall, Glen... | 27,823 |
NBER | 2,009 | NBER WORKING PAPER SERIES
CONFIDENCE, CRASHES AND ANIMAL SPIRITS
Roger E.A. Farmer
Working Paper 14846
http://www.nber.org/papers/w14846
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
The author gratefully acknowledges the support of NSF grant 0720839. I wish to thank Katr... | 13,889 |
NBER | 2,009 | NBER WORKING PAPER SERIES
FLUCTUATIONS IN OVERSEAS TRAVEL BY AMERICANS, 1820 TO 2000
Brandon Dupont
Alka Gandhi
Thomas Weiss
Working Paper 14847
http://www.nber.org/papers/w14847
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We wish to thank the Center for Economic Studie... | 17,748 |
NBER | 2,009 | NBER WORKING PAPER SERIES
CONSUMPTION AND REAL EXCHANGE RATES IN PROFESSIONAL FORECASTS
Michael B. Devereux
Gregor W. Smith
James Yetman
Working Paper 14795
http://www.nber.org/papers/w14795
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
March 2009
We thank the Social Sciences Resear... | 18,833 |
NBER | 2,009 | NBER WORKING PAPER SERIES
MENSTRUATION AND EDUCATION IN NEPAL
Emily Oster
Rebecca Thornton
Working Paper 14853
http://www.nber.org/papers/w14853
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
The Menstruation and Education in Nepal Project is supported by grants from the U... | 13,638 |
NBER | 2,009 | NBER WORKING PAPER SERIES
WHY IS MOBILITY IN INDIA SO LOW? SOCIAL INSURANCE, INEQUALITY,
AND GROWTH
Kaivan Munshi
Mark Rosenzweig
Working Paper 14850
http://www.nber.org/papers/w14850
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are very grateful to Andrew Foster for ... | 31,194 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE OLYMPIC EFFECT
Andrew K. Rose
Mark M. Spiegel
Working Paper 14854
http://www.nber.org/papers/w14854
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank Christopher Candelaria for excellent research assistance, and Keith Head for help with ... | 25,337 |
NBER | 2,009 | NBER WORKING PAPER SERIES
PROCUREMENT CONTRACTING WITH TIME INCENTIVES:
THEORY AND EVIDENCE
Patrick Bajari
Gregory Lewis
Working Paper 14855
http://www.nber.org/papers/w14855
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful to Saeid Asgari, Bassem Barsoum, Ma... | 28,923 |
NBER | 2,009 | NBER WORKING PAPER SERIES
CHARTER SCHOOLS IN NEW YORK CITY:
WHO ENROLLS AND HOW THEY AFFECT THEIR STUDENTS' ACHIEVEMENT
Caroline M. Hoxby
Sonali Murarka
Working Paper 14852
http://www.nber.org/papers/w14852
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
This research was s... | 34,488 |
NBER | 2,009 | NBER WORKING PAPER SERIES
A THEORY OF OUTSOURCING AND WAGE DECLINE
Thomas J. Holmes
Julia Thornton Snider
Working Paper 14856
http://www.nber.org/papers/w14856
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Holmes acknowledges NSF Grant 0551062 for support of this research... | 30,047 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE GOVERNANCE AND PERFORMANCE OF RESEARCH UNIVERSITIES:
EVIDENCE FROM EUROPE AND THE U.S.
Philippe Aghion
Mathias Dewatripont
Caroline M. Hoxby
Andreu Mas-Colell
André Sapir
Working Paper 14851
http://www.nber.org/papers/w14851
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Ca... | 27,724 |
NBER | 2,009 | NBER WORKING PAPER SERIES
COMPLEMENTARITY AND AGGREGATE IMPLICATIONS OF ASSORTATIVE MATCHING:
A NONPARAMETRIC ANALYSIS
Bryan S. Graham
Guido W. Imbens
Geert Ridder
Working Paper 14860
http://www.nber.org/papers/w14860
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Financia... | 81,123 |
NBER | 2,009 | NBER WORKING PAPER SERIES
HOW DOES SIMPLIFIED DISCLOSURE AFFECT INDIVIDUALS' MUTUAL FUND
CHOICES?
