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Gentzkow: Stanford University and NBER. gentzkow@stanford.edu. Song: New York University. lena.song@nyu.edu. We thank Dan Acland, Matthew Levy, Peter Maxted, Matthew Rabin, Dmitry Taubinsky, and seminar participants at the Behavioral Economics Annual Meeting, the Berkeley-Chicago Behavioral Economics Workshop, Bocconi, Boston University, Chicago Harris, Columbia Business School, Cornell, Di Tella University, the Federal Trade Commission Microeconomics Conference, Harvard, HBS, London Business School, London School of Economics, the Marketplace Innovation Workshop, Microsoft Research, MIT, the National Association for Business Economics Tech Economics Conference, the New York City Media Seminar, the New York Fed, NYU, Paris School of Economics, Princeton, Stanford Institute for Theoretical Economics, Trinity College Dublin, University of British Columbia, University College London, USC, Wharton, and Yale for helpful comments. We thank Michael Butler, Zong Huang, Zane Kashner, Uyseok Lee, Ana Carolina Paixao de Queiroz, Houda Nait El Barj, Bora Ozaltun, Ahmad Rahman, Andres Rodriguez, Eric Tang, and Sherry Yan for exceptional research assistance. We thank Chris Karr and Audacious Software for dedicated work on the Phone Dashboard app. We are grateful to the Sloan Foundation for generous support. Research was also supported by the Army Research Office under Grant Number W911NF-20-1-0252. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. The study was approved by Institutional Review Boards at Stanford (eProtocol #50759) and NYU (IRB-FY2020-3618). This experiment was registered in the American Economic Association Registry for randomized control trials; the pre-analysis plan is available from https://www.socialscienceregistry.org/trials/5796. Replication files and survey instruments are available from https://sites.google. com/site/allcott/research. Disclosures: Gentzkow does paid consulting work for Amazon, has done litigation consulting for clients including Facebook, and has been a member of the Toulouse Network for Information Technology, a research group funded by Microsoft. Both Allcott and Gentzkow are unpaid members of Facebook's 2020 Election Research Project. \end_layout \end_inset \begin_inset Newline newline \end_inset \series bold \size normal \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Abstract Many have argued that digital technologies such as smartphones and social media are addictive. We develop an economic model of digital addiction and estimate it using a randomized experiment. Temporary incentives to reduce social media use have persistent effects, suggesting social media are habit forming. Allowing people to set limits on their future screen time substantially reduces use, suggesting self-control problems. Additional evidence suggests people are inattentive to habit formation and partially unaware of self-control problems. Looking at these facts through the lens of our model suggests that self-control problems cause \begin_inset Formula $\pctreductiontemptationresnice$ \end_inset percent of social media use. \end_layout \begin_layout Abstract \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Abstract \series bold JEL Codes: \series default D12, D61, D90, D91, I31, L86, O33. \end_layout \begin_layout Abstract \series bold Keywords: \series default Habit formation, projection bias, self-control, temptation, naivete, commitment devices, randomized experiments, social media. \end_layout \begin_layout Standard \begin_inset Note Comment status open \begin_layout Plain Layout Use latex comment fields for documenting sources of all claims and numbers in the text. These never get removed. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \begin_inset Newpage newpage \end_inset \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard Digital technologies occupy a large and growing share of leisure time for people around the world. The average person with internet access spends 2.5 hours each day on social media, and there are now 3.8 billion social media users ( \begin_inset CommandInset citation LatexCommand citealt key "Kemp2020" literal "false" \end_inset ). In a 57-country survey, people now say they spend more time consuming online media than they do watching television ( \begin_inset CommandInset citation LatexCommand citealt key "zenith2019" literal "false" \end_inset ). \begin_inset Note Comment status collapsed \begin_layout Plain Layout 170 minutes total: 130 minutes per day on smartphones, 40 on desktop, vs. 167 minutes on TV. This is media consumption only, not work on the internet \end_layout \end_inset Americans check their smartphones 50 to 80 times each day ( \begin_inset CommandInset citation LatexCommand citealt key "deloitte_2018" literal "false" \end_inset ; \begin_inset CommandInset citation LatexCommand citealt key "vox_2020" literal "false" \end_inset ; \begin_inset CommandInset citation LatexCommand citealt key "nyp_2017" literal "false" \end_inset ). \end_layout \begin_layout Standard A natural interpretation of these facts is that digital technologies provide tremendous consumer surplus. However, an increasingly popular alternative view is that habit formation and self-control problems—what we call \begin_inset Quotes eld \end_inset digital addiction \begin_inset Quotes erd \end_inset —play a substantial role. Many argue that smartphones, video games, and social media apps may be harmful and addictive in the same ways as cigarettes, drugs, or gambling ( \begin_inset CommandInset citation LatexCommand citealt key "Alter2018" literal "false" \end_inset ; \begin_inset CommandInset citation LatexCommand citealt key "Newport2019" literal "false" \end_inset ; \begin_inset CommandInset citation LatexCommand citealt key "eyal_2020" literal "false" \end_inset ). The World Health Organization ( \begin_inset CommandInset citation LatexCommand citeyear key "who" literal "false" \end_inset ) has listed digital gaming disorder as an official medical condition. Recent experimental studies find that social media use can decrease subjective well-being (e.g. \begin_inset CommandInset citation LatexCommand citealt key "Mosqueraetal2018" literal "false" \end_inset ; \begin_inset CommandInset citation LatexCommand citealt* key "AllcottBraghieri" literal "false" \end_inset ). Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:OnlineOfflineTemptation" plural "false" caps "false" noprefix "false" \end_inset shows that social media and smartphone use are two of the top five activities that a sample of Americans think they do \begin_inset Quotes eld \end_inset too little \begin_inset Quotes erd \end_inset or \begin_inset Quotes eld \end_inset too much. \begin_inset Quotes erd \end_inset Compared to the other three top activities ordered at left (exercise, retiremen t savings, and healthy eating), digital self-control problems have received much less attention from economists. \begin_inset Foot status open \begin_layout Plain Layout Among many important examples, see \begin_inset CommandInset citation LatexCommand citet key "CharnessGneezy2009" literal "false" \end_inset and \begin_inset CommandInset citation LatexCommand citet key "Carrera2019" literal "false" \end_inset on exercise, \begin_inset CommandInset citation LatexCommand citet key "MadrianShea2001" literal "false" \end_inset and \begin_inset CommandInset citation LatexCommand citet key "Carroll2009" literal "false" \end_inset on retirement savings, and \begin_inset CommandInset citation LatexCommand citet key "sadoff2020" literal "false" \end_inset on healthy eating. \end_layout \end_inset \end_layout \begin_layout Standard The nature and magnitude of digital addiction matter for a number of important questions. Should people take steps to limit the amount of time they and their children spend on their smartphones and social media? What is the best way to design digital self-control tools? How can companies that make video games, social media, and smartphones best align their products with consumer welfare? Are proposed regulations such as the Social Media Addiction Reduction Technolog y (SMART) Act a good idea? \begin_inset Foot status open \begin_layout Plain Layout This bill, introduced in 2019 by Republican Senator Josh Hawley, proposed to prohibit the use of design features such as infinite scroll and autoplay believed to make social media more addictive, and to require companies to default users into a limit of 30 minutes per day of social media use. See \begin_inset CommandInset citation LatexCommand citet key "Hawley2019" literal "false" \end_inset . \end_layout \end_inset \end_layout \begin_layout Standard In this paper, we formalize an economic model of digital addiction, use a randomized experiment to provide model-free evidence and estimate model parameters, and use the model to simulate the effects of habit formation and self-control problems on smartphone use. We focus on six apps that account for much of smartphone screen time and that participants report to be especially tempting: Facebook, Instagram, Twitter, Snapchat, web browsers, and YouTube. We refer to these apps as \begin_inset Quotes eld \end_inset FITSBY. \begin_inset Quotes erd \end_inset \end_layout \begin_layout Standard Our model follows \begin_inset CommandInset citation LatexCommand citet key "GruberKoszegi2001" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "GulPesendorfers2007" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "BernheimRangel2004" literal "false" \end_inset , and others in defining addiction as the combination of two key forces: habit formation and self-control problems. As in \begin_inset CommandInset citation LatexCommand citet key "BeckerMurphy1988" literal "false" \end_inset , habit formation means that today's consumption increases tomorrow's demand. As in \begin_inset CommandInset citation LatexCommand citet key "Laibson1997" literal "false" \end_inset and others, self-control problems mean that people consume more today than they would have chosen for themselves in advance. These two forces are central to classic addictive goods such as cigarettes, drugs, and alcohol. \end_layout \begin_layout Standard Our model allows for projection bias ( \begin_inset CommandInset citation LatexCommand citealt* key "LowensteinODonoghueRabin2003" literal "false" \end_inset ), where people choose as if they are inattentive to habit formation, as well as naivete about self-control problems. As in \begin_inset CommandInset citation LatexCommand citet key "BeckerMurphy1988" literal "false" \end_inset , people who perceive at least some habit formation would reduce consumption if they know the price will increase in the future, while projection bias would dampen that effect. As in many other models (see \begin_inset CommandInset citation LatexCommand citealt key "EricsonLaibson2018" literal "false" \end_inset ), people who are at least partially aware of self-control problems might want commitment devices to restrict future consumption, and people who are at least partially unaware will underestimate future consumption. \end_layout \begin_layout Standard For our experiment, we used Facebook and Instagram ads to recruit about 2,000 American adults with Android smartphones and asked them to install Phone Dashboard, an app designed for our experiment that records smartphone screen time and allows participants to set screen time limits. Participants completed four surveys at three-week intervals—a baseline (survey 1) and three follow-ups (surveys 2, 3, and 4)—that included survey measures of smartphone addiction and subjective well-being as well as predictio ns of future FITSBY use. Participants answered three text message survey questions per week and kept Phone Dashboard installed for six weeks after survey 4. \end_layout \begin_layout Standard We independently randomized two treatments. The \emph on bonus treatment \emph default was a temporary subsidy of $2.50 per hour for reducing FITSBY use during the three weeks between surveys 3 and 4. We informed people whether or not they were assigned to the bonus treatment in advance, on survey 2. The \emph on limit treatment \emph default made available screen time limit functionality in Phone Dashboard. Participants in this group could set personalized daily time limits for each app on their phone, with changes effective the next day. These limits forced participants to stop using the relevant app and in most cases could not be immediately overridden, unlike the flexible limits in existing tools such as Android's Digital Wellbeing and iOS's Screen Time. The surveys encouraged participants to set limits in line with their self-repor ted ideal screen time, but doing so was entirely optional. We used multiple price lists (MPLs) to elicit participants' valuations of the bonus treatment and the limit functionality. \end_layout \begin_layout Standard The bonus treatment had persistent effects that are consistent with habit formation. The bonus reduced FITSBY use by \begin_inset Formula $\bonusthree$ \end_inset minutes per day during the three weeks when the incentives were in effect, a \begin_inset Formula $\bonusthreepct$ \end_inset percent reduction from the control group average. In the three weeks after the incentive had ended, the bonus treatment group still used \begin_inset Formula $\bonusfour$ \end_inset minutes less per day. In the three weeks after that, they used \begin_inset Formula $\bonusfive$ \end_inset minutes less per day. \end_layout \begin_layout Standard Participants correctly predict habit formation: the effects of the bonus on predicted post-incentive FITSBY use line up closely with the effects on actual use. However, in the three weeks between when the bonus was announced and when it took effect, there was only a modest (and possibly zero) anticipatory response, which is only \begin_inset Formula $\pcttaubtwonice$ \end_inset percent of what our model would predict for forward-looking habit formation without projection bias. These results are consistent with a form of projection bias in which consumers are aware of habit formation while consuming as if they are inattentive to it. \begin_inset Foot status open \begin_layout Plain Layout This distinction between awareness and attention raises interesting questions about other evidence of projection bias. For example, \begin_inset CommandInset citation LatexCommand citet key "Busseetal2015" literal "false" \end_inset find that people are more likely to buy a convertible on sunny days. On sunny days, do people have different beliefs about future weather or how much they would drive a convertible? \end_layout \end_inset \end_layout \begin_layout Standard We also find clear evidence that people have self-control problems and are at least partly aware of them. The limit treatment reduced FITSBY screen time by \begin_inset Formula $\limiteffectnice$ \end_inset minutes per day ( \begin_inset Formula $\limiteffectpct$ \end_inset percent) over 12 weeks. The effects decline slightly over the course of the experiment; this decline is consistent with some loss of motivation, but the fact that the decline is slight means that the effects are unlikely to be driven by confusion or temporary novelty. Although the experiment offered no incentive to set limits, \begin_inset Formula $\percentpositivetightnesspfive$ \end_inset percent of participants set binding limits and continued using them through the final weeks of the experiment. This far exceeds takeup of almost all commitment devices studied in the literature reviewed by \begin_inset CommandInset citation LatexCommand citet after "Table 1" key "Schilbach2019" literal "false" \end_inset . \begin_inset Note Comment status open \begin_layout Plain Layout Column 1 of Table 1. Chow (2011) and Casaburi and Macchiavello (2019) have higher takeup rates. \end_layout \end_inset On average, participants were willing to give up $ \begin_inset Formula $\valuelimit$ \end_inset for three weeks of access to the limit functionality, and when trading off the bonus versus a fixed payment, \begin_inset Formula $\MPLwishreduce$ \end_inset percent said they valued the bonus more highly because they wanted to give themselves an incentive to reduce consumption. These distinct measures of commitment demand are correlated with each other and with survey measures of addiction and desire to reduce screen time. \end_layout \begin_layout Standard Notwithstanding their demand for commitment, participants seem to slightly underestimate their self-control problems. The control group modestly but repeatedly underestimated their future FITSBY use in all of our surveys, even though use is fairly steady over time and we reminded them of recent past use before asking them to predict. On average, the control group underestimated next-period FITSBY use by \begin_inset Formula $\mispredictnicenice$ \end_inset minutes per day, or about \begin_inset Formula $\mispredictpctnice$ \end_inset percent. \end_layout \begin_layout Standard To further evaluate whether our interventions reduced addiction in a way that participants perceive to be beneficial, we examine effects on a variety of survey outcomes. On both the main surveys and text messages, the bonus and limit treatments significantly reduced an index of smartphone addiction adapted from the psychology literature. For example, both treatment groups reported being less likely to use their phone longer than intended, use their phone to distract from anxiety or fall asleep, have difficulty putting down their phone, lose sleep from phone use, procrastinate by using their phone, and use their phone mindlessly. \begin_inset Note Comment status open \begin_layout Plain Layout This is all outcomes that are significant on the appendix figures for Addition and SMS Addiction. \end_layout \end_inset Both treatment groups reported improved alignment between ideal and actual screen time. The bonus treatment group also scored higher on an index of subjective well-being, with statistically significant increases in components related to concentration and avoiding distraction and statistically insignificant changes in measures of happiness, life satisfaction, anxiety, and depression. Finally, both treatments are well-targeted in the sense that effects were more positive for people who report more interest in reducing their use and who score higher on our addiction measures at baseline. \end_layout \begin_layout Standard In the final section of the paper, we look at these results through the lens of our structural model. The model allows us to translate our short-run experimental estimates into effects on long-run steady state behavior, to quantify the magnitude of the effects we observe in terms of economically meaningful parameters, and to decompose the role of different behavioral forces through counterfactual s. We first estimate the model parameters by matching key moments from the experiment. We model the limit treatment as eliminating share \begin_inset Formula $\omega$ \end_inset of self-control problems, and for our primary estimates we conservatively assume \begin_inset Formula $\omega=1$ \end_inset . The estimates reflect our experimental results: substantial habit formation and self-control problems, substantial projection bias, and slight naivete about self-control problems. We then evaluate how steady-state consumption would change in counterfactuals where we eliminate self-control problems. Without habit formation, a conservative estimate of the effect of self-control problems is the effect of giving people screen time limit functionality: \begin_inset Formula $\limiteffectnice$ \end_inset minutes per day. But habit formation amplifies the effect of self-control problems, as the increase in current consumption also increases future marginal utility. In the presence of habit formation, our primary model prediction is that eliminating self-control problems would reduce FITSBY use by \begin_inset Formula $\deltaxtemptationresnice$ \end_inset minutes per day, or \begin_inset Formula $\pctreductiontemptationresnice$ \end_inset percent of baseline use. Alternative assumptions mostly imply more self-control problems, more attention to habit formation, and larger effects on use. \end_layout \begin_layout Standard Our results should be interpreted with caution for several reasons. First, our experiment took place during the beginning of the coronavirus pandemic. Our survey evidence suggests that this increased screen time but did not have clear effects on the magnitude of self-control problems. Furthermore, even as the pandemic evolved over the three-month experiment, average screen time and the treatment effects of the limit were fairly stable. Second, our estimates apply to the 2,000 people who selected into our experimen t, and these people are not representative of U.S. adults. When we reweight our estimates to more closely approximate national average demographic characteristics, the modeled effect of self-control problems increases. Third, our model's predictions of FITSBY use without self-control problems depend on assumptions such as linear demand and geometric decay of habit stock. Fourth, our analysis is partial equilibrium in the sense that we do not model network effects and other externalities across users. If one person's social media use increases others' use, such positive network externalities would magnify the effects of self-control problems on population- wide social media use. Finally, our surveys walked participants through a process of setting optional screen time limits that implemented their self-reported ideal screen time, and we hypothesize that simply offering time limit functionality without walking through that process would have had smaller effects. \begin_inset Foot status open \begin_layout Plain Layout While \begin_inset CommandInset citation LatexCommand citet key "Carrera2019" literal "false" \end_inset show that takeup of commitment devices can be driven by experimenter demand effects or decisionmaking noise instead of perceived self-control problems, there are three reasons why their concerns are less likely to apply to our experiment. First, while \begin_inset CommandInset citation LatexCommand citet key "Carrera2019" literal "false" \end_inset studied one-time takeup of an unfamiliar commitment contract, our participants repeatedly set and continually kept screen time limits over a 12-week period. Second, we estimate even larger perceived self-control problems using participa nts' valuations of the bonus treatment, which leverages an alternative methodolo gy favored by \begin_inset CommandInset citation LatexCommand citet key "Carrera2019" literal "false" \end_inset as well as \begin_inset CommandInset citation LatexCommand citet key "AclandLevy2012" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "AugenblickRabin2019" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "Chaloupka2019" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "allcottkim2020" literal "false" \end_inset , and \begin_inset CommandInset citation LatexCommand citet key "StrackTaubinsky" literal "false" \end_inset . Third, unlike \begin_inset CommandInset citation LatexCommand citet key "Carrera2019" literal "false" \end_inset , we find strong correlations between use of screen time limits and other measures of perceived self-control problems. \end_layout \end_inset \end_layout \begin_layout Standard Our work builds on several existing literatures. We extend a distinguished literature documenting present focus in diverse settings including exercise, healthy eating, consumption-savings decisions, and laboratory tasks ( \begin_inset CommandInset citation LatexCommand citealt key "EricsonLaibson2018" literal "false" \end_inset ). \begin_inset Foot status open \begin_layout Plain Layout This includes \begin_inset CommandInset citation LatexCommand citet key "ReadVanLeeuwen1998" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "FangSilverman2004" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Shapiro2005" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "ShuiAnsubel2005" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "AshrafKarlanYin2006" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "DellaVignaMalmendier2006" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Paserman2008" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "GineKarlanZinman2010" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "DufloKremerRobinson2011" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "AclandLevy2012" literal "false" \end_inset , Andreoni and Sprenger \begin_inset CommandInset citation LatexCommand citeyearpar key "AndreoniSprenger2012a,AndreoniSprenger2012b" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "AugenblickNiederleSprenger2015" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Beshears2015" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Goda2015" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Kaur2015" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Laibson2015" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "RoyerStehrSydnor2015" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "exley2017" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Auglenblick2018" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "KuchlerPagel2018" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Toussaert2018" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "AugenblickRabin2019" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "casaburi2019" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Schilbach2019" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "John" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Toussaert2018" literal "false" \end_inset , and \begin_inset CommandInset citation LatexCommand citet* key "sadoff2020" literal "false" \end_inset . \end_layout \end_inset Ours is one of a small handful of papers that estimate the parameters of a present focus model with partial naivete using field (instead of laboratory) behavior. \begin_inset Foot status open \begin_layout Plain Layout To our knowledge, these are \begin_inset CommandInset citation LatexCommand citet* key "allcottkim2020" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Bai2018" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Carrera2019" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "Chaloupka2019" literal "false" \end_inset , and \begin_inset CommandInset citation LatexCommand citet key "SkibaTobacman2018" literal "false" \end_inset . \end_layout \end_inset The digital self-control problems we study are particularly interesting because this is one of the few domains where market forces have created commitment devices, such as blockers for smartphone apps, email, and websites ( \begin_inset CommandInset citation LatexCommand citealt key "Laibson2018" literal "false" \end_inset ). Our results suggest additional unmet demand for these commitment devices. \end_layout \begin_layout Standard We also extend a distinguished literature on habit formation. One set of papers documents persistent impacts of temporary interventions in settings such as academic performance, energy use, exercise, hand washing, political protest, smoking, recycling, voting, water use, and weight loss. \begin_inset Foot status open \begin_layout Plain Layout This includes \begin_inset CommandInset citation LatexCommand citet* key "gerber2003voting" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "CharnessGneezy2009" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "GineKarlanZinman2010" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "ferraro2011" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "john2011financial" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "allcottrodgers2014" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "bernedo2014persistent" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "AclandLevy2015" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "RoyerStehrSydnor2015" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "FujiwaraMengVogl2016" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "levittlistsadoff2016" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "beshears2017" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "brandonetal2017" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "carerra2018" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "AllcottBraghieri" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "bursztyn2020misinformation" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "gosnell2020impact" literal "false" \end_inset , and \begin_inset CommandInset citation LatexCommand citet key "vansoest2019" literal "false" \end_inset . \end_layout \end_inset We provide evidence in an important new domain. A second set of papers tests for forward-looking habit formation using belief elicitation or advance responses to future price changes, sometimes interpreting such forward-looking behavior as support for \begin_inset Quotes eld \end_inset rational \begin_inset Quotes erd \end_inset models of addiction. \begin_inset Foot status open \begin_layout Plain Layout This includes \begin_inset CommandInset citation LatexCommand citet key "chaloupka1991" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "beckergrossman1994" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "GruberKoszegi2001" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "AclandLevy2015" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "hussam2019" literal "false" \end_inset , and \begin_inset CommandInset citation LatexCommand citet key "do2019" literal "false" \end_inset . \end_layout \end_inset We estimate anticipatory responses using an experimental approach that, like the one in \begin_inset CommandInset citation LatexCommand citet key "hussam2019" literal "false" \end_inset , addresses many confounds that arise in observational data ( \begin_inset CommandInset citation LatexCommand citealt key "chaloupkawarner1999" literal "false" \end_inset ; \begin_inset CommandInset citation LatexCommand citealt key "GruberKoszegi2001" literal "false" \end_inset ; \begin_inset CommandInset citation LatexCommand citealt key "auld2004empirical" literal "false" \end_inset ; \begin_inset CommandInset citation LatexCommand citealt key "ReesJones2020" literal "false" \end_inset ). Furthermore, we use our model to actually estimate the magnitude of projection bias, which is important because earlier studies that reject a null hypothesis of fully myopic habit formation could still be consistent with substantial projection bias. \end_layout \begin_layout Standard Finally, we extend three literatures that speak directly to digital addiction. The first literature includes theoretical papers modeling temptation in digital networks ( \begin_inset CommandInset citation LatexCommand citealt key "Makarov2011" literal "false" \end_inset ; \begin_inset CommandInset citation LatexCommand citealt* key "LiuSockinXiong2020" literal "false" \end_inset ). The second includes experimental papers studying the effects of social media use on outcomes such as subjective well-being and academic performance. \begin_inset Foot status open \begin_layout Plain Layout This includes \begin_inset CommandInset citation LatexCommand citet key "SagiogluGreitemeyer2014" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Trumholt2016" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Huntetal2018" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "Vanmanetal2018" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "Mosqueraetal2018" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "AllcottBraghieri" literal "false" \end_inset , and \begin_inset CommandInset citation LatexCommand citet key "collis2020" literal "false" \end_inset . \end_layout \end_inset The third studies the effects of digital self-control tools. \begin_inset Foot status open \begin_layout Plain Layout This includes \begin_inset CommandInset citation LatexCommand citet key "MarottaAcquisti2017" literal "false" \end_inset and \begin_inset CommandInset citation LatexCommand citet key "acland2018self" literal "false" \end_inset . \end_layout \end_inset \begin_inset CommandInset citation LatexCommand citet key "Hoong_2021" literal "false" \end_inset is particularly related, and is an important antecedent to our study. In a smaller-scale experiment, she pioneers the use of encouragement to adopt self-control tools, compares predicted and ideal use to actual use, and shows results consistent with significant self-control problems. Our paper helps to unify the previous empirical literature with a formal model of digital addiction, relatively large sample, multiple treatment arms that convincingly identify habit formation and self-control problems using several different strategies, and robust measurement of screen time and survey outcomes. \end_layout \begin_layout Standard Section \begin_inset CommandInset ref LatexCommand ref reference "sec:Model" plural "false" caps "false" noprefix "false" \end_inset sets up the model. Sections \begin_inset CommandInset ref LatexCommand ref reference "sec:Design" \end_inset – \begin_inset CommandInset ref LatexCommand ref reference "sec:ModelFree" plural "false" caps "false" noprefix "false" \end_inset detail the experimental design, data, and model-free results. Section \begin_inset CommandInset ref LatexCommand ref reference "sec:StructuralEstimation" plural "false" caps "false" noprefix "false" \end_inset presents the model estimation strategy and parameter estimates, and Section \begin_inset CommandInset ref LatexCommand ref reference "sec:Counterfactuals" plural "false" caps "false" noprefix "false" \end_inset presents the modeled effects of temptation on time use. \end_layout \begin_layout Section Model \begin_inset CommandInset label LatexCommand label name "sec:Model" \end_inset \end_layout \begin_layout Standard The goal of the model is to formalize the meaning of \begin_inset Quotes eld \end_inset digital addiction \begin_inset Quotes erd \end_inset and foreshadow how we identify the model parameters using our experiment. \end_layout \begin_layout Standard In each period \begin_inset Formula $t\leq T$ \end_inset , consumers choose consumption of a good \begin_inset Formula $x_{t}$ \end_inset sold at price \begin_inset Formula $p_{t}$ \end_inset that delivers flow utility \begin_inset Formula $u_{t}\left(x_{t};s_{t},p_{t}\right)$ \end_inset . To model habit formation, utility depends on a stock \begin_inset Formula $s_{t}$ \end_inset of past consumption that evolves according to \begin_inset Formula \begin{equation} s_{t+1}=\rho\left(s_{t}+x_{t}\right),\label{eq:evolution} \end{equation} \end_inset where \begin_inset Formula $\rho\in[0,1)$ \end_inset captures the strength of habit formation. Habit formation captures why temporary price changes generate persistent effects in our experiment. \end_layout \begin_layout Standard To model self-control problems, we follow \begin_inset CommandInset citation LatexCommand citet key "banerjee2010" literal "false" \end_inset in modeling \begin_inset Formula $x$ \end_inset as a temptation good. Before period \begin_inset Formula $t$ \end_inset , consumers consider period \begin_inset Formula $t$ \end_inset flow utility to be \begin_inset Formula $u_{t}\left(x_{t};s_{t},p_{t}\right)$ \end_inset . In period \begin_inset Formula $t$ \end_inset , however, consumers choose as if period \begin_inset Formula $t$ \end_inset flow utility is \begin_inset Formula $u_{t}\left(x_{t};s_{t},p_{t}\right)+\gamma x_{t}$ \end_inset , where \begin_inset Formula $\gamma\geq0$ \end_inset reflects the amount of temptation. If \begin_inset Formula $\gamma>0$ \end_inset , consumers choose more \begin_inset Formula $x_{t}$ \end_inset in period \begin_inset Formula $t$ \end_inset than they would choose in advance. This temptation good framework generates similar predictions to the quasi-hyper bolic model from \begin_inset CommandInset citation LatexCommand citet key "Laibson1997" literal "false" \end_inset and \begin_inset CommandInset citation LatexCommand citet key "GruberKoszegi2001" literal "false" \end_inset , but it naturally matches our application to a single addictive good and yields simpler estimating equations where temptation is additively separable. \end_layout \begin_layout Standard Consumers may misperceive temptation: before period \begin_inset Formula $t$ \end_inset , consumers predict that in period \begin_inset Formula $t$ \end_inset , they will consider flow utility to be \begin_inset Formula $u_{t}\left(x_{t};s_{t},p_{t}\right)+\tilde{\gamma}x_{t}$ \end_inset . We say that consumers are fully naive if \begin_inset Formula $\tilde{\gamma}=0$ \end_inset , and fully sophisticated if \begin_inset Formula $\tilde{\gamma}=\gamma$ \end_inset . Partial naivete captures why our experiment participants underestimate \begin_inset Formula $x_{t}$ \end_inset when asked to predict in advance. Partial sophistication captures why our participants want commitment devices to change their future behavior. \end_layout \begin_layout Standard Following \begin_inset CommandInset citation LatexCommand citet* key "LowensteinODonoghueRabin2003" literal "false" \end_inset , we allow the possibility of projection bias, in which consumers choose as if to maximize a weighted average of utility given the current habit stock \begin_inset Formula $s_{t}$ \end_inset and utility given the predicted habit stock \begin_inset Formula $\tilde{s}_{r}$ \end_inset in future period \begin_inset Formula $r>t$ \end_inset . We let \begin_inset Formula $\alpha$ \end_inset denote the weight on the current habit stock, and thus the magnitude of projection bias. Projection bias captures why consumers in our experiment might not reduce consumption in anticipation of a known future price change. We assume that consumers are fully naive about projection bias; sophistication would introduce strategic incentives to adjust current consumption to offset future bias. \begin_inset Foot status open \begin_layout Plain Layout \begin_inset CommandInset citation LatexCommand citet* after "page 1219" key "LowensteinODonoghueRabin2003" literal "false" \end_inset also assume naivete about projection bias, writing that “because this time inconsistency derives solely from misprediction of future utilities, it would make little sense to assume that the person is fully aware of it. \begin_inset Quotes erd \end_inset We note that our formulation of projection bias is slightly different than in \begin_inset CommandInset citation LatexCommand citet* key "LowensteinODonoghueRabin2003" literal "false" \end_inset : while their consumers' predictions of future consumption are biased due to projection bias, our consumers predict consumption accounting for habit formation, but choose as if they are inattentive to it. This matches our empirical results. \end_layout \end_inset \end_layout \begin_layout Standard Following \begin_inset CommandInset citation LatexCommand citet key "odonoghue1999" literal "false" \end_inset and others, we solve for perception-perfect equilibrium strategies, where consumers maximize current utility given predictions of future behavior. Let \begin_inset Formula $x_{t}\left(s_{t},\gamma,\boldsymbol{p}_{t}\right)$ \end_inset denote a strategy of the period- \begin_inset Formula $t$ \end_inset self, which depends on habit stock, temptation, and the vector of future prices \begin_inset Formula $\boldsymbol{p}_{t}=\{p_{t},p_{t+1},...,p_{T}\}$ \end_inset . Let \begin_inset Formula $\tilde{x}_{r}\left(s_{r},\tilde{\gamma},\boldsymbol{p}_{r}\right)$ \end_inset be a consumer's \emph on prediction, \emph default as of period \begin_inset Formula $t0$ \end_inset ). \end_layout \begin_layout Subsection Bonus and Limit Valuations \begin_inset CommandInset label LatexCommand label name "sec:Valuations" \end_inset \end_layout \begin_layout Standard We used incentive-compatible multiple price list mechanisms to elicit valuations of the Screen Time Bonus and the limit functionality. Because both the bonus and the limit functionality reduce future social media use, these valuations help identify perceived temptation \begin_inset Formula $\tilde{\gamma}$ \end_inset . \end_layout \begin_layout Standard All multiple price lists included a table with a series of choices between \begin_inset Quotes eld \end_inset Option A \begin_inset Quotes erd \end_inset and \begin_inset Quotes eld \end_inset Option B \begin_inset Quotes erd \end_inset in separate rows. Option B was the same in each row, while Option A included an amount of money that decreased monotonically from top to bottom. Participants would typically choose Option A at the top and Option B at the bottom, and we infer their valuation of Option B from the row where they switch. To encourage valid answers, participants who did not switch between Option A and Option B exactly once were alerted to this fact and given a chance to change their answers. All MPLs were incentivized, as described below. To help participants become familiar with MPLs, survey 1 included an incentiviz ed practice MPL that asked participants to choose between receiving different survey completion payments at different times. \end_layout \begin_layout Standard Our approach to valuing the Screen Time Bonus builds on \begin_inset CommandInset citation LatexCommand citet* key "allcottkim2020" literal "false" \end_inset and \begin_inset CommandInset citation LatexCommand citet key "Carrera2019" literal "false" \end_inset . Survey 2 informed participants of their average daily FITSBY screen time over the past three weeks and asked them to predict their screen time over the next three weeks. The survey then introduced the Screen Time Bonus and asked participants to predict how much they would reduce their FITSBY screen time relative to their original prediction if they were selected for the bonus. \end_layout \begin_layout Standard After these two predictions, we asked participants to make a hypothetical choice between the Screen Time Bonus and a payment equal to their expected earnings from the bonus. The survey described potential considerations as follows: \end_layout \begin_layout Itemize \emph on You might prefer $[expected earnings] instead of the Screen Time Bonus if you don’t want any pressure to reduce your screen time. \end_layout \begin_layout Itemize \emph on You might prefer the Screen Time Bonus instead of $[expected earnings] if you want to give yourself extra incentive to use your phone less. \end_layout \begin_layout Standard Participants then completed an MPL where Option B was receiving the Screen Time Bonus, and Option A was receiving a payment ranging from $150 to $0. \end_layout \begin_layout Standard To make the MPL incentive compatible, participants were told, \begin_inset Quotes eld \end_inset Last week, the computer randomly selected some participants to receive what they choose on the multiple price list below, and also randomly selected one of the rows to be `the question that counts.' If you were randomly selected to participate, you will be paid based on what you choose in that row. \begin_inset Quotes erd \end_inset 0.2 percent of participants were randomly assigned to the MPL group that received what they chose on a randomly selected row. \end_layout \begin_layout Standard On survey 3, the Limit group completed an MPL that elicited valuations of the Phone Dashboard limit functions. Option B was retaining access to the Phone Dashboard limit functions, and Option A was having those functions disabled for the following three weeks in exchange for a dollar payment that ranged from $20 to -$1. The MPL group received what they chose on a randomly selected row. \end_layout \begin_layout Subsection Predicted Use \begin_inset CommandInset label LatexCommand label name "sec:Predictions" \end_inset \end_layout \begin_layout Standard At the end of surveys 2, 3 and 4, we elicited predictions of future FITSBY use. These predictions help identify the degree of naivete or sophistication about temptation—the difference between \begin_inset Formula $\gamma$ \end_inset and \begin_inset Formula $\tilde{\gamma}$ \end_inset . \end_layout \begin_layout Standard Before each elicitation, we told each participant their average FITSBY screen time over the previous three weeks. Surveys 2 and 3 also reminded the Bonus and Limit groups about the bonus and limits. Survey 2 then elicited predictions of FITSBY screen time for the next three weeks (period 2), the three weeks after that (period 3), and the three weeks after that (period 4). Survey 3 elicited separate predictions for periods 3, 4, and 5. Survey 4 elicited separate predictions for periods 4 and 5. \end_layout \begin_layout Standard Predictions were incentivized. Survey 2 told participants, \begin_inset Quotes eld \end_inset Answer carefully, because you might earn a Prediction Reward. After the study ends, we will pick a prediction question at random and check how close your prediction is. If your predicted daily screen time is within 15 minutes of your actual screen time, we will pay you an additional $X. \begin_inset Quotes erd \end_inset We randomized the prediction reward X to be $1 or $5, each with 50 percent probability. \end_layout \begin_layout Subsection Survey Outcome Variables \begin_inset CommandInset label LatexCommand label name "sec:SurveyOutcomeVariables" \end_inset \end_layout \begin_layout Standard Surveys 1, 3, and 4 asked questions designed to measure participants' perception s of their addiction and subjective well-being (SWB). For the nine weeks between survey 1 and survey 4, we also sent three text messages per week with a subset of questions that we thought were important to ask in real time instead of retrospectively. Using these questions, we construct five pre-specified outcome variables. Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:VariableDefinitions" plural "false" caps "false" noprefix "false" \end_inset presents details on the survey questions. \end_layout \begin_layout Standard \series bold \emph on Ideal use change. \series default \emph default The survey said, \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Standard \emph on Some people say they use their smartphone too much and ideally would use it less. Other people are happy with their usage or would ideally use it more. How do you feel about your smartphone use over the past 3 weeks? \end_layout \begin_layout Itemize \emph on I use my smartphone too much. \end_layout \begin_layout Itemize \emph on I use my smartphone the right amount. \end_layout \begin_layout Itemize \emph on I use my smartphone too little. \end_layout \begin_layout Standard For people who said they used their smartphone \begin_inset Quotes eld \end_inset too much \begin_inset Quotes erd \end_inset or \begin_inset Quotes eld \end_inset too little, \begin_inset Quotes erd \end_inset we then asked, \emph on Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your smartphone use? \emph default The \emph on ideal use change \emph default variable is the answer to this question, in percent. \end_layout \begin_layout Standard \series bold \emph on Addiction scale \series default \emph default . Our addiction scale is a battery of 16 questions modified from two well-establi shed survey scales, the Mobile Phone Problem Use Scale ( \begin_inset CommandInset citation LatexCommand citealt key "bianchi2005psychological" literal "false" \end_inset ) and the Bergen Facebook Addiction Scale ( \begin_inset CommandInset citation LatexCommand citealt key "andreassen2012development" literal "false" \end_inset ). The questions attempt to measure the six core components of addition identified in the addiction literature: salience, tolerance, mood modification, relapse, withdrawal, and conflict ( \begin_inset CommandInset citation LatexCommand citealt key "griffiths2005" literal "false" \end_inset ). \end_layout \begin_layout Standard The survey asked, \emph on In the past three weeks, how often have you ... \emph default , with a matrix of 16 questions, such as \end_layout \begin_layout Itemize \emph on used your phone longer than intended? \end_layout \begin_layout Itemize \emph on felt anxious when you don’t have your phone? \end_layout \begin_layout Itemize \emph on lost sleep due to using your phone late at night? \end_layout \begin_layout Standard Possible answers were Never, Rarely, Sometimes, Often, and Always, which we coded as 0, 0.25, 0.5, 0.75, and 1, respectively. \emph on Addiction scale \emph default is the sum of these numerical scores for the 16 questions. \end_layout \begin_layout Standard \series bold \emph on SMS addiction scale. \series default \emph default The SMS addiction scale includes shortened versions of nine questions from the addiction scale. Examples include: \end_layout \begin_layout Itemize \emph on In the past day, did you feel like you had an easy time controlling your screen time? \end_layout \begin_layout Itemize \emph on In the past day, did you use your phone mindlessly? \end_layout \begin_layout Itemize \emph on When you woke up today, did you immediately check social media, text messages, or email? \end_layout \begin_layout Standard People were instructed to text back their answers on a scale from 1 (not at all) to 10 (definitely). \emph on SMS addiction scale \emph default is the sum of these scores for the nine questions. \end_layout \begin_layout Standard \series bold \emph on Phone makes life better \series default \emph default . The survey asked, \emph on To what extent do you think your smartphone use makes your life better or worse? \emph default Responses were on a scale from -5 ( \begin_inset Quotes eld \end_inset Makes my life worse \begin_inset Quotes erd \end_inset ) through 0 ( \begin_inset Quotes eld \end_inset Neutral \begin_inset Quotes erd \end_inset ) to +5 ( \begin_inset Quotes eld \end_inset Makes my life better \begin_inset Quotes erd \end_inset ). \end_layout \begin_layout Standard \series bold \emph on Subjective well-being. \series default \emph default We use standard measures from the subjective well-being literature, mostly following the measures from our own earlier work ( \begin_inset CommandInset citation LatexCommand citealt* key "AllcottBraghieri" literal "false" \end_inset ). The survey asked, \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Standard \emph on Please tell us the extent to which you agree or disagree with each of the following statements. \series bold Over the last three weeks, \series default \emph default with a matrix of seven questions: \end_layout \begin_layout Itemize \emph on … I was a happy person \end_layout \begin_layout Itemize \emph on … I was satisfied with my life \end_layout \begin_layout Itemize \emph on … I felt anxious \end_layout \begin_layout Itemize \emph on … I felt depressed \end_layout \begin_layout Itemize \emph on … I could concentrate on what I was doing \end_layout \begin_layout Itemize \emph on … I was easily distracted \end_layout \begin_layout Itemize \emph on … I slept well \end_layout \begin_layout Standard Possible answers were on a seven-point scale from \begin_inset Quotes eld \end_inset strongly disagree \begin_inset Quotes erd \end_inset through \begin_inset Quotes eld \end_inset neutral \begin_inset Quotes erd \end_inset to \begin_inset Quotes eld \end_inset strongly agree, \begin_inset Quotes erd \end_inset which were coded as -1, -2/3, -1/3, 0, 1/3, 2/3, and 1, respectively. The variable \emph on subjective well-being \emph default is the sum of these numerical scores for the seven questions, after reversing \emph on anxious \emph default , \emph on depressed \emph default , and \emph on easily distracted \emph default so that more positive reflects better subjective well-being. \end_layout \begin_layout Standard \series bold \emph on Indices. \series default \emph default We define the \emph on survey index \emph default to be the sum of the five survey outcome variables described above, weighted by the baseline inverse covariance matrix as described by \begin_inset CommandInset citation LatexCommand citet key "Anderson2008" literal "false" \end_inset . When presenting results and constructing this index, we orient the variables so that more positive values imply normatively better outcomes. Thus, we multiply \emph on addiction scale \emph default and \emph on SMS addiction scale \emph default by (-1). \end_layout \begin_layout Standard We define the \emph on restriction index \emph default to be the sum of \emph on interest in limits \emph default (with the four categorical answers coded as 0, 1, 2, and 3) and \emph on ideal use change \emph default , after normalizing each into standard deviation units. We define the \emph on addiction index \emph default to be the sum of \emph on addiction scale \emph default and \emph on phone makes life better \emph default after normalizing each into standard deviation units. We use these two indices for stratified randomization and as moderators when testing for heterogeneous treatment effects. \end_layout \begin_layout Subsection Pre-Analysis Plan \end_layout \begin_layout Standard We submitted our pre-analysis plan (PAP) on May 4th, the day that post-treatment data collection began. The PAP specified (i) the equation for treatment effect estimation (equation \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset below); (ii) the construction of the survey outcome variables and indices described in Section \begin_inset CommandInset ref LatexCommand ref reference "sec:SurveyOutcomeVariables" plural "false" caps "false" noprefix "false" \end_inset , the \emph on limit tightness \emph default variable, and the winsorization of predicted FITSBY use; and (iii) the analysis of heterogeneous treatment effects by splitting the sample on above- versus below-median values of six moderators: education, age, gender, baseline FITSBY use, \emph on restriction index \emph default , and \emph on addiction index \emph default . The PAP also included shells of Tables \begin_inset CommandInset ref LatexCommand ref reference "tab:SampleSizes" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "tab:SampleDemographics" plural "false" caps "false" noprefix "false" \end_inset , and \begin_inset CommandInset ref LatexCommand ref reference "tab:Attrition" plural "false" caps "false" noprefix "false" \end_inset – \begin_inset CommandInset ref LatexCommand ref reference "tab:DescriptiveStats" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset Note Comment status open \begin_layout Plain Layout full list: \begin_inset CommandInset ref LatexCommand ref reference "tab:Attrition" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "tab:Balance" plural "false" caps "false" noprefix "false" \end_inset , and \begin_inset CommandInset ref LatexCommand ref reference "tab:DescriptiveStats" plural "false" caps "false" noprefix "false" \end_inset \end_layout \end_inset as well as Figures \begin_inset CommandInset ref LatexCommand ref reference "fig:OnlineOfflineTemptation" plural "false" caps "false" noprefix "false" \end_inset – \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualControl" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset Note Comment status open \begin_layout Plain Layout full list: \begin_inset CommandInset ref LatexCommand ref reference "fig:OnlineOfflineTemptation" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "fig:Design" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "fig:Qualitative" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "fig:UsageEffects" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "fig:UsageEffectsByApp" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualControl" plural "false" caps "false" noprefix "false" \end_inset \end_layout \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:RecruitmentAds" plural "false" caps "false" noprefix "false" \end_inset – \begin_inset CommandInset ref LatexCommand ref reference "fig:FITSBYUsageDist" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset Note Comment status open \begin_layout Plain Layout full list: \begin_inset CommandInset ref LatexCommand ref reference "fig:RecruitmentAds" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "fig:PDScreenshots" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "fig:PopularApps" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "fig:FITSBYUsageDist" plural "false" caps "false" noprefix "false" \end_inset \end_layout \end_inset \begin_inset CommandInset ref LatexCommand ref reference "fig:IdealChangeByApp" plural "false" caps "false" noprefix "false" \end_inset , and \begin_inset CommandInset ref LatexCommand ref reference "fig:LATEs" plural "false" caps "false" noprefix "false" \end_inset – \begin_inset CommandInset ref LatexCommand ref reference "fig:HetAddictionIndex" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Standard We deviate from the PAP in five ways. First, the bottom left panel of Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Qualitative" plural "false" caps "false" noprefix "false" \end_inset includes results from each addiction scale question, whereas the PAP figure shell presented the sum across all questions. Second, we clarify that our analysis sample includes only the balanced panel of people who completed the study. Results are essentially identical if we use an unbalanced panel that includes data from attriters before they attritted, but the balanced panel is helpful in ensuring that our habit formation results are not spuriously driven by attrition. Third, three figures from the PAP are not included here, as we plan to study them in a separate paper. \begin_inset Note Comment status open \begin_layout Plain Layout PAP Figures 6, 8, A4. \end_layout \end_inset Fourth, Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualControl" plural "false" caps "false" noprefix "false" \end_inset includes predicted FITSBY use from all surveys before period \begin_inset Formula $t$ \end_inset , whereas the PAP figure shell presented predictions from only the survey immediately before period \begin_inset Formula $t$ \end_inset . Fifth, we use equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ) for subgroup analysis, whereas the PAP specified that we would use an instrumental variables regression. We present the pre-specified instrumental variables estimates in Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:LATEs" plural "false" caps "false" noprefix "false" \end_inset . The results are similar, and we decided that equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ) was simpler. \end_layout \begin_layout Section Data \end_layout \begin_layout Standard The analysis sample for all results reported below is the balanced panel of \begin_inset Formula $\kepttoendanalysis$ \end_inset participants who were assigned to either Bonus or Bonus Control (not the MPL group), completed all four surveys, and kept Phone Dashboard installed until the end of the study on July 26. This group's attrition rate after being informed of treatment was \begin_inset Formula $(1-\kepttoendanalysis/\informedtreatanalysis)\times100\%\approx\attritionratenicenice$ \end_inset percent. Attrition rates and observable characteristics are balanced across the bonus and limit treatment conditions; see Appendix Tables \begin_inset CommandInset ref LatexCommand ref reference "tab:Attrition" plural "false" caps "false" noprefix "false" \end_inset and \begin_inset CommandInset ref LatexCommand ref reference "tab:Balance" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Standard Table \begin_inset CommandInset ref LatexCommand ref reference "tab:SampleDemographics" plural "false" caps "false" noprefix "false" \end_inset quantifies the representativeness of our analysis sample on observables, by comparing their demographics to the U.S. adult population. Our sample is more educated, more heavily female, younger, and slightly lower-income than the U.S. population. We estimate an alternative specification of our structural model with sample weights to adjust for these observable differences. \end_layout \begin_layout Standard Table \begin_inset CommandInset ref LatexCommand ref reference "tab:SampleDemographics" plural "false" caps "false" noprefix "false" \end_inset also shows that the average participant had \begin_inset Formula $\avgOverall$ \end_inset minutes per day of screen time during the baseline period, of which \begin_inset Formula $\avgFITSBY$ \end_inset minutes ( \begin_inset Formula $\avgFITSBYpct$ \end_inset percent) was on FITSBY apps. Different sources report very different estimates of average social media use and smartphone screen time for U.S. adults, so we do not report nationwide averages in the table. \begin_inset CommandInset citation LatexCommand citet key "Kemp2020" literal "false" \end_inset reports that internet users in the U.S. and worldwide, respectively, spend an average of 123 and 144 minutes per day on social media, mostly on mobile devices. \begin_inset CommandInset citation LatexCommand citet key "Wurmser2020" literal "false" \end_inset and \begin_inset CommandInset citation LatexCommand citet key "Brown2020" literal "false" \end_inset report national averages of 186 and 324 minutes of total smartphone screen time per day, respectively. The comparisons suggest that the heavy use in our sample may not be far from the national average. \end_layout \begin_layout Standard During the baseline period, the average participant used Facebook, browsers, YouTube, Instagram, Snapchat, and Twitter for \begin_inset Formula $\avgFB$ \end_inset , \begin_inset Formula $\avgBR$ \end_inset , \begin_inset Formula $\avgYT$ \end_inset , \begin_inset Formula $\avgIN$ \end_inset , \begin_inset Formula $\avgSC$ \end_inset , and \begin_inset Formula $\avgTW$ \end_inset minutes per day, respectively; see Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:PopularApps" plural "false" caps "false" noprefix "false" \end_inset . Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:FITSBYUsageDist" plural "false" caps "false" noprefix "false" \end_inset presents the distribution of baseline FITSBY use. Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:DescriptiveStats" plural "false" caps "false" noprefix "false" \end_inset presents descriptive statistics for the survey outcome variables. \end_layout \begin_layout Section Model-Free Results \begin_inset CommandInset label LatexCommand label name "sec:ModelFree" \end_inset \end_layout \begin_layout Subsection Treatment Effect Estimating Equation \end_layout \begin_layout Standard To estimate treatment effects, define \begin_inset Formula $Y_{it}$ \end_inset as an outcome for participant \begin_inset Formula $i$ \end_inset for period \begin_inset Formula $t$ \end_inset . \begin_inset Formula $Y_{it}$ \end_inset could represent a survey outcome variable measured on survey \begin_inset Formula $t\in\{3,4\}$ \end_inset , or period \begin_inset Formula $t$ \end_inset FITSBY use. Define \begin_inset Formula $L_{i}$ \end_inset and \begin_inset Formula $B_{i}$ \end_inset as Limit and Bonus group indicators. Define \begin_inset Formula $\boldsymbol{X}_{i1}$ \end_inset as a vector of baseline covariates: baseline FITSBY use and, if and only if \begin_inset Formula $Y$ \end_inset is a survey outcome variable, the baseline value \begin_inset Formula $Y_{i1}$ \end_inset and the baseline value of \emph on survey index. \emph default Define \begin_inset Formula $\nu_{it}$ \end_inset as a vector of the eight randomization stratum indicators, allowing separate coefficients for each period \begin_inset Formula $t$ \end_inset . We estimate the effects of the limit and bonus treatments using the following regression: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} Y_{it}=\tau_{t}^{B}B_{i}+\tau_{t}^{L}L_{i}+\beta_{t}\boldsymbol{X}_{i1}+\nu_{it}+\varepsilon_{it}.\label{eq:ATE} \end{equation} \end_inset When combining data across multiple periods, we cluster standard errors by participant. \end_layout \begin_layout Subsection Baseline Qualitative Evidence \begin_inset CommandInset label LatexCommand label name "subsec:PreliminaryEvidence" \end_inset \end_layout \begin_layout Standard Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Qualitative" plural "false" caps "false" noprefix "false" \end_inset presents qualitative evidence on digital addiction from the baseline survey. The top two panels present the variables in the \emph on restriction index \emph default . The top left panel shows that \begin_inset Formula $\percentlimitinterested$ \end_inset percent of people reported being \begin_inset Quotes eld \end_inset moderately \begin_inset Quotes erd \end_inset or \begin_inset Quotes eld \end_inset very \begin_inset Quotes erd \end_inset interested in setting time use limits on their smartphone apps, while \begin_inset Formula $\percentlimitnot$ \end_inset percent reported being \begin_inset Quotes eld \end_inset not at all \begin_inset Quotes erd \end_inset interested. The top right panel presents the distribution of responses to the \emph on ideal use change \emph default question. \begin_inset Formula $\percentuseright$ \end_inset percent of people said that they used their smartphone the right amount over the past three weeks, and only \begin_inset Formula $\percentuselittle$ \end_inset percent said that they used it too little. Among people who said they used their smartphone too much, the average ideal reduction was \begin_inset Formula $\idealreduction$ \end_inset percent. \end_layout \begin_layout Standard Survey 1 also asked people to report their ideal use change for specific apps or categories. FITSBY, games, video streaming, and messaging are the nine apps on which people want to reduce screen time the most; see Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:IdealChangeByApp" plural "false" caps "false" noprefix "false" \end_inset . Facebook is by far the most tempting app: the average participant would ideally reduce Facebook use by \begin_inset Formula $\idealreductionfacebook$ \end_inset percent. The average participant did not want to change their use of email, news, and maps and wanted to slightly increase use of phone, music, and podcast apps. \end_layout \begin_layout Standard The bottom two panels present the variables in the \emph on addiction index \emph default . The bottom left panel presents the share of participants who responded \begin_inset Quotes eld \end_inset often \begin_inset Quotes erd \end_inset or \begin_inset Quotes eld \end_inset always \begin_inset Quotes erd \end_inset on each question in the \emph on addiction scale \emph default . The top seven questions capture three components of moderate addictions (salience, tolerance, and mood modification); \begin_inset Formula $\meantopsevenaddiction$ \end_inset percent of participants often or always experience each of these, and \begin_inset Formula $\meantopsevenanyaddiction$ \end_inset percent often or always experience at least one. The bottom nine questions capture three components of more severe addictions (relapse, withdrawal, or conflict); \begin_inset Formula $\meanbottomnineaddiction$ \end_inset percent of participants often or always experience each of these, and \begin_inset Formula $\meanbottomnineanyaddiction$ \end_inset percent often or always experience at least one. The bottom right panel shows that while most people think that their smartphone use makes their life better, \begin_inset Formula $\percentlifeworse$ \end_inset percent think that it makes their life worse. Taken together, these results suggest substantial heterogeneity: many people report experiences consistent with addiction, while many others do not. \end_layout \begin_layout Standard Our experiment took place during the coronavirus pandemic, which significantly disrupted people's daily routines. To understand how this might affect our results, we included several baseline survey questions, which we report in Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:2019" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset Formula $\covidmorefree$ \end_inset percent of people reported having more free time as a result of the pandemic, and \begin_inset Formula $\morephoneuse$ \end_inset percent of people reported that the pandemic had increased their phone use. However, it is not clear that the pandemic affected the extent of self-control problems: the means and distributions of key qualitative measures of addiction that we also asked for 2019, \emph on ideal use change \emph default and \emph on phone makes life better \emph default , were statistically different but economically similar. \emph on Ideal use change \emph default is closer to zero in 2020 compared to in 2019, suggesting less perceived self-control problems, but \emph on phone makes life better \emph default is also less positive, suggesting more perceived self-control problems. \end_layout \begin_layout Subsection Bonus Treatment and Habit Formation \begin_inset CommandInset label LatexCommand label name "subsec:BonusEffects" \end_inset \end_layout \begin_layout Standard The darker coefficients in Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:UsageEffects" plural "false" caps "false" noprefix "false" \end_inset present the effect of the bonus on FITSBY use, estimated using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). Recall that the bonus provides an incentive to reduce FITSBY use in period 3, but we informed participants about whether or not they were offered the bonus at the beginning of period 2. The incentive is $50 per \emph on average \emph default hour measured over the 20-day period, or $2.50 per hour of consumption. \end_layout \begin_layout Standard In period 3 (while the incentive was in effect), the Bonus group reduced FITSBY use by \begin_inset Formula $\bonusthree$ \end_inset minutes per day, or \begin_inset Formula $\bonusthreepct$ \end_inset percent relative to the Control group. This is a striking price response: it implies that participants value a substantial share of smartphone FITSBY use at less than $2.50 per hour. \end_layout \begin_layout Standard In periods 4 and 5 (after the incentive had ended), the Bonus group still reduced FITSBY use by \begin_inset Formula $\taubfournicenice$ \end_inset and \begin_inset Formula $\taubfivenicenice$ \end_inset minutes per day, respectively. This persistent effect suggests substantial habit formation, implying \begin_inset Formula $\zeta>0$ \end_inset in our model. The decay of the effect in period 5 relative to period 4 provides information about the habit stock decay parameter \begin_inset Formula $\rho$ \end_inset in our model. \end_layout \begin_layout Standard In period 2 (before the incentive was in effect), the Bonus group reduced FITSBY use by \begin_inset Formula $\taubtwofullnicenice$ \end_inset minutes per day, which is marginally statistically significant. This is consistent with the model's prediction that a consumer who perceives habit formation should reduce period 2 consumption in order to reduce period 3 marginal utility, which makes it easier to reduce period 3 consumption in response to the financial incentive. However, additional evidence suggests some caution about interpreting the period 2 effect as forward-looking habit formation. Appendix Figures \begin_inset CommandInset ref LatexCommand ref reference "fig:BonusEffectsByDay" plural "false" caps "false" noprefix "false" \end_inset and \begin_inset CommandInset ref LatexCommand ref reference "fig:BonusEffectsByWeek" plural "false" caps "false" noprefix "false" \end_inset break out the period 2 effect separately by day and week, showing that it loads mostly on the first half of the period. If anything, forward-looking habit formation would predict the opposite pattern, with larger anticipatory effects closer to the beginning of the incentive period. Possible explanations include intertemporal substitution, a temporary idiosyncr atic effect, and the salience of the bonus after its introduction on survey 2. \begin_inset Foot status open \begin_layout Plain Layout Although we stratified randomization on period 1 FITSBY use and also control for period 1 use when estimating equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ), some idiosyncratic factor could temporarily affect consumption in Bonus versus Bonus Control at the beginning of period 2. Some evidence supports this possibility: Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:BonusEffectsByDay" plural "false" caps "false" noprefix "false" \end_inset shows that consumption is slightly lower in the Bonus group compared to Bonus Control in the final 11 days of period 1. Salience could also play a role, although as described in Section \begin_inset CommandInset ref LatexCommand ref reference "sec:BonusIntro" plural "false" caps "false" noprefix "false" \end_inset , we took many steps to eliminate confusion about the timing of the bonus incentive period, and participants likely would have emailed our team if they were confused. \end_layout \end_inset \end_layout \begin_layout Subsection Limit Treatment and Temptation \begin_inset CommandInset label LatexCommand label name "subsec:LimitEffects" \end_inset \end_layout \begin_layout Standard The Limit group made extensive use of the limit functionality. To summarize the stringency of time limits, we define the variable \emph on limit tightness \emph default to be the amount by which a user's limits would have hypothetically reduced screen time if applied to their baseline use. \begin_inset Foot status collapsed \begin_layout Plain Layout Specifically, define \begin_inset Formula $x_{iadt}$ \end_inset as the screen time of person \begin_inset Formula $i$ \end_inset on app \begin_inset Formula $a$ \end_inset on day \begin_inset Formula $d$ \end_inset in period \begin_inset Formula $t$ \end_inset . Define \begin_inset Formula $h_{iat}$ \end_inset as the average screen time limit in place in period \begin_inset Formula $t$ \end_inset , and define \begin_inset Formula $N_{d\in t=1}$ \end_inset as the number of days in the baseline period. \emph on Limit tightness \emph default is \end_layout \begin_layout Plain Layout \begin_inset Formula \begin{equation} H_{iat}=\frac{1}{N_{d\in t=1}}\sum_{d\in t=1}\max\left\{ 0,x_{iad1}-h_{iat}\right\} .\label{eq:tightness} \end{equation} \end_inset If the daily limit \begin_inset Formula $h_{iat}$ \end_inset would not have been binding in baseline day \begin_inset Formula $d$ \end_inset , the \begin_inset Formula $\max$ \end_inset function returns 0. If \begin_inset Formula $h_{iat}$ \end_inset would have been binding in day \begin_inset Formula $d$ \end_inset , then the \begin_inset Formula $\max$ \end_inset function returns the excess screen time on that day. We aggregate over apps to construct user-level limit tightness \begin_inset Formula $H_{it}=\sum_{a}H_{iat}$ \end_inset . \end_layout \end_inset \emph on Limit tightness \emph default equals zero (instead of missing) for an app if the participant doesn't have the app or doesn't set a limit, so this variable speaks to what apps contribute the most to aggregate temptation. About \begin_inset Formula $\percentpositivetightness$ \end_inset percent of the Limit group had positive \emph on limit tightness \emph default at some point during the experiment, suggesting that they set binding screen time limits, and \begin_inset Formula $\percentpositivetightnesspfive$ \end_inset had positive \emph on limit tightness \emph default in period 5, meaning that they kept those limits for more than three weeks after the final survey. Participants most wanted to restrict Facebook, web browsers, YouTube, and Instagram: \emph on limit tightness \emph default averaged \begin_inset Formula $\fblimittight$ \end_inset , \begin_inset Formula $\browserlimittight$ \end_inset , \begin_inset Formula $\youtubelimittight$ \end_inset , and \begin_inset Formula $\instalimittight$ \end_inset minutes per day on those apps, respectively, across periods 2–5. Across all apps, the Limit group's average \emph on limit tightness \emph default was \begin_inset Formula $\averagelimittightness$ \end_inset minutes per day. See Appendix Figures \begin_inset CommandInset ref LatexCommand ref reference "fig:TightnessDist" plural "false" caps "false" noprefix "false" \end_inset and \begin_inset CommandInset ref LatexCommand ref reference "fig:TightnessByApp" plural "false" caps "false" noprefix "false" \end_inset for details. \end_layout \begin_layout Standard The lighter coefficients on Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:UsageEffects" plural "false" caps "false" noprefix "false" \end_inset present the effect of the limit on FITSBY use. These actual effects are smaller than the \emph on limit tightness \emph default values in the previous paragraph primarily because users snooze the limits. Access to the limit functionality reduced use in periods 2–5 by an average of \begin_inset Formula $\limiteffectnice$ \end_inset minutes per day, or \begin_inset Formula $\limiteffectpct$ \end_inset percent relative to the Control group. The effects attenuate only slightly as the experiment continues, and the effect is still \begin_inset Formula $\limiteffectlastweeknicenice$ \end_inset minutes per day in the last week of period 5. This is notable because while surveys 2 and 3 walked people through a limit-set ting process and survey 4 included an optional review of the limits, the end of period 5 is nine weeks after survey 3 and six weeks after survey 4. These continuing effects suggest that while motivation might decrease over time, use of the limits is not primarily driven by confusion or temporary novelty. Furthermore, Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:Validation" plural "false" caps "false" noprefix "false" \end_inset shows that \emph on limit tightness \emph default is correlated in expected ways with bonus and limit valuations and with survey measures of addiction and desire to reduce screen time. This evidence consistently points toward perceived self-control problems, implying \begin_inset Formula $\tilde{\gamma}>0$ \end_inset in our model. \end_layout \begin_layout Standard When we add the interaction between Bonus and Limit group indicators to equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ), the main effects are similar and the interaction terms are not statistically significant; see Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Interactions" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Subsection Substitution \begin_inset CommandInset label LatexCommand label name "subsec:Substitution" \end_inset \end_layout \begin_layout Standard Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:UsageEffectsByApp" plural "false" caps "false" noprefix "false" \end_inset presents usage effects of the bonus (in period 3 only) and the limit (across periods 2–5) separately by app. Among the FITSBY apps, Facebook sees the largest reductions, followed by web browsers, YouTube, Instagram, Twitter, and Snapchat. The effects on other apps (the right-most coefficients) provide evidence on the extent to which participants substituted FITSBY time to alternative apps. The bonus has no statistically detectable effect on use of other apps in period 3, and the confidence intervals rule out any substantial substitution relative to the \begin_inset Formula $\bonusthree$ \end_inset minutes per day reduction in FITSBY use. The limit induces substitution of \begin_inset Formula $\limitotherfitsbynice$ \end_inset minutes per day, so that roughly half of the FITSBY screen time that the limit eliminates moves to other apps where people had been less likely to set limits. \end_layout \begin_layout Standard One important limitation is that we cannot directly monitor FITSBY use on devices other than the participant's smartphone. We screened out potential participants who reported using more than one smartphone regularly, but our remaining participants may still have used desktops, tablets, or other devices. To provide some evidence on this substitution, survey 4 asked participants to estimate their FITSBY use on other devices in period 3 compared to the three weeks before they joined the study. The results, shown in Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:SubstitutionEffect" plural "false" caps "false" noprefix "false" \end_inset , imply that the limit increased FITSBY use on other devices by a marginally significant \begin_inset Formula $\limitsubstitution$ \end_inset minutes per day. The bonus \emph on reduced \emph default the amount of time they spent on FITSBY on other devices by \begin_inset Formula $\bonussubstitution$ \end_inset minutes per day, suggesting that time on other devices was a mild complement in this case. \end_layout \begin_layout Standard The differences in substitution induced by the bonus versus limit are notable. In a simple model where other apps and devices are either complements or substitutes for smartphone FITSBY use, the substitution effects described above might have the same sign for both the bonus and limit and might be in proportion to their direct effects on smartphone FITSBY use. In contrast, a much smaller share of the effect on FITSBY use is substituted to other smartphone apps for the bonus compared to the limit, and the self-repo rted effects on FITSBY use on other devices have opposite signs for the bonus versus the limit. This is an interesting result to understand in future work. \end_layout \begin_layout Subsection Predicted versus Actual Use \end_layout \begin_layout Standard Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualControl" plural "false" caps "false" noprefix "false" \end_inset presents predicted and actual FITSBY use in the Control condition, where participants had neither the bonus nor the limit functionality. As specified in our pre-analysis plan, we winsorize predicted use at no more than 60 minutes per day more or less than actual use in the corresponding period. Within each period, the left-most spike is actual average use. The spikes to the right are average predictions. The point estimates show that people consistently underestimate their use in all future periods, even though actual use is fairly stable throughout the experiment and the surveys had reminded them of their past use before eliciting predictions. This is consistent with naivete, implying \begin_inset Formula $\tilde{\gamma}<\gamma$ \end_inset in our model. \end_layout \begin_layout Standard Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualHabitFormation" plural "false" caps "false" noprefix "false" \end_inset presents predicted versus actual habit formation. Within each period, the left-most point is the treatment effect of the bonus on actual use, reproduced from Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:UsageEffects" plural "false" caps "false" noprefix "false" \end_inset . Recall that before the multiple price list for the Screen Time Bonus on survey 2, we asked people to report the percent by which they thought the bonus would reduce their FITSBY use. Their estimates (translated into minutes using their status quo predictions) are almost exactly correct on average: \begin_inset Formula $\MPLStwonicenice$ \end_inset minutes per day. Then on survey 3, we asked people to predict their use in future periods. Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualHabitFormation" plural "false" caps "false" noprefix "false" \end_inset also presents treatment effects of the bonus on predicted use, estimated from equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). The figure shows that people correctly predict that the bonus will reduce their consumption in period 3 and that this reduction will persist even after the incentive is no longer in effect. If anything, comparing the time path of actual versus predicted effects suggests that people overestimate the extent of habit formation. Overall, these results suggest that people are well aware of habit formation. \end_layout \begin_layout Standard Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:Validation" plural "false" caps "false" noprefix "false" \end_inset presents additional results that validate that the usage predictions are meaningful. Predicted use lines up well with actual use, and the higher ($5 instead of $1) prediction accuracy reward slightly reduces the absolute value of the prediction error but has tightly estimated zero effects on predicted use, actual use, and the level of the prediction error. \end_layout \begin_layout Subsection Bonus and Limit Valuations \begin_inset CommandInset label LatexCommand label name "subsec:ModelFree_LimitBonusValuations" \end_inset \end_layout \begin_layout Standard On the survey 3 multiple price list, the average Limit group participant was willing to give up a $ \begin_inset Formula $\valuelimit$ \end_inset fixed payment for three weeks of access to the limit functionality. About \begin_inset Formula $\positivelimit$ \end_inset percent of participants were willing to give up at least some money for the limits, and \begin_inset Formula $\tenlimit$ \end_inset percent were willing to give up more than $10; see Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:LimitValuationDist" plural "false" caps "false" noprefix "false" \end_inset . This willingness to pay for a commitment device is consistent with perceived self-control problems ( \begin_inset Formula $\tilde{\gamma}>0$ \end_inset ) and unmet market demand for digital self-control tools. \end_layout \begin_layout Standard On the survey 2 multiple price list, people who perceive self-control problems should prefer the Screen Time Bonus over higher fixed payments, as the incentive helps bring future use in line with current preferences. We show in Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:AltTemptationEstimates" plural "false" caps "false" noprefix "false" \end_inset that participants' average valuation of the bonus is consistent with perceived self-control problems ( \begin_inset Formula $\tilde{\gamma}>0$ \end_inset ). \end_layout \begin_layout Standard Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:Validation" plural "false" caps "false" noprefix "false" \end_inset presents additional results that validate that the MPL responses are meaningful. First, participants' valuations of the bonus are correlated with the amount of money they could expect to earn. Second, the bonus and limit valuations are correlated with each other and with \emph on limit tightness \emph default , \emph on ideal use change \emph default , \emph on addiction scale \emph default , \emph on SMS addiction scale \emph default , and other variables in expected ways. Third, after the bonus MPL, we asked people to \begin_inset Quotes eld \end_inset select the statement that best describes your thinking when trading off the Screen Time Bonus against the fixed payment. \begin_inset Quotes erd \end_inset \begin_inset Formula $\MPLwishreduce$ \end_inset percent responded that \begin_inset Quotes eld \end_inset I wanted to give myself an incentive to use my phone less over the next three weeks, even though it might result in a smaller payment, \begin_inset Quotes erd \end_inset and this group had a higher average valuation. \end_layout \begin_layout Subsection Effects on Survey Outcomes \end_layout \begin_layout Standard Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Effects_SurveyOutcomes" plural "false" caps "false" noprefix "false" \end_inset presents the effects of the bonus and limit treatments on the survey outcomes described in Section \begin_inset CommandInset ref LatexCommand ref reference "sec:SurveyOutcomeVariables" plural "false" caps "false" noprefix "false" \end_inset . The outcome variables are signed so more positive effects always correspond to less addiction and/or higher subjective well-being. Following our pre-analysis plan, when estimating effects on survey outcomes, we constrain the limit effect to be the same for surveys 3 and 4 (because we correctly anticipated similar \begin_inset Quotes eld \end_inset first stage \begin_inset Quotes erd \end_inset effects on FITSBY use in both periods 2 and 3) and we report the bonus effect only for survey 4 (because we correctly anticipated negligible \begin_inset Quotes eld \end_inset first stage \begin_inset Quotes erd \end_inset effects on FITSBY use in period 2). \begin_inset Foot status open \begin_layout Plain Layout Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Effects_SurveyOutcomes_Null" plural "false" caps "false" noprefix "false" \end_inset presents the treatment effects on survey outcomes separately for surveys 3 and 4. The limit effects on surveys 3 and 4 are statistically indistinguishable. Although the bonus did not substantially affect consumption in period 2, the Bonus group reported more ideal use reduction and more addiction on survey 3. One potential explanation is that the Bonus group hoped to reduce FITSBY use in anticipation of the period 3 incentive, and these survey responses reflect their failure to do so. \end_layout \end_inset \end_layout \begin_layout Standard Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Effects_SurveyOutcomes" plural "false" caps "false" noprefix "false" \end_inset shows that both interventions significantly reduced self-reported measures of addiction. Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:LATEs" plural "false" caps "false" noprefix "false" \end_inset presents coefficient estimates and p-values. The bonus effect is larger than the limit effect for five of the six variables, consistent with the bonus's larger effects on FITSBY use. The bonus decreased \emph on ideal use change \emph default by \begin_inset Formula $\bonusidealcoefn$ \end_inset standard deviations (about \begin_inset Formula $\bonusidealcoefone$ \end_inset percentage points), while the limit decreased it by \begin_inset Formula $\limitidealcoefn$ \end_inset standard deviations (about \begin_inset Formula $\limitidealcoefone$ \end_inset percentage points). Both interventions reduced \emph on addiction scale \emph default and \emph on SMS addiction scale \emph default by \begin_inset Formula $\limitaddictioncoefnone$ \end_inset to \begin_inset Formula $\bonusaddictioncoefn$ \end_inset standard deviations, or about \begin_inset Formula $\limitaddictioncoef$ \end_inset – \begin_inset Formula $\text{\bonusaddictioncoef}$ \end_inset points on the 16-point \emph on addiction scale \emph default . Both interventions statistically significantly reduced the chance that people reported using their smartphones to relax to go to sleep, losing sleep from use, using longer than intended, using to distract from anxiety, having difficulty putting down their phone, using mindlessly, and other specific measures from the addiction scales; see Appendix Figures \begin_inset CommandInset ref LatexCommand ref reference "fig:AddictionEffects" plural "false" caps "false" noprefix "false" \end_inset and \begin_inset CommandInset ref LatexCommand ref reference "fig:SMSAddictionEffects" plural "false" caps "false" noprefix "false" \end_inset . The limit treatment statistically significantly increased the extent to which people thought their smartphone use made their life better, while the bonus did not. \end_layout \begin_layout Standard The bonus and limit treatments increased subjective well-being (SWB) by \begin_inset Formula $\bonusswbindexcoefnone$ \end_inset standard deviations ( \begin_inset Formula $p\approx\pvaluebonusswbindex$ \end_inset ) and \begin_inset Formula $\limitswbindexcoefnone$ \end_inset standard deviations ( \begin_inset Formula $p\approx\pvallimitswbindex$ \end_inset ) respectively. The sharpened False Discovery Rate-adjusted p-values (see \begin_inset CommandInset citation LatexCommand citealt key "Benjamini1995" literal "false" \end_inset ) are \begin_inset Formula $\qadjbonusswbindex$ \end_inset and \begin_inset Formula $\qadjlimitswbindex$ \end_inset , respectively. These SWB effects appear to be driven particularly by improved concentration and reduced distraction; see Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:SWBEffects" plural "false" caps "false" noprefix "false" \end_inset . The effects of the bonus and limit on happiness, life satisfaction, depression, and anxiety are individually and collectively insignificant, while the effects of the bonus (but not the limit) on concentration, distraction, and sleep quality are collectively significant. Both interventions improved \emph on survey index \emph default , the inverse covariance-weighted average of the five survey outcome variables, by about \begin_inset Formula $\bonusindexwellcoefnone$ \end_inset standard deviations. \end_layout \begin_layout Standard One point of comparison for the SWB effects is \begin_inset CommandInset citation LatexCommand citet* key "AllcottBraghieri" literal "false" \end_inset . They find that deactivating subjects' Facebook accounts for a four week period increased an index of SWB by a statistically significant 0.09 standard deviations. Although the two interventions had similar effects on time use—deactivation in \begin_inset CommandInset citation LatexCommand citet* key "AllcottBraghieri" literal "false" \end_inset reduced Facebook use by 60 minutes per day for 27 days, while our Screen Time Bonus reduced FITSBY use by \begin_inset Formula $\taubthreenicenice$ \end_inset minutes per day for 20 days—they differed on a number of dimensions including the apps affected and the time period in which the study took place. \end_layout \begin_layout Standard Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Heterogeneity_SurveyOutcomes" plural "false" caps "false" noprefix "false" \end_inset presents effects on \emph on survey index \emph default in subgroups with above- and below-median values of our six pre-specified moderators. There is little heterogeneity with respect to the first four moderators, other than that the limit seems to have larger effects on women. However, the effects of both interventions are 2–3 times larger for people with above-median baseline values of \emph on restriction index \emph default , which measures interest in restricting smartphone time use, and \emph on addiction index. \emph default This implies that the interventions are well-targeted: they have larger effects for people who report wanting and needing them the most. Consistent with this, point estimates suggest that the bonus and limit both have larger effects on FITSBY use for people with higher \emph on restriction index \emph default and \emph on addiction index \emph default , although the differences are not as significant; see Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Heterogeneity_Usage" plural "false" caps "false" noprefix "false" \end_inset . This targeting result need not have been the case: for example, it could have been that more addicted people were less likely to feel that the limit functionality worked well for them. \end_layout \begin_layout Section Estimating the Model \begin_inset CommandInset label LatexCommand label name "sec:StructuralEstimation" \end_inset \end_layout \begin_layout Subsection Setup \end_layout \begin_layout Standard We now turn to our model to simulate the effect of temptation on steady-state FITSBY use. In the model from Section \begin_inset CommandInset ref LatexCommand ref reference "sec:Model" plural "false" caps "false" noprefix "false" \end_inset , temptation and habit formation interact, because the current consumption increase caused by temptation also increases future consumption. The long-run effect of temptation could therefore be different than any effects identified during the experiment. In this section, we estimate the model's structural parameters. In the next section, we simulate steady-state FITSBY use with counterfactual self-control and habit formation parameters. \end_layout \begin_layout Standard We estimate the model using indirect inference: we derive equations that characterize how a consumer from our model would behave in our experiment, and we solve for the structural parameters consistent with the data. In our baseline estimates, we assume that all parameters other than \begin_inset Formula $\xi$ \end_inset are homogeneous across consumers, although we relax this assumption in an extension that allows heterogeneity in temptation \begin_inset Formula $\gamma$ \end_inset and perceived temptation \begin_inset Formula $\tilde{\gamma}$ \end_inset . \end_layout \begin_layout Standard In describing the estimation strategy, we focus on a \begin_inset Quotes eld \end_inset restricted model \begin_inset Quotes erd \end_inset where we set the anticipatory bonus effect \begin_inset Formula $\tau_{2}^{B}$ \end_inset to zero. This implies full projection bias ( \begin_inset Formula $\alpha=1$ \end_inset ), and thus that consumption decisions maximize current-period flow utility with no dynamic considerations. This substantially simplifies the exposition and, as we will show, has little impact on the results. Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:UnrestrictedModel" plural "false" caps "false" noprefix "false" \end_inset presents our \begin_inset Quotes eld \end_inset unrestricted model, \begin_inset Quotes erd \end_inset in which we use the empirical \begin_inset Formula $\tau_{2}^{B}$ \end_inset and allow partial projection bias. \end_layout \begin_layout Standard In the restricted model, consumers maximize current-period flow utility from equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:QuadraticUtility" plural "false" caps "false" noprefix "false" \end_inset ), giving equilibrium consumption \begin_inset Formula \begin{equation} x_{t}^{*}\left(s_{t},\gamma,\boldsymbol{p}_{t}\right)=\frac{\zeta s_{t}+\xi_{t}-p_{t}+\gamma}{-\eta}.\label{eq:x* static} \end{equation} \end_inset We define \begin_inset Formula $\lambda\coloneqq\frac{\partial x_{t}^{*}}{\partial s_{t}}$ \end_inset as the effect of habit stock on consumption; \begin_inset Formula $\lambda=-\zeta/\eta$ \end_inset in the restricted model. In a steady state with constant \begin_inset Formula $s$ \end_inset , \begin_inset Formula $\xi$ \end_inset , and \begin_inset Formula $p$ \end_inset , we must have \begin_inset Formula $s_{ss}=\rho\left(s_{ss}+x_{ss}\right)$ \end_inset , and thus \begin_inset Formula $s_{ss}=\frac{\rho}{1-\rho}x_{ss}$ \end_inset . \end_layout \begin_layout Standard We model the Screen Time Bonus as a price \begin_inset Formula $p^{B}=$ \end_inset $2.50 per hour in period 3 plus a fixed payment. \begin_inset Foot status collapsed \begin_layout Plain Layout Modeling the bonus as a linear price simplifies the model substantially, although it is an approximation: \begin_inset Formula $\PercentBoundDrop$ \end_inset percent of the Bonus group hit the $150 payment limit because they reduced period 3 FITSBY use by more than three hours per day relative to their Bonus Benchmark, and \begin_inset Formula $\PercentExceedBonus$ \end_inset percent used more than their Bonus Benchmark. These two subgroups in practice faced zero subsidy for marginal screen time reductions, although they may not have known that. \end_layout \end_inset We model the limit functionality as an intervention that elimates share \begin_inset Formula $\omega$ \end_inset of perceived and actual temptation. We conservatively assume \begin_inset Formula $\omega=1$ \end_inset in our primary estimates, and we consider alternative assumptions below. We assume that when predicting period \begin_inset Formula $t$ \end_inset consumption on the survey at the beginning of period \begin_inset Formula $t$ \end_inset , consumers use perceived temptation \begin_inset Formula $\tilde{\gamma}$ \end_inset but are aware of projection bias, so the prediction is denoted \begin_inset Formula $x_{t}^{*}\left(s_{t},\tilde{\gamma},\boldsymbol{p}_{t}\right)$ \end_inset . \end_layout \begin_layout Standard Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:TemptationIdentification" plural "false" caps "false" noprefix "false" \end_inset illustrates temptation, naivete, and our identification strategies. The three demand curves are desired demand \begin_inset Formula $x_{t}^{*}(s_{t},0,\boldsymbol{p}_{t})$ \end_inset according to preferences before period \begin_inset Formula $t$ \end_inset , predicted demand \begin_inset Formula $x_{t}^{*}(s_{t},\tilde{\gamma},\boldsymbol{p}_{t})$ \end_inset as of survey \begin_inset Formula $t$ \end_inset , and actual demand \begin_inset Formula $x_{t}^{*}(s_{t},\gamma,\boldsymbol{p}_{t})$ \end_inset . The actual equilibrium at \begin_inset Formula $p=0$ \end_inset is point \emph on L \emph default , and the predicted equilibrium is at point \emph on C \emph default , so the distance \emph on CL \emph default is Control group misprediction \begin_inset Formula $m^{C}\coloneqq x_{t}^{*}(s_{t},\gamma,\boldsymbol{p}_{t})-x_{t}^{*}(s_{t},\tilde{\gamma},\boldsymbol{p}_{t})$ \end_inset . \begin_inset Note Comment status open \begin_layout Plain Layout Note that this second term is \begin_inset Formula $x^{*}$ \end_inset instead of \begin_inset Formula $\tilde{x}^{*}$ \end_inset , as we assume that people become aware of projection bias before predicting, per the previous paragraph. \end_layout \end_inset The bonus moves the equilibrium to point \emph on J \emph default in period 3, so the contemporaneous bonus effect \begin_inset Formula $\tau_{3}^{B}$ \end_inset is the distance \emph on JK \emph default . The limit treatment moves the equilibrium to point \emph on G \emph default , so the limit treatment effect \begin_inset Formula $\tau^{L}$ \end_inset is the distance \emph on GL \emph default . \end_layout \begin_layout Subsection Estimating Equations \begin_inset CommandInset label LatexCommand label name "subsec:Estimating Equations" \end_inset \end_layout \begin_layout Standard Unlike many applications of indirect inference, we derive equations that allow us to directly solve for the model parameters, so we do not need to use an optimization routine to search for parameters that fit the data. We estimate the parameters in stages, as parameters estimated in the first few equations are used as inputs to subsequent equations. We estimate confidence intervals by bootstrapping. Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:EstimatingEquationDerivations" plural "false" caps "false" noprefix "false" \end_inset presents formal derivations and additional details. \end_layout \begin_layout Subsubsection* Habit Formation \end_layout \begin_layout Standard We first estimate \begin_inset Formula $\rho$ \end_inset from the decay of the bonus treatment effects. Taking the expectations over \begin_inset Formula $\xi$ \end_inset in the Bonus and Bonus Control groups, we can write the average treatment effect of the bonus on period 4 consumption as the result of the decayed period 3 effect. Similarly, the average treatment effect in period 5 results from the cumulative decayed effects from periods 3 and 4: \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \tau_{4}^{B} & =\lambda\rho\tau_{3}^{B}\label{eq:tauB4 restricted}\\ \tau_{5}^{B} & =\lambda\left(\rho\tau_{4}^{B}+\rho^{2}\tau_{3}^{B}\right). \end{align} \end_inset Dividing those two equations gives \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \rho & =\frac{\tau_{5}^{B}}{\tau_{4}^{B}}-\frac{\tau_{4}^{B}}{\tau_{3}^{B}}.\label{eq:rho restricted} \end{align} \end_inset This equation shows that if the bonus effect is more persistent between periods 4 and 5, we infer that habit stock is more persistent (a larger \begin_inset Formula $\rho$ \end_inset ). \end_layout \begin_layout Standard In the unrestricted model in Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:UnrestrictedModel" plural "false" caps "false" noprefix "false" \end_inset , we also estimate \begin_inset Formula $\lambda$ \end_inset , because it is useful in estimating the other parameters. To provide a comparison, we also estimate \begin_inset Formula $\lambda$ \end_inset in the restricted model by rearranging equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:tauB4 restricted" plural "false" caps "false" noprefix "false" \end_inset ) and inserting the \begin_inset Formula $\rho$ \end_inset from equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:rho restricted" plural "false" caps "false" noprefix "false" \end_inset ): \begin_inset Formula $\lambda=\frac{\tau_{4}^{B}}{\rho\tau_{3}^{B}}$ \end_inset . \end_layout \begin_layout Subsubsection* Price Response and Habit Stock Effect on Marginal Utility \end_layout \begin_layout Standard After estimating \begin_inset Formula $\rho$ \end_inset , we estimate \begin_inset Formula $\eta$ \end_inset and \begin_inset Formula $\zeta$ \end_inset from the magnitude and decay of the bonus treatment effects. For each of periods 3 and 4, we take the expectations over \begin_inset Formula $\xi$ \end_inset of equilibrium consumption in the Bonus and Bonus Control groups, difference the two, and rearrange, giving \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \eta & =\frac{p^{B}}{\tau_{3}^{B}}\label{eq:eta restricted}\\ \zeta & =\frac{-\eta\tau_{4}^{B}}{\rho\tau_{3}^{B}}.\label{eq:zeta restricted} \end{align} \end_inset Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:TemptationIdentification" plural "false" caps "false" noprefix "false" \end_inset illustrates the first equation: the inverse demand slope \begin_inset Formula $\eta$ \end_inset is just the ratio of \begin_inset Formula $p^{B}$ \end_inset (the vertical distance \emph on KL \emph default ) to \begin_inset Formula $\tau_{3}^{B}$ \end_inset (the horizontal distance \emph on JK \emph default ). The second equation shows that if the bonus effect is more persistent between periods 3 and 4, we infer that habit stock has a larger effect on marginal utility (a higher \begin_inset Formula $\zeta$ \end_inset ). \end_layout \begin_layout Subsubsection* Naivete about Temptation \end_layout \begin_layout Standard Next, we estimate naivete about temptation \begin_inset Formula $\gamma-\tilde{\gamma}$ \end_inset using misprediction in the Control group. To solve for \begin_inset Formula $\gamma-\tilde{\gamma}$ \end_inset , we take the expectations over \begin_inset Formula $\xi$ \end_inset of actual consumption and consumption as predicted at the beginning of the period, difference the two, and rearrange, giving \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \gamma-\tilde{\gamma}=-\eta m^{C}.\label{eq:Naivete restricted} \end{equation} \end_inset Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:TemptationIdentification" plural "false" caps "false" noprefix "false" \end_inset illustrates: naivete \begin_inset Formula $\gamma-\tilde{\gamma}$ \end_inset is the vertical distance \emph on HC \emph default between actual and predicted marginal utility, and this can be inferred by multiplying Control group average misprediction \begin_inset Formula $m^{C}$ \end_inset (the horizontal distance \emph on CL \emph default between actual and predicted demand) by the inverse demand slope \begin_inset Formula $\eta$ \end_inset . \end_layout \begin_layout Subsubsection* Temptation \end_layout \begin_layout Standard To estimate temptation \begin_inset Formula $\gamma$ \end_inset , we take the expectations over \begin_inset Formula $\xi$ \end_inset of equilibrium consumption in the Limit and Limit Control groups, difference the two, and rearrange, giving \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \gamma=\eta\tau_{2}^{L}.\label{eq:gamma_Limit restricted} \end{equation} \end_inset Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:TemptationIdentification" plural "false" caps "false" noprefix "false" \end_inset illustrates: temptation \begin_inset Formula $\gamma$ \end_inset is the vertical distance \emph on LM \emph default between desired and actual demand, and this can be inferred by multiplying the effect of removing temptation ( \begin_inset Formula $\tau_{2}^{L}$ \end_inset , the horizontal distance \emph on GL \emph default between long-run and present demand) by the inverse demand slope \begin_inset Formula $\eta$ \end_inset . We then substitute the estimated \begin_inset Formula $\gamma$ \end_inset into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Naivete restricted" plural "false" caps "false" noprefix "false" \end_inset ) to infer \begin_inset Formula $\tilde{\gamma}$ \end_inset . \end_layout \begin_layout Subsubsection* Intercept \end_layout \begin_layout Standard Finally, we back out the distribution of \begin_inset Formula $\xi$ \end_inset that fits the distribution of baseline consumption. We assume that participant \begin_inset Formula $i$ \end_inset 's baseline consumption \begin_inset Formula $x_{i1}$ \end_inset was in a steady state. Substituting \begin_inset Formula $s_{ss}=\frac{\rho}{1-\rho}x_{ss}$ \end_inset into equilibrium consumption from equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:x* static" plural "false" caps "false" noprefix "false" \end_inset ) and rearranging gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \xi_{i}=p-\gamma+x_{i1}\left(-\eta-\zeta\frac{\rho}{1-\rho}\right).\label{eq:xi} \end{equation} \end_inset This equation shows that we infer larger \begin_inset Formula $\xi_{i}$ \end_inset for people with higher baseline consumption \begin_inset Formula $x_{i1}$ \end_inset . \end_layout \begin_layout Standard In the unrestricted model in Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:UnrestrictedModel" plural "false" caps "false" noprefix "false" \end_inset , equilibrium consumption also depends on \begin_inset Formula $\phi$ \end_inset , the direct effect of habit stock on utility. Our data do not allow us to separately identify \begin_inset Formula $\phi$ \end_inset from \begin_inset Formula $\xi$ \end_inset , so we estimate an \begin_inset Quotes eld \end_inset intercept \begin_inset Quotes erd \end_inset \begin_inset Formula $\kappa_{i}\coloneqq(1-\alpha)\delta\rho(\phi-\xi_{i})+\xi_{i}$ \end_inset that includes both of these structural parameters. In the restricted model with \begin_inset Formula $\alpha=1$ \end_inset , this simplifies to \begin_inset Formula $\kappa_{i}=\xi_{i}$ \end_inset . \end_layout \begin_layout Subsection Empirical Moments \begin_inset CommandInset label LatexCommand label name "subsec:EstimationMomentsDetails restricted" \end_inset \end_layout \begin_layout Standard Table \begin_inset CommandInset ref LatexCommand ref reference "tab:Moments" plural "false" caps "false" noprefix "false" \end_inset presents the moments used to estimate the restricted model. The bonus and limit effects \begin_inset Formula $\tau_{t}^{B}$ \end_inset and \begin_inset Formula $\tau_{2}^{L}$ \end_inset are as displayed in Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:UsageEffects" plural "false" caps "false" noprefix "false" \end_inset . Control group misprediction \begin_inset Formula $m^{C}$ \end_inset is the average across periods 2–4 of the difference between actual period \begin_inset Formula $t$ \end_inset FITSBY use and the prediction for period \begin_inset Formula $t$ \end_inset elicited on survey \begin_inset Formula $t$ \end_inset , as displayed in Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualControl" plural "false" caps "false" noprefix "false" \end_inset . The unrestricted model and our robustness checks also use the anticipatory bonus effect \begin_inset Formula $\tau_{2}^{B}$ \end_inset and additional parameters presented in Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:Moments-Full" plural "false" caps "false" noprefix "false" \end_inset . In light of the discussion in Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:BonusEffects" plural "false" caps "false" noprefix "false" \end_inset , we omit the first half of period 2 when we estimate \begin_inset Formula $\tau_{2}^{B}$ \end_inset . \begin_inset Foot status open \begin_layout Plain Layout Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:StructuralEstimates-fulltaub2" plural "false" caps "false" noprefix "false" \end_inset presents parameter estimates when we use all of period 2 to estimate \begin_inset Formula $\tau_{2}^{B}$ \end_inset . The estimated projection bias \begin_inset Formula $\alpha$ \end_inset is smaller, as expected, but the other parameter estimates are very similar. \end_layout \end_inset \end_layout \begin_layout Subsection Parameter Estimates \begin_inset CommandInset label LatexCommand label name "subsec:ParameterEstimates" \end_inset \end_layout \begin_layout Standard Table \begin_inset CommandInset ref LatexCommand ref reference "tab:StructuralEstimates" plural "false" caps "false" noprefix "false" \end_inset presents our point estimates and bootstrapped 95 percent confidence intervals. Column 1 presents the restricted model described above (fixing \begin_inset Formula $\tau_{2}^{B}=0$ \end_inset and \begin_inset Formula $\alpha=1$ \end_inset ), while column 2 presents the unrestricted model described in Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:UnrestrictedModel" plural "false" caps "false" noprefix "false" \end_inset . Since the estimated \begin_inset Formula $\tau_{2}^{B}$ \end_inset is close to zero and \begin_inset Formula $\hat{\alpha}$ \end_inset is close to one, the estimates in the two columns are very similar. \end_layout \begin_layout Standard In column 1, we estimate \begin_inset Formula $\hat{\lambda}\approx\lambdareshat$ \end_inset and \begin_inset Formula $\hat{\rho}\approx\rhoreshat$ \end_inset . In our model, this implies that an exogenous consumption increase of 1 minute per day over a three week period will cause consumption to increase by \begin_inset Formula $\hat{\lambda}\hat{\rho}\approx\lambdarhonice$ \end_inset minutes per day in the next three-week period, and \begin_inset Formula $\hat{\lambda}\hat{\rho}^{2}\approx\lambdarhosquaredtwodigits$ \end_inset minutes per day in the period after that. \end_layout \begin_layout Standard Consistent with the small and statistically insignificant anticipatory bonus effect \begin_inset Formula $\tau_{2}^{B}$ \end_inset in the second half of period 2, we estimate \begin_inset Formula $\hat{\alpha}\approx\alphahat$ \end_inset in the unrestricted model in column 2, which is marginally significantly different from one. The point estimate suggests that participants were attentive to only \begin_inset Formula $(1-\hat{\alpha})\times100\%\approx\oneminusalphapct$ \end_inset percent of habit formation. Inserting the estimates of \begin_inset Formula $\lambda$ \end_inset , \begin_inset Formula $\rho$ \end_inset , \begin_inset Formula $\eta$ \end_inset , and \begin_inset Formula $\zeta$ \end_inset into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:alpha" plural "false" caps "false" noprefix "false" \end_inset ) in Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:UnrestrictedModel" plural "false" caps "false" noprefix "false" \end_inset , we calculate that \begin_inset Formula $\tau_{2}^{B}$ \end_inset would have needed to be \begin_inset Formula $\tauBtwodesired$ \end_inset minutes per day (compared to the actual point estimate of \begin_inset Formula $\tauBtwo$ \end_inset minutes per day in the second half of period 2) to estimate zero projection bias ( \begin_inset Formula $\alpha=0$ \end_inset ). In other words, the anticipatory bonus effect is only \begin_inset Formula $\pcttaubtwonice$ \end_inset percent of what our model would predict with fully forward-looking ( \begin_inset Quotes eld \end_inset rational \begin_inset Quotes erd \end_inset ) habit formation. This is striking when combined with the evidence from Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualHabitFormation" plural "false" caps "false" noprefix "false" \end_inset that participants correctly predicted habit formation. It is consistent with a model in which people are intellectually aware of habit formation but consume as if they are inattentive to it. \end_layout \begin_layout Standard Since the restricted model estimating equations are so simple, one can easily calculate the point estimates in column 1 with the moments from Table \begin_inset CommandInset ref LatexCommand ref reference "tab:Moments" plural "false" caps "false" noprefix "false" \end_inset . For example, the Control group underestimated FITSBY use by an average of \begin_inset Formula $\mispredict$ \end_inset minutes per day on surveys 2–4. Inserting that into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Naivete restricted" plural "false" caps "false" noprefix "false" \end_inset ) gives a naivete of \begin_inset Formula $\widehat{\gamma-\tilde{\gamma}}=-\hat{\eta}\cdot m^{C}\approx-(\etareshour)\cdot(\mispredict/60)\approx\naivetereshour$ \end_inset $/hour in column 1. \end_layout \begin_layout Standard The limit changed period 2 FITSBY use by \begin_inset Formula $\tauLtwo$ \end_inset minutes per day. Inserting that into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gamma_Limit restricted" plural "false" caps "false" noprefix "false" \end_inset ) gives temptation \begin_inset Formula $\hat{\gamma}=\hat{\eta}\tau_{2}^{L}\approx(\etareshour)\cdot(\text{\tauLtwo}/60)\approx\gammaLeffectreshour$ \end_inset $/hour in column 1. This estimate implies that a tax on FITSBY use of $ \begin_inset Formula $\gammaLeffectreshour$ \end_inset per hour would reduce consumption to the level our participants would choose for themselves in advance. Dividing estimated naivete \begin_inset Formula $\widehat{\gamma-\tilde{\gamma}}$ \end_inset by this \begin_inset Formula $\hat{\gamma}$ \end_inset suggests that our participants underestimate temptation by \begin_inset Formula $\naivetereshour/\gammaLeffectreshour\times100\%\approx\underestimatetempresnice$ \end_inset percent. \end_layout \begin_layout Standard Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:AltTemptationEstimates" plural "false" caps "false" noprefix "false" \end_inset presents alternative estimates of temptation \begin_inset Formula $\gamma$ \end_inset in the restricted and unrestricted models. First, we infer perceived temptation using participants' valuations of the limit functionality and the Screen Time Bonus, following \begin_inset CommandInset citation LatexCommand citet key "AclandLevy2012" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "AugenblickRabin2019" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "Chaloupka2019" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "allcottkim2020" literal "false" \end_inset , and \begin_inset CommandInset citation LatexCommand citet key "Carrera2019" literal "false" \end_inset . Second, we generalize the model to include multiple temptation goods, using the self-reports of substitution to FITSBY use on other devices discussed in Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:Substitution" plural "false" caps "false" noprefix "false" \end_inset . Third, we assume that the limit treatment eliminates share \begin_inset Formula $\omega\in[0,1]$ \end_inset of temptation, relaxing the assumption of \begin_inset Formula $\omega=1$ \end_inset in our primary estimates; we estimate \begin_inset Formula $\omega$ \end_inset from differences in self-reported \emph on ideal use change \emph default between the Limit and Limit Control groups. Finally, we allow for individual-specific heterogeneity in \begin_inset Formula $\gamma$ \end_inset , using the distribution of \emph on limit tightness \emph default set by Limit group participants. These alternative approaches all imply temptation \begin_inset Formula $\gamma$ \end_inset between about $1 and $3 per hour, and our primary estimate of $ \begin_inset Formula $\ensuremath{\gammaLeffectreshour}$ \end_inset per hour is relatively conservative. \end_layout \begin_layout Section Counterfactuals: Effects of Temptation on Time Use \begin_inset CommandInset label LatexCommand label name "sec:Counterfactuals" \end_inset \end_layout \begin_layout Subsection Methodology \end_layout \begin_layout Standard Using the parameter estimates from the previous section, we can predict the effects of changes in temptation and habit formation on steady-state FITSBY use. Equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS" plural "false" caps "false" noprefix "false" \end_inset ) in Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:UnrestrictedModel" plural "false" caps "false" noprefix "false" \end_inset characterizes steady-state consumption in the unrestricted model. Using that equation, we can predict participant \begin_inset Formula $i$ \end_inset 's steady-state FITSBY use at \begin_inset Formula $p=0$ \end_inset as a function of any values of habit formation, temptation, and steady-state misprediction parameters \begin_inset Formula $\{\zeta,\gamma,\tilde{\gamma},m_{ss}\}$ \end_inset : \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \hat{x}_{i,ss}\left(\zeta,\gamma,\tilde{\gamma},m_{ss}\right)=\frac{\hat{\kappa}_{i}+(1-\hat{\alpha})\delta\hat{\rho}\left[\left(\zeta-\hat{\eta}\right)m_{ss}-\left(1+\hat{\tilde{\lambda}}\right)\tilde{\gamma}\right]+\gamma}{-\hat{\eta}-(1-\hat{\alpha})\delta\hat{\rho}(\zeta-\hat{\eta})-\zeta\frac{\hat{\rho}-(1-\hat{\alpha})\delta\hat{\rho}^{2}}{1-\hat{\rho}}}.\label{eq:xSS counterfactual} \end{equation} \end_inset The sample average prediction is denoted \begin_inset Formula $\bar{\hat{x}}_{ss}\left(\zeta,\gamma,\tilde{\gamma},m_{ss}\right)$ \end_inset . As discussed in Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Estimating Equations Unrestricted" plural "false" caps "false" noprefix "false" \end_inset , we assume that the predicted \begin_inset Formula $\tilde{\lambda}$ \end_inset equals the estimated \begin_inset Formula $\hat{\lambda}$ \end_inset , that steady-state misprediction \begin_inset Formula $m_{ss}$ \end_inset equals observed Control group misprediction \begin_inset Formula $m^{C}$ \end_inset , and that the discount factor is \begin_inset Formula $\delta=0.997$ \end_inset per three-week period, consistent with a five percent annual discount rate. \end_layout \begin_layout Standard Since we can't identify \begin_inset Formula $\phi$ \end_inset (the direct effect of habit stock on utility), we must hold constant each participant's intercept \begin_inset Formula $\kappa_{i}\coloneqq(1-\alpha)\delta\rho(\phi-\xi_{i})+\xi_{i}$ \end_inset across counterfactuals in the restricted model. Since this intercept contains \begin_inset Formula $\rho$ \end_inset and \begin_inset Formula $\alpha$ \end_inset , we can't predict consumption with counterfactual values of \begin_inset Formula $\rho$ \end_inset or \begin_inset Formula $\alpha$ \end_inset . \end_layout \begin_layout Standard In the restricted model with \begin_inset Formula $\alpha=1$ \end_inset , equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS counterfactual" plural "false" caps "false" noprefix "false" \end_inset ) simplifies to \begin_inset Formula \begin{equation} \hat{x}_{i,ss}\left(\rho,\gamma\right)=\frac{\hat{\xi}_{i}+\gamma}{-\hat{\eta}-\hat{\zeta}\frac{\rho}{1-\rho}},\label{eq:xSS counterfactual restricted} \end{equation} \end_inset which could also be derived from substituting \begin_inset Formula $s_{ss}=\frac{\rho}{1-\rho}x_{ss}$ \end_inset into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:x* static" plural "false" caps "false" noprefix "false" \end_inset ). Steady-state misprediction \begin_inset Formula $m_{ss}$ \end_inset and perceived temptation \begin_inset Formula $\tilde{\gamma}$ \end_inset do not affect steady-state consumption in the restricted model because consumers simply maximize current-period flow utility. \end_layout \begin_layout Subsection Counterfactual Results \end_layout \begin_layout Standard Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Counterfactuals" plural "false" caps "false" noprefix "false" \end_inset presents point estimates and bootstrapped 95 percent confidence intervals for predicted average FITSBY use at counterfactual parameter values. For each counterfactual, we present predictions from the restricted model ( \begin_inset Formula $\alpha=1$ \end_inset ) and unrestricted model ( \begin_inset Formula $\alpha=\hat{\alpha}$ \end_inset ). We label the restricted model predictions as our primary results, because they are simpler and more conservative. \end_layout \begin_layout Standard The first \begin_inset Quotes eld \end_inset counterfactual \begin_inset Quotes erd \end_inset is the baseline at our point estimates: \begin_inset Formula $\hat{x}_{ss}\left(\hat{\zeta},\hat{\gamma},\hat{\tilde{\gamma}},\hat{m}^{C}\right)$ \end_inset . This mechanically matches baseline average FITSBY use of \begin_inset Formula $\xss$ \end_inset minutes per day. The second counterfactual removes naivete: \begin_inset Formula $\bar{\hat{x}}_{ss}\left(\hat{\zeta},\hat{\gamma},\hat{\gamma},0\right)$ \end_inset . \begin_inset Foot status open \begin_layout Plain Layout Since Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualHabitFormation" plural "false" caps "false" noprefix "false" \end_inset shows that participants predicted habit formation fairly accurately, we attribute all of steady-state misprediction \begin_inset Formula $m_{ss}$ \end_inset to naivete about temptation. \end_layout \end_inset As described above, naivete has no effect when \begin_inset Formula $\alpha=1$ \end_inset . Because naivete is so small and projection bias is so strong, the point estimate with \begin_inset Formula $\alpha=\hat{\alpha}$ \end_inset is very close to the baseline. \end_layout \begin_layout Standard The third counterfactual removes temptation: \begin_inset Formula $\bar{\hat{x}}_{ss}\left(\hat{\zeta},0,0,0\right)$ \end_inset . Relative to baseline, removing temptation reduces predicted FITSBY use by \begin_inset Formula $\deltaxtemptationresnice$ \end_inset minutes per day ( \begin_inset Formula $\pctreductiontemptationresnice$ \end_inset percent) with \begin_inset Formula $\alpha=1$ \end_inset . Thus, our primary estimate is that smartphone FITSBY use would be \begin_inset Formula $\pctreductiontemptationresnice$ \end_inset percent lower without self-control problems. \end_layout \begin_layout Standard The fourth and fifth counterfactuals remove habit formation, first with temptation and then without: \begin_inset Formula $\bar{\hat{x}}_{ss}\left(0,\hat{\gamma},\hat{\tilde{\gamma}},\hat{m}^{C}\right)$ \end_inset and then \begin_inset Formula $\bar{\hat{x}}_{ss}\left(0,0,0,0\right)$ \end_inset . We emphasize that habit formation on its own is not a departure from rationalit y ( \begin_inset CommandInset citation LatexCommand citealt key "BeckerMurphy1988" literal "false" \end_inset ), and it could capture forces such as learning and investment that increase consumer welfare. Relative to baseline, removing habit formation reduces predicted FITSBY use by \begin_inset Formula $\deltaxhabitresnice$ \end_inset minutes per day with \begin_inset Formula $\alpha=1$ \end_inset . Without habit formation, the effect of removing temptation (going from the fourth to the fifth counterfactual) is just the limit treatment effect ( \begin_inset Formula $\tau_{2}^{L}\approx\text{\tauLtwo}$ \end_inset minutes per day), which is about half of the effect of removing temptation with habit formation ( \begin_inset Formula $\deltaxtemptationres$ \end_inset minutes per day with \begin_inset Formula $\alpha=1$ \end_inset ). \begin_inset Foot status open \begin_layout Plain Layout Without habit formation, the effect of removing temptation on \begin_inset Formula $\bar{\hat{x}}_{ss}$ \end_inset is \begin_inset Formula $\frac{\hat{\gamma}}{-\hat{\eta}}$ \end_inset , which equals \begin_inset Formula $\tau_{2}^{L}$ \end_inset after substituting \begin_inset Formula $\hat{\gamma}=\tau_{2}^{L}\hat{\eta}$ \end_inset . \end_layout \end_inset This quantifies how habit formation magnifies the effects of temptation, because current temptation increases current consumption and thus future demand. \end_layout \begin_layout Standard We highlight one important tension in our results: Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:UsageEffects" plural "false" caps "false" noprefix "false" \end_inset shows that the limit effects decay slightly over periods 2–5, while our model predicts that the limit effects should grow over time as the Limit group's habit stock diminishes. One potential explanation is that habit formation works differently in response to prices versus self-control tools. Another potential explanation is that motivation to use the limit functionality decays enough that it outweighs the habit stock effect. \end_layout \begin_layout Standard Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:AltAssumptionsTable" plural "false" caps "false" noprefix "false" \end_inset presents 19 alternative estimates of the effects of temptation on steady-state FITSBY use across the restricted and unrestricted models. Consistent with the fact that our primary estimates of \begin_inset Formula $\gamma$ \end_inset are smaller than most alternative estimates, our primary estimates of the steady-state temptation effects are also relatively conservative. Furthermore, weighting our sample on observables to look more like the U.S. adult population also increases the predicted effects of temptation on consumption. This means that while our sample may still be non-representative on unobservabl e characteristics, sample selection bias captured by observables causes us to \emph on understate \emph default the effects of temptation on FITSBY use. \begin_inset Foot status open \begin_layout Plain Layout Appendix Tables \begin_inset CommandInset ref LatexCommand ref reference "tab:SampleDemographicsWeighted" plural "false" caps "false" noprefix "false" \end_inset – \begin_inset CommandInset ref LatexCommand ref reference "tab:StructuralEstimatesWeighted" plural "false" caps "false" noprefix "false" \end_inset present the demographics, moments, and parameter estimates in the weighted sample. Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:CounterfactualsTable" plural "false" caps "false" noprefix "false" \end_inset presents the numbers plotted in Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Counterfactuals" plural "false" caps "false" noprefix "false" \end_inset . Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:IndividualTemptation" plural "false" caps "false" noprefix "false" \end_inset presents the distribution of modeled temptation effects across participants, using the Limit group's distribution of \emph on limit tightness \emph default to identify heterogeneity in temptation. The effect is less than 10 minutes per day for \begin_inset Formula $\temptationeffectbelowten$ \end_inset percent of participants, and over 100 minutes per day for \begin_inset Formula $\temptationeffectabovehundred$ \end_inset percent. \end_layout \end_inset \end_layout \begin_layout Standard Since we don't identify \begin_inset Formula $\phi$ \end_inset (the direct effect of habit stock on utility), we can't do a full welfare analysis. The relatively elastic demand—from Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:BonusEffects" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset Formula $\bonusthreepct$ \end_inset percent of consumption is worth less than $2.50 per hour—suggests that participa nts do not have strong preferences over how to spend this marginal time, so the welfare losses from self-control problems might be limited. On the other hand, even small individual-level losses might be substantial when aggregated over many social media users. In a static model, the deadweight loss from temptation would be the triangle \emph on GLM \emph default on Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:TemptationIdentification" plural "false" caps "false" noprefix "false" \end_inset : \begin_inset Formula $-\tau^{L}\gamma/2\approx-(\text{\tauLtwo}/60)\times\gammaLeffectreshour/2\approx\$\DWLStaticnice$ \end_inset per day, or \begin_inset Formula $\$\DWLStaticThreeWeeks$ \end_inset per three-week period. This is closely consistent with the average valuation of $ \begin_inset Formula $\valuelimit$ \end_inset for three weeks of access to the limit functionality. Aggregated across 240 million American social media users ( \begin_inset CommandInset citation LatexCommand citealt key "pewresearch2021" literal "false" \end_inset ) \begin_inset Note Comment status open \begin_layout Plain Layout 72% of US population (from Pew) \begin_inset Formula $\times$ \end_inset 333 million \begin_inset Formula $\approx$ \end_inset 240 million \end_layout \end_inset , this would be \begin_inset Formula $\$\DWLStaticThreeWeeks$ \end_inset \begin_inset Formula $\times(52/3)\times0.24\approx\$\YearlyWelfare$ \end_inset billion per year in welfare losses from overuse of social media caused by self-control problems. For comparison, Facebook's total global profits in 2020 were $29 billion ( \begin_inset CommandInset citation LatexCommand citealt key "SECFB2020" literal "false" \end_inset ). However, we don't know how these effects would cumulate over time, as represent ed by \begin_inset Formula $\phi$ \end_inset : for example, after a longer period of reduced screen time, people might find more peace of mind or regret the loss of online interactions with friends and family. \end_layout \begin_layout Section Conclusion \end_layout \begin_layout Standard While digital technologies provide important benefits, some argue that they can be addictive and harmful. We formalize this argument in an economic model and transparently estimate the parameters using data from a field experiment. The Screen Time Bonus intervention had persistent effects after the incentives ended, suggesting that smartphone social media use is habit forming. Participants predicted these persistent effects on surveys but did not reduce FITSBY use before the bonus was in effect, suggesting that they are aware of but inattentive to habit formation. Participants used the screen time limit functionality when we offered it in the experiment, and this functionality reduced FITSBY use by over 20 minutes per day, suggesting that social media use involves self-control problems. The Control group repeatedly underestimated future use, suggesting slight naivete. Many participants reported indicators of smartphone addiction on surveys, and both the bonus and limit interventions reduced this self-reported addiction. Looking at these facts through the lens of our economic model implies that self-control problems magnified by habit formation might be responsible for \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none \begin_inset Formula $\pctreductiontemptationresnice$ \end_inset \family default \series default \shape default \size default \emph default \bar default \strikeout default \xout default \uuline default \uwave default \noun default \color inherit percent of social media use. These results suggest that better aligning digital technologies with well-being should be an important goal of users, parents, technology workers, investors, and regulators. \end_layout \begin_layout Standard Our results raise many additional questions; here are two. First, what are the underlying mechanisms and microfoundations that generate the persistent bonus treatment effects? We model this persistence simply through a capital stock of past consumption, but it could be driven by learning (followed by forgetting), network investments (e.g. connections with friends ebb and flow if maintained or neglected), or more nuanced habit formation mechanisms involving cues or automaticity (e.g. \begin_inset CommandInset citation LatexCommand citealt key "laibson2001" literal "false" \end_inset ; \begin_inset CommandInset citation LatexCommand citealt key "BernheimRangel2004" literal "false" \end_inset ; \begin_inset CommandInset citation LatexCommand citealt key "steinywellsjo2021" literal "false" \end_inset ). Second, if so many of our participants perceive self-control problems and use (and are willing to pay for) the Phone Dashboard time limit functionality, why isn't there higher demand for commercial digital self-control tools? Only \begin_inset Formula $\OtherBlockerUse$ \end_inset percent of our sample reported using any apps to limit their smartphone use at baseline. Potential explanations include that our experimental setting or selected set of participants overstates demand for commitment, that commercial self-cont rol tools are too expensive or are ineffective because it's too easy to evade them or substitute across devices, that people aren't aware of existing tools, that the time misallocated due to temptation is not very valuable, or that the commitment and flexibility features we built into Phone Dashboard were better suited to people's needs. We leave these questions for future work. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash leftskip=0em \end_layout \begin_layout Plain Layout \backslash parindent=-2em \end_layout \begin_layout Plain Layout \backslash onehalfspacing \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash begin{singlespace} \end_layout \end_inset \begin_inset CommandInset bibtex LatexCommand bibtex btprint "btPrintCited" bibfiles "SocialMediaEffects_Bibliography" options "jpe" \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash end{singlespace} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash parindent=2em \end_layout \begin_layout Plain Layout \backslash leftskip=0em \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Experiment Timeline and Sample Sizes \series default \begin_inset CommandInset label LatexCommand label name "tab:SampleSizes" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout Phase \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Date \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Sample size \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Recruitment and intake \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout March 22 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\shownad}$ \end_inset shown ads \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout - April 8 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\clickedonad}$ \end_inset clicked on ads \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\passedprescreen}$ \end_inset passed screen \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\consented}$ \end_inset consented \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\finishedintake}$ \end_inset finished intake survey \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Survey 1 (baseline) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout April 12 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\beganbaseline}$ \end_inset began Survey 1 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\finishedbaseline}$ \end_inset finished Survey 1 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\randomized}$ \end_inset were randomized \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Survey 2 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout May 3 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\beganmidline}$ \end_inset began Survey 2 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\informedtreat}$ \end_inset informed of treatment, of which: \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\ \ \ $ \end_inset \begin_inset Formula $\text{\informedtreatanalysis}$ \end_inset were not in MPL group \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\finishedmidline}$ \end_inset finished Survey 2 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Survey 3 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout May 24 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\beganendline}$ \end_inset began Survey 3 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\finishedendline}$ \end_inset finished Survey 3 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Survey 4 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout June 14 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\beganpostendline}$ \end_inset began Survey 4 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\finishedpostendline}$ \end_inset finished Survey 4 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Completion \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout July 26 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\kepttoend}$ \end_inset kept Phone Dashboard through July 26, of which: \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\ \ \ $ \end_inset \begin_inset Formula $\text{\kepttoendanalysis}$ \end_inset were not in MPL group ( \begin_inset Quotes eld \end_inset analysis sample \begin_inset Quotes erd \end_inset ) \end_layout \end_inset \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Sample Demographics \series default \begin_inset CommandInset label LatexCommand label name "tab:SampleDemographics" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset CommandInset include LatexCommand include filename "../input/descriptive/sample_demographics.tex" \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: Column 1 presents average demographics for our analysis sample, and column 2 presents average demographics of American adults using data from the 2018 American Community Survey. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Empirical Moments for Restricted Model Estimation \begin_inset CommandInset label LatexCommand label name "tab:Moments" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (2) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Point \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Confidence \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Parameter \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Description \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout estimate \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout interval \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{3}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Contemporaneous bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBthree}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBthreeboot}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{4}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Long-term bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfour}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfourboot}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{5}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Long-term bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfive}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfiveboot}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{2}^{L}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Limit effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauLtwo}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauLtwoboot}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $m^{C}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Control group misprediction (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\mispredict$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\mispredictboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{x}_{1}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average baseline use (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xss$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssboot$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table presents point estimates and bootstrapped 95 percent confidenc e intervals for the empirical moments used for our primary estimates of the restricted model. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Primary Parameter Estimates \begin_inset CommandInset label LatexCommand label name "tab:StructuralEstimates" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (2) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Restricted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Unrestricted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Parameter \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Description (units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout ( \begin_inset Formula $\tau_{2}^{B}=0$ \end_inset , \begin_inset Formula $\alpha=1$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout ( \begin_inset Formula $\alpha=\hat{\alpha}$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\lambda$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Habit stock effect on consumption (unitless) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\lambdareshat$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\lambdahat$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\lambdareshatboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\lambdahatboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rho$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Habit formation (unitless) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rhoreshat$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rhohat$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rhoreshatboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rhohatboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\alpha$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Projection bias (unitless) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $1$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\alphahat$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\alphahatboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\eta$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Price coefficient ($-day/hour \begin_inset Formula $^{2}$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\etareshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\etahour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\etareshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\etahourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\zeta$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Habit stock effect on marginal utility ($-day/hour \begin_inset Formula $^{2}$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\zetareshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\zetahour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\zetareshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\zetahourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gamma-\tilde{\gamma}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Naivete about temptation ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\naivetereshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\naivetehour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\naivetereshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\naivetehourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gamma$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Temptation ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectreshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammahour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectreshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammahourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{\kappa}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average intercept ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\intercepthetLeffectreshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\intercepthetLeffecthour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\intercepthetLeffectreshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\intercepthetLeffecthourboot$ \end_inset \end_layout \end_inset \end_inset \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table presents point estimates and bootstrapped 95 percent confidenc e intervals from the estimation strategy described in Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:Estimating Equations" plural "false" caps "false" noprefix "false" \end_inset and Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Estimating Equations Unrestricted" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Newpage newpage \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Online and Offline Temptation \series default \begin_inset CommandInset label LatexCommand label name "fig:OnlineOfflineTemptation" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/online_and_offline_temptation_scatter.pdf scale 115 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents responses to the following question, which we asked participants in our experiment during the baseline survey, \begin_inset Quotes eld \end_inset For each of the activities below, please tell us whether you think you do it too little, too much, or the right amount. \begin_inset Quotes erd \end_inset The bars are ordered from left to right in order of largest to smallest absolute value of (share \begin_inset Quotes eld \end_inset too little \begin_inset Quotes erd \end_inset – share \begin_inset Quotes eld \end_inset too much \begin_inset Quotes erd \end_inset ). \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Experimental Design \series default \begin_inset CommandInset label LatexCommand label name "fig:Design" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../figures/Randomization.pdf lyxscale 50 scale 50 \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Baseline Qualitative Evidence of Self-Control Problems \series default \begin_inset CommandInset label LatexCommand label name "fig:Qualitative" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/hist_limits_interest.pdf lyxscale 50 scale 50 \end_inset \begin_inset Graphics filename ../input/descriptive/hist_phone_use.pdf lyxscale 50 scale 50 \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/addiction_large.pdf lyxscale 50 scale 50 \end_inset \begin_inset Graphics filename ../input/descriptive/hist_life_betterworse.pdf lyxscale 50 scale 50 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the distributions of four measures of smartphone addiction from the baseline survey. \shape italic Interest in limits \shape default is the answer to, \begin_inset Quotes eld \end_inset How interested are you to set limits on your phone use? \begin_inset Quotes erd \end_inset \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset The bottom left panel presents the share of participants who responded \begin_inset Quotes eld \end_inset often \begin_inset Quotes erd \end_inset or \begin_inset Quotes eld \end_inset always \begin_inset Quotes erd \end_inset to each of 16 questions modified from the Mobile Phone Problem Use Scale and the Bergen Facebook Addiction Scale. \shape italic Phone use makes life worse or better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \begin_inset Quotes erd \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Treatment Effects on FITSBY Use \series default \begin_inset CommandInset label LatexCommand label name "fig:UsageEffects" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/treatment_effects_periods_limit_bonus.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents effects of the bonus and limit treatments on FITSBY use using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects on Smartphone Use by App \series default \begin_inset CommandInset label LatexCommand label name "fig:UsageEffectsByApp" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_usage_by_app.pdf scale 75 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents effects of the bonus and limit treatments on smartphone use by app using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). The bonus effects are measured in period 3, while the limit effects are measured in periods 2–5. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. FITSBY apps are in order of decreasing period 1 use. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Predicted vs. Actual FITSBY Use \series default \series bold in Control Conditions \series default \begin_inset CommandInset label LatexCommand label name "fig:PredictedVsActualControl" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/cispike_naivete_BcontrolxLcontrol_W60.pdf \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents average actual FITSBY use by period and average predicted FITSBY use for that period, for participants in the intersection of the Bonus Control and Limit Control groups. Period \begin_inset Formula $t$ \end_inset is the three weeks immediately after survey \begin_inset Formula $t$ \end_inset , so \begin_inset Quotes eld \end_inset survey \begin_inset Formula $t$ \end_inset prediction \begin_inset Quotes erd \end_inset is the prediction for period \begin_inset Formula $t$ \end_inset made just prior to period \begin_inset Formula $t$ \end_inset . FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Predicted vs. Actual Habit Formation \series default \begin_inset CommandInset label LatexCommand label name "fig:PredictedVsActualHabitFormation" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_belief_bonus_effect.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the treatment effects of the bonus on FITSBY use and on predicted FITSBY use from survey 3 using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ), as well as the average predicted bonus treatment effect elicited on survey 2 before the bonus multiple price list. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects of Limits and Bonus \series default \series bold on Survey Outcome Variables \series default \begin_inset CommandInset label LatexCommand label name "fig:Effects_SurveyOutcomes" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_self_control.pdf \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents effects of the bonus and limit treatments on survey outcome variables using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). The bonus effect is measured on survey 4, while the limit effect is measured on both surveys 3 and 4. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset \shape italic Addiction scale \shape default is answers to a battery of 16 questions modified from the Mobile Phone Problem Use Scale and the Bergen Facebook Addiction Scale. \emph on SMS addiction scale \emph default is answers to shortened versions of the addiction scale questions delivered via text message. \shape italic Phone makes life better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \begin_inset Quotes erd \end_inset \emph on Subjective well-being \emph default is answers to seven questions reflecting happiness, life satisfaction, anxiety, depression, concentration, distraction, and sleep quality; anxiety, depression, and distraction are re-oriented so that more positive reflects better subjective well-being. \emph on Survey index \emph default combines the previous five variables, weighting by the inverse of their covariance at baseline. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Model Identification \series default \begin_inset CommandInset label LatexCommand label name "fig:TemptationIdentification" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../figures/TemptationIdentification 1.pdf lyxscale 60 scale 55 \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects of Temptation and Habit Formation on FITSBY Use \series default \begin_inset CommandInset label LatexCommand label name "fig:Counterfactuals" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/structural/structural_decomposition_plot_boot.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents point estimates and bootstrapped 95 percent confidence intervals for predicted steady-state FITSBY use with different parameter assumptions, using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS counterfactual" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Newpage pagebreak \end_inset \end_layout \begin_layout Standard \start_of_appendix \align center \begin_inset ERT status open \begin_layout Plain Layout \backslash doparttoc \end_layout \begin_layout Plain Layout \backslash faketableofcontents \end_layout \begin_layout Plain Layout \backslash part{Online Appendix} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset VSpace bigskip \end_inset \end_layout \begin_layout Standard \align center \size larger Digital Addiction \end_layout \begin_layout Standard \align center \emph on Hunt Allcott, Matthew Gentzkow, and Lena Song \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash setcounter{figure}{0} \backslash renewcommand{ \backslash thefigure}{A \backslash arabic{figure}} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash setcounter{table}{0} \backslash renewcommand{ \backslash thetable}{A \backslash arabic{table}} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash pagestyle{fancy} \end_layout \begin_layout Plain Layout \backslash lhead{Online Appendix} \backslash rhead{Allcott, Gentzkow, and Song} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Newpage newpage \end_inset \end_layout \begin_layout Standard \paragraph_spacing single \begin_inset ERT status open \begin_layout Plain Layout \backslash parttoc \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Newpage newpage \end_inset \end_layout \begin_layout Section Experimental Design Appendix \begin_inset CommandInset label LatexCommand label name "app:Design" \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Facebook Recruitment Ads \series default \begin_inset CommandInset label LatexCommand label name "fig:RecruitmentAds" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../figures/Recruitment ad.png lyxscale 50 scale 50 \end_inset \begin_inset Formula $\ \ \ $ \end_inset \begin_inset Graphics filename ../figures/Recruitment ad - older.png lyxscale 50 scale 50 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: The ads at left and right were shown to users aged 18–34 and 35–64, respectively. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Phone Dashboard Screenshots \series default \begin_inset CommandInset label LatexCommand label name "fig:PDScreenshots" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../figures/PDScreenshots1.pdf lyxscale 50 scale 50 \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../figures/PDScreenshots2_X.pdf lyxscale 50 scale 50 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents screenshots of the Phone Dashboard app. The top left presents the day's total usage by app. The top middle shows how a user can set daily a daily usage limit for each app, effective tomorrow. The top right shows the usage limits set for each app. The bottom left shows the warning users receive when they are within five minutes or one minute of their limit. The bottom middle shows the message users receive when they reach the limit. Users with the snooze functionality can resume using an app after a delay of \begin_inset Formula $X\in\{0,2,5,20\}$ \end_inset minutes. The bottom right shows the option for a user to choose how many additional minutes to add to the daily limit after the snooze delay. All participants had the usage information in the top left panel, while only the Limit group had the time limit functionalities in the other panels. \end_layout \end_inset \end_layout \begin_layout Subsection Variable Definitions \begin_inset CommandInset label LatexCommand label name "app:VariableDefinitions" \end_inset \end_layout \begin_layout Standard \series bold \emph on Ideal use change \series default \emph default . Some people say they use their smartphone too much and ideally would use it less. Other people are happy with their usage or would ideally use it more. How do you feel about your overall smartphone use over the past 3 weeks? \end_layout \begin_layout Itemize I used my smartphone too much. \end_layout \begin_layout Itemize I used my smartphone the right amount. \end_layout \begin_layout Itemize I used my smartphone too little. \end_layout \begin_layout Standard Relative to your actual use over the past 3 weeks, by how much would you ideally have [if “too much”: reduced. If “too little”: increased] your smartphone use? Please give a number in percent. ____ % \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold \emph on Addiction scale. \emph default \series default Over the past 3 weeks, how often have you… \end_layout \begin_layout Itemize Been worried about missing out on things online when not checking your phone? \end_layout \begin_layout Itemize Checked social media, text messages, or email immediately after waking up? \end_layout \begin_layout Itemize Used your phone longer than intended? \end_layout \begin_layout Itemize Found yourself saying “just a few more minutes” when using your phone? \end_layout \begin_layout Itemize Used your phone to distract yourself from personal problems? \end_layout \begin_layout Itemize Used your phone to distract yourself from feelings of guilt, anxiety, helplessne ss, or depression? \end_layout \begin_layout Itemize Used your phone to relax in order to go to sleep? \end_layout \begin_layout Itemize Tried to reduce your phone use without success? \end_layout \begin_layout Itemize Experienced that people close to you are concerned about the amount of time you use your phone? \end_layout \begin_layout Itemize Felt anxious when you don’t have your phone? \end_layout \begin_layout Itemize Found it difficult to switch off or put down your phone? \end_layout \begin_layout Itemize Been annoyed or bothered when people interrupt you while you use your phone? \end_layout \begin_layout Itemize Felt your performance in school or at work suffers because of the amount of time you use your phone? \end_layout \begin_layout Itemize Lost sleep due to using your phone late at night? \end_layout \begin_layout Itemize Preferred to use your phone rather than interacting with your partner, friends, or family? \end_layout \begin_layout Itemize Put off things you have to do by using your phone? \end_layout \begin_layout Standard \emph on Never, Rarely, Sometimes, Often, Always \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold \emph on SMS addiction scale. \end_layout \begin_layout Itemize In the past 24 hours, did you use your phone longer than intended? \end_layout \begin_layout Itemize In the past 24 hours, did your performance at school or work suffer because of the amount of time you used your phone? \end_layout \begin_layout Itemize In the past 24 hours, did you feel like you had an easy time controlling your screen time? \end_layout \begin_layout Itemize In the past 24 hours, did you use your phone mindlessly? \end_layout \begin_layout Itemize In the past 24 hours, did you use your phone because you were feeling down? \end_layout \begin_layout Itemize In the past 24 hours, did using your phone keep you from working on something you needed to do? \end_layout \begin_layout Itemize In the past 24 hours, would you ideally have used your phone less? \end_layout \begin_layout Itemize Last night, did you lose sleep because of using your phone late at night? \end_layout \begin_layout Itemize When you woke up today, did you immediately check social media, text messages, or email? \end_layout \begin_layout Standard \emph on Please text back your answer on a scale from 1 (not at all) to 10 (definitely). \emph default \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold \emph on Phone makes life better. \series default \emph default To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \end_layout \begin_layout Standard \emph on 11-point scale from -5 (Makes my life worse) to 0 (Neutral) to 5 (Makes my life better) \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold \emph on Subjective well-being. \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none Please tell us the extent to which you agree or disagree with each of the following statements. Over the past 3 weeks, ... \end_layout \begin_layout Itemize \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none … I was a happy person \end_layout \begin_layout Itemize \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none … I was satisfied with my life \end_layout \begin_layout Itemize \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none … I felt anxious \end_layout \begin_layout Itemize \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none … I felt depressed \end_layout \begin_layout Itemize \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none … I could concentrate on what I was doing \end_layout \begin_layout Itemize \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none … I was easily distracted \end_layout \begin_layout Itemize \family roman \series medium \shape up \size normal \emph off \bar no \strikeout off \xout off \uuline off \uwave off \noun off \color none … I slept well \end_layout \begin_layout Standard \emph on 7-point scale from strongly disagree to neutral to strongly agree \end_layout \begin_layout Section Data Appendix \begin_inset CommandInset label LatexCommand label name "app:DataAppendix" \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Response Rates \begin_inset CommandInset label LatexCommand label name "tab:Attrition" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \size small (a) \series bold Limit \series default \begin_inset CommandInset include LatexCommand include filename "../input/descriptive/attrition_limit.tex" \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \align center \size small (b) \series bold Bonus \end_layout \begin_layout Plain Layout \align center \size small \begin_inset CommandInset include LatexCommand include filename "../input/descriptive/attrition_bonus.tex" \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: Columns 1 and 2 of Panel (a) present present response rates for Limit and Limit Control groups. Columns 3–7 present response rates for each of the snooze delay conditions within the Limit group. Column 8 presents the p-value of an F-test of differences between the Limit Control and the separate snooze delay conditions. Columns 1 and 2 of Panel (b) present response rates for Bonus and Bonus Control groups. Column 3 presents the p-value of a t-test of differences between the Bonus and Bonus Control groups. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Covariate Balance \begin_inset CommandInset label LatexCommand label name "tab:Balance" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \shape italic \size small \emph on (a) \series bold Limit \end_layout \begin_layout Plain Layout \align center \size small \begin_inset CommandInset include LatexCommand input filename "../input/descriptive/balance_limit.tex" \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \align center \shape italic \size small \emph on (b) \series bold Bonus \end_layout \begin_layout Plain Layout \align center \size small \begin_inset CommandInset include LatexCommand input filename "../input/descriptive/balance_bonus.tex" \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: Panels (a) and (b) present tests of covariate balance for the Limit and Bonus treatment and control groups. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Most Popular Apps \series default \begin_inset CommandInset label LatexCommand label name "fig:PopularApps" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/bar_share_use_by_app.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the share of users that have each app and the average daily screen time in period 1 (baseline). Period 1 use is across all users, not conditioning on whether or not they have the app. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Distribution of Baseline FITSBY Use \series default \begin_inset CommandInset label LatexCommand label name "fig:FITSBYUsageDist" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/hist_baseline_usage_fitsby.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents a distribution of FITSBY use in period 1 (baseline). FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Descriptive Statistics for Survey Outcome Variables \begin_inset CommandInset label LatexCommand label name "tab:DescriptiveStats" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \size small \begin_inset CommandInset include LatexCommand input filename "../input/descriptive/baseline_welfare.tex" \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table present descriptive statistics for the survey outcome variables at baseline. \end_layout \end_inset \end_layout \begin_layout Section Differences Between 2019 and the Study Period \begin_inset CommandInset label LatexCommand label name "app:2019" \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects of Coronavirus Outbreak on Free Time \series default \begin_inset CommandInset label LatexCommand label name "fig:covid_FreeTime" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/hist_covid.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the distribution of responses to the baseline survey question, \begin_inset Quotes eld \end_inset To what extent has the recent coronavirus outbreak changed how much free time you have? \begin_inset Quotes erd \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects of Coronavirus on Smartphone Use \series default \begin_inset CommandInset label LatexCommand label name "fig:covid_PhoneUse" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/hist_covid_reason.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: The baseline survey asked, \begin_inset Quotes eld \end_inset How has the recent coronavirus outbreak changed how you use your smartphone? \begin_inset Quotes erd \end_inset We coded the responses as to whether they indicated increased, decreased, or unchanged smartphone use. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Self-Control Problems in 2019 versus Now \series default \begin_inset CommandInset label LatexCommand label name "fig:BonusValuationDist-1-2" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/cispike_covid.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the mean (dots) and 25th and 75th percentiles (spikes) of responses to \emph on ideal use change \emph default and \emph on phone use makes life better \emph default for 2019 and for the past 3 weeks, as reported on the baseline survey. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use [in 2019 / over the past 3 weeks], by how much would you ideally have [reduced/increased] your screen time? \shape italic Phone use makes life better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse [in 2019 / over the past 3 weeks]? \begin_inset Quotes erd \end_inset \end_layout \end_inset \end_layout \begin_layout Section Model-Free Results Appendix \begin_inset CommandInset label LatexCommand label name "app:ModelFree" \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Ideal Use Change by App or Category \series default \begin_inset CommandInset label LatexCommand label name "fig:IdealChangeByApp" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/overuse_by_app.pdf scale 75 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents mean \shape italic ideal use change \shape default by app or app category at baseline. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset We code \begin_inset Quotes eld \end_inset I don't use this app at all \begin_inset Quotes erd \end_inset as 0, so these results reflect how much each app contributes to overall temptation, not how tempting each app is for the subset of people who use it. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects of Bonus on FITSBY Use by Day for Periods 1 and 2 \series default \begin_inset CommandInset label LatexCommand label name "fig:BonusEffectsByDay" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_usage_simple_daily_p12_fitsby.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents differences in average FITSBY use between the Bonus and Bonus Control group for each day of periods 1 and 2. The vertical line indicates the day of survey 2, when the bonus was announced. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects of Bonus on FITSBY Use by Week \series default \begin_inset CommandInset label LatexCommand label name "fig:BonusEffectsByWeek" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/treatment_effects_weeks_bonus.pdf scale 85 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents effects of the bonus treatment on FITBSY use by week using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Distribution of User-Level Limit Tightness \series default \begin_inset CommandInset label LatexCommand label name "fig:TightnessDist" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/hist_limit_tight.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents mean \shape italic user-level limit tightness \shape default over periods 2–5. \shape italic User-level limit tightness \shape default is the amount by which a user's limits would have hypothetically reduced overall screen time if applied to their baseline use without snoozes; see equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:tightness" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Average Limit Tightness by App \series default \begin_inset CommandInset label LatexCommand label name "fig:TightnessByApp" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/cispike_limit_tight_combined_by_app.