John Beshears
James J. Choi
David Laibson
Brigitte C. Madrian
Working Paper 14859
http://www.nber.org/papers/w14859
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
This publica... | 12,896 |
NBER | 2,009 | NBER WORKING PAPER SERIES
CREDIT MARKET SHOCKS AND ECONOMIC FLUCTUATIONS:
EVIDENCE FROM CORPORATE BOND AND STOCK MARKETS
Simon Gilchrist
Vladimir Yankov
Egon Zakrajsek
Working Paper 14863
http://www.nber.org/papers/w14863
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We t... | 38,539 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE INVESTMENT STRATEGIES OF SOVEREIGN WEALTH FUNDS
Shai Bernstein
Josh Lerner
Antoinette Schoar
Working Paper 14861
http://www.nber.org/papers/w14861
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank Harvard Business School’s Division of Re... | 25,755 |
NBER | 2,009 | NBER WORKING PAPER SERIES
STRATEGY-PROOFNESS VERSUS EFFICIENCY IN MATCHING WITH INDIFFERENCES:
REDESIGNING THE NEW YORK CITY HIGH SCHOOL MATCH
Atila Abdulkadiroglu
Parag A. Pathak
Alvin E. Roth
Working Paper 14864
http://www.nber.org/papers/w14864
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge... | 27,025 |
NBER | 2,009 | NBER WORKING PAPER SERIES
TECHNOLOGY DIFFUSION AND PRODUCTIVITY GROWTH IN HEALTH CARE
Jonathan Skinner
Douglas Staiger
Working Paper 14865
http://www.nber.org/papers/w14865
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful to seminar participants at Brown, Chi... | 22,516 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE RIDDLE OF THE GREAT PYRAMIDS
Randall Morck
Working Paper 14858
http://www.nber.org/papers/w14858
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
I am grateful for numerous helpful suggestions from participants at the “Evolutionary Dynamics of
B... | 19,241 |
NBER | 2,009 | NBER WORKING PAPER SERIES
FORCED SALES AND HOUSE PRICES
John Y. Campbell
Stefano Giglio
Parag Pathak
Working Paper 14866
http://www.nber.org/papers/w14866
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful to Tuomo Vuolteenaho and Paul Willen for early conversa... | 26,024 |
NBER | 2,009 | NBER WORKING PAPER SERIES
BUSINESS AND FINANCIAL METHOD PATENTS, INNOVATION, AND POLICY
Bronwyn H. Hall
Working Paper 14868
http://www.nber.org/papers/w14868
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
This is a substantially revised and expanded version of a paper prep... | 19,589 |
NBER | 2,009 | NBER WORKING PAPER SERIES
OUTSIDE AND INSIDE LIQUIDITY
Patrick Bolton
Tano Santos
Jose A. Scheinkman
Working Paper 14867
http://www.nber.org/papers/w14867
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank Rafael Repullo, Lasse Pedersen, Raghu Sundaram, and Doug Diamo... | 35,421 |
NBER | 2,009 | NBER WORKING PAPER SERIES
VALUING TOXIC ASSETS:
AN ANALYSIS OF CDO EQUITY
Francis A. Longstaff
Brett Myers
Working Paper 14871
http://www.nber.org/papers/w14871
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
The authors are grateful for helpful discussions with Navneet Aro... | 25,507 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE FINNISH GREAT DEPRESSION:
FROM RUSSIA WITH LOVE
Yuriy Gorodnichenko
Enrique G. Mendoza
Linda L. Tesar
Working Paper 14874
http://www.nber.org/papers/w14874
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
¸˛The authors would like to thank partic... | 38,529 |
NBER | 2,009 | NBER WORKING PAPER SERIES
STICKY PRICES VERSUS MONETARY FRICTIONS:
AN ESTIMATION OF POLICY TRADE-OFFS
S. Boragan Aruoba
Frank Schorfheide
Working Paper 14870
http://www.nber.org/papers/w14870
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Previous versions of this paper ci... | 34,473 |
NBER | 2,009 | NBER WORKING PAPER SERIES
DSGE MODEL-BASED FORECASTING OF NON-MODELLED VARIABLES
Frank Schorfheide
Keith Sill
Maxym Kryshko
Working Paper 14872
http://www.nber.org/papers/w14872
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank seminar participants at the Board of Go... | 27,524 |
NBER | 2,009 | NBER WORKING PAPER SERIES
PAYING FOR PROGRESS:
CONDITIONAL GRANTS AND THE DESEGREGATION OF SOUTHERN SCHOOLS
Elizabeth Cascio
Nora Gordon
Ethan Lewis
Sarah Reber
Working Paper 14869
http://www.nber.org/papers/w14869
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
For their h... | 25,090 |
NBER | 2,009 | NBER WORKING PAPER SERIES
GENDER ROLES AND MEDICAL PROGRESS
Stefania Albanesi
Claudia Olivetti
Working Paper 14873
http://www.nber.org/papers/w14873
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
This paper first appeared under the title "GENDER ROLES AND TECHNOLOGICAL PRO... | 33,853 |
NBER | 2,009 | NBER WORKING PAPER SERIES
AN ELABORATED GLOBAL CLIMATE POLICY ARCHITECTURE:
SPECIFIC FORMULAS AND EMISSION TARGETS FOR ALL COUNTRIES IN ALL DECADES
Jeffrey A. Frankel
Working Paper 14876
http://www.nber.org/papers/w14876
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
The a... | 29,139 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE ALCHEMY OF CDO CREDIT RATINGS
Efraim Benmelech
Jennifer Dlugosz
Working Paper 14878
http://www.nber.org/papers/w14878
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
This paper has being prepared for the Carnegie-Rochester conference series. We... | 20,907 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE GREAT INFLATION DRIFT
Marvin Goodfriend
Robert G. King
Working Paper 14862
http://www.nber.org/papers/w14862
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
The paper was prepared for "The Great Inflation Conference" sponsored by the National B... | 17,480 |
NBER | 2,009 | NBER WORKING PAPER SERIES
BAYESIAN AND FREQUENTIST INFERENCE IN PARTIALLY IDENTIFIED MODELS
Hyungsik Roger Moon
Frank Schorfheide
Working Paper 14882
http://www.nber.org/papers/w14882
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank seminar participants at the 2007 ... | 21,660 |
NBER | 2,009 | NBER WORKING PAPER SERIES
A NEW TEST OF BORROWING CONSTRAINTS FOR EDUCATION
Meta Brown
John Karl Scholz
Ananth Seshadri
Working Paper 14879
http://www.nber.org/papers/w14879
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful to the National Institutes of Aging ... | 30,712 |
NBER | 2,009 | NBER WORKING PAPER SERIES
CARBON MOTIVATED REGIONAL TRADE ARRANGEMENTS:
ANALYTICS AND SIMULATIONS
Yan Dong
John Whalley
Working Paper 14880
http://www.nber.org/papers/w14880
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank the Center for International Governance Inn... | 16,231 |
NBER | 2,009 | NBER WORKING PAPER SERIES
INFORMATION ASYMMETRY, INFORMATION PRECISION, AND THE COST
OF CAPITAL
Richard A. Lambert
Christian Leuz
Robert E. Verrecchia
Working Paper 14881
http://www.nber.org/papers/w14881
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
The views expressed h... | 21,329 |
NBER | 2,009 | NBER WORKING PAPER SERIES
SOPHISTICATED MONETARY POLICIES
Andrew Atkeson
V. V. Chari
Patrick Kehoe
Working Paper 14883
http://www.nber.org/papers/w14883
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Atkeson, Chari, and Kehoe thank the National Science Foundation for finan... | 33,697 |
NBER | 2,009 | NBER WORKING PAPER SERIES
DOES CORPORATE GOVERNANCE MATTER IN COMPETITIVE INDUSTRIES?