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents average \shape italic limit tightness \shape default by app over periods 2–5. \emph on Limit tightness \emph default is the amount by which a user's limits would have hypothetically reduced screen time if applied to their baseline use without snoozes; see equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:tightness" plural "false" caps "false" noprefix "false" \end_inset ). FITSBY apps are in order of decreasing period 1 use. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Interaction Effects of Bonus and Limit by Period \series default \begin_inset CommandInset label LatexCommand label name "fig:Interactions" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/interaction_treatment_effects.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents effects of bonus and limit treatments on FITSBY use using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ) with an additional interaction term for participants in the intersection of the Limit and Bonus groups. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects on Self-Reported FITSBY Use Change on Other Devices \series default \begin_inset CommandInset label LatexCommand label name "fig:SubstitutionEffect" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_self_reported_substitution_w.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the effects of bonus and limit treatments on self-reported change in FITSBY use on other devices relative to the three weeks before the study using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. Self-reported changes are winsorized at 150 minutes. \end_layout \end_inset \end_layout \begin_layout Subsection Validation of Predicted Use and Multiple Price List Responses \begin_inset CommandInset label LatexCommand label name "app:Validation" \end_inset \end_layout \begin_layout Standard Predicted use lines up well with actual use; see Appendix Figures \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedActualControl" plural "false" caps "false" noprefix "false" \end_inset and \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictVsActualHist" plural "false" caps "false" noprefix "false" \end_inset . The $5 (instead of $1) prediction accuracy reward slightly reduces the absolute value of the prediction error but has tightly estimated zero effects on predicted use, actual use, and the level of the prediction error; see Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:AccuracyReward" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Standard Multiple price lists are cognitively challenging, so we carry out several additional analyses to validate that these valuations are informative about people's preferences. First, participants' valuations of the bonus are correlated with the amount of money they could expect to earn; see Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:BonusValuationVsPredictedEarnings" plural "false" caps "false" noprefix "false" \end_inset . Second, the limit valuation and the behavior change premium (defined in Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:Estimating Equations Unrestricted" plural "false" caps "false" noprefix "false" \end_inset ) are correlated with each other and with \emph on limit tightness \emph default , \emph on ideal use change \emph default , \emph on addiction scale \emph default , \emph on SMS addiction scale \emph default , and other variables in expected ways; see Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:motivation_correlation" plural "false" caps "false" noprefix "false" \end_inset . Third, after the bonus MPL, we asked people to \begin_inset Quotes eld \end_inset select the statement that best describes your thinking when trading off the Screen Time Bonus against the fixed payment. \begin_inset Quotes erd \end_inset \begin_inset Formula $\MPLwishreduce$ \end_inset percent responded that \begin_inset Quotes eld \end_inset I wanted to give myself an incentive to use my phone less over the next three weeks, even though it might result in a smaller payment, \begin_inset Quotes erd \end_inset and this group had a substantially higher average behavior change premium; see Appendix Figures \begin_inset CommandInset ref LatexCommand ref reference "fig:BonusMPLReasoning" plural "false" caps "false" noprefix "false" \end_inset and \begin_inset CommandInset ref LatexCommand ref reference "fig:BehaviorChangePremiumbyReasoning" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Predicted vs. Actual FITSBY Use in Control \series default \begin_inset CommandInset label LatexCommand label name "fig:PredictedActualControl" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/heatmap_usage.pdf \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the number of Control group participants in each cell of actual and predicted FITSBY use across periods 2–4, using predictions from the survey just before each period. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Histogram of Actual Minus Predicted FITSBY Use in Control Group \series default \begin_inset CommandInset label LatexCommand label name "fig:PredictVsActualHist" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/histogram_predicted_actual_p24.pdf \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the distribution of the difference between actual and predicted FITSBY use across periods 2–4 in the Control group, using predictions from the survey just before each period. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effect of Prediction Accuracy Reward \series default \begin_inset CommandInset label LatexCommand label name "tab:AccuracyReward" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset CommandInset include LatexCommand include filename "../input/treatment_effects/high_reward_reg.tex" \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \align left \size small Notes: This table presents the effects of being offered the higher Prediction Reward ($5 instead of $1 for predicting within 15 minutes of actual screen time) on predicted and actual FITSBY use in minutes per day. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. Standard errors are in parentheses. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Valuation of Limit Functionality \series default \begin_inset CommandInset label LatexCommand label name "fig:LimitValuationDist" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/hist_limit_wtp.pdf \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the distribution of valuations of access to the limit functionality for the next three weeks, as elicited in a multiple price list on survey 3. Valuations above $20 are plotted at $25, and valuations below $-1 are plotted at $-5. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Valuation of Screen Time Bonus \series default \begin_inset CommandInset label LatexCommand label name "fig:BonusValuationDist" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/hist_rsi_wtp.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the distribution of valuations of the Screen Time Bonus incentive, as elicited on survey 2. Valuations above $150 are plotted at $175. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Valuation of Bonus vs. Predicted Bonus Earnings \series default \begin_inset CommandInset label LatexCommand label name "fig:BonusValuationVsPredictedEarnings" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/heatmap_wtp_prediction.pdf \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the number of participants in each cell of predicted earnings from the Screen Time Bonus (given the participant's Bonus Benchmark and predicted FITSBY use) and valuation of the bonus, as elicited on survey 2. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset include LatexCommand include filename "../output/motivation_correlation_filled.lyx" \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Reported Reasoning on Screen Time Bonus Multiple Price List \series default \begin_inset CommandInset label LatexCommand label name "fig:BonusMPLReasoning" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/hist_motivation_mpl.pdf \end_inset \end_layout \begin_layout Plain Layout \size small Notes: After the bonus multiple price list, survey 2 asked participants to \begin_inset Quotes eld \end_inset select the statement that best describes your thinking when trading off the Screen Time Bonus against the fixed payment. \begin_inset Quotes erd \end_inset This figure presents the share of participants who selected each answer. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Behavior Change Premium by Reported Reasoning \series default \begin_inset CommandInset label LatexCommand label name "fig:BehaviorChangePremiumbyReasoning" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/descriptive/cispike_motivation_reason.pdf \end_inset \end_layout \begin_layout Plain Layout \size small Notes: The behavior change premium is the difference between the valuation of the Screen Time Bonus and the modeled valuation if the consumer believed herself to be time consistent. After the bonus multiple price list, survey 2 asked participants to \begin_inset Quotes eld \end_inset select the statement that best describes your thinking when trading off the Screen Time Bonus against the fixed payment. \begin_inset Quotes erd \end_inset This figure presents means and 95 percent confidence intervals of the behavior change premium by responses to that question. \end_layout \end_inset \end_layout \begin_layout Subsection Additional Estimates of Effects on Survey Outcome Variables \begin_inset CommandInset label LatexCommand label name "app:SurveyOutcomeEffects" \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects of Limits and Bonus \series default \series bold on Survey Outcomes on Surveys 3 and 4 \series default \begin_inset CommandInset label LatexCommand label name "fig:Effects_SurveyOutcomes_Null" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_self_control_null.pdf \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents effects of the bonus and limit treatment on survey outcome variables using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ), allowing separate coefficients for effects on surveys 3 vs. 4. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset \shape italic Addiction scale \shape default is answers to a battery of 16 questions modified from the Mobile Phone Problem Use Scale and the Bergen Facebook Addiction Scale. \emph on SMS addiction scale \emph default is answers to shortened versions of the addiction scale questions delivered via text message. \shape italic Phone makes life better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \begin_inset Quotes erd \end_inset \emph on Subjective well-being \emph default is answers to seven questions reflecting happiness, life satisfaction, anxiety, depression, concentration, distraction, and sleep quality; anxiety, depression, and distraction are re-oriented so that more positive reflects better subjective well-being. \emph on Survey index \emph default combines the previous five variables, weighting by the inverse of their covariance at baseline. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Treatment Effects \series default \begin_inset CommandInset label LatexCommand label name "tab:LATEs" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \shape italic \size small \emph on (a) \series bold Bonus \end_layout \begin_layout Plain Layout \align center \begin_inset VSpace smallskip \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (2) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (3) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (4) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (5) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (6) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Treatment \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Treatment \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout P-value \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Sharpened \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout effect \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout error \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout effect \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout error \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout FDR- \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (original \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (original \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (SD \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (SD \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout adjusted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout q-value \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Ideal use change \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bonusidealcoef$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bonusidealse$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bonusidealcoefn$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bonusidealsen$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\pvaluebonusideal$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\qadjbonusphonechange$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Addiction scale x (-1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bonusaddictioncoef$ \end_inset 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\end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \align center \shape italic \size small \emph on (b) \series bold Limit \end_layout \begin_layout Plain Layout \align center \begin_inset VSpace smallskip \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (2) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (3) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (4) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (5) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (6) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 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\begin_inset Text \begin_layout Plain Layout (original \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (original \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (SD \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (SD \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout adjusted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout q-value \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Ideal use change \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitidealcoef$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitidealse$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitidealcoefn$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitidealsen$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\pvallimitideal$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\qadjlimitphonechange$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Addiction scale x (-1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitaddictioncoef$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitaddictionse$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitaddictioncoefn$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitaddictionsen$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\pvallimitaddict$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\qadjlimitaddictionindex$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout SMS addiction scale x (-1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitsmscoef$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitsmsse$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitsmscoefn$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitsmssen$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\pvallimitsmsindex$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\qadjlimitsmsindex$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Phone makes life better \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitlifebettercoef$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitlifebetterse$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitlifebettercoefn$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitlifebettersen$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\pvallimitlifebetter$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\qadjlimitlifebetter$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Subjective well-being \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitswbindexcoef$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitswbindexse$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitswbindexcoefn$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitswbindexsen$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\pvallimitswbindex$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\qadjlimitswbindex$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Survey index \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitindexwellcoef$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitindexwellse$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitindexwellcoefn$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\limitindexwellsen$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\pvallimitindexwell$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\qadjlimitindexwell$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table presents effects of the bonus and limit treatments on survey outcome variables using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). The bonus effect is measured on survey 4, while the limit effect is measured on both surveys 3 and 4. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset \shape italic Addiction scale \shape default is answers to a battery of 16 questions modified from the Mobile Phone Problem Use Scale and the Bergen Facebook Addiction Scale. \emph on SMS addiction scale \emph default is answers to shortened versions of the addiction scale questions delivered via text message. \shape italic Phone makes life better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \begin_inset Quotes erd \end_inset \emph on Subjective well-being \emph default is answers to seven questions reflecting happiness, life satisfaction, anxiety, depression, concentration, distraction, and sleep quality; anxiety, depression, and distraction are re-oriented so that more positive reflects better subjective well-being. \emph on Survey index \emph default combines the previous five variables, weighting by the inverse of their covariance at baseline. The effects in standard deviation units in column 3 match those reported on Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Effects_SurveyOutcomes" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects on Addiction Responses \series default \begin_inset CommandInset label LatexCommand label name "fig:AddictionEffects" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_addiction_simple.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the effects of the bonus and limit treatments on individual items in the \emph on addiction scale \emph default variable using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). The bonus effect is measured on survey 4, while the limit effect is measured on both surveys 3 and 4. The direction of the effects in this figure are opposite those in the main figures, because \emph on addiction scale \emph default is multiplied by -1 in those figures. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects on SMS Addiction Responses \series default \begin_inset CommandInset label LatexCommand label name "fig:SMSAddictionEffects" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_sms_addiction_simple.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the effects of the bonus and limit treatments on individual items in the \emph on SMS addiction scale \emph default variable using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). The bonus effect is measured on survey 4, while the limit effect is measured on both surveys 3 and 4. The direction of the effects in this figure are opposite those in the main figures, because \emph on SMS addiction scale \emph default is multiplied by -1 in those figures. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects on Subjective Well-Being Responses \series default \begin_inset CommandInset label LatexCommand label name "fig:SWBEffects" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_swb_simple.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents the effects of the bonus and limit treatments on individual items in the \emph on subjective well-being \emph default variable using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). The bonus effect is measured on survey 4, while the limit effect is measured on both surveys 3 and 4. \end_layout \end_inset \end_layout \begin_layout Subsection Heterogeneous Treatment Effects \begin_inset CommandInset label LatexCommand label name "app:Heterogeneity" \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Heterogeneous Effects of Limits and Bonus \series default \series bold on Survey Index \series default \begin_inset CommandInset label LatexCommand label name "fig:Heterogeneity_SurveyOutcomes" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Bonus \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_heterogenous_bonus_itt_welfare.pdf scale 52 \end_inset \end_layout \end_inset \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Limit \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_heterogenous_limit_itt_welfare.pdf scale 52 \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents heterogeneous effects of the bonus and limit treatments on \emph on survey index \emph default , the inverse-covariance weighted average of five measures of smartphone addiction and subjective well-being, using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). The bonus effect is measured on survey 4, while the limit effect is measured on both surveys 3 and 4. Above-median education includes people with a college degree or more, above-med ian age includes people \begin_inset Formula $\text{\MedianAge}$ \end_inset and older, and median baseline FITSBY use is \begin_inset Formula $\text{\MedianFITSBYUsage}$ \end_inset minutes per day. \emph on Restriction index \emph default is a combination of \emph on interest in limits \emph default and \emph on ideal use change \emph default . \emph on Addiction index \emph default is a combination of \emph on addiction scale \emph default and \emph on phone makes life better \emph default . \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Heterogeneous Effects of Limits and Bonus on FITSBY Use \series default \begin_inset CommandInset label LatexCommand label name "fig:Heterogeneity_Usage" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align left \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Bonus \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_bonus_usage_by_heterogeneity_P3_fitsby.pdf scale 52 \end_inset \end_layout \end_inset \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Limit \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_limit_usage_by_heterogeneity_fitsby.pdf scale 52 \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents heterogeneous effects of the bonus and limit treatments on FITSBY use using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ). The bonus effects are measured in period 3, while the limit effects are measured in periods 2–5. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. Above-median education includes people with a college degree or more, above-med ian age includes people \begin_inset Formula $\text{\MedianAge}$ \end_inset and older, and median baseline FITSBY use is \begin_inset Formula $\text{\MedianFITSBYUsage}$ \end_inset minutes per day. \emph on Restriction index \emph default is a combination of \emph on interest in limits \emph default and \emph on ideal use change \emph default . \emph on Addiction index \emph default is a combination of \emph on addiction scale \emph default and \emph on phone makes life better \emph default . \end_layout \end_inset \end_layout \begin_layout Subsection Local Average Treatment Effects on Survey Outcomes \begin_inset CommandInset label LatexCommand label name "app:LATEs" \end_inset \end_layout \begin_layout Standard Our pre-analysis plan specified that we would also estimate instrumental variables (IV) regressions with previous period FITSBY use \begin_inset Formula $x_{i,t-1}$ \end_inset as the endogenous variable: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} Y_{it}=\tau x_{i,t-1}+\beta_{t}\boldsymbol{X}_{i1}+\nu_{it}+\varepsilon_{i},\label{eq:LATE} \end{equation} \end_inset instrumenting for \begin_inset Formula $x_{i,t-1}$ \end_inset with \begin_inset Formula $B_{i}$ \end_inset and \begin_inset Formula $L_{i}$ \end_inset interacted with \begin_inset Formula $t=3$ \end_inset and \begin_inset Formula $t=4$ \end_inset indicators. We combine data from surveys 3 and 4 and let all coefficients other than \begin_inset Formula $\tau$ \end_inset vary across the two periods. Conceptually, this regression combines the effects of the bonus and limit intervention, weighting the interventions by their effects on FITSBY use. Because the limit treatment could affect survey outcomes through channels other than reduced FITSBY use—for example, by giving people an increased feeling of control over their screen time—we do not claim that the IV exclusion restriction necessarily holds. \end_layout \begin_layout Standard Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:LATEs" plural "false" caps "false" noprefix "false" \end_inset presents local average treatment effects estimated using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:LATE" plural "false" caps "false" noprefix "false" \end_inset ), combining effects from both treatments. Appendix Figures \begin_inset CommandInset ref LatexCommand ref reference "fig:HetEduc" plural "false" caps "false" noprefix "false" \end_inset – \begin_inset CommandInset ref LatexCommand ref reference "fig:HetAddictionIndex" plural "false" caps "false" noprefix "false" \end_inset study heterogeneity along the six pre-specified moderators. The results are qualitatively similar to Figures \begin_inset CommandInset ref LatexCommand ref reference "fig:Effects_SurveyOutcomes" plural "false" caps "false" noprefix "false" \end_inset and \begin_inset CommandInset ref LatexCommand ref reference "fig:Heterogeneity_SurveyOutcomes" plural "false" caps "false" noprefix "false" \end_inset , except that the estimates are slightly more precise, as would be expected from combining effects of two interventions. Note that since the average effects of both interventions are about the same for people with low versus high baseline use (Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Heterogeneity_SurveyOutcomes" plural "false" caps "false" noprefix "false" \end_inset ), the local average treatment effects of reduced use are much larger for people with low baseline use (Appendix Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:HetUsage" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Local Average Treatment Effects of FITSBY Use on Survey Outcome Variables \series default \begin_inset CommandInset label LatexCommand label name "fig:LATEs" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_iv_self_control.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents local average treatment effects of FITSBY use on survey outcome variables using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:LATE" plural "false" caps "false" noprefix "false" \end_inset ). We instrument for FITSBY use with Bonus and Limit group indicators interacted with period indicators. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset \shape italic Addiction scale \shape default is answers to a battery of 16 questions modified from the Mobile Phone Problem Use Scale and the Bergen Facebook Addiction Scale. \emph on SMS addiction scale \emph default is answers to shortened versions of the addiction scale questions delivered via text message. \shape italic Phone makes life better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \begin_inset Quotes erd \end_inset \emph on Subjective well-being \emph default is answers to seven questions reflecting happiness, life satisfaction, anxiety, depression, concentration, distraction, and sleep quality; anxiety, depression, and distraction are re-oriented so that more positive reflects better subjective well-being. \emph on Survey index \emph default combines the previous five variables, weighting by the inverse of their covariance at baseline. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Heterogeneous Effects on Survey Outcome Variables by Education \series default \begin_inset CommandInset label LatexCommand label name "fig:HetEduc" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_iv_self_control_by_educ.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents local average treatment effects of FITSBY use on survey outcome variables using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:LATE" plural "false" caps "false" noprefix "false" \end_inset ), for above- and below-median education. We instrument for FITSBY use with Bonus and Limit group indicators interacted with period indicators. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset \shape italic Addiction scale \shape default is answers to a battery of 16 questions modified from the Mobile Phone Problem Use Scale and the Bergen Facebook Addiction Scale. \emph on SMS addiction scale \emph default is answers to shortened versions of the addiction scale questions delivered via text message. \shape italic Phone makes life better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \begin_inset Quotes erd \end_inset \emph on Subjective well-being \emph default is answers to seven questions reflecting happiness, life satisfaction, anxiety, depression, concentration, distraction, and sleep quality; anxiety, depression, and distraction are re-oriented so that more positive reflects better subjective well-being. \emph on Survey index \emph default combines the previous five variables, weighting by the inverse of their covariance at baseline. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Heterogeneous Effects on Survey Outcome Variables by Age \series default \begin_inset CommandInset label LatexCommand label name "fig:HetAge" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_iv_self_control_by_age.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents local average treatment effects of FITSBY use on survey outcome variables using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:LATE" plural "false" caps "false" noprefix "false" \end_inset ), for above- and below-median age. We instrument for FITSBY use with Bonus and Limit group indicators interacted with period indicators. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset \shape italic Addiction scale \shape default is answers to a battery of 16 questions modified from the Mobile Phone Problem Use Scale and the Bergen Facebook Addiction Scale. \emph on SMS addiction scale \emph default is answers to shortened versions of the addiction scale questions delivered via text message. \shape italic Phone makes life better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \begin_inset Quotes erd \end_inset \emph on Subjective well-being \emph default is answers to seven questions reflecting happiness, life satisfaction, anxiety, depression, concentration, distraction, and sleep quality; anxiety, depression, and distraction are re-oriented so that more positive reflects better subjective well-being. \emph on Survey index \emph default combines the previous five variables, weighting by the inverse of their covariance at baseline. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Heterogeneous Effects on Survey Outcome Variables by Gender \series default \begin_inset CommandInset label LatexCommand label name "fig:HetGender" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_iv_self_control_by_gender.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents local average treatment effects of FITSBY use on survey outcome variables using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:LATE" plural "false" caps "false" noprefix "false" \end_inset ), for men versus women. We instrument for FITSBY use with Bonus and Limit group indicators interacted with period indicators. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset \shape italic Addiction scale \shape default is answers to a battery of 16 questions modified from the Mobile Phone Problem Use Scale and the Bergen Facebook Addiction Scale. \emph on SMS addiction scale \emph default is answers to shortened versions of the addiction scale questions delivered via text message. \shape italic Phone makes life better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \begin_inset Quotes erd \end_inset \emph on Subjective well-being \emph default is answers to seven questions reflecting happiness, life satisfaction, anxiety, depression, concentration, distraction, and sleep quality; anxiety, depression, and distraction are re-oriented so that more positive reflects better subjective well-being. \emph on Survey index \emph default combines the previous five variables, weighting by the inverse of their covariance at baseline. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Heterogeneous Effects on Survey Outcome Variables by Baseline FITSBY Use \series default \begin_inset CommandInset label LatexCommand label name "fig:HetUsage" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_iv_self_control_by_usage.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents local average treatment effects of FITSBY use on survey outcome variables using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:LATE" plural "false" caps "false" noprefix "false" \end_inset ), for above- and below-median baseline FITSBY use. We instrument for FITSBY use with Bonus and Limit group indicators interacted with period indicators. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset \shape italic Addiction scale \shape default is answers to a battery of 16 questions modified from the Mobile Phone Problem Use Scale and the Bergen Facebook Addiction Scale. \emph on SMS addiction scale \emph default is answers to shortened versions of the addiction scale questions delivered via text message. \shape italic Phone makes life better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \begin_inset Quotes erd \end_inset \emph on Subjective well-being \emph default is answers to seven questions reflecting happiness, life satisfaction, anxiety, depression, concentration, distraction, and sleep quality; anxiety, depression, and distraction are re-oriented so that more positive reflects better subjective well-being. \emph on Survey index \emph default combines the previous five variables, weighting by the inverse of their covariance at baseline. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Heterogeneous Effects on Survey Outcome Variables by Restriction Index \series default \begin_inset CommandInset label LatexCommand label name "fig:HetRestrictionIndex" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_iv_self_control_by_restrict.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents local average treatment effects of FITSBY use on survey outcome variables using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:LATE" plural "false" caps "false" noprefix "false" \end_inset ), for above- and below-median values of \emph on restriction index \emph default , a combination of \emph on interest in limits \emph default and \emph on ideal use change \emph default . We instrument for FITSBY use with Bonus and Limit group indicators interacted with period indicators. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset \shape italic Addiction scale \shape default is answers to a battery of 16 questions modified from the Mobile Phone Problem Use Scale and the Bergen Facebook Addiction Scale. \emph on SMS addiction scale \emph default is answers to shortened versions of the addiction scale questions delivered via text message. \shape italic Phone makes life better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \begin_inset Quotes erd \end_inset \emph on Subjective well-being \emph default is answers to seven questions reflecting happiness, life satisfaction, anxiety, depression, concentration, distraction, and sleep quality; anxiety, depression, and distraction are re-oriented so that more positive reflects better subjective well-being. \emph on Survey index \emph default combines the previous five variables, weighting by the inverse of their covariance at baseline. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Heterogeneous Effects on Survey Outcome Variables by Addiction Index \series default \begin_inset CommandInset label LatexCommand label name "fig:HetAddictionIndex" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/coef_iv_self_control_by_life.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This figure presents local average treatment effects of FITSBY use on survey outcome variables using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:LATE" plural "false" caps "false" noprefix "false" \end_inset ), for above- and below-median values of \emph on addiction index \emph default , a combination of \emph on addiction scale \emph default and \emph on phone makes life better \emph default . We instrument for FITSBY use with Bonus and Limit group indicators interacted with period indicators. FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browsers, and YouTube. \shape italic Ideal use change \emph on is the answer to \shape default \emph default , \begin_inset Quotes eld \end_inset Relative to your actual use over the past 3 weeks, by how much would you ideally have [reduced/increased] your screen time? \begin_inset Quotes erd \end_inset \shape italic Addiction scale \shape default is answers to a battery of 16 questions modified from the Mobile Phone Problem Use Scale and the Bergen Facebook Addiction Scale. \emph on SMS addiction scale \emph default is answers to shortened versions of the addiction scale questions delivered via text message. \shape italic Phone makes life better \shape default is the answer to, \begin_inset Quotes eld \end_inset To what extent do you think your smartphone use made your life better or worse over the past 3 weeks? \begin_inset Quotes erd \end_inset \emph on Subjective well-being \emph default is answers to seven questions reflecting happiness, life satisfaction, anxiety, depression, concentration, distraction, and sleep quality; anxiety, depression, and distraction are re-oriented so that more positive reflects better subjective well-being. \emph on Survey index \emph default combines the previous five variables, weighting by the inverse of their covariance at baseline. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Newpage newpage \end_inset \end_layout \begin_layout Standard \begin_inset Newpage clearpage \end_inset \end_layout \begin_layout Section Unrestricted Model and Alternative Temptation Estimates \begin_inset CommandInset label LatexCommand label name "app:UnrestrictedModel" \end_inset \end_layout \begin_layout Standard In this appendix, we estimate the unrestricted model and present alternative estimates of the temptation parameter \begin_inset Formula $\gamma$ \end_inset . \end_layout \begin_layout Subsection Key Theoretical Results \begin_inset CommandInset label LatexCommand label name "subsec:EulerLinearitySS" \end_inset \end_layout \begin_layout Standard Three theoretical results are key to our estimation strategy: the Euler equation, linear policy functions, and the steady state. \end_layout \begin_layout Standard \series bold Euler equation. \series default The first-order conditions of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:equilibrium" plural "false" caps "false" noprefix "false" \end_inset ) for periods \begin_inset Formula $t$ \end_inset and \begin_inset Formula $t+1$ \end_inset can be re-arranged into an Euler equation characterizing the equilibrium relationship between consumption in periods \begin_inset Formula $t$ \end_inset and \begin_inset Formula $t+1$ \end_inset . To simplify notation, define \begin_inset Formula $u_{t}\coloneqq u_{t}(x_{t}^{*};s_{t},p_{t})$ \end_inset as current utility, define \begin_inset Formula $\tilde{x}_{r}\coloneqq\tilde{x}_{r}^{*}\left(\tilde{s}_{r},\tilde{\gamma},\boldsymbol{p}_{r}\right)$ \end_inset and \begin_inset Formula $\tilde{u}_{r}\coloneqq u_{r}\left(\tilde{x}_{r};\tilde{s}_{r},p_{r}\right)$ \end_inset as predicted consumption and utility for future periods \begin_inset Formula $r>t$ \end_inset , and define \begin_inset Formula $\tilde{\lambda}_{r}\coloneqq\frac{\partial\tilde{x}_{r}}{\partial\tilde{s}_{r}}$ \end_inset as the predicted effect of habit stock on consumption\SpecialChar endofsentence \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Proposition \begin_inset CommandInset label LatexCommand label name "prop:Euler" \end_inset Suppose \begin_inset Formula $u_{t}(x_{t};s_{t},p_{t})$ \end_inset is given by equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:QuadraticUtility" plural "false" caps "false" noprefix "false" \end_inset ) and \begin_inset Formula $\left(x_{0}^{*},...,x_{T}^{*}\right)$ \end_inset is a perception-perfect strategy profile with differentiable strategies. Then for each \begin_inset Formula $t(1-\alpha)\delta\rho\left[\left(\zeta-\eta\right)\left(1+\rho\tilde{\lambda}_{t+1}\right)-\rho\zeta\right]$ \end_inset . \end_layout \begin_layout Proof See Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:LinearityProof" plural "false" caps "false" noprefix "false" \end_inset . That appendix also provides an explicit condition that guarantees concavity. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold Steady state. \series default Over a period of time when strategies are well approximated by the limiting values \begin_inset Formula $\lambda$ \end_inset and \begin_inset Formula $\mu$ \end_inset , consumption converges to a steady state. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Lemma \begin_inset CommandInset label LatexCommand label name "lemma:SteadyStateConvergence" \end_inset Suppose that strategies in all periods take the form \begin_inset Formula $x_{t}^{*}\left(s_{t},\gamma,\boldsymbol{p}_{t}\right)=\lambda s_{t}+\mu$ \end_inset , where \begin_inset Formula $\lambda$ \end_inset and \begin_inset Formula $\mu$ \end_inset are constant. If \begin_inset Formula $\rho\left(1+\lambda\right)<1$ \end_inset , both \begin_inset Formula $x_{t}^{*}$ \end_inset and \begin_inset Formula $s_{t}$ \end_inset converge monotonically over time to steady-state values \begin_inset Formula $x_{ss}$ \end_inset and \begin_inset Formula $s_{ss}$ \end_inset . \end_layout \begin_layout Proof See Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:SteadyStateConvergenceProof" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard If consumption has reached a steady state, we can use the Euler equation to characterize its level in closed form. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Proposition \begin_inset CommandInset label LatexCommand label name "prop:SteadyStateX" \end_inset Suppose that \begin_inset Formula $p_{t}$ \end_inset and \begin_inset Formula $\xi_{t}$ \end_inset are constant and that consumption and habit stock are in steady state with \begin_inset Formula $s_{t}=s_{ss}$ \end_inset , \begin_inset Formula $x_{t}=x_{ss}$ \end_inset , and \begin_inset Formula $x_{ss}=\rho\left(s_{ss}+x_{ss}\right)$ \end_inset . Then consumption can be written as \end_layout \begin_layout Proposition \begin_inset Formula \begin{equation} x_{ss}=\frac{\kappa-\left(1-(1-\alpha)\delta\rho\right)p+(1-\alpha)\delta\rho\left[(\zeta-\eta)m_{ss}-\left(1+\tilde{\lambda}\right)\tilde{\gamma}\right]+\gamma}{-\eta-(1-\alpha)\delta\rho(\zeta-\eta)-\zeta\frac{\rho-(1-\alpha)\delta\rho^{2}}{1-\rho}},\label{eq:xSS} \end{equation} \end_inset where \begin_inset Formula $\kappa\coloneqq(1-\alpha)\delta\rho(\phi-\xi)+\xi$ \end_inset and \begin_inset Formula $m_{ss}\coloneqq\tilde{x}_{t+1}-x_{ss}$ \end_inset is steady-state misprediction. \end_layout \begin_layout Proof See Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:SteadyStateXProof" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard The parameter restrictions required for Proposition \begin_inset CommandInset ref LatexCommand ref reference "prop:Linearity" plural "false" caps "false" noprefix "false" \end_inset and Lemma \begin_inset CommandInset ref LatexCommand ref reference "lemma:SteadyStateConvergence" plural "false" caps "false" noprefix "false" \end_inset (including concavity) essentially amount to requiring that perceived and actual habit formation are not too strong. We have confirmed that these restrictions hold at the parameter estimates presented in Table \begin_inset CommandInset ref LatexCommand ref reference "tab:StructuralEstimates" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Subsection Modeling the Experiment \begin_inset CommandInset label LatexCommand label name "subsec:ModelingTheExperiment Unrestricted" \end_inset \end_layout \begin_layout Standard We need additional notation to map the experiment's treatments and data into the model and estimation. We define \begin_inset Formula $x_{it}$ \end_inset to be participant \begin_inset Formula $i$ \end_inset 's daily average FITSBY screen time during period \begin_inset Formula $t$ \end_inset , \begin_inset Formula $\tilde{x}_{it}$ \end_inset to be participant \begin_inset Formula $i$ \end_inset 's predicted screen time elicited on a survey, and \begin_inset Formula $m_{it}=x_{it}-\tilde{x}_{it}$ \end_inset to be the difference between the two. The Bonus and Bonus Control groups are denoted \begin_inset Formula $g\in\{B,BC\}$ \end_inset , the Limit and Limit Control groups are \begin_inset Formula $g\in\{L,LC\}$ \end_inset , and the intersection of Bonus Control and Limit Control is \begin_inset Formula $g=C$ \end_inset . We define \begin_inset Formula $\bar{y}\coloneqq\mathbb{E}_{i}y_{i}$ \end_inset as the expectatation over participants of variable \begin_inset Formula $y$ \end_inset , and \begin_inset Formula $y^{g}\coloneqq\mathbb{E}_{i\in g}y_{i}$ \end_inset as the expectation over group \begin_inset Formula $g$ \end_inset . \begin_inset Formula $\tau_{t}^{g}\coloneqq x_{t}^{g}-x_{t}^{gC}$ \end_inset and \begin_inset Formula $\tilde{\tau}_{t}^{g}\coloneqq\tilde{x}_{t}^{g}-\tilde{x}_{t}^{gC}$ \end_inset are the actual and predicted average treatment effects. \end_layout \begin_layout Standard We model the Screen Time Bonus as a price \begin_inset Formula $p^{B}=$ \end_inset $2.50 per hour in period 3 plus a fixed payment \begin_inset Formula $F_{i}^{B}=\$50\times\text{ceil}(x_{i1}\ \frac{\text{hours}}{\text{day}})$ \end_inset , where \begin_inset Formula $\text{ceil}(\cdot)$ \end_inset rounds up to the nearest integer, giving participant \begin_inset Formula $i$ \end_inset 's Bonus Benchmark. In this appendix, we generalize the primary model from Section \begin_inset CommandInset ref LatexCommand ref reference "sec:StructuralEstimation" plural "false" caps "false" noprefix "false" \end_inset by modeling the limit as an intervention that eliminates share \begin_inset Formula $\omega$ \end_inset of temptation. \end_layout \begin_layout Standard We define \begin_inset Formula $v_{i}^{B}$ \end_inset as the valuation of the bonus elicited on survey 2, and we define \begin_inset Formula $v_{i}^{L}$ \end_inset as the valuation of access to the limit functionality elicited on survey 3. We assume that on survey \begin_inset Formula $t$ \end_inset , consumers are aware of period \begin_inset Formula $t$ \end_inset projection bias when predicting period \begin_inset Formula $t$ \end_inset consumption and are projection biased when determining their bonus and limit valuations. This assumption means that misprediction of period- \begin_inset Formula $t$ \end_inset consumption is driven only by naivete about temptation, and that bonus and limit valuations are driven only by perceived temptation, not by an additional desire to offset projection bias. We acknowledge that alternative assumptions could be made. \end_layout \begin_layout Subsection Estimating Equations \begin_inset CommandInset label LatexCommand label name "subsec:Estimating Equations Unrestricted" \end_inset \end_layout \begin_layout Standard Using the theoretical results from Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:EulerLinearitySS" plural "false" caps "false" noprefix "false" \end_inset , we can now derive equations that characterize how a consumer from our unrestricted model would behave in our experiment. These equations parallel the equation in Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:Estimating Equations" plural "false" caps "false" noprefix "false" \end_inset , with additional terms that account for perceived habit formation. We assume that the discount factor is \begin_inset Formula $\delta=0.997$ \end_inset per three-week period, consistent with a five percent annual discount rate. We estimate the remaining parameters in stages, as described below. Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:EstimatingEquationDerivations" plural "false" caps "false" noprefix "false" \end_inset presents formal derivations and additional details. \end_layout \begin_layout Subsubsection* Habit Formation \end_layout \begin_layout Standard We first estimate \begin_inset Formula $\lambda$ \end_inset and \begin_inset Formula $\rho$ \end_inset from the decay of the bonus treatment effects. Even though \begin_inset Formula $\lambda$ \end_inset is not a structural parameter, it is easily identified and useful in estimating the other parameters. Using the habit stock evolution formula and the linearity result in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Linearity" plural "false" caps "false" noprefix "false" \end_inset ), we can write the period 4 bonus effect as the result of decayed effects from periods 2 and 3: \begin_inset Formula $\tau_{4}^{B}=\lambda\left(\rho\tau_{3}^{B}+\rho^{2}\tau_{2}^{B}\right)$ \end_inset . Similarly, the period 5 effect results from the cumulative decayed effects from periods 2–4: \begin_inset Formula $\tau_{5}^{B}=\lambda\left(\rho\tau_{4}^{B}+\rho^{2}\tau_{3}^{B}+\rho^{3}\tau_{2}^{B}\right)$ \end_inset . Rearranging gives a system of two equations for \begin_inset Formula $\lambda$ \end_inset and \begin_inset Formula $\rho$ \end_inset : \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \lambda & =\frac{\tau_{4}^{B}}{\rho\tau_{3}^{B}+\rho^{2}\tau_{2}^{B}}\label{eq:lambda}\\ \rho & =\frac{\tau_{5}^{B}}{\tau_{4}^{B}(1+\lambda)}.\label{eq:rho} \end{align} \end_inset This non-linear system has two solutions when \begin_inset Formula $\tau_{2}^{B}\neq0$ \end_inset , but in our data there is only one solution that satisfies the requirement that \begin_inset Formula $\rho\geq0$ \end_inset . \end_layout \begin_layout Standard For estimation, we assume \begin_inset Formula $\tilde{\lambda}=\lambda$ \end_inset . This is reasonable because Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualHabitFormation" plural "false" caps "false" noprefix "false" \end_inset shows that participants predicted the time path of bonus effects with reasonabl e accuracy, so calibrating equations ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambda" plural "false" caps "false" noprefix "false" \end_inset ) and ( \begin_inset CommandInset ref LatexCommand ref reference "eq:rho" plural "false" caps "false" noprefix "false" \end_inset ) with predicted \begin_inset Formula $\tau_{t}^{B}$ \end_inset would not change the estimates much. To the extent that predictions differ from actual behavior, we prefer to err on the side of using actual behavior instead of beliefs to estimate the model. \end_layout \begin_layout Subsubsection* Perceived Habit Formation, Price Response, and Habit Stock Effect on Marginal Utility \end_layout \begin_layout Standard After estimating \begin_inset Formula $\lambda$ \end_inset and \begin_inset Formula $\rho$ \end_inset , we estimate \begin_inset Formula $\alpha$ \end_inset , \begin_inset Formula $\eta$ \end_inset , and \begin_inset Formula $\zeta$ \end_inset from the magnitude and decay of the bonus treatment effects. For each of periods 2, 3, and 4, we difference the Euler equations for the Bonus and Bonus Control groups and rearrange, giving a system of three equations for \begin_inset Formula $(1-\alpha)$ \end_inset , \begin_inset Formula $\eta$ \end_inset , and \begin_inset Formula $\zeta$ \end_inset : \end_layout \begin_layout Standard \begin_inset Formula \begin{align} (1-\alpha) & =\text{\ensuremath{\frac{\eta\tau_{2}^{B}}{\delta\rho\left[-p^{B}+(\eta-\zeta)\tilde{\tau}_{3}^{B}+\zeta\rho\tau_{2}^{B}\right]}}}.\label{eq:alpha}\\ \eta & =\frac{p^{B}-\zeta\rho\tau_{2}^{B}+(1-\alpha)\delta\rho^{2}\zeta(1-\tilde{\lambda})\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)}{\tau_{3}^{B}-(1-\alpha)\delta\rho^{2}\tilde{\lambda}\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)}\label{eq:eta}\\ \zeta & =\frac{-\eta\tau_{4}^{B}+(1-\alpha)\delta\rho^{2}\eta\tilde{\lambda}\left(\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}+\tau_{4}^{B}\right)}{\rho\tau_{3}^{B}+\rho^{2}\tau_{2}^{B}-(1-\alpha)\delta\rho^{2}\left(1-\tilde{\lambda}\right)\left(\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}+\tau_{4}^{B}\right)}.\label{eq:zeta} \end{align} \end_inset The first equation shows that as the anticipatory demand response in period 2 grows compared to the predicted demand response in period 3 (making \begin_inset Formula $\tau_{2}^{B}/\tilde{\tau}_{3}^{B}$ \end_inset larger), we infer more perceived habit formation (smaller \begin_inset Formula $\alpha$ \end_inset ). \end_layout \begin_layout Subsubsection* Naivete about Temptation \end_layout \begin_layout Standard Next, we estimate naivete about temptation \begin_inset Formula $\gamma-\tilde{\gamma}$ \end_inset using the Control group's difference between perceived and actual consumption. To solve for \begin_inset Formula $\gamma-\tilde{\gamma}$ \end_inset , we difference the actual versus perceived Euler equations for group \begin_inset Formula $C$ \end_inset , giving \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \gamma-\tilde{\gamma}=m_{t}^{C}\cdot\left[-\eta+(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)\right].\label{eq:Naivete} \end{equation} \end_inset \end_layout \begin_layout Subsubsection* Temptation \end_layout \begin_layout Standard We estimate temptation \begin_inset Formula $\gamma$ \end_inset using three different strategies: the limit treatment effect and valuations of the bonus and limit. Each strategy delivers an equation that we combine with equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Naivete" plural "false" caps "false" noprefix "false" \end_inset ) to form a system of two equations for \begin_inset Formula $\gamma$ \end_inset and \begin_inset Formula $\tilde{\gamma}$ \end_inset . \end_layout \begin_layout Standard \emph on Limit effect. \emph default Recall that we model the limit as an intervention that eliminates share \begin_inset Formula $\omega$ \end_inset of temptation, starting in period 2. Thus, we can identify \begin_inset Formula $\gamma$ \end_inset using an assumed \begin_inset Formula $\omega$ \end_inset plus the effect of the limit on consumption. To solve for \begin_inset Formula $\gamma$ \end_inset , we difference the Euler equations for periods 2 versus 3 for the Limit group compared to Limit Control and rearrange, giving \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \gamma=\eta\tau_{2}^{L}/\omega-(1-\alpha)\delta\rho\left[(\eta-\zeta)\tilde{\tau}_{3}^{L}/\omega+\zeta\rho\tau_{2}^{L}/\omega-\tilde{\gamma}-\tilde{\gamma}\tilde{\lambda}\right].\label{eq:gamma_Limit} \end{equation} \end_inset Our primary estimates in Section \begin_inset CommandInset ref LatexCommand ref reference "sec:StructuralEstimation" plural "false" caps "false" noprefix "false" \end_inset use this equation, after setting \begin_inset Formula $\omega=1$ \end_inset and \begin_inset Formula $\alpha=1$ \end_inset . \end_layout \begin_layout Standard \emph on Bonus valuation. \emph default Since the bonus is like a commitment device that reduces future use, people with perceived self-control problems will place higher value on the bonus. We can estimate perceived temptation \begin_inset Formula $\tilde{\gamma}$ \end_inset from participants' valuations. Our derivation follows \begin_inset CommandInset citation LatexCommand citet* key "allcottkim2020" literal "false" \end_inset , and the approach also follows \begin_inset CommandInset citation LatexCommand citet key "AclandLevy2012" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet key "AugenblickRabin2019" literal "false" \end_inset , \begin_inset CommandInset citation LatexCommand citet* key "Chaloupka2019" literal "false" \end_inset , and \begin_inset CommandInset citation LatexCommand citet key "Carrera2019" literal "false" \end_inset . \end_layout \begin_layout Standard Let \begin_inset Formula $V_{t}\left(\tilde{s}_{t},\cdot\right)$ \end_inset be the period \begin_inset Formula $t$ \end_inset continuation value function conditional on \begin_inset Formula $\tilde{s}_{t}$ \end_inset , according to predicted consumption and preferences before period \begin_inset Formula $t$ \end_inset . This reflects preferences of a consumer filling out the multiple price list on a survey before period \begin_inset Formula $t$ \end_inset . Since utility is quasilinear in money, \begin_inset Formula $V_{t}\left(s_{t},\cdot\right)$ \end_inset is in units of period \begin_inset Formula $t$ \end_inset dollars. \end_layout \begin_layout Standard The effect of a period 3 price increase from 0 to \begin_inset Formula $p_{3}^{B}$ \end_inset on the period 3 continuation value is \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \Delta V_{3}\left(p^{B}\right)\coloneqq V_{3}\left(\tilde{s}_{3},p_{3}=p_{3}^{B}\right)-V_{3}\left(\tilde{s}_{3},p_{3}=0\right) & =-p_{3}^{B}\cdot\frac{1}{2}\left(\tilde{x}_{3}(p_{3}^{B})+\tilde{x}_{3}(0)\right)-\tilde{\gamma}\cdot\left(\tilde{x}_{3}(p_{3}^{B})-\tilde{x}_{3}(0)\right),\label{eq:DeltaV(pB)} \end{align} \end_inset where \begin_inset Formula $\tilde{x}_{3}(p_{3})=\tilde{x}_{3}^{*}(\tilde{s}_{3},\tilde{\gamma},\boldsymbol{p}_{3})$ \end_inset is shorthand for predicted period 3 consumption as a function of period 3 price. Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:TemptationIdentification" plural "false" caps "false" noprefix "false" \end_inset illustrates. The trapezoid \emph on ABCD \emph default is \begin_inset Formula $p_{3}^{B}\cdot\frac{1}{2}\left(\tilde{x}_{3}(p_{3}^{B})+\tilde{x}_{3}(0)\right)$ \end_inset : the survey taker's prediction of the consumer surplus loss from the price increase from the period 3 self's perspective. The parallelogram \emph on BCEF \emph default is \begin_inset Formula $-\tilde{\gamma}\cdot\left(\tilde{x}_{3}(p_{3}^{B})-\tilde{x}_{3}(0)\right)$ \end_inset : the predicted additional temptation reduction benefit from the survey taker's perspective. \end_layout \begin_layout Standard The Screen Time Bonus combines a price change with a fixed payment of \begin_inset Formula $F^{B}$ \end_inset . Thus, the model predicts that people filling out the bonus MPL would be indifferent between the bonus and a fixed payment of \begin_inset Formula $v^{B}=F^{B}+\Delta V_{3}(p^{B})$ \end_inset . Taking the expectation over participants to allow mean-zero survey noise, substituting \begin_inset Formula $\tilde{\tau}_{3}^{B}\coloneqq\mathbb{E}_{i}\left[\tilde{x}_{i3}(p_{3}^{B})-\tilde{x}_{i3}(0)\right]$ \end_inset and \begin_inset Formula $\bar{\tilde{x}}_{3}^{B+BC}\coloneqq\mathbb{E}_{i}\left[\frac{1}{2}\left(\tilde{x}_{i3}(p_{3}^{B})+\tilde{x}_{i3}(0)\right)\right]$ \end_inset , and rearranging gives perceived temptation: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \tilde{\gamma}=\frac{\bar{v}^{B}-\bar{F}^{B}+p_{3}^{B}\bar{\tilde{x}}_{3}^{B+BC}}{-\tilde{\tau}_{3}^{B}}.\label{eq:gammatildeB} \end{equation} \end_inset The model predicts that if consumers perceive themselves to be time consistent ( \begin_inset Formula $\tilde{\gamma}=0$ \end_inset ), the average bonus valuation would equal the average valuation from the period 3 self's perspective, \begin_inset Formula $\bar{F}^{B}-p_{3}^{B}\bar{\tilde{x}}_{3}^{B+BC}$ \end_inset . We refer to the difference between the observed average valuation and the modeled time-consistent valuation (the numerator of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gammatildeB" plural "false" caps "false" noprefix "false" \end_inset )) as \begin_inset Quotes eld \end_inset behavior change premium. \begin_inset Quotes erd \end_inset We infer more perceived temptation \begin_inset Formula $\tilde{\gamma}$ \end_inset from a larger behavior change premium. \end_layout \begin_layout Standard \emph on Limit valuation. \emph default People who perceive future temptation value the limit, as they perceive that it eliminates share \begin_inset Formula $\omega$ \end_inset of temptation. We can estimate perceived temptation \begin_inset Formula $\tilde{\gamma}$ \end_inset using an assumed \begin_inset Formula $\omega$ \end_inset plus the valuation the limit functionality. We solve for the modeled valuation similarly to how we solved for the bonus valuation above. \end_layout \begin_layout Standard The effect of a period 3 temptation reduction from \begin_inset Formula $\tilde{\gamma}$ \end_inset to \begin_inset Formula $(1-\omega)\tilde{\gamma}$ \end_inset on the period 3 continuation value is \end_layout \begin_layout Standard \begin_inset Formula \begin{align} v^{L}=V_{3}\left(s_{3},\tilde{\gamma}_{3}=(1-\omega)\tilde{\gamma}\right)-V_{3}\left(s_{3},\tilde{\gamma}_{3}=\tilde{\gamma}\right) & =\tilde{\gamma}\cdot\left(x_{3}^{*}(\tilde{\gamma})-x_{3}^{*}((1-\omega)\tilde{\gamma})\right)\cdot\frac{2-\omega}{2},\label{eq:vL} \end{align} \end_inset where \begin_inset Formula $x_{3}^{*}(\tilde{\gamma}_{3})$ \end_inset is now shorthand for predicted period 3 consumption as a function of predicted period 3 temptation. Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:TemptationIdentification" plural "false" caps "false" noprefix "false" \end_inset illustrates. With \begin_inset Formula $\omega=1$ \end_inset , the limit valuation is the deadweight loss reduction \emph on CEG \emph default from the survey taker's perspective from consuming the desired amount ( \begin_inset Formula $x_{3}^{*}(0)$ \end_inset , point \emph on G \emph default ) instead of the predicted amount ( \begin_inset Formula $x_{3}^{*}(\tilde{\gamma})$ \end_inset , point \emph on C \emph default ). The height of this triangle is \begin_inset Formula $\tilde{\gamma}$ \end_inset and the width is \begin_inset Formula $x_{3}^{*}(\tilde{\gamma})-x_{3}^{*}(0)$ \end_inset , and thus the area is \begin_inset Formula $\tilde{\gamma}\cdot\left(x_{3}^{*}(\tilde{\gamma})-x_{3}^{*}(0)\right)\cdot\frac{1}{2}$ \end_inset . With \begin_inset Formula $\omega<1$ \end_inset , the valuation \begin_inset Formula $v^{L}$ \end_inset equals the deadweight loss reduction trapezoid starting to the right of point \emph on G \emph default and bounded by segment \emph on CE \emph default . \end_layout \begin_layout Standard Taking the expectation over participants, substituting \begin_inset Formula $\tilde{\tau}_{3}^{L}\coloneqq\mathbb{E}_{i}\left[x_{3}^{*}((1-\omega)\tilde{\gamma})-x_{3}^{*}(\tilde{\gamma})\right]$ \end_inset , and rearranging gives perceived temptation: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \tilde{\gamma}=\frac{\bar{v}^{L}}{-\tilde{\tau}_{3}^{L}(2-\omega)/2}.\label{eq:gammatildeL} \end{equation} \end_inset We infer more perceived temptation \begin_inset Formula $\tilde{\gamma}$ \end_inset from higher valuation \begin_inset Formula $\bar{v}^{L}$ \end_inset . \end_layout \begin_layout Subsubsection* Intercept \end_layout \begin_layout Standard Finally, we back out a heterogeneous intercept \series bold \begin_inset Formula $\kappa_{i}$ \end_inset \series default that explains observed consumption heterogeneity. Our data do not allow us to separately identify \begin_inset Formula $\phi$ \end_inset (the direct effect of habit stock on utility) from \begin_inset Formula $\xi$ \end_inset (the marginal utility shifter), so \begin_inset Formula $\kappa_{i}$ \end_inset includes both of these structural parameters. We assume that participant \begin_inset Formula $i$ \end_inset 's observed baseline consumption \begin_inset Formula $x_{i1}$ \end_inset is in a steady state characterized by equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS" plural "false" caps "false" noprefix "false" \end_inset ). Rearranging that equation gives \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \kappa_{i}\coloneqq(1-\alpha)\delta\rho(\phi-\xi_{i})+\xi_{i}= & \left(1-(1-\alpha)\delta\rho\right)p-(1-\alpha)\delta\rho\left[\left(\zeta-\eta\right)m_{ss}-\left(1+\tilde{\lambda}\right)\tilde{\gamma}\right]\nonumber \\ & \ \ -\gamma+x_{i1}\left[-\eta-(1-\alpha)\delta\rho(\zeta-\eta)-\zeta\frac{\rho-(1-\alpha)\delta\rho^{2}}{1-\rho}\right].\label{eq:Intercept} \end{align} \end_inset \end_layout \begin_layout Subsection Empirical Moments and Estimation Details \begin_inset CommandInset label LatexCommand label name "subsec:EstimationMomentsDetails Unrestricted" \end_inset \end_layout \begin_layout Standard Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:Moments-Full" plural "false" caps "false" noprefix "false" \end_inset presents the full set of moments and fixed parameter values that are inputs to our unrestricted model and alternative specifications. In light of the discussion in Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:BonusEffects" plural "false" caps "false" noprefix "false" \end_inset , we omit the first half of period 2 when we estimate the anticipatory bonus effect \begin_inset Formula $\tau_{2}^{B}$ \end_inset . \begin_inset Foot status open \begin_layout Plain Layout Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:StructuralEstimates-fulltaub2" plural "false" caps "false" noprefix "false" \end_inset presents parameter estimates when we use all of period 2 to estimate \begin_inset Formula $\tau_{2}^{B}$ \end_inset . The estimated \begin_inset Formula $\rho$ \end_inset is larger, as expected, but the other parameter estimates are very similar. \end_layout \end_inset The average of predicted use with and without the bonus \begin_inset Formula $\bar{\tilde{x}}_{3}^{B,BC}$ \end_inset and the predicted contemporaneous bonus effect \begin_inset Formula $\tilde{\tau}_{3}^{B}$ \end_inset are the predictions before the bonus MPL on survey 2, as displayed in Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualHabitFormation" plural "false" caps "false" noprefix "false" \end_inset . Because we do not have an explicit elicitation of the predicted limit effect, we use the actual limit effect \begin_inset Formula $\tau_{3}^{L}$ \end_inset to proxy for the predicted limit effect \begin_inset Formula $\tilde{\tau}_{3}^{L}$ \end_inset . \begin_inset Foot status open \begin_layout Plain Layout The average difference in predicted FITSBY use between Limit and Limit Control on survey 3 is \begin_inset Formula $\tilde{\tau}_{3}^{L}\approx\truetautildeL$ \end_inset minutes per day, much smaller than the actual limit effect of \begin_inset Formula $\tau_{3}^{L}\approx\tautildeL$ \end_inset minutes per day. In the limit effect strategy in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gamma_Limit" plural "false" caps "false" noprefix "false" \end_inset ), \begin_inset Formula $\tilde{\tau}_{3}^{L}$ \end_inset makes little difference because it is multiplied by \begin_inset Formula $(1-\alpha)$ \end_inset , which is small. However, in the limit valuation strategy in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gammatildeL" plural "false" caps "false" noprefix "false" \end_inset ), \begin_inset Formula $\tilde{\gamma}$ \end_inset is inversely proportional to \begin_inset Formula $\tilde{\tau}_{3}^{L}$ \end_inset , so a much smaller \begin_inset Formula $\tilde{\tau}_{3}^{L}$ \end_inset would make the estimated \begin_inset Formula $\ensuremath{\tilde{\gamma}}$ \end_inset much larger. \end_layout \end_inset Since Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:PredictedVsActualControl" plural "false" caps "false" noprefix "false" \end_inset shows that the average prediction error for period \begin_inset Formula $t$ \end_inset consumption is similar when elicited on survey \begin_inset Formula $t$ \end_inset versus survey \begin_inset Formula $t-1$ \end_inset , we let observed Control group misprediction \begin_inset Formula $m^{C}$ \end_inset proxy for steady-state misprediction \begin_inset Formula $m_{ss}$ \end_inset . \end_layout \begin_layout Standard We winsorize the anticipatory bonus effect at \begin_inset Formula $\tau_{2}^{B}\leq0$ \end_inset , which affects \begin_inset Formula $\percenttaubtwopre$ \end_inset percent of draws. We also drop the \begin_inset Formula $\NegSSDenom$ \end_inset percent of bootstrap draws in which the denominator of steady-state consumption in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS counterfactual" plural "false" caps "false" noprefix "false" \end_inset ) is not positive. \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Empirical Moments and Additional Parameters \begin_inset CommandInset label LatexCommand label name "tab:Moments-Full" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (2) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Point \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Confidence \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Parameter \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Description \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout estimate \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout interval \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\delta$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Three-week discount factor (unitless) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.997$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{2}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Anticipatory bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tauBtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tauBtwoboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{3}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Contemporaneous bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBthree}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBthreeboot}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{4}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Long-term bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfour}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfourboot}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{5}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Long-term bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfive}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfiveboot}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{2}^{L}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Limit effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauLtwo}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauLtwoboot}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $m^{C}$ \end_inset , \begin_inset Formula $m_{ss}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Control group misprediction (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\mispredict$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\mispredictboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{\tilde{x}}_{3}^{B+BC}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Predicted use with/without bonus (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xtildetwoB$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xtildetwoBboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tilde{\tau}_{3}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Predicted bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tautildeBthreetwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tautildeBthreetwoboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tilde{\tau}_{3}^{L}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Predicted limit effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tautildeL$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tautildeLboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\omega$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Temptation reduction from limit \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 1 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{v}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average bonus valuation ($/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\vB$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\vBboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{v}^{L}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average limit valuation ($/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\vL}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\vLboot}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $p^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Bonus price ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 2.5 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{F}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average bonus fixed payment ($/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\FB$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\FBboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{x}_{1}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average baseline use (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xss$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssboot$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table presents point estimates and bootstrapped 95 percent confidenc e intervals for the empirical moments used for estimation. We winsorize at \begin_inset Formula $\tau_{2}^{B}\leq0$ \end_inset , and we drop the \begin_inset Formula $\NegSSDenom$ \end_inset percent of draws in which the denominator of steady-state consumption in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS counterfactual" plural "false" caps "false" noprefix "false" \end_inset ) is not positive. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Primary Parameter Estimates Using \begin_inset Formula $\tau_{2}^{B}$ \end_inset for All of Period 2 \begin_inset CommandInset label LatexCommand label name "tab:StructuralEstimates-fulltaub2" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Unrestricted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Parameter \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Description (units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout ( \begin_inset Formula $\alpha=\hat{\alpha}$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\lambda$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Habit stock effect on consumption (unitless) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\lambdahatfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\lambdahatbootfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rho$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Habit formation (unitless) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rhohatfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rhohatbootfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\alpha$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Projection bias (unitless) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\alphahatfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\alphahatbootfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\eta$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Price coefficient ($-day/hour \begin_inset Formula $^{2}$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\etahourfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\etahourbootfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\zeta$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Habit stock effect on marginal utility ($-day/hour \begin_inset Formula $^{2}$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\zetahourfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\zetahourbootfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gamma-\tilde{\gamma}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Naivete about temptation ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\naivetehourfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\naivetehourbootfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gamma$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Temptation ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammahourfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammahourbootfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{\kappa}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average intercept ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\intercepthetLeffecthourfulltaubtwo$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\intercepthetLeffecthourbootfulltaubtwo$ \end_inset \end_layout \end_inset \end_inset \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table presents point estimates and bootstrapped 95 percent confidenc e intervals from the estimation strategy described in Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:Estimating Equations Unrestricted" plural "false" caps "false" noprefix "false" \end_inset . We winsorize at \begin_inset Formula $\tau_{2}^{B}\leq0$ \end_inset , and we drop the \begin_inset Formula $\NegSSDenom$ \end_inset percent of draws in which the denominator of steady-state consumption in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS counterfactual" plural "false" caps "false" noprefix "false" \end_inset ) is not positive. Temptation \begin_inset Formula $\gamma$ \end_inset is from the limit effect strategy, using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gamma_Limit" plural "false" caps "false" noprefix "false" \end_inset ). This parallels column 2 of Table \begin_inset CommandInset ref LatexCommand ref reference "tab:StructuralEstimates" plural "false" caps "false" noprefix "false" \end_inset , except using all of period 2 (instead of only the second half of period 2) to estimate the anticipatory bonus effect \begin_inset Formula $\tau_{2}^{B}$ \end_inset . \end_layout \end_inset \end_layout \begin_layout Subsection \series bold Alternative Temptation Estimates \series default \begin_inset CommandInset label LatexCommand label name "app:AltTemptationEstimates" \end_inset \end_layout \begin_layout Standard Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:Alternative_gamma" plural "false" caps "false" noprefix "false" \end_inset presents alternative estimates of temptation \begin_inset Formula $\gamma$ \end_inset in the restricted and unrestricted models. After repeating the primary limit effect estimate, the table reports the bonus valuation estimate. Before the bonus MPL on survey 2, the average participant predicted that they would use FITSBY \begin_inset Formula $\text{\meanpredictuse}$ \end_inset and \begin_inset Formula $\meanpredictbonus$ \end_inset hours per day without and with the bonus, respectively. Thus, the average survey taker would have predicted that the price increase would cause a consumer surplus loss from their period 3 self's perspective of \begin_inset Formula $p_{3}^{B}\bar{\tilde{x}}_{3}\approx\$\p\times\frac{1}{2}\left(\text{\meanpredictuse}+\meanpredictbonus\right)\approx\$\abcd$ \end_inset per day of period 3. This is the trapezoid \emph on ABCD \emph default on Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:TemptationIdentification" plural "false" caps "false" noprefix "false" \end_inset . The average bonus fixed payment was \begin_inset Formula $\bar{F}^{B}\approx\$\FB$ \end_inset per day. Thus, if the average participant perceived herself to be time consistent, she would have been indifferent between the bonus and a certain payment of \begin_inset Formula $\$\FB-\$\abcd\approx\$\MPL$ \end_inset per day. \end_layout \begin_layout Standard In reality, the average participant was indifferent between the bonus and a certain payment of $ \begin_inset Formula $\MPLvalued$ \end_inset , or \begin_inset Formula $\bar{v}^{B}\approx\$\MPLvalue/20\approx\$\vB$ \end_inset per day over the 20-day period. This excess valuation implies a behavior change premium of \begin_inset Formula $\$\vB-\$\MPL\approx\$\behaviourpremium$ \end_inset per day. This is the parallelogram \emph on BCEF \emph default on Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:TemptationIdentification" plural "false" caps "false" noprefix "false" \end_inset : the additional temptation reduction benefit that the period 2 survey taker perceives from the reduced FITSBY use caused by the bonus. Rearranging this logic into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gammatildeB" plural "false" caps "false" noprefix "false" \end_inset ) gives perceived temptation \begin_inset Formula $\hat{\tilde{\gamma}}\approx\gammatildeBhour$ \end_inset $/hour. Using the estimated naivete of \begin_inset Formula $\widehat{\gamma-\tilde{\gamma}}\approx\naivetereshour$ \end_inset gives \begin_inset Formula $\hat{\gamma}\approx\gammaBreshour$ \end_inset for the bonus valuation strategy in column 1. \end_layout \begin_layout Standard The average Limit group participant was indifferent between access to the limit functionality for period 3 and a certain payment of $ \begin_inset Formula $\valuelimit$ \end_inset , or \begin_inset Formula $\bar{v}^{L}\approx\$\ensuremath{\valuelimit}/20\approx\$\vL$ \end_inset per day over the 20-day period. This is the triangle on Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:TemptationIdentification" plural "false" caps "false" noprefix "false" \end_inset : the perceived deadweight loss reduction from the reduced FITSBY use caused by the limit. Inserting this into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gammatildeL" plural "false" caps "false" noprefix "false" \end_inset ) with \begin_inset Formula $\omega=1$ \end_inset gives perceived temptation \begin_inset Formula $\hat{\tilde{\gamma}}=\frac{\bar{v}^{L}}{-\tilde{\tau}_{3}^{L}/2}\approx\frac{\vL}{(-(\tautildeL)/60)/2}\approx\gammatildeLhour$ \end_inset $/hour. Using \begin_inset Formula $\widehat{\gamma-\tilde{\gamma}}\approx\naivetereshour$ \end_inset gives \begin_inset Formula $\hat{\gamma}\approx\gammaLreshour$ \end_inset for the limit valuation strategy in column 1. \end_layout \begin_layout Standard So far, we have modeled FITSBY screen time on other devices as part of an outside option that is not affected by self-control problems. In Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:TemptationMultipleGoods" plural "false" caps "false" noprefix "false" \end_inset , we generalize the model to include multiple temptation goods. As discussed in Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:Substitution" plural "false" caps "false" noprefix "false" \end_inset , self-reports suggest that the limit increased FITSBY use on other devices by \begin_inset Formula $\limitsubstitution$ \end_inset minutes per day, while the bonus reduced FITSBY use on other devices by \begin_inset Formula $\bonussubstitution$ \end_inset minutes per day. We use these additional moments to identify the multiple-good model. \end_layout \begin_layout Standard The next three rows in Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:Alternative_gamma" plural "false" caps "false" noprefix "false" \end_inset present estimates from the multiple-good model. The limit effect estimate increases to \begin_inset Formula $\hat{\gamma}\approx\gammaLeffectmultiplereshour$ \end_inset $/hour, because in the multiple-good model, more temptation is needed to explain the observed limits when consumers setting the limits think they'll evade the limits through substitution to other devices. The bonus valuation estimate decreases to \begin_inset Formula $\hat{\gamma}\approx\text{\gammaBmultiplereshour}$ \end_inset $/hour, because in the multiple-good model, less temptation is needed to explain the observed bonus valuation when consumers think the bonus will also reduce FITSBY use on other devices. The limit valuation estimate increases to to \begin_inset Formula $\hat{\gamma}\approx\text{\gammaLmultiplereshour}$ \end_inset $/hour, because in the multiple-good model, more temptation is needed to explain the observed limit valuation when consumers think the limit will also increase FITSBY use on other devices. \end_layout \begin_layout Standard Next, we return to the single-good model and consider an alternative specificati on where we estimate \begin_inset Formula $\omega$ \end_inset from differences in self-reported \emph on ideal use change \emph default between the Limit and Limit Control groups. Intuitively, if the Limit group reports on survey 3 that looking back over period 2, they ideally would not have further reduced their screen time, this suggests that the limit functionality fully eliminated temptation ( \begin_inset Formula $\omega=1$ \end_inset ). Extending this intuition, we estimate \begin_inset Formula $\omega$ \end_inset as the share of the Limit Control group's \emph on ideal use change \emph default that is eliminated in the Limit treatment group. If \begin_inset Formula $d_{2}^{g}$ \end_inset is group \begin_inset Formula $g$ \end_inset 's average \emph on ideal use change \emph default reported on survey 3 retrospectively about period 2, this is: \begin_inset Formula \begin{equation} \omega=\frac{d_{2}^{L}-d_{2}^{LC}}{-d_{2}^{LC}}.