Xavier Giroud
Holger M. Mueller
Working Paper 14877
http://www.nber.org/papers/w14877
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We especially thank an anonymous referee whose excelle... | 29,867 |
NBER | 2,009 | NBER WORKING PAPER SERIES
SIXTY YEARS AFTER THE MAGIC CARPET RIDE:
THE LONG-RUN EFFECT OF THE EARLY CHILDHOOD ENVIRONMENT ON SOCIAL AND ECONOMIC OUTCOMES
Eric D. Gould
Victor Lavy
M. Daniele Paserman
Working Paper 14884
http://www.nber.org/papers/w14884
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cam... | 39,235 |
NBER | 2,009 | NBER WORKING PAPER SERIES
DOES AFFIRMATIVE ACTION LEAD TO MISMATCH? A NEW TEST AND EVIDENCE
Peter Arcidiacono
Esteban M. Aucejo
Hanming Fang
Kenneth I. Spenner
Working Paper 14885
http://www.nber.org/papers/w14885
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We would lik... | 21,125 |
NBER | 2,009 | NBER WORKING PAPER SERIES
GOVERNMENT FORM AND PUBLIC SPENDING:
THEORY AND EVIDENCE FROM U.S. MUNICIPALITIES
Stephen Coate
Brian Knight
Working Paper 14857
http://www.nber.org/papers/w14857
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are particularly indebted to Ruben... | 39,551 |
NBER | 2,009 | NBER WORKING PAPER SERIES
GLOBAL FINANCIAL STRUCTURE AND CLIMATE CHANGE
John Whalley
Yufei Yuan
Working Paper 14888
http://www.nber.org/papers/w14888
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank the Center for International Governance Innovation (CIGI) and the A... | 8,757 |
NBER | 2,009 | NBER WORKING PAPER SERIES
FINANCIAL REGULATION, FINANCIAL GLOBALIZATION AND THE SYNCHRONIZATION
OF ECONOMIC ACTIVITY
Sebnem Kalemli-Ozcan
Elias Papaioannou
José Luis Peydró
Working Paper 14887
http://www.nber.org/papers/w14887
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009... | 139,257 |
NBER | 2,009 | NBER WORKING PAPER SERIES
CAPM FOR ESTIMATING THE COST OF EQUITY CAPITAL:
INTERPRETING THE EMPIRICAL EVIDENCE
Zhi Da
Re-Jin Guo
Ravi Jagannathan
Working Paper 14889
http://www.nber.org/papers/w14889
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank for helpful commen... | 34,569 |
NBER | 2,009 | NBER WORKING PAPER SERIES
RISK MATTERS: THE REAL EFFECTS OF VOLATILITY SHOCKS
Jesús Fernández-Villaverde
Pablo A. Guerrón-Quintana
Juan Rubio-Ramírez
Martín Uribe
Working Paper 14875
http://www.nber.org/papers/w14875
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank ... | 29,673 |
NBER | 2,009 | NBER WORKING PAPER SERIES
DO REGULATIONS BASED ON CREDIT RATINGS AFFECT A FIRM'S COST
OF CAPITAL?
Darren J. Kisgen
Philip E. Strahan
Working Paper 14890
http://www.nber.org/papers/w14890
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank seminar participants at Boston... | 15,860 |
NBER | 2,009 | NBER WORKING PAPER SERIES
GENDER INTERACTIONS WITHIN HIERARCHIES:
EVIDENCE FROM THE POLITICAL ARENA
Stefano Gagliarducci
M. Daniele Paserman
Working Paper 14893
http://www.nber.org/papers/w14893
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank seminar participants a... | 34,461 |
NBER | 2,009 | NBER WORKING PAPER SERIES
REGIONAL TRADE INTEGRATION AND MULTINATIONAL FIRM STRATEGIES
Pol Antràs
C. Fritz Foley
Working Paper 14891
http://www.nber.org/papers/w14891
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
This paper was prepared for the Workshop on "Quantifying th... | 22,676 |
NBER | 2,009 | NBER WORKING PAPER SERIES
CAN YOU GET WHAT YOU PAY FOR? PAY-FOR-PERFORMANCE AND THE
QUALITY OF HEALTHCARE PROVIDERS
Kathleen J. Mullen
Richard G. Frank
Meredith B. Rosenthal
Working Paper 14886
http://www.nber.org/papers/w14886
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 20... | 27,080 |
NBER | 2,009 | NBER WORKING PAPER SERIES
EURO MEMBERSHIP AS A U.K. MONETARY POLICY OPTION:
RESULTS FROM A STRUCTURAL MODEL
Riccardo DiCecio
Edward Nelson
Working Paper 14894
http://www.nber.org/papers/w14894
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Presented at the NBER conference ... | 18,827 |
NBER | 2,009 | NBER WORKING PAPER SERIES
FUND MANAGERS, CAREER CONCERNS, AND ASSET PRICE VOLATILITY
Veronica Guerrieri
Péter Kondor
Working Paper 14898
http://www.nber.org/papers/w14898
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
¸˛ We are grateful to Fernando Alvarez, Cristina Arella... | 27,233 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE GREAT INFLATION IN THE UNITED STATES AND THE UNITED KINGDOM:
RECONCILING POLICY DECISIONS AND DATA OUTCOMES
Riccardo DiCecio
Edward Nelson
Working Paper 14895
http://www.nber.org/papers/w14895
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
An ... | 32,821 |
NBER | 2,009 | NBER WORKING PAPER SERIES
ANTICIPATED ALTERNATIVE INSTRUMENT-RATE PATHS IN POLICY SIMULATIONS
Stefan Laséen
Lars E.O. Svensson
Working Paper 14902
http://www.nber.org/papers/w14902
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank Malin Adolfson, Andrew Blake, Jordi ... | 19,498 |
NBER | 2,009 | NBER WORKING PAPER SERIES
IMMIGRATION AND PRODUCT DIVERSITY
Francesca Mazzolari
David Neumark
Working Paper 14900
http://www.nber.org/papers/w14900
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Both authors are research fellows at IZA. Neumark is also a research associate... | 22,977 |
NBER | 2,009 | NBER WORKING PAPER SERIES
WALL STREET'S FIRST CORPORATE GOVERNANCE CRISIS:
THE PANIC OF 1826
Eric Hilt
Working Paper 14892
http://www.nber.org/papers/w14892
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
I would like to thank Carola Frydman, Bert Huang, Naomi Lamoreaux, Wi... | 26,715 |
NBER | 2,009 | NBER WORKING PAPER SERIES
WAGE RISK AND EMPLOYMENT RISK OVER THE LIFE CYCLE
Hamish Low
Costas Meghir
Luigi Pistaferri
Working Paper 14901
http://www.nber.org/papers/w14901
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful to the editor, two anonymous referees,... | 39,888 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE IMPORTANCE OF HISTORY FOR ECONOMIC DEVELOPMENT
Nathan Nunn
Working Paper 14899
http://www.nber.org/papers/w14899
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Forthcoming in the Annual Review of Economics. I have benefitted from useful discus... | 24,086 |
NBER | 2,009 | NBER WORKING PAPER SERIES
MOTIVATING POLITICIANS:
THE IMPACTS OF MONETARY INCENTIVES ON QUALITY AND PERFORMANCE
Claudio Ferraz
Frederico Finan
Working Paper 14906
http://www.nber.org/papers/w14906
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank Daron Acemoglu, Simo... | 38,993 |
NBER | 2,009 | NBER WORKING PAPER SERIES
HOW THE SUBPRIME CRISIS WENT GLOBAL:
EVIDENCE FROM BANK CREDIT DEFAULT SWAP SPREADS
Barry Eichengreen
Ashoka Mody
Milan Nedeljkovic
Lucio Sarno
Working Paper 14904
http://www.nber.org/papers/w14904
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Th... | 16,292 |
NBER | 2,009 | NBER WORKING PAPER SERIES
EFFICIENT SEARCH ON THE JOB AND THE BUSINESS CYCLE
Guido Menzio
Shouyong Shi
Working Paper 14905
http://www.nber.org/papers/w14905
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful for comments received from participants at the Search... | 31,597 |
NBER | 2,009 | NBER WORKING PAPER SERIES
BLOCK RECURSIVE EQUILIBRIA FOR STOCHASTIC MODELS OF SEARCH
ON THE JOB
Guido Menzio
Shouyong Shi
Working Paper 14907
http://www.nber.org/papers/w14907
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful for comments received from partici... | 40,488 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE LONG OR SHORT OF IT:
DETERMINANTS OF FOREIGN CURRENCY EXPOSURE IN EXTERNAL BALANCE SHEETS
Philip R. Lane
Jay C. Shambaugh
Working Paper 14909
http://www.nber.org/papers/w14909
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Prepared for the IMF... | 24,798 |
NBER | 2,009 | NBER WORKING PAPER SERIES
INEQUALITY AND THE MEASUREMENT OF RESIDENTIAL SEGREGATION
BY INCOME IN AMERICAN NEIGHBORHOODS
Tara Watson
Working Paper 14908
http://www.nber.org/papers/w14908
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
I gratefully acknowledge financial suppo... | 20,640 |
NBER | 2,009 | NBER WORKING PAPER SERIES
TRADE AND INCOME -- EXPLOITING TIME SERIES IN GEOGRAPHY
James Feyrer
Working Paper 14910
http://www.nber.org/papers/w14910
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Many thanks to Alan Taylor and Reuven Glick for sharing their bilateral trade... | 24,025 |