\label{eq:omega} \end{equation} \end_inset In the data, the Limit and Limit Control groups report that they ideally would have changed use by \begin_inset Formula $\dLnice$ \end_inset and \begin_inset Formula $\dCLnice$ \end_inset percent, respectively. This gives \begin_inset Formula $\hat{\omega}\approx\frac{\dLpercentnice-(\dCLpercentnice)}{-(\dCLpercentnice)}\approx\omegahat$ \end_inset . \end_layout \begin_layout Standard If we assume that the limit only eliminates share \begin_inset Formula $\omega<1$ \end_inset of temptation, the limit effect strategy will deliver larger \begin_inset Formula $\gamma$ \end_inset , because we infer that the true effect of temptation on consumption is larger. By contrast, the limit valuation strategy will deliver smaller \begin_inset Formula $\gamma$ \end_inset , because a smaller \begin_inset Formula $\gamma$ \end_inset is needed to explain a given valuation \begin_inset Formula $\bar{v}^{L}$ \end_inset when temptation has a larger effect on consumption. Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:Alternative_gamma" plural "false" caps "false" noprefix "false" \end_inset shows that in the restricted model ( \begin_inset Formula $\alpha=1$ \end_inset ), the limit effect \begin_inset Formula $\hat{\gamma}$ \end_inset increases from \begin_inset Formula $\gammaLeffectreshour$ \end_inset to \begin_inset Formula $\gammaLeffectomegareshour$ \end_inset , while the limit valuation strategy \begin_inset Formula $\hat{\gamma}$ \end_inset decreases from \begin_inset Formula $\gammaLreshour$ \end_inset to \begin_inset Formula $\gammaLomegareshour$ \end_inset . \end_layout \begin_layout Standard Finally, we extend the limit effect strategy to allow for individual-specific heterogeneity in \begin_inset Formula $\gamma$ \end_inset . To do this, we exploit the facts that we observe each participant's period 2 \emph on limit tightness \emph default \begin_inset Formula $H_{i2}$ \end_inset and that tightness is closely related to the limit treatment effect. We estimate heterogeneous period 2 and 3 limit effects as a function of period 2 \emph on limit tightness \emph default by adding an interaction term \begin_inset Formula $\tau^{HL}H_{i2}L_{i}$ \end_inset to the treatment effect estimation in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ); see Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:linear_specific" plural "false" caps "false" noprefix "false" \end_inset . \begin_inset Foot status open \begin_layout Plain Layout \begin_inset Formula $H_{i}$ \end_inset is missing for the Limit Control group, so we are not able to include the main effect of \begin_inset Formula $H_{i2}$ \end_inset in this regression. In theory, this could generate omitted variable bias if period 2 or 3 control group consumption varies with the tightness that they would have set. Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:linear_specific" plural "false" caps "false" noprefix "false" \end_inset shows that \begin_inset Formula $H_{i2}$ \end_inset is associated with the Limit group's consumption in the second half of period 1 (before the limit functionality was turned on). However, the association is small compared to the association in periods 2 and 3, which suggests that the potential omitted variables bias is relatively small. \end_layout \end_inset For each participant, we insert the fitted limit effect \begin_inset Formula $\hat{\tau}_{it}^{L}=\hat{\tau}_{t}^{L}+\hat{\tau}^{HL}H_{i2}$ \end_inset into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gamma_Limit" plural "false" caps "false" noprefix "false" \end_inset ) to infer \begin_inset Formula $\gamma_{i}$ \end_inset . The final row of Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:Alternative_gamma" plural "false" caps "false" noprefix "false" \end_inset shows that although this allows substantial heterogeneity, the average temptation \begin_inset Formula $\bar{\gamma}$ \end_inset is essentially the same as the homogeneous \begin_inset Formula $\gamma$ \end_inset from the limit effect strategy, as one would expect. \end_layout \begin_layout Standard These alternative approaches imply temptation \begin_inset Formula $\gamma$ \end_inset is between about $1 and $3 per hour. Our primary strategy (the limit effect) is relatively conservative. \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Alternative Temptation Parameter Estimates \begin_inset CommandInset label LatexCommand label name "tab:Alternative_gamma" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (2) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Restricted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Unrestricted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Parameter \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Description (units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout ( \begin_inset Formula $\tau_{2}^{B}=0$ \end_inset , \begin_inset Formula $\alpha=1$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout ( \begin_inset Formula $\alpha=\hat{\alpha}$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gamma$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Temptation ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \emph on \begin_inset Formula $\ \ \ \ \ \ $ \end_inset Limit effect (primary) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectreshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammahour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectreshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammahourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \emph on \begin_inset Formula $\ \ \ \ \ \ $ \end_inset Bonus valuation \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaBreshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaBhour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaBreshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaBhourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \emph on \begin_inset Formula $\ \ \ \ \ \ $ \end_inset Limit valuation \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLreshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLhour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLreshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLhourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \emph on \begin_inset Formula $\ \ \ \ \ \ $ \end_inset Limit effect, multiple-good model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectmultiplereshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectmultiplereshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \emph on \begin_inset Formula $\ \ \ \ \ \ $ \end_inset Bonus valuation, multiple-good model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaBmultiplereshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaBmultiplehour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaBmultiplereshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaBmultiplehourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \emph on \begin_inset Formula $\ \ \ \ \ \ $ \end_inset Limit valuation, multiple-good model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLmultiplereshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLmultiplehour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLmultiplereshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLmultiplehourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \emph on \begin_inset Formula $\ \ \ \ \ \ $ \end_inset Limit effect, \begin_inset Formula $\omega=\hat{\omega}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectomegareshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectomegahour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectomegareshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectomegahourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \emph on \begin_inset Formula $\ \ \ \ \ \ $ \end_inset Limit valuation, \begin_inset Formula $\omega=\hat{\omega}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLomegareshour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLomegahour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLomegareshourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLomegahourboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{\gamma}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average temptation ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaspechourres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaspechour$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \emph on \begin_inset Formula $\ \ \ \ \ \ $ \end_inset Heterogeneous limit effect \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaspechourresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaspechourboot$ \end_inset \end_layout \end_inset \end_inset \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table presents point estimates and bootstrapped 95 percent confidenc e intervals for alternative estimates of temptation \begin_inset Formula $\gamma$ \end_inset . Each row reflects estimates from a different specification. \begin_inset Formula $\gamma$ \end_inset for the limit effect, bonus valuation, and limit valuation strategies is from equations ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gamma_Limit" plural "false" caps "false" noprefix "false" \end_inset ), ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gammatildeB" plural "false" caps "false" noprefix "false" \end_inset ), and ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gammatildeL" plural "false" caps "false" noprefix "false" \end_inset ), respectively, combined with naivete \begin_inset Formula $\gamma-\tilde{\gamma}$ \end_inset from equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Naivete" plural "false" caps "false" noprefix "false" \end_inset ). \begin_inset Formula $\gamma$ \end_inset for the multiple-good model is from equations ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gamma_Limit_y" plural "false" caps "false" noprefix "false" \end_inset ), ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gammatildeB_y" plural "false" caps "false" noprefix "false" \end_inset ), and ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gammatildeL_y" plural "false" caps "false" noprefix "false" \end_inset ) in Appendix \begin_inset CommandInset ref LatexCommand ref reference "app:TemptationMultipleGoods" plural "false" caps "false" noprefix "false" \end_inset ; we do not have a limit effect estimate for the unrestricted multiple-good model. \begin_inset Formula $\hat{\omega}$ \end_inset is from equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:omega" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Heterogeneity in Limit Effect by Limit Tightness \begin_inset CommandInset label LatexCommand label name "tab:linear_specific" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \align center \size small \begin_inset CommandInset include LatexCommand input filename "../input/treatment_effects/heterogeneity_reg.tex" \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table presents the effects of bonus and limit treatments on FITSBY use in periods 1, 2, and 3 using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:ATE" plural "false" caps "false" noprefix "false" \end_inset ), including an additional interaction between the Limit group indicator and period 2 \emph on limit tightness. \shape italic \emph default Limit tightness \shape default is the amount by which a user's limits would have hypothetically reduced overall screen time if applied to their baseline use without snoozes; see equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:tightness" plural "false" caps "false" noprefix "false" \end_inset ). FITSBY use refers to screen time on Facebook, Instagram, Twitter, Snapchat, browser, and YouTube. \end_layout \end_inset \end_layout \begin_layout Subsection Model Estimates with Sample Weights \begin_inset CommandInset label LatexCommand label name "app:ModelwithSampleWeights" \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Demographics in Weighted Sample \series default \begin_inset CommandInset label LatexCommand label name "tab:SampleDemographicsWeighted" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset CommandInset include LatexCommand include filename "../input/descriptive/sample_demographics_balance.tex" \end_inset \end_layout \begin_layout Plain Layout \size small Notes: Column 1 presents average demographics for our analysis sample, column 2 presents average demographics for our weighted sample, and column 3 presents average demographics of American adults using data from the 2018 American Community Survey. The sample weights are initially calculated to make the sample nationally representative on these five demographics but are then winsorized at \begin_inset Formula $[1/3,3]$ \end_inset to reduce precision loss. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Empirical Moments and Additional Parameters in Weighted Sample \begin_inset CommandInset label LatexCommand label name "tab:MomentsWeighted" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (2) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Point \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Confidence \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Parameter \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Description \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout estimate \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout interval \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\delta$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Three-week discount factor (unitless) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $0.997$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{2}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Anticipatory bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tauBtwobalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tauBtwobalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{3}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Contemporaneous bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBthreebalancedmedian}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBthreebalanced}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{4}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Long-term bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfourbalancedmedian}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfourbalanced}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{5}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Long-term bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfivebalancedmedian}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauBfivebalanced}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tau_{2}^{L}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Limit effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauLtwobalancedmedian}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\tauLtwobalanced}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $m^{C}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Control group misprediction (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\mispredictbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\mispredictbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{\tilde{x}}_{3}^{B+BC}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Predicted use with/without bonus (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xtildetwoBbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xtildetwoBbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tilde{\tau}_{3}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Predicted bonus effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tautildeBthreetwobalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tautildeBthreetwobalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tilde{\tau}_{3}^{L}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Predicted limit effect (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tautildeLbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tautildeLbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\omega$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Temptation reduction from limit \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 1 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{v}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average bonus valuation ($/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\vBbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\vBbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{v}^{L}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average limit valuation ($/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\vLbalancedmedian}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\text{\vLbalanced}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $p^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Bonus price ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 2.5 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{F}^{B}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average bonus fixed payment ($/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\FBbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\FBbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{x}_{1}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average baseline use (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssbalanced$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table presents point estimates and bootstrapped 95 percent confidenc e intervals for the empirical moments used for estimation. We winsorize at \begin_inset Formula $\tau_{2}^{B}\leq0$ \end_inset , and we drop the \begin_inset Formula $\NegSSDenom$ \end_inset percent of draws in which the denominator of steady-state consumption in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS counterfactual" plural "false" caps "false" noprefix "false" \end_inset ) is not positive. This parallels Table \begin_inset CommandInset ref LatexCommand ref reference "tab:Moments" plural "false" caps "false" noprefix "false" \end_inset , except using the weighted sample. The sample weights are initially calculated to make the sample nationally representative on the five demographics in Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:SampleDemographicsWeighted" plural "false" caps "false" noprefix "false" \end_inset but are then winsorized at \begin_inset Formula $[1/3,3]$ \end_inset to reduce precision loss. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Model Parameter Estimates in Weighted Sample \begin_inset CommandInset label LatexCommand label name "tab:StructuralEstimatesWeighted" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Restricted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Parameter \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Description (units) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout ( \begin_inset Formula $\tau_{2}^{B}=0$ \end_inset , \begin_inset Formula $\alpha=1$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\lambda$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Habit stock effect on consumption (unitless) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\lambdareshatbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\lambdareshatbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rho$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Habit formation (unitless) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rhoreshatbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\rhoreshatbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\alpha$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Projection bias (unitless) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 1 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\eta$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Price coefficient ($-day/hour \begin_inset Formula $^{2}$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\etareshourbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\etareshourbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\zeta$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Habit stock effect on marginal utility ($-day/hour \begin_inset Formula $^{2}$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\zetareshourbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\zetareshourbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gamma-\tilde{\gamma}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Naivete about temptation ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\naivetereshourbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\naivetereshourbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gamma$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Temptation ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectreshourbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\gammaLeffectreshourbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\bar{\kappa}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Average intercept ($/hour) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\intercepthetLeffectreshourbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\intercepthetLeffectreshourbalanced$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table presents point estimates and bootstrapped 95 percent confidenc e intervals from the estimation strategy described in Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:Estimating Equations Unrestricted" plural "false" caps "false" noprefix "false" \end_inset . We winsorize at \begin_inset Formula $\tau_{2}^{B}\leq0$ \end_inset , and we drop the \begin_inset Formula $\NegSSDenom$ \end_inset percent of draws in which the denominator of steady-state consumption in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS counterfactual" plural "false" caps "false" noprefix "false" \end_inset ) is not positive. Temptation \begin_inset Formula $\gamma$ \end_inset is from the limit effect strategy, using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gamma_Limit" plural "false" caps "false" noprefix "false" \end_inset ). This parallels Table \begin_inset CommandInset ref LatexCommand ref reference "tab:StructuralEstimates" plural "false" caps "false" noprefix "false" \end_inset , except using the weighted sample. The sample weights are initially calculated to make the sample nationally representative on the five demographics in Appendix Table \begin_inset CommandInset ref LatexCommand ref reference "tab:SampleDemographicsWeighted" plural "false" caps "false" noprefix "false" \end_inset but are then winsorized at \begin_inset Formula $[1/3,3]$ \end_inset to reduce precision loss. \end_layout \end_inset \end_layout \begin_layout Section Proofs of Propositions in Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:EulerLinearitySS" plural "false" caps "false" noprefix "false" \end_inset \begin_inset CommandInset label LatexCommand label name "app:ModelAppendix" \end_inset \end_layout \begin_layout Standard Given naivete about projection bias, the predicted continuation value function given predicted consumption and habit stock is \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} V_{t+1}\left(\tilde{s}_{t+1}\right)=\sum_{r=t+1}^{T}\delta^{r-t}u_{r}\left(\tilde{x}_{r}^{*}\left(\tilde{s}_{r},\tilde{\gamma},\boldsymbol{p}_{r}\right);\tilde{s}_{r},p_{r}\right).\label{eq:V} \end{equation} \end_inset The consumer's predicted objective function in future period \begin_inset Formula $t$ \end_inset can thus be written as \begin_inset Formula \begin{equation} \tilde{U}_{t}\left(x_{t};\tilde{s}_{t}\right)=u_{t}\left(x_{t};\tilde{s}_{t},p_{t}\right)+\tilde{\gamma}x_{t}+\delta V_{t+1}\left(\tilde{s}_{t+1}\right),\label{eq:Utilde} \end{equation} \end_inset and the consumer's actual period \begin_inset Formula $t$ \end_inset objective function from equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:equilibrium" plural "false" caps "false" noprefix "false" \end_inset ) can be written as \begin_inset Formula \begin{equation} U_{t}\left(x_{t};s_{t}\right)=u_{t}\left(x_{t};s_{t},p_{t}\right)+\gamma x_{t}+\begin{array}{c} \alpha\sum_{r=t+1}^{T}\delta^{r-t}u_{r}\left(\tilde{x}_{r}^{*}\left(s_{t},\tilde{\gamma},\boldsymbol{p}_{r}\right);s_{t},p_{r}\right)\\ +(1-\alpha)\delta V_{t+1}\left(\tilde{s}_{t+1}\right) \end{array}.\label{eq:U} \end{equation} \end_inset Recall that we defined \begin_inset Formula $u_{t}\coloneqq u_{t}\left(x_{t}^{*};s_{t},p_{t}\right)$ \end_inset , \begin_inset Formula $\tilde{x}_{r}\coloneqq\tilde{x}_{r}^{*}\left(\tilde{s}_{r},\tilde{\gamma},\boldsymbol{p}_{r}\right)$ \end_inset , and \begin_inset Formula $\tilde{u}_{r}\coloneqq u_{r}\left(\tilde{x}_{r};\tilde{s}_{r},p_{r}\right)$ \end_inset . \end_layout \begin_layout Subsection Proof of Proposition \begin_inset CommandInset ref LatexCommand ref reference "prop:Euler" plural "false" caps "false" noprefix "false" \end_inset : Euler Equation \begin_inset CommandInset label LatexCommand label name "app:EulerEqnProof" \end_inset \end_layout \begin_layout Standard In this section, we derive the Euler equation (equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Euler" plural "false" caps "false" noprefix "false" \end_inset )), proving Proposition \begin_inset CommandInset ref LatexCommand ref reference "prop:Euler" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Proof The time \begin_inset Formula $t$ \end_inset first-order condition from maximizing utility (equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:U" plural "false" caps "false" noprefix "false" \end_inset )) is \end_layout \begin_layout Proof \begin_inset Formula \begin{align} \frac{\partial u_{t}}{\partial x_{t}}+\gamma & =-(1-\alpha)\delta\frac{d\tilde{s}_{t+1}}{dx_{t}}\frac{dV_{t+1}\left(\tilde{s}_{t+1}\right)}{d\tilde{s}_{t+1}}\\ & =-(1-\alpha)\delta\frac{d\tilde{s}_{t+1}}{dx_{t+1}}\left[\frac{\partial\tilde{u}_{t+1}}{\partial\tilde{x}_{t+1}}\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}+\frac{\partial\tilde{u}_{t+1}}{\partial\tilde{s}_{t+1}}\right]-(1-\alpha)\delta^{2}\frac{d\tilde{s}_{t+2}}{dx_{t}}\frac{dV_{t+2}\left(\tilde{s}_{t+2}\right)}{d\tilde{s}_{t+2}}\\ & =-(1-\alpha)\delta\rho\left[\frac{\partial\tilde{u}_{t+1}}{\partial\tilde{x}_{t+1}}\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}+\frac{\partial\tilde{u}_{t+1}}{\partial\tilde{s}_{t+1}}\right]-(1-\alpha)\left(\delta\ensuremath{\rho}\right)^{2}\left(1+\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}\right)\frac{dV_{t+2}\left(\tilde{s}_{t+2}\right)}{d\tilde{s}_{t+2}},\label{eq:t FOC} \end{align} \end_inset where the third line uses the fact that the total derivative of predicted period \begin_inset Formula $t+2$ \end_inset habit stock with respect to period \begin_inset Formula $t$ \end_inset consumption is \end_layout \begin_layout Proof \begin_inset Formula \begin{align} \frac{d\tilde{s}_{t+2}}{dx_{t}} & =\frac{\partial\tilde{s}_{t+2}}{\partial\tilde{s}_{t+1}}\frac{\partial\tilde{s}_{t+1}}{\partial x_{t}}+\frac{\partial\tilde{s}_{t+2}}{\partial\tilde{x}_{t+1}}\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}\frac{\partial\tilde{s}_{t+1}}{\partial x_{t}}\nonumber \\ & =\text{\ensuremath{\rho^{2}}\left(1+\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}\right)}\label{eq:TimeShift} \end{align} \end_inset \end_layout \begin_layout Proof The time \begin_inset Formula $t$ \end_inset self predicts that the time \begin_inset Formula $t+1$ \end_inset self will maximize equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Utilde" plural "false" caps "false" noprefix "false" \end_inset ), setting \begin_inset Formula $x_{t+1}$ \end_inset according to the following first-order condition: \end_layout \begin_layout Proof \begin_inset Formula \begin{align} 0= & \frac{\partial\tilde{u}_{t+1}}{\partial\tilde{x}_{t+1}}+\tilde{\gamma}+\delta\frac{d\tilde{s}_{t+2}}{dx_{t+1}}\frac{dV_{t+2}\left(\tilde{s}_{t+2}\right)}{d\tilde{s}_{t+2}}\\ & =\frac{\partial\tilde{u}_{t+1}}{\partial\tilde{x}_{t+1}}+\tilde{\gamma}+\delta\rho\frac{dV_{t+2}\left(\tilde{s}_{t+2}\right)}{d\tilde{s}_{t+2}}\label{eq:t+1 FOC} \end{align} \end_inset \end_layout \begin_layout Proof Multiplying the predicted time \begin_inset Formula $t+1$ \end_inset first-order condition by \begin_inset Formula $(1-\alpha)\delta\rho\left(1+\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}\right)$ \end_inset gives \end_layout \begin_layout Proof \begin_inset Formula \begin{align} 0=(1-\alpha)\delta\rho\left(1+\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}\right) & \left(\frac{\partial\tilde{u}_{t+1}}{\partial\tilde{x}_{t+1}}+\tilde{\gamma}\right)+(1-\alpha)\left(\delta\rho\right)^{2}\left(1+\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}\right)\frac{dV_{t+2}\left(\tilde{s}_{t+2}\right)}{d\tilde{s}_{t+2}} \end{align} \end_inset \end_layout \begin_layout Proof The last term is the same as the last term in the time \begin_inset Formula $t$ \end_inset first-order condition. Adding this equation to the time \begin_inset Formula $t$ \end_inset first-order condition yields \end_layout \begin_layout Proof \begin_inset Formula \begin{align} \frac{\partial u_{t}}{\partial x_{t}}+\gamma & =(1-\alpha)\delta\rho\left(1+\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}\right)\left(\frac{\partial\tilde{u}_{t+1}}{\partial\tilde{x}_{t+1}}+\tilde{\gamma}\right)-(1-\alpha)\delta\rho\left[\frac{\partial\tilde{u}_{t+1}}{\partial\tilde{x}_{t+1}}\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}+\frac{\partial\tilde{u}_{t+1}}{\partial\tilde{s}_{t+1}}\right]\\ \frac{\partial u_{t}}{\partial x_{t}}+\gamma & =(1-\alpha)\delta\rho\left[\frac{\partial\tilde{u}_{t+1}}{\partial\tilde{x}_{t+1}}+\tilde{\gamma}+\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}\tilde{\gamma}-\frac{\partial\tilde{u}_{t+1}}{\partial\tilde{s}_{t+1}}\right].\label{eq:Euler_general} \end{align} \end_inset \end_layout \begin_layout Proof We now derive the Euler equation with our quadratic functional form. The partial derivatives are \end_layout \begin_layout Proof \begin_inset Formula \begin{align} \frac{\partial u_{t}}{\partial x_{t}}= & \eta x_{t}^{*}+\zeta s_{t}+\xi_{t}-p_{t}\\ \frac{\partial\tilde{u}_{t+1}}{\partial\tilde{x}_{t+1}}= & \eta\tilde{x}_{t+1}+\zeta\tilde{s}_{t+1}+\xi_{t+1}-p_{t+1}\\ \tilde{\lambda}_{t+1}\coloneqq & \frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}\\ \frac{\partial\tilde{u}_{t+1}}{\partial\tilde{s}_{t+1}}= & \zeta\tilde{x}_{t+1}+\phi. \end{align} \end_inset Substituting these into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Euler_general" plural "false" caps "false" noprefix "false" \end_inset ) yields equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Euler" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Subsection Proof of Proposition \begin_inset CommandInset ref LatexCommand ref reference "prop:Linearity" plural "false" caps "false" noprefix "false" \end_inset : Linear Policy Functions \begin_inset CommandInset label LatexCommand label name "app:LinearityProof" \end_inset \end_layout \begin_layout Standard In this section, we first show that the policy function is linear in habit stock. We then show that if the objective function is concave, \begin_inset Formula $\lambda$ \end_inset converges to a constant far from the time horizon. We then show the conditions under which utility is concave. Finally, we show the condition required for \begin_inset Formula $\mu$ \end_inset to converge to a constant far from the time horizon. Our proof strategy follows \begin_inset CommandInset citation LatexCommand citet key "GruberKoszegi2001" literal "false" \end_inset . \end_layout \begin_layout Lemma \begin_inset CommandInset label LatexCommand label name "lemma:Linearity" \end_inset Suppose \begin_inset Formula $u_{t}(x_{t};s_{t},p_{t})$ \end_inset is given by equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:QuadraticUtility" plural "false" caps "false" noprefix "false" \end_inset ) and \begin_inset Formula $\left(x_{0}^{*},...,x_{T}^{*}\right)$ \end_inset is a perception-perfect strategy profile. Then for any \begin_inset Formula $t$ \end_inset , \end_layout \begin_layout Lemma \begin_inset Formula \begin{align} x_{t}^{*}(s_{t},\gamma,\boldsymbol{p}_{t}) & =\lambda_{t}s_{t}+\mu_{t}(\gamma)\\ \tilde{x}_{t}^{*}(s_{t},\tilde{\gamma},\boldsymbol{p}_{t}) & =\tilde{\lambda}_{t}s_{t}+\mu_{t}(\tilde{\gamma}) \end{align} \end_inset where \begin_inset Formula $\lambda_{t}$ \end_inset is a function of only \begin_inset Formula $\left\{ \eta,\zeta,\delta,\rho,\alpha\right\} $ \end_inset , \begin_inset Formula $\tilde{\lambda}_{t}$ \end_inset is a function of only \begin_inset Formula $\left\{ \eta,\zeta,\delta,\rho\right\} $ \end_inset , and \begin_inset Formula $\mu_{t}$ \end_inset is linear in \begin_inset Formula $p_{t}$ \end_inset . \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Proof We prove by backwards induction. First, we show that the result holds for period \begin_inset Formula $T$ \end_inset . Given our functional form, the period \begin_inset Formula $T$ \end_inset first-order condition is \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} \eta x_{T}^{*}+\zeta s_{T}+\xi_{T}-p_{T}+\gamma=0, \end{equation} \end_inset and thus \begin_inset Formula \begin{equation} x_{T}^{*}=\frac{\zeta s_{T}+\xi_{T}-p_{T}+\gamma}{-\eta}. \end{equation} \end_inset Thus, \begin_inset Formula $x_{T}^{*}$ \end_inset can be written as \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} x_{T}^{*}=\lambda_{T}s_{T}+\mu_{T}(\gamma), \end{equation} \end_inset with \begin_inset Formula $\lambda_{T}=$ \end_inset \begin_inset Formula $\frac{\zeta}{-\eta}$ \end_inset and \begin_inset Formula $\mu_{T}(\gamma)=\frac{\xi_{T}-p_{T}+\gamma}{-\eta}$ \end_inset . \end_layout \begin_layout Proof Analogously, predicted consumption is \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} \tilde{x}_{T}=\frac{\zeta\tilde{s}_{T}+\xi_{T}-p_{T}+\tilde{\gamma}}{-\eta}, \end{equation} \end_inset so \begin_inset Formula $\tilde{x}_{T}$ \end_inset can be written as \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} \tilde{x}_{T}=\lambda_{T}\tilde{s}_{T}+\mu_{T}(\tilde{\gamma}), \end{equation} \end_inset with \begin_inset Formula $\mu_{T}(\tilde{\gamma})=\frac{\xi_{T}-p_{T}+\tilde{\gamma}}{-\eta}$ \end_inset . The function \begin_inset Formula $\mu_{T}$ \end_inset is linear in \begin_inset Formula $p_{T}$ \end_inset . \end_layout \begin_layout Proof Now, we use the Euler equation to show that if the result holds for \begin_inset Formula $t+1$ \end_inset , it holds for \begin_inset Formula $t$ \end_inset . The Euler equation is \end_layout \begin_layout Proof \begin_inset Formula \[ \begin{aligned}\eta x_{t}^{*}+\zeta s_{t}+\xi_{t}-p_{t}+\gamma & =(1-\alpha)\delta\rho\left[\eta\tilde{x}_{t+1}+\zeta\tilde{s}_{t+1}+\xi_{t+1}-p_{t+1}+\left(1+\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}\right)\tilde{\gamma}-\zeta\tilde{x}_{t+1}-\phi\right]\\ & =(1-\alpha)\delta\rho\left[(\eta-\zeta)\tilde{x}_{t+1}+\zeta\tilde{s}_{t+1}+\xi_{t+1}-p_{t+1}+\left(1+\frac{\partial\tilde{x}_{t+1}}{\partial\tilde{s}_{t+1}}\right)\tilde{\gamma}-\phi\right] \end{aligned} \] \end_inset \end_layout \begin_layout Proof Substituting \begin_inset Formula $\tilde{x}_{t+1}=\tilde{\lambda}_{t+1}\tilde{s}_{t+1}+\mu_{t+1}(\tilde{\gamma})$ \end_inset , \begin_inset Formula $\tilde{s}_{t+1}=\rho\left(s_{t}+x_{t}^{*}\right)$ \end_inset , and \begin_inset Formula $\tilde{\lambda}_{t+1}=\frac{\partial\tilde{x}_{t+1}^{*}}{\partial\tilde{s}_{t+1}}$ \end_inset gives \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} \eta x_{t}^{*}+\zeta\tilde{s}_{t}+\xi_{t}-p_{t}+\gamma=(1-\alpha)\delta\rho\left[(\eta-\zeta)\left(\tilde{\lambda}_{t+1}\rho\left(x_{t}^{*}+s_{t}\right)+\mu_{t+1}(\tilde{\gamma})\right)+\zeta\rho(s_{t}+x_{t}^{*})+\xi_{t+1}-p_{t+1}+\tilde{\gamma}+\tilde{\gamma}\tilde{\lambda}_{t+1}-\phi\right]. \end{equation} \end_inset \end_layout \begin_layout Proof Solving for \begin_inset Formula $x_{t}^{*}$ \end_inset gives \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} x_{t}^{*}=\frac{s_{t}\left[\zeta-(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}_{t+1}+\zeta\right)\right]+\xi_{t}-p_{t}+\gamma-(1-\alpha)\delta\rho\left[(\eta-\zeta)\mu_{t+1}(\tilde{\gamma})+\xi_{t+1}-p_{t+1}+\tilde{\gamma}+\tilde{\gamma}\tilde{\lambda}_{t+1}-\phi\right]}{-\eta+(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}_{t+1}+\zeta\right)}. \end{equation} \end_inset \end_layout \begin_layout Proof Thus, \begin_inset Formula $x_{t}^{*}=\lambda_{t}s_{t}+\mu_{t}(\gamma)$ \end_inset , with \begin_inset Formula \begin{equation} \lambda_{t}=\frac{\zeta-(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}_{t+1}+\zeta\right)}{-\eta+(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}_{t+1}+\zeta\right)},\label{eq:lambdat} \end{equation} \end_inset \end_layout \begin_layout Proof and \begin_inset Formula \begin{equation} \mu_{t}(\gamma)=\frac{\xi_{t}-p_{t}+\gamma-(1-\alpha)\delta\rho\left[\xi_{t+1}-p_{t+1}+\tilde{\gamma}+\tilde{\gamma}\tilde{\lambda}_{t+1}-\phi\right]+(1-\alpha)\delta\rho\left(\zeta-\eta\right)\mu_{t+1}(\tilde{\gamma})}{-\eta+(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}_{t+1}+\zeta\right)}.\label{eq:mu_t} \end{equation} \end_inset \end_layout \begin_layout Proof We can analogously begin with the period \begin_inset Formula $t$ \end_inset Euler equation as \emph on predicted \emph default before period \begin_inset Formula $t$ \end_inset , which has \begin_inset Formula $\tilde{\gamma}$ \end_inset and \begin_inset Formula $\tilde{s}_{t}$ \end_inset instead of \begin_inset Formula $\gamma$ \end_inset and \begin_inset Formula $s_{t}$ \end_inset on the left-hand side, and does not have the \begin_inset Formula $(1-\alpha)$ \end_inset term. This gives \begin_inset Formula $\tilde{x}_{t}=\tilde{\lambda}_{t}\tilde{s}_{t}+\mu_{t}(\tilde{\gamma})$ \end_inset , with \begin_inset Formula \begin{equation} \tilde{\lambda}_{t}=\frac{\zeta-\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}_{t+1}+\zeta\right)}{-\eta+\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}_{t+1}+\zeta\right)}.\label{eq:lambdatildet} \end{equation} \end_inset and \begin_inset Formula $\mu_{t}(\tilde{\gamma})$ \end_inset given by equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:mu_t" plural "false" caps "false" noprefix "false" \end_inset ) except that, as as implied by writing \begin_inset Formula $\mu_{t}(\tilde{\gamma})$ \end_inset instead of \begin_inset Formula $\mu_{t}(\gamma)$ \end_inset , the third term in the numerator is \begin_inset Formula $\tilde{\gamma}$ \end_inset instead of \begin_inset Formula $\gamma$ \end_inset . \begin_inset Foot status open \begin_layout Plain Layout Equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambdat" plural "false" caps "false" noprefix "false" \end_inset ) is much simpler than equation (25) of \begin_inset CommandInset citation LatexCommand citet key "GruberKoszegi2001" literal "false" \end_inset , and our expression for \begin_inset Formula $\lambda_{t}$ \end_inset does not depend on actual or perceived temptation \begin_inset Formula $\gamma$ \end_inset or \begin_inset Formula $\tilde{\gamma}$ \end_inset , while theirs depends on present focus \begin_inset Formula $\beta$ \end_inset . This is because in their quasi-hyperbolic framework, \begin_inset Formula $1-\beta$ \end_inset multiplies \begin_inset Formula $\lambda_{t+1}$ \end_inset parameters in the Euler equation and doesn't drop out. \end_layout \end_inset Thus, \begin_inset Formula $\lambda_{t}$ \end_inset is not correctly perceived in advance of period \begin_inset Formula $t$ \end_inset . \end_layout \begin_layout Proof \begin_inset Formula $\lambda_{t}$ \end_inset depends only on \begin_inset Formula $\{\eta,\zeta,\delta,\rho,\alpha\}$ \end_inset , and \begin_inset Formula $\tilde{\lambda}_{t}$ \end_inset depends only on \begin_inset Formula $\{\eta,\zeta,\delta,\rho\}$ \end_inset , as long as \begin_inset Formula $\tilde{\lambda}_{t+1}$ \end_inset depends only on \begin_inset Formula $\{\eta,\zeta,\delta,\rho\}$ \end_inset . Because consumers misperceive \begin_inset Formula $\gamma$ \end_inset , \begin_inset Formula $\mu_{r}$ \end_inset is also misperceived for \begin_inset Formula $r>t$ \end_inset . The function \begin_inset Formula $\mu_{t}$ \end_inset is linear in \begin_inset Formula $p_{t}$ \end_inset . \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard We now show that with concave utility, \begin_inset Formula $\lambda_{t}$ \end_inset and \begin_inset Formula $\tilde{\lambda}_{t}$ \end_inset are constant in \begin_inset Formula $t$ \end_inset far from the time horizon. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Lemma \begin_inset CommandInset label LatexCommand label name "lemma:Constantlambda" \end_inset Suppose the conditions for Lemma \begin_inset CommandInset ref LatexCommand ref reference "lemma:Linearity" plural "false" caps "false" noprefix "false" \end_inset hold and utility is concave. Then for any fixed \begin_inset Formula $t$ \end_inset , \end_layout \begin_layout Lemma \begin_inset Formula \begin{equation} \lambda=\lim_{T\rightarrow\infty}\lambda_{t}=\frac{\zeta-(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)}{-\eta+(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)}, \end{equation} \end_inset and \end_layout \begin_layout Lemma \begin_inset Formula \begin{equation} \tilde{\lambda}=\lim_{T\rightarrow\infty}\tilde{\lambda_{t}}=\frac{-\eta-\sqrt{\eta^{2}-4\frac{\delta\rho^{2}\left(\zeta-\eta\right)}{\left(1-\delta\rho^{2}\right)}\zeta}}{\frac{2\delta\rho^{2}\left(\zeta-\eta\right)}{\left(1-\delta\rho^{2}\right)}}. \end{equation} \end_inset \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Proof To show that \begin_inset Formula $\lambda_{t}$ \end_inset is constant in \begin_inset Formula $t$ \end_inset far from the time horizon, it suffices to prove the convergence of \begin_inset Formula $\tilde{\lambda_{t}}$ \end_inset to the steady state, since \begin_inset Formula $\lambda_{t}$ \end_inset is a function of \begin_inset Formula $\tilde{\lambda}_{t+1}$ \end_inset and other deterministic parameters. We define the function \begin_inset Formula $f(\tilde{{\lambda}})$ \end_inset according to Equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambdatildet" plural "false" caps "false" noprefix "false" \end_inset ) that describes the recursion \begin_inset Formula $\tilde{\lambda}_{t}=f(\tilde{\lambda}_{t+1})$ \end_inset . We first find the values of \begin_inset Formula $\tilde{\lambda}$ \end_inset that could be fixed points. Assuming constant \begin_inset Formula $\tilde{\lambda}$ \end_inset and rearranging Equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambdat" plural "false" caps "false" noprefix "false" \end_inset ) gives \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} -\eta\tilde{\lambda}+\delta\rho^{2}\left(\left(\eta-\zeta\right)\tilde{\lambda}^{2}+\zeta\tilde{\lambda}\right)=\zeta+\delta\rho^{2}\left(\left(\zeta-\eta\right)-\zeta\right). \end{equation} \end_inset Collecting terms gives \end_layout \begin_layout Proof \begin_inset Formula \begin{align} \tilde{\lambda}^{2}\delta\rho^{2}\left(\eta-\zeta\right)+\tilde{\lambda}\eta\left(\delta\rho^{2}-1\right)+\zeta\left(\delta\rho^{2}-1\right) & =0\\ \tilde{\lambda}^{2}\frac{\delta\rho^{2}\left(\zeta-\eta\right)}{\left(1-\delta\rho^{2}\right)}+\tilde{\lambda}\eta+\zeta & =0. \end{align} \end_inset Using the quadratic formula gives \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} \tilde{\lambda}=\frac{-\eta\pm\sqrt{\eta^{2}-4\frac{\delta\rho^{2}\left(\zeta-\eta\right)}{\left(1-\delta\rho^{2}\right)}\zeta}}{\frac{2\delta\rho^{2}\left(\zeta-\eta\right)}{\left(1-\delta\rho^{2}\right)}}.