NBER | 2,009 | NBER WORKING PAPER SERIES
ABILITY-GROUPING AND ACADEMIC INEQUALITY:
EVIDENCE FROM RULE-BASED STUDENT ASSIGNMENTS
C. Kirabo Jackson
Working Paper 14911
http://www.nber.org/papers/w14911
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
The author is grateful for feedback recei... | 29,597 |
NBER | 2,009 | NBER WORKING PAPER SERIES
BETTER LATE THAN NOTHING:
SOME COMMENTS ON DEATON (2009) AND HECKMAN AND URZUA (2009)
Guido W. Imbens
Working Paper 14896
http://www.nber.org/papers/w14896
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
I have benefitted from discussions with Josh... | 18,919 |
NBER | 2,009 | NBER WORKING PAPER SERIES
U.S. STOCK MARKET CRASH RISK, 1926-2006
David S. Bates
Working Paper 14913
http://www.nber.org/papers/w14913
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
I am grateful for comments on earlier versions of the paper from seminar participants at Io... | 24,127 |
NBER | 2,009 | NBER WORKING PAPER SERIES
FINANCE AND DEVELOPMENT:
A TALE OF TWO SECTORS
Francisco J. Buera
Joseph Kaboski
Yongseok Shin
Working Paper 14914
http://www.nber.org/papers/w14914
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
Buera and Shin gratefully acknowledge the support o... | 29,294 |
NBER | 2,009 | NBER WORKING PAPER SERIES
STATE DEPENDENCE AND ALTERNATIVE EXPLANATIONS FOR CONSUMER
INERTIA
Jean-Pierre Dubé
Günter J. Hitsch
Peter E. Rossi
Working Paper 14912
http://www.nber.org/papers/w14912
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank Wes Hutchinson, Ariel... | 17,270 |
NBER | 2,009 | NBER WORKING PAPER SERIES
RISK SHIFTING AND MUTUAL FUND PERFORMANCE
Jennifer Huang
Clemens Sialm
Hanjiang Zhang
Working Paper 14903
http://www.nber.org/papers/w14903
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We thank Keith Brown, Jie Cao, Joe Chen, Jonathan Cohn, Ken ... | 27,433 |
NBER | 2,009 | NBER WORKING PAPER SERIES
RUGGEDNESS: THE BLESSING OF BAD GEOGRAPHY IN AFRICA
Nathan Nunn
Diego Puga
Working Paper 14918
http://www.nber.org/papers/w14918
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful for valuable comments from Ann Carlos, Edward Glaeser, ... | 27,367 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THE LONG TERM CONSEQUENCES OF FAMINE ON SURVIVORS:
EVIDENCE FROM A UNIQUE NATURAL EXPERIMENT USING CHINA'S GREAT FAMINE
Xin Meng
Nancy Qian
Working Paper 14917
http://www.nber.org/papers/w14917
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
This p... | 27,722 |
NBER | 2,009 | NBER WORKING PAPER SERIES
THRESHOLDS IN THE PROCESS OF INTERNATIONAL FINANCIAL INTEGRATION
M. Ayhan Kose
Eswar S. Prasad
Ashley D. Taylor
Working Paper 14916
http://www.nber.org/papers/w14916
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
The views expressed in this paper ... | 27,763 |
NBER | 2,009 | NBER WORKING PAPER SERIES
DISCLOSURE AND THE COST OF CAPITAL:
EVIDENCE FROM FIRMS’ RESPONSES TO THE ENRON SHOCK
Christian Leuz
Catherine Schrand
Working Paper 14897
http://www.nber.org/papers/w14897
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
¸˛We appreciate helpful com... | 38,186 |
NBER | 2,009 | NBER WORKING PAPER SERIES
INTERVIEWING IN TWO-SIDED MATCHING MARKETS
Robin S. Lee
Michael Schwarz
Working Paper 14922
http://www.nber.org/papers/w14922
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009, Revised March 2017
We thank Neil Thakral for exceptional research assista... | 24,385 |
NBER | 2,009 | NBER WORKING PAPER SERIES
ADVERSE SELECTION IN COMPETITIVE SEARCH EQUILIBRIUM
Veronica Guerrieri
Robert Shimer
Randall Wright
Working Paper 14915
http://www.nber.org/papers/w14915
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We are grateful for comments from Daron Acemog... | 27,355 |
NBER | 2,009 | NBER WORKING PAPER SERIES
EXPERTS AND THEIR RECORDS
Alexander Frankel
Michael Schwarz
Working Paper 14921
http://www.nber.org/papers/w14921
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2009
We would like to thank Heski Bar-Isaac, Martin Cripps, Drew Fudenberg, Andrzej Skrzypa... | 22,589 |
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