\label{eq:lambda_quadratic} \end{equation} \end_inset \end_layout \begin_layout Proof We now prove convergence. The function \begin_inset Formula $f(\lambda)$ \end_inset has the following properties. First, \begin_inset Formula $f(\lambda)$ \end_inset is always increasing as \end_layout \begin_layout Proof \series bold \begin_inset Formula \begin{align} f'(\tilde{\lambda}) & =\dfrac{-\delta\rho^{2}\left(\eta-\zeta\right)\left(-\eta+\delta\rho^{2}\left(\left(\eta-\zeta\right)\tilde{{\lambda}}+\zeta\right)\right)-\delta\rho^{2}\left(\eta-\zeta\right)\left(\zeta-\delta\rho^{2}\left(\left(\eta-\zeta\right)\tilde{\lambda}+\zeta\right)\right)}{\left(-\eta+\delta\rho^{2}\left(\left(\eta-\zeta\right)\tilde{\lambda}+\zeta\right)\right)^{2}}\\ & =\dfrac{\delta\rho^{2}\left(\zeta-\eta\right)^{2}}{\left(-\eta+\delta\rho^{2}\left(\left(\eta-\zeta\right)\tilde{\lambda}+\zeta\right)\right)^{2}}>0. \end{align} \end_inset \end_layout \begin_layout Proof Second, \begin_inset Formula $f$ \end_inset is convex on \begin_inset Formula $(-\infty,\bar{\tilde{\lambda}}),$ \end_inset where \begin_inset Formula $\bar{\tilde{\lambda}}=\dfrac{-\eta+\delta\rho^{2}\zeta}{\delta\rho^{2}(\zeta-\eta)}>0$ \end_inset . This comes from the sign of its second derivative \series bold \begin_inset Formula \begin{equation} f''(\tilde{\lambda})=\dfrac{2\delta^{2}\rho^{4}\left(-\eta+\zeta\right)^{3}}{\left(-\eta+\delta\rho^{2}\left(\left(\eta-\zeta\right)\tilde{\lambda}+\zeta\right)\right)^{3}}, \end{equation} \end_inset \series default which \series bold \series default is determined by the sign of the denominator. \end_layout \begin_layout Proof Third, for \begin_inset Formula $\mathbf{\tilde{\lambda}}>\bar{\tilde{\lambda}}$ \end_inset , \begin_inset Formula $f(\tilde{\lambda})$ \end_inset is always negative due to the denominator in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambdatildet" plural "false" caps "false" noprefix "false" \end_inset ), hence none of the solutions for a constant \begin_inset Formula $\mathbf{\tilde{\lambda}}_{t}$ \end_inset are in this region. \end_layout \begin_layout Proof Fourth, \begin_inset Formula $f(0)>0$ \end_inset since \begin_inset Formula $\delta\rho^{2}<1$ \end_inset and \begin_inset Formula \begin{equation} f(0)=\dfrac{\zeta(1-\delta\rho^{2})}{-\eta+\delta\rho^{2}\zeta}. \end{equation} \end_inset \end_layout \begin_layout Proof Fifth, \begin_inset Formula $f(\tilde{\lambda})$ \end_inset is continuous on \begin_inset Formula $[0,\bar{\tilde{\lambda}})$ \end_inset and \begin_inset Formula $\lim_{\tilde{\lambda}\to\bar{\tilde{\lambda}}}f(\tilde{\lambda})=\infty$ \end_inset as the denominator in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambdat" plural "false" caps "false" noprefix "false" \end_inset ) goes to 0. \end_layout \begin_layout Proof The properties highlighted above imply that both candidate solutions for a constant \begin_inset Formula $\tilde{\lambda}_{t}$ \end_inset in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambda_quadratic" plural "false" caps "false" noprefix "false" \end_inset ) are positive. To see this, denote the two candidate solutions as \begin_inset Formula $(\tilde{\lambda}_{1},\mathbf{\tilde{\lambda}}_{2})$ \end_inset , with \begin_inset Formula $\mathbf{\tilde{\lambda}}_{1}<\tilde{\lambda}_{2}$ \end_inset . Since \begin_inset Formula $f(0)>0$ \end_inset , we know that at least one solution for \begin_inset Formula $\tilde{\lambda}$ \end_inset is positive given \begin_inset Formula $-\eta>0$ \end_inset . Furthermore, since \begin_inset Formula $f(\tilde{\lambda})>0$ \end_inset on \begin_inset Formula $(-\infty,\bar{\tilde{\lambda}}]$ \end_inset , it cannot be true that an increasing, continuous, and convex function that diverges to infinity at \begin_inset Formula $\bar{\tilde{\lambda}}$ \end_inset only crosses the identity function once in \begin_inset Formula $[0,\bar{\tilde{\lambda}})$ \end_inset . Hence, both solutions are in \begin_inset Formula $[0,\bar{\tilde{{\lambda}}}]$ \end_inset . \end_layout \begin_layout Proof Given this result and the convex shape of this function, it must be true that \begin_inset Formula $\tilde{\lambda}_{1}$ \end_inset is a stable constant solution for the recursion while \begin_inset Formula $\tilde{\lambda}_{2}$ \end_inset is unstable. For any point in \begin_inset Formula $[0,\tilde{\lambda}_{1}]$ \end_inset the recursion implies an increase in \begin_inset Formula $\tilde{\lambda}_{t}$ \end_inset ( \begin_inset Formula $f(\tilde{\lambda})>\tilde{\lambda})$ \end_inset , for any point in \begin_inset Formula $[\tilde{{\lambda}}_{1},\tilde{{\lambda}}_{2}]$ \end_inset the recursion implies a decrease in \begin_inset Formula $\lambda_{t}$ \end_inset ( \begin_inset Formula $f(\tilde{{\lambda}})<\tilde{{\lambda}})$ \end_inset , and for any point in \begin_inset Formula $[\tilde{{\lambda}}_{2},\bar{\tilde{{\lambda}}}]$ \end_inset the recursion implies an increase in \begin_inset Formula $\tilde{{\lambda}}_{t}$ \end_inset ( \begin_inset Formula $f(\tilde{{\lambda}})>\tilde{{\lambda}})$ \end_inset . Overall, this means that for any starting value of \begin_inset Formula $\tilde{{\lambda}}_{t}\in[0,\tilde{{\lambda}}_{2})$ \end_inset the recursion converges to \begin_inset Formula $\tilde{{\lambda}}_{1}.$ \end_inset \end_layout \begin_layout Proof To complete the proof, we begin with \begin_inset Formula $\tilde{{\lambda}}_{T}$ \end_inset and then prove that far away from the time horizon, \begin_inset Formula $\tilde{{\lambda}}_{t}$ \end_inset is constant. To do this, we need to show that this initial value, given by \begin_inset Formula $\tilde{{\lambda}}_{T}=\dfrac{\zeta}{-\eta}$ \end_inset , is less than \begin_inset Formula $\tilde{{\lambda}}_{2}$ \end_inset . To show this, notice that the two solutions \begin_inset Formula $(\tilde{{\lambda}}_{1},\tilde{{\lambda}}_{2})$ \end_inset are symmetrically placed around \begin_inset Formula $\tilde{\lambda_{s}}=\dfrac{-\eta(1-\delta\rho^{2})}{2\delta\rho^{2}(\zeta-\eta)}$ \end_inset . Given this value, by the parametric assumption that guarantees the existence of the two constant solutions for the recursion, we know that \begin_inset Formula \begin{equation} \eta^{2}-4\frac{\delta\rho^{2}\left(\zeta-\eta\right)}{\left(1-\delta\rho^{2}\right)}\zeta>0, \end{equation} \end_inset and since \begin_inset Formula \begin{equation} \eta^{2}>2\frac{\delta\rho^{2}\left(\zeta-\eta\right)\zeta}{\left(1-\delta\tilde{\rho}^{2}\right)}\iff\dfrac{\zeta}{-\eta}<\dfrac{-\eta(1-\delta\rho^{2})}{2\delta\rho^{2}(\zeta-\eta)}, \end{equation} \end_inset we have that \begin_inset Formula $\tilde{\lambda}_{T}<\tilde{\lambda_{s}}$ \end_inset . Then \begin_inset Formula $\tilde{\lambda}_{T}<\tilde{\lambda_{s}}<\lambda_{2}$ \end_inset , and hence the backward recursion starting from \begin_inset Formula $\tilde{\lambda}_{T}$ \end_inset converges far from the time horizon to a stationary value \begin_inset Formula $\tilde{\lambda}^{*}=\tilde{\lambda}_{1}$ \end_inset . Moreover, \begin_inset Formula $f(\tilde{\lambda}_{T})$ \end_inset can be written as \begin_inset Formula $\frac{\zeta-X}{-\eta+X}$ \end_inset , and we know that \begin_inset Formula $\frac{\zeta-X}{-\eta+X}>\frac{\zeta}{-\eta}$ \end_inset whenever \begin_inset Formula $X<0$ \end_inset . Then, given that \begin_inset Formula \[ X=(1-\alpha)\delta\rho^{2}\left(\left(\eta-\zeta\right)\tilde{{\lambda}}_{T}+\zeta\right)<0\iff\left(\eta-\zeta\right)\frac{\zeta}{-\eta}+\zeta<0\iff\dfrac{\zeta^{2}}{\eta}<0\iff\eta<0, \] \end_inset we have \begin_inset Formula $X<0$ \end_inset . Thus we can conclude that \begin_inset Formula $f(\tilde{\lambda}_{T})>\tilde{\lambda}_{T}$ \end_inset and therefore, \begin_inset Formula $\tilde{\lambda}_{T}<\tilde{\lambda}_{1}$ \end_inset . Thus, we have proved that the backward recursion converges to an stationary value of \begin_inset Formula $\tilde{\lambda}^{*}=\tilde{\lambda}_{1}$ \end_inset , and it does so as an increasing sequence. \end_layout \begin_layout Proof Finally, we demonstrate that \begin_inset Formula $\lambda_{t}$ \end_inset also converges to a steady-state in a decreasing manner. We note that \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} \lambda=g(\tilde{\lambda})=\frac{\zeta-(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)}{-\eta+(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)} \end{equation} \end_inset \end_layout \begin_layout Proof Which we can rewrite as \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} \lambda=g(\tilde{\lambda})=\frac{\zeta+(1-\alpha)\delta\rho^{2}\left((\zeta-\eta)\tilde{\lambda}+\zeta\right)}{-\eta-(1-\alpha)\delta\rho^{2}\left((\zeta-\eta)\tilde{\lambda}+\zeta\right)} \end{equation} \end_inset \end_layout \begin_layout Proof Note that \begin_inset Formula $(1-\alpha)\delta\rho^{2}(\zeta-\eta)$ \end_inset is positive, so the numerator decreases when \begin_inset Formula $\tilde{\lambda}$ \end_inset decreases, whereas the denominator increases, since \begin_inset Formula $-(1-\alpha)\delta\rho^{2}(\zeta-\eta)\tilde{\lambda}$ \end_inset becomes less negative. Hence, \begin_inset Formula $g(\tilde{\lambda})=\lambda$ \end_inset also decreases when \begin_inset Formula $\tilde{\lambda}$ \end_inset decreases. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard We now show that utility is concave in \begin_inset Formula $x_{t}$ \end_inset as long as there is not too much habit formation in a specific sense. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Lemma \begin_inset CommandInset label LatexCommand label name "lemma:ConcaveU" \end_inset Suppose the conditions for Lemma \begin_inset CommandInset ref LatexCommand ref reference "lemma:Linearity" plural "false" caps "false" noprefix "false" \end_inset hold and \begin_inset Formula $U_{t}$ \end_inset is given by equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:U" plural "false" caps "false" noprefix "false" \end_inset ). Then for any \begin_inset Formula $t$ \end_inset , \begin_inset Formula $\frac{dU_{t}}{dx_{t}}$ \end_inset is continuous in \begin_inset Formula $x_{t}$ \end_inset . Furthermore, if \begin_inset Formula $\tilde{\lambda}^{b}$ \end_inset is an upper bound on \begin_inset Formula $\tilde{\lambda}_{t}$ \end_inset and \begin_inset Formula $\frac{(1-\alpha)\tilde{\lambda}^{b}}{\left(1+\tilde{\lambda}^{b}\right)-\delta\rho^{2}\left(1+\tilde{\lambda}^{b}\right)^{2}}<\frac{-\eta}{\zeta}$ \end_inset , then \begin_inset Formula $\frac{\partial^{2}U_{t}}{\partial x_{t}^{2}}<0$ \end_inset for all \begin_inset Formula $t\geq0$ \end_inset . \end_layout \begin_layout Lemma \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Proof The period \begin_inset Formula $t$ \end_inset decisionmaker maximizes equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:U" plural "false" caps "false" noprefix "false" \end_inset ). The derivative of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:U" plural "false" caps "false" noprefix "false" \end_inset ) can be written as \end_layout \begin_layout Proof \begin_inset Formula \begin{align} \frac{dU_{t}\left(x_{t};s_{t}\right)}{dx_{t}} & =\frac{\partial u_{t}}{\partial x_{t}}+\gamma+(1-\alpha)\sum_{r=t+1}^{T}\delta^{r-t}\frac{\partial\tilde{s}_{r}}{\partial x_{t}}\left[\underbrace{\frac{\partial\tilde{u}_{r}}{\partial\tilde{x}_{r}}\frac{\partial\tilde{x}_{r}}{\partial\tilde{s}_{r}}+\frac{\partial\tilde{u}_{r}}{\partial\tilde{s}_{r}}}_{\text{effect of \ensuremath{\tilde{s}_{r}} on period \ensuremath{r} utility}}+\underbrace{\delta\rho\frac{\partial V_{r+1}}{\partial\tilde{s}_{r+1}}\frac{\partial\tilde{x}_{r}}{\partial\tilde{s}_{r}}}_{\text{partial effect on future utility }}\right].\label{eq:dUdf concavity} \end{align} \end_inset The summation term in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dUdf concavity" plural "false" caps "false" noprefix "false" \end_inset ) is the effect on future utility from the change in habit stock brought into future periods. \begin_inset Formula $\frac{\partial\tilde{s}_{r}}{\partial x_{t}}=\rho^{r-t}$ \end_inset is the predicted direct effect of consumption \begin_inset Formula $\tilde{x}_{t}$ \end_inset on stock in period \begin_inset Formula $r$ \end_inset . The first two terms inside brackets are the effect of that change on period \begin_inset Formula $r$ \end_inset utility. The final term inside brackets accounts for the fact that the resulting change in \begin_inset Formula $\tilde{x}_{r}$ \end_inset will affect utility in later periods. \end_layout \begin_layout Proof The period \begin_inset Formula $t$ \end_inset decisionmaker predicts that her period \begin_inset Formula $r>t$ \end_inset self will maximize equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Utilde" plural "false" caps "false" noprefix "false" \end_inset ). The predicted period \begin_inset Formula $r$ \end_inset first-order condition is \begin_inset Formula \begin{equation} \left.\frac{d\tilde{U}_{r}\left(x_{r};\tilde{s}_{r}\right)}{d\tilde{x}_{r}}\right|_{_{\tilde{x}_{r}}}=0=\frac{\partial\tilde{u}_{r}}{\partial\tilde{x}_{r}}+\tilde{\gamma}+\delta\rho\frac{\partial V_{r+1}}{\partial\tilde{s}_{r+1}}. \end{equation} \end_inset \end_layout \begin_layout Proof Multiplying this FOC by \begin_inset Formula $\tilde{\lambda}_{r}\coloneqq\frac{\partial\tilde{x}_{r}}{\partial\tilde{s}_{r}}$ \end_inset and subtracting it from the term inside brackets in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dUdf concavity" plural "false" caps "false" noprefix "false" \end_inset ) gives \end_layout \begin_layout Proof \begin_inset Formula \begin{align} \frac{dU_{t}}{dx_{t}} & =\frac{\partial u_{t}}{\partial x_{t}}+\gamma+(1-\alpha)\sum_{r=t+1}^{T}\delta^{r-t}\rho^{r-t}\left[\begin{array}{c} \frac{\partial\tilde{u}_{r}}{\partial\tilde{x}_{r}}\tilde{\lambda}_{r}+\frac{\partial\tilde{u}_{r}}{\partial\tilde{s}_{r}}+\delta\rho\frac{\partial V_{r+1}}{\partial\tilde{s}_{r+1}}\tilde{\lambda}_{r}\\ -\left[\frac{\partial\tilde{u}_{r}}{\partial\tilde{x}_{r}}\tilde{\lambda}_{r}+\tilde{\gamma}\tilde{\lambda}_{r}+\delta\rho\frac{\partial V_{r+1}}{\partial\tilde{s}_{r+1}}\tilde{\lambda}_{r}\right] \end{array}\right]\\ & =\frac{\partial u_{t}}{\partial x_{t}}+\gamma+(1-\alpha)\sum_{r=t+1}^{T}\left(\delta\rho\right)^{r-t}\left[\frac{\partial\tilde{u}_{r}}{\partial\tilde{s}_{r}}-\tilde{\gamma}\tilde{\lambda}_{r}\right] \end{align} \end_inset \end_layout \begin_layout Proof With the quadratic functional form, this becomes \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} \frac{dU_{t}}{dx_{t}}=\eta x_{t}+\zeta s_{t}+\xi_{t}-p_{t}+\gamma+(1-\alpha)\sum_{r=t+1}^{T}\left(\delta\rho\right){}^{r-t}\left[\zeta\tilde{x}_{r}+\phi-\tilde{\gamma}\tilde{\lambda}\right].\label{eq:FOC_Envelope} \end{equation} \end_inset \end_layout \begin_layout Proof In this equation, two terms ( \begin_inset Formula $x_{t}$ \end_inset and \begin_inset Formula $\tilde{x}_{r}$ \end_inset ) depend on \begin_inset Formula $x_{t}$ \end_inset . \begin_inset Formula $x_{t}$ \end_inset is by definition continuous in \begin_inset Formula $x_{t}$ \end_inset , and \begin_inset Formula $\tilde{x}_{r}$ \end_inset is continuous in past consumption \begin_inset Formula $x_{t}$ \end_inset due to the evolution of habit stock and Lemma \begin_inset CommandInset ref LatexCommand ref reference "lemma:Linearity" plural "false" caps "false" noprefix "false" \end_inset . Thus, \begin_inset Formula $\frac{dU_{t}}{dx_{t}}$ \end_inset is continuous in \begin_inset Formula $x$ \end_inset . \end_layout \begin_layout Proof We now turn to concavity. The derivative of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:FOC_Envelope" plural "false" caps "false" noprefix "false" \end_inset ) is \end_layout \begin_layout Proof \begin_inset Formula \begin{align} \frac{d^{2}U_{t}}{dx_{t}^{2}} & =\eta+(1-\alpha)\sum_{r=t+1}^{\infty}\left(\delta\rho\right){}^{r-t}\zeta\frac{d\tilde{x}_{r}}{dx_{t}}.\\ & =\eta+(1-\alpha)\sum_{r=t+1}^{\infty}\left(\delta\rho\right){}^{r-t}\zeta\tilde{\lambda}_{r}\left[\rho^{r-t}\prod_{j=t+1}^{r-1}\left(1+\tilde{\lambda}_{j}\right)\right] \end{align} \end_inset \end_layout \begin_layout Proof Intuitively, \begin_inset Formula $\frac{d^{2}U_{t}}{dx_{t}^{2}}<0$ \end_inset requires that the diminishing marginal utility in period \begin_inset Formula $t$ \end_inset outweighs the incentive to increase current consumption for the purpose of increasing future utility through \begin_inset Formula $\zeta$ \end_inset . This will tend to be true when projection bias \begin_inset Formula $\alpha$ \end_inset is large and/or habit formation \begin_inset Formula $\rho$ \end_inset is small. A small \begin_inset Formula $\rho$ \end_inset has a direct effect by causing the habit stock from \begin_inset Formula $dx_{t}$ \end_inset to decay faster. It also has an indirect effect by reducing \begin_inset Formula $\frac{d\tilde{x}_{r}}{dx_{t}}$ \end_inset , the perceived effect of current consumption on future consumption. \end_layout \begin_layout Proof If we know an upper bound \begin_inset Formula $\tilde{\lambda}^{b}$ \end_inset such that \begin_inset Formula $\tilde{\lambda}^{b}>\tilde{\lambda}_{t}$ \end_inset for all \begin_inset Formula $t$ \end_inset , we can write a simpler necessary condition for concavity: \begin_inset Formula $\frac{d^{2}U_{t}}{dx_{t}^{2}}<0$ \end_inset for all \begin_inset Formula $t\geq0$ \end_inset if \begin_inset Formula \begin{align} (1-\alpha)\sum_{r=t+1}^{\infty}\left(\delta\rho\right){}^{r-t}\tilde{\lambda}_{r}\left[\rho^{r-t}\prod_{j=t+1}^{r-1}\left(1+\tilde{\lambda}_{j}\right)\right] & <\frac{-\eta}{\zeta}\\ (1-\alpha)\sum_{r=t+1}^{\infty}\left(\delta\rho\right){}^{r-t}\tilde{\lambda}^{b}\left[\rho^{r-t}\left(1+\tilde{\lambda}^{b}\right)^{r-t-1}\right] & <\frac{-\eta}{\zeta}\\ (1-\alpha)\frac{\tilde{\lambda}^{b}}{1+\tilde{\lambda}^{b}}\cdot\sum_{r=1}^{\infty}\left(\delta\rho^{2}\left(1+\tilde{\lambda}^{b}\right)\right){}^{r-1} & <\frac{-\eta}{\zeta}\\ (1-\alpha)\frac{\tilde{\lambda}^{b}}{1+\tilde{\lambda}^{b}}\cdot\left[\frac{1}{1-\left(\delta\rho^{2}\left(1+\tilde{\lambda}^{b}\right)\right)}\right] & <\frac{-\eta}{\zeta}\\ \frac{(1-\alpha)\tilde{\lambda}^{b}}{\left(1+\tilde{\lambda}^{b}\right)-\delta\rho^{2}\left(1+\tilde{\lambda}^{b}\right)^{2}} & <\frac{-\eta}{\zeta}. \end{align} \end_inset \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard From the proof of Lemma \begin_inset CommandInset ref LatexCommand ref reference "lemma:Constantlambda" plural "false" caps "false" noprefix "false" \end_inset , we know that \begin_inset Formula $\tilde{\lambda}_{t}$ \end_inset decreases as \begin_inset Formula $t\rightarrow T$ \end_inset . \end_layout \begin_layout Standard Finally, we show the conditions under which \begin_inset Formula $\mu_{t}$ \end_inset converges to a constant far from the time horizon. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Lemma \begin_inset CommandInset label LatexCommand label name "lemma:Constantmu" \end_inset Suppose the conditions for Lemma \begin_inset CommandInset ref LatexCommand ref reference "lemma:Linearity" plural "false" caps "false" noprefix "false" \end_inset hold, and \begin_inset Formula $-\eta>(1-\alpha)\delta\rho\left[\left(\zeta-\eta\right)\left(1+\rho\tilde{\lambda}_{t+1}\right)-\rho\zeta\right]$ \end_inset . Then \begin_inset Formula $\lim_{(T-t)\rightarrow\infty}\mu_{t}=\mu$ \end_inset . \end_layout \begin_layout Proof Since \begin_inset Formula $\mu_{t}(\gamma)$ \end_inset is a function of only constants, \begin_inset Formula $\tilde{\lambda}_{t+1}$ \end_inset (which converges per Lemma \begin_inset CommandInset ref LatexCommand ref reference "lemma:Constantlambda" plural "false" caps "false" noprefix "false" \end_inset ), and \begin_inset Formula $\mu_{t+1}(\tilde{\gamma})$ \end_inset , it is sufficient to show that the sequence \begin_inset Formula $\mu_{t}(\tilde{\gamma})$ \end_inset converges. The coefficient on \begin_inset Formula $\mu_{t+1}(\tilde{\gamma})$ \end_inset in equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:mu_t" plural "false" caps "false" noprefix "false" \end_inset ) is \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} \frac{(1-\alpha)\delta\rho\left(\zeta-\eta\right)}{-\eta+(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}_{t+1}+\zeta\right)}. \end{equation} \end_inset The sequence \begin_inset Formula $\mu_{t+1}(\tilde{\gamma})$ \end_inset will converge if and only if \begin_inset Formula \begin{equation} \frac{(1-\alpha)\delta\rho\left(\zeta-\eta\right)}{-\eta+(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}_{t+1}+\zeta\right)}<1. \end{equation} \end_inset The denominator is positive at our parameter values, so this inequality requires \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} -\eta>(1-\alpha)\delta\rho\left[\left(\zeta-\eta\right)\left(1+\rho\tilde{\lambda}_{t+1}\right)-\rho\zeta\right]. \end{equation} \end_inset In words, this requires that perceived habit formation \begin_inset Formula $(1-\alpha)\rho$ \end_inset is small relative to the demand slope parameter \begin_inset Formula $\eta$ \end_inset . \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard Proposition \begin_inset CommandInset ref LatexCommand ref reference "prop:Linearity" plural "false" caps "false" noprefix "false" \end_inset combines Lemmas \begin_inset CommandInset ref LatexCommand ref reference "lemma:Linearity" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "lemma:Constantlambda" plural "false" caps "false" noprefix "false" \end_inset , \begin_inset CommandInset ref LatexCommand ref reference "lemma:ConcaveU" plural "false" caps "false" noprefix "false" \end_inset , and \begin_inset CommandInset ref LatexCommand ref reference "lemma:Constantmu" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Subsection Proof of Lemma \begin_inset CommandInset ref LatexCommand ref reference "lemma:SteadyStateConvergence" plural "false" caps "false" noprefix "false" \end_inset : Steady-State Convergence \begin_inset CommandInset label LatexCommand label name "app:SteadyStateConvergenceProof" \end_inset \end_layout \begin_layout Proof Capital stock evolves according to \begin_inset Formula $s_{t}=\rho\left(s_{t-1}+x_{t-1}\right)$ \end_inset . Substituting in the stable equilibrium strategy \begin_inset Formula $x_{t}^{*}=\lambda s_{t}+\mu$ \end_inset gives \begin_inset Formula \begin{align} s_{t} & =\rho\left(s_{t-1}+\lambda s_{t-1}+\mu\right)\\ & =\rho\mu+\rho\left(1+\lambda\right)s_{t-1}\\ & =\rho\mu+\rho\left(1+\lambda\right)\left(\rho\mu+\rho\left(1+\lambda\right)s_{t-2}\right)\\ & =\rho\mu+\rho^{2}\left(1+\lambda\right)\mu+\rho^{2}\left(1+\lambda\right)^{2}s_{t-2}\\ & =\rho\mu+\rho^{2}\left(1+\lambda\right)\mu+\rho^{3}\left(1+\lambda\right)^{2}\mu+\rho^{3}\left(1+\lambda\right)^{3}s_{t-3}. \end{align} \end_inset Thus \begin_inset Formula \begin{equation} s_{t}=\frac{\mu}{1+\lambda}\left(\iota+\iota^{2}+...+\iota^{k}\right)+\iota^{k}s_{t-k}, \end{equation} \end_inset where \begin_inset Formula $\iota=\left(1+\lambda\right)\rho$ \end_inset . Thus, provided that \begin_inset Formula $\iota<1$ \end_inset , in the limit as \begin_inset Formula $k\rightarrow\infty$ \end_inset we have \begin_inset Formula \begin{align} s_{t} & =\frac{\mu}{1+\lambda}\cdot\frac{\iota}{1-\iota}\\ & =\frac{\mu\rho}{1-\left(1+\lambda\right)\rho}. \end{align} \end_inset \end_layout \begin_layout Proof We can then check that this is indeed a steady state: \begin_inset Formula \begin{align} s_{t} & =\rho\left(\frac{\mu\rho}{1-\left(1+\lambda\right)\rho}+\mu+\lambda\left(\frac{\mu\rho}{1-\left(1+\lambda\right)\rho}\right)\right)\\ & =\rho\left(\frac{\mu\rho+\mu\left(1-\left(1+\lambda\right)\rho\right)+\lambda\mu\rho}{1-\left(1+\lambda\right)\rho}\right)\\ & =\rho\left(\frac{\mu\rho+\mu-\mu\rho-\mu\lambda\rho+\lambda\mu\rho}{1-\left(1+\lambda\right)\rho}\right)\\ & =\frac{\mu\rho}{1-\left(1+\lambda\right)\rho} \end{align} \end_inset \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Subsection Proof of Proposition \begin_inset CommandInset ref LatexCommand ref reference "prop:SteadyStateX" plural "false" caps "false" noprefix "false" \end_inset : Steady-State Consumption \begin_inset CommandInset label LatexCommand label name "app:SteadyStateXProof" \end_inset \end_layout \begin_layout Proof We assume steady state implies constant consumption and habit stock, but not necessarily constant \emph on predicted \emph default consumption and habit stock. In steady state, \begin_inset Formula $p_{t}=p$ \end_inset , \begin_inset Formula $\xi_{t}=\xi$ \end_inset , \begin_inset Formula $s_{t}=s_{ss}$ \end_inset , and \begin_inset Formula $x_{t}=x_{ss}$ \end_inset . By equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:evolution" plural "false" caps "false" noprefix "false" \end_inset ) governing the evolution of habit stock, \begin_inset Formula $s_{ss}=\rho(s_{ss}+x_{ss})$ \end_inset , and re-arranging this equation gives \begin_inset Formula $s_{ss}=\frac{\rho}{1-\rho}x_{ss}$ \end_inset . Earlier, we defined steady-state misprediction as \begin_inset Formula $m_{ss}\coloneqq\tilde{x}_{t+1}-x_{ss}$ \end_inset . \end_layout \begin_layout Proof We substitute \begin_inset Formula $p_{t}=p$ \end_inset , \begin_inset Formula $\xi_{t}=\xi$ \end_inset , \begin_inset Formula $s_{t}=s_{ss}$ \end_inset , and \begin_inset Formula $x_{t}=x_{ss}$ \end_inset into the Euler equation (equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Euler" plural "false" caps "false" noprefix "false" \end_inset )), giving \end_layout \begin_layout Proof \begin_inset Formula \begin{equation} \eta x_{ss}+\zeta s_{ss}+\xi-p+\gamma=(1-\alpha)\delta\rho\left[\eta\tilde{x}_{t+1}+\zeta\rho\left(x_{ss}+s_{ss}\right)+\xi-p+\left(1+\tilde{\lambda}\right)\tilde{\gamma}-\zeta\tilde{x}_{t+1}-\phi\right]. \end{equation} \end_inset \end_layout \begin_layout Proof Substituting in \begin_inset Formula $s_{ss}=\frac{\rho}{1-\rho}x_{ss}$ \end_inset and also writing predicted consumption as a deviation from the actual value gives \begin_inset Formula \begin{equation} \begin{aligned}\eta x_{ss}+\xi-p+\frac{\rho\zeta}{1-\rho}x_{ss}+\gamma= & (1-\alpha)\delta\rho\left[(\eta-\zeta)\left(\left(\tilde{x}_{t+1}-x_{ss}\right)+x_{ss}\right)+\zeta\rho\left(\frac{1}{1-\rho}x_{ss}\right)+\xi-p+\left(1+\tilde{\lambda}\right)\tilde{\gamma}-\phi\right].\end{aligned} \end{equation} \end_inset \end_layout \begin_layout Proof Substituting \begin_inset Formula $m_{ss}\coloneqq\tilde{x}_{t+1}-x_{ss}$ \end_inset and collecting terms gives \end_layout \begin_layout Proof \begin_inset Formula \begin{align} x_{ss}\left[\eta+\frac{\rho\zeta}{1-\rho}-(1-\alpha)\delta\rho\left((\eta-\zeta)+\frac{\zeta\rho}{1-\rho}\right)\right]= & p-\xi-\gamma+(1-\alpha)\delta\rho\bigg[(\eta-\zeta)m_{ss}\nonumber \\ & \ \ +\xi-p+\left(1+\tilde{\lambda}\right)\tilde{\gamma}-\phi\bigg]\\ x_{ss}\left[\eta-(1-\alpha)\delta\rho(\eta-\zeta)+\zeta\frac{\rho-(1-\alpha)\delta\rho^{2}}{1-\rho}\right]= & \left(1-(1-\alpha)\delta\rho\right)\left(p-\xi\right)+(1-\alpha)\delta\rho\bigg[(\eta-\zeta)m_{ss}\nonumber \\ & \ \ +\left(1+\tilde{\lambda}\right)\tilde{\gamma}-\phi\bigg]-\gamma. \end{align} \end_inset \end_layout \begin_layout Proof Multiplying both sides by \begin_inset Formula $(-1)$ \end_inset , setting \begin_inset Formula $\kappa\coloneqq(1-\alpha)\delta\rho(\phi-\xi)+\xi$ \end_inset , and dividing through gives equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Proof \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Section Derivations of Estimating Equations in Appendix \begin_inset CommandInset ref LatexCommand ref reference "subsec:Estimating Equations Unrestricted" plural "false" caps "false" noprefix "false" \end_inset \begin_inset CommandInset label LatexCommand label name "app:EstimatingEquationDerivations" \end_inset \end_layout \begin_layout Standard We define \begin_inset Formula $y^{g}\coloneqq\mathbb{E}_{i\in g}y_{i}$ \end_inset as the expectation over individuals in group \begin_inset Formula $g$ \end_inset of parameter \begin_inset Formula $y$ \end_inset . Due to random assignment, \begin_inset Formula $\xi_{t}^{g}=\xi_{t}^{g'}$ \end_inset and \begin_inset Formula $s_{2}^{g}=s_{2}^{g'}$ \end_inset for all \begin_inset Formula $\{g,g'\}$ \end_inset , and \begin_inset Formula $\mu_{t}^{B}=\mu_{t}^{BC}$ \end_inset for \begin_inset Formula $t\in\{2,4,5\}$ \end_inset . The estimating equations for the restricted model in Section \begin_inset CommandInset ref LatexCommand ref reference "subsec:Estimating Equations" plural "false" caps "false" noprefix "false" \end_inset are the below equations with the additional assumptions that \begin_inset Formula $\tau_{2}^{B}=0$ \end_inset and \begin_inset Formula $\alpha=1$ \end_inset . \end_layout \begin_layout Subsection Habit Formation \end_layout \begin_layout Standard \series bold Derivation of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambda" plural "false" caps "false" noprefix "false" \end_inset ). \series default From equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Linearity" plural "false" caps "false" noprefix "false" \end_inset ) and the evolution of habit stock, we have \end_layout \begin_layout Standard \begin_inset Formula \begin{align} x_{4}^{*} & =\lambda s_{4}+\mu_{4}\\ & =\lambda\rho\left(s_{3}+x_{3}^{*}\right)+\mu_{4}\\ & =\text{\ensuremath{\lambda\rho\left(\rho\left(s_{2}+x_{2}^{*}\right)+x_{3}^{*}\right)}+\mu_{4}}. \end{align} \end_inset \end_layout \begin_layout Standard Thus, group average consumption is \begin_inset Formula $x_{4}^{g}=\lambda\left(\rho^{2}\left(s_{2}^{g}+x_{2}^{g}\right)+\rho x_{3}^{g}\right)+\mu_{4}^{g}$ \end_inset , and the period 4 bonus effect is \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \tau_{4}^{B}=\lambda\left(\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}\right).\label{eq:tauB4} \end{equation} \end_inset \end_layout \begin_layout Standard Re-arranging gives equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambda" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold Derivation of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:rho" plural "false" caps "false" noprefix "false" \end_inset ). \series default Similarly, we have \begin_inset Formula \begin{align} x_{5}^{*} & =\lambda s_{5}+\mu_{5}\\ & =\lambda\rho\left(s_{4}+x_{4}^{*}\right)+\mu_{5}\\ & =\lambda\rho\left(\rho\left(s_{3}+x_{3}^{*}\right)+x_{4}^{*}\right)+\mu_{5}\\ & =\lambda\rho\left(\rho\left(\rho\left(s_{2}+x_{2}^{*}\right)+x_{3}^{*}\right)+x_{4}^{*}\right)+\mu_{5}. \end{align} \end_inset \end_layout \begin_layout Standard Thus, group average consumption is \begin_inset Formula $x_{5}^{g}=\lambda\left(\rho^{3}\left(s_{2}^{g}+x_{2}^{g}\right)+\rho^{2}x_{3}^{g}+\rho x_{4}^{g}\right)+\mu_{5}^{g}$ \end_inset , and the period 5 bonus effect is \begin_inset Formula \begin{equation} \tau_{5}^{B}=\lambda\left(\rho^{3}\tau_{2}^{B}+\rho^{2}\tau_{3}^{B}+\rho\tau_{4}^{B}\right).\label{eq:tauB5} \end{equation} \end_inset \end_layout \begin_layout Standard Multiplying equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:tauB4" plural "false" caps "false" noprefix "false" \end_inset ) by \begin_inset Formula $\rho$ \end_inset and subtracting from equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:tauB5" plural "false" caps "false" noprefix "false" \end_inset ) gives \begin_inset Formula $\tau_{5}^{B}-\tau_{4}^{B}\rho=\lambda\rho\tau_{4}^{B}$ \end_inset , and re-arranging gives equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:rho" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold System of equations for \begin_inset Formula $\lambda$ \end_inset and \begin_inset Formula $\rho$ \end_inset . \series default Re-arranging equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:rho" plural "false" caps "false" noprefix "false" \end_inset ) gives \begin_inset Formula \begin{equation} \begin{aligned}\lambda & =\frac{\tau_{5}^{B}}{\tau_{4}^{B}\rho}-1.\end{aligned} \end{equation} \end_inset \end_layout \begin_layout Standard Substituting this into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambda" plural "false" caps "false" noprefix "false" \end_inset ) gives: \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{\tau_{5}^{B}-\tau_{4}^{B}\rho}{\tau_{4}^{B}\rho} & =\frac{\tau_{4}^{B}}{\rho\tau_{3}^{B}+\rho^{2}\tau_{2}^{B}}\\ \left(\tau_{4}^{B}\right)^{2} & =\left(\tau_{5}^{B}-\tau_{4}^{B}\rho\right)\left(\tau_{3}^{B}+\rho\tau_{2}^{B}\right)\\ 0 & =\left[\tau_{2}^{B}\tau_{4}^{B}\right]\rho^{2}+\left[\tau_{3}^{B}\tau_{4}^{B}-\tau_{2}^{B}\tau_{5}^{B}\right]\rho+\left[\left(\tau_{4}^{B}\right)^{2}-\tau_{3}^{B}\tau_{5}^{B}\right]. \end{align} \end_inset \end_layout \begin_layout Standard The quadratic formula gives \begin_inset Formula \begin{equation} \rho=\frac{-\left[\tau_{3}^{B}\tau_{4}^{B}-\tau_{2}^{B}\tau_{5}^{B}\pm\sqrt{\left[\tau_{3}^{B}\tau_{4}^{B}-\tau_{2}^{B}\tau_{5}^{B}\right]^{2}-4\left[\tau_{2}^{B}\tau_{4}^{B}\right]\left[\left(\tau_{4}^{B}\right)^{2}-\tau_{3}^{B}\tau_{5}^{B}\right]}\right]}{2\left[\tau_{2}^{B}\tau_{4}^{B}\right]}. \end{equation} \end_inset \end_layout \begin_layout Standard In all bootstrap draws in our data, only one of the two solutions satisfies the requirement that \begin_inset Formula $\rho\geq0$ \end_inset . \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold Special case with \begin_inset Formula $\tau_{2}^{B}=0$ \end_inset . \series default If there is no anticipatory demand response ( \begin_inset Formula $\tau_{2}^{B}=0$ \end_inset ), we have \begin_inset Formula $\tau_{4}^{B}=\lambda\rho\tau_{3}^{B}$ \end_inset and \begin_inset Formula $\tau_{5}^{B}=\lambda\rho^{2}\tau_{3}^{B}+\lambda\rho\tau_{4}^{B}$ \end_inset . Dividing the two equations gives \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{\tau_{5}^{B}}{\tau_{4}^{B}} & =\rho+\frac{\tau_{4}^{B}}{\tau_{3}^{B}}\nonumber \\ \rho & =\frac{\tau_{5}^{B}}{\tau_{4}^{B}}-\frac{\tau_{4}^{B}}{\tau_{3}^{B}}.\label{eq:rho alt appendix} \end{align} \end_inset We then solve for \begin_inset Formula $\lambda$ \end_inset by inserting this \begin_inset Formula $\rho$ \end_inset into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:lambda" plural "false" caps "false" noprefix "false" \end_inset ) with \begin_inset Formula $\tau_{2}^{B}=0$ \end_inset . \end_layout \begin_layout Subsection Perceived Habit Formation, Price Response, and Habit Stock Effect on Marginal Utility \end_layout \begin_layout Standard The expectation over \begin_inset Formula $i$ \end_inset of the Euler equations for group \begin_inset Formula $g$ \end_inset is \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \begin{aligned}\eta x_{t}^{g}+\zeta s_{t}^{g}+\xi_{t}^{g}-p_{t}+\gamma & =(1-\alpha)\delta\rho\left[\eta\tilde{x}_{t+1}^{g}+\zeta\tilde{s}_{t+1}^{g}+\xi_{t+1}^{g}-p_{t+1}+\tilde{\gamma}+\tilde{\gamma}\tilde{\lambda}_{t+1}-\left(\zeta\tilde{x}_{t+1}^{g}+\phi\right)\right]\end{aligned} .\label{eq:Euler expectation} \end{equation} \end_inset \end_layout \begin_layout Standard \series bold Derivation of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:alpha" plural "false" caps "false" noprefix "false" \end_inset ). \series default Differencing the Euler equations for periods 2 versus 3 for the Bonus and Bonus Control groups gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \eta\tau_{2}^{B}=(1-\alpha)\delta\rho\left[-p^{B}+(\eta-\zeta)\left(\tilde{x}_{3}^{B}-\tilde{x}_{3}^{BC}\right)+\zeta\left(\tilde{s}_{3}^{B}-\tilde{s}_{3}^{BC}\right)\right]. \end{equation} \end_inset \end_layout \begin_layout Standard Substituting \begin_inset Formula $\tilde{x}_{3}^{B}-\tilde{x}_{3}^{BC}=\tilde{\tau}_{3}^{B}$ \end_inset and \begin_inset Formula $\tilde{s}_{3}^{B}-\tilde{s}_{3}^{BC}=\rho\tau_{2}^{B}$ \end_inset gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \eta\tau_{2}^{B}=(1-\alpha)\delta\rho\left[-p^{B}+(\eta-\zeta)\tilde{\tau}_{3}^{B}+\zeta\rho\tau_{2}^{B}\right]. \end{equation} \end_inset \end_layout \begin_layout Standard Rearranging gives equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:alpha" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Standard If \begin_inset Formula $\tilde{\gamma}\neq\gamma$ \end_inset , then people update their predictions of \begin_inset Formula $\tilde{x}_{3}$ \end_inset as they set \begin_inset Formula $x_{2}^{*}$ \end_inset , and thus the predictions of \begin_inset Formula $\tilde{x}_{3}$ \end_inset from survey 2 are inconsistent with \begin_inset Formula $x_{2}^{*}$ \end_inset . However, there is only limited misprediction in our data, so this is not very consequential. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold Derivation of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:eta" plural "false" caps "false" noprefix "false" \end_inset ). \series default Differencing the Euler equations for periods 3 versus 4 for the Bonus and Bonus Control groups gives \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \left(-p^{B}-0\right)+\eta\tau_{3}^{B}+\zeta\left(s_{3}^{B}-s_{3}^{BC}\right) & =(1-\alpha)\delta\rho\left[(\eta-\zeta)\left(\tilde{x}_{4}^{B}-\tilde{x}_{4}^{BC}\right)+\zeta\left(\tilde{s}_{4}^{B}-\tilde{s}_{4}^{BC}\right)\right]. \end{align} \end_inset \end_layout \begin_layout Standard Habit stock evolution implies \begin_inset Formula $s_{3}^{B}-s_{3}^{BC}=\rho(s_{2}^{B}-s_{2}^{BC}+x_{2}^{B}-x_{2}^{BC})=\rho\tau_{2}^{B}$ \end_inset and \begin_inset Formula $\tilde{s}_{4}^{B}-\tilde{s}_{4}^{BC}=\rho\left(s_{3}^{B}-s_{3}^{BC}+x_{3}^{B}-x_{3}^{BC}\right)=\rho\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)$ \end_inset . Linear policy functions imply \begin_inset Formula $\tilde{x}_{4}=\tilde{\lambda}\tilde{s}_{4}+\tilde{\mu}_{4}$ \end_inset , so \begin_inset Formula $\tilde{x}_{4}^{B}-\tilde{x}_{4}^{BC}=\tilde{\lambda}\left(\tilde{s}_{4}^{B}-\tilde{s}_{4}^{BC}\right)$ \end_inset . Substituting these equations gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \left(-p^{B}-0\right)+\eta\tau_{3}^{B}+\zeta\rho\tau_{2}^{B}=(1-\alpha)\delta\rho\left[\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)\rho\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)\right]. \end{equation} \end_inset \end_layout \begin_layout Standard Rearranging gives \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \eta\left(\tau_{3}^{B}-(1-\alpha)\delta\rho^{2}\tilde{\lambda}\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)\right) & =p^{B}-\zeta\rho\tau_{2}^{B}+(1-\alpha)\delta\rho^{2}\zeta\left(1-\tilde{\lambda}\right)\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right). \end{align} \end_inset Solving for \begin_inset Formula $\eta$ \end_inset gives equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:eta" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold Derivation of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:zeta" plural "false" caps "false" noprefix "false" \end_inset ). \series default Differencing the Euler equations for periods 4 versus 5 for the Bonus and Bonus Control groups gives \begin_inset Formula \begin{align} \eta\left(x_{4}^{B}-x_{4}^{BC}\right)+\zeta\left(s_{4}^{B}-s_{4}^{BC}\right) & =(1-\alpha)\delta\rho\left[(\eta-\zeta)\left(\tilde{x}_{5}^{B}-\tilde{x}_{5}^{BC}\right)+\zeta\left(\tilde{s}_{5}^{B}-\tilde{s}_{5}^{BC}\right)\right] \end{align} \end_inset \end_layout \begin_layout Standard Habit stock evolution implies \begin_inset Formula $s_{4}^{B}-s_{4}^{BC}=\rho\left(s_{3}^{B}-s_{3}^{BC}+x_{3}^{B}-x_{3}^{BC}\right)=\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}$ \end_inset and \begin_inset Formula $\tilde{s}_{5}^{B}-\tilde{s}_{5}^{BC}=\rho\left(s_{4}^{B}-s_{4}^{BC}+x_{4}^{B}-x_{4}^{BC}\right)=\rho\left(\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}+\tau_{4}^{B}\right)$ \end_inset . Linear policy functions imply \begin_inset Formula $\tilde{x}_{5}=\tilde{\lambda}\tilde{s}_{5}+\tilde{\mu}_{5}$ \end_inset , so \begin_inset Formula $\tilde{x}_{5}^{B}-\tilde{x}_{5}^{BC}=\tilde{\lambda}\left(\tilde{s}_{5}^{B}-\tilde{s}_{5}^{BC}\right).$ \end_inset Substituting these equations gives \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \eta\tau_{4}^{B}+\zeta\left(\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}\right) & =(1-\alpha)\delta\rho\left[\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)\rho\left(\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}+\tau_{4}^{B}\right)\right]\\ & =(1-\alpha)\delta\rho^{2}\left[\left(\eta\lambda+\zeta\left(1-\tilde{\lambda}\right)\right)\left(\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}+\tau_{4}^{B}\right)\right]. \end{align} \end_inset \end_layout \begin_layout Standard Collecting \begin_inset Formula $\zeta$ \end_inset terms gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \zeta\left(\rho\tau_{3}^{B}+\rho^{2}\tau_{2}^{B}\right)-(1-\alpha)\delta\rho^{2}\left[\zeta\left(1-\tilde{\lambda}\right)\left(\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}+\tau_{4}^{B}\right)\right]=-\eta\tau_{4}^{B}+(1-\alpha)\delta\rho^{2}\eta\tilde{\lambda}\left(\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}+\tau_{4}^{B}\right). \end{equation} \end_inset \end_layout \begin_layout Standard Solving for \begin_inset Formula $\zeta$ \end_inset gives equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:zeta" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold System of equations for \begin_inset Formula $(1-\alpha)$ \end_inset , \begin_inset Formula $\eta$ \end_inset , and \begin_inset Formula $\zeta$ \end_inset . \series default \end_layout \begin_layout Standard First, we solve explicitly for \series bold \begin_inset Formula $(1-\alpha)$ \end_inset \series default before substituting it back in Equations ( \begin_inset CommandInset ref LatexCommand ref reference "eq:eta" plural "false" caps "false" noprefix "false" \end_inset ) and ( \begin_inset CommandInset ref LatexCommand ref reference "eq:zeta" plural "false" caps "false" noprefix "false" \end_inset ) to solve for \begin_inset Formula $\eta$ \end_inset and \begin_inset Formula $\zeta$ \end_inset . \end_layout \begin_layout Standard We define \begin_inset Formula \begin{equation} y\coloneqq\frac{-\tau_{4}^{B}+(1-\alpha)\delta\rho^{2}\lambda\left[\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}+\tau_{4}^{B}\right]}{\rho\tau_{3}^{B}+\rho^{2}\tau_{2}^{B}-(1-\alpha)\delta\rho^{2}(1-\lambda)\left[\rho^{2}\tau_{2}^{B}+\rho\tau_{3}^{B}+\tau_{4}^{B}\right]}. \end{equation} \end_inset Observe that \begin_inset Formula \begin{equation} \zeta=\eta\cdot y. \end{equation} \end_inset We can use this observation to rearrange Equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:eta" plural "false" caps "false" noprefix "false" \end_inset ): \begin_inset Formula \begin{align} \eta & =\frac{p^{B}-\zeta\rho\tau_{2}^{B}+(1-\alpha)\delta\rho^{2}\zeta(1-\lambda)\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)}{\tau_{3}^{B}-(1-\alpha)\delta\rho^{2}\lambda\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)}\\ \eta\left[\tau_{3}^{B}-(1-\alpha)\delta\rho^{2}\lambda\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)\right] & =p^{B}-\zeta\left(\rho\tau_{2}^{B}-(1-\alpha)\delta\rho^{2}(1-\lambda)\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)\right)\\ & =p^{B}-\eta\cdot y\left(\rho\tau_{2}^{B}-(1-\alpha)\delta\rho^{2}(1-\lambda)\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)\right)\\ p^{B} & =\eta\left[\tau_{3}^{B}-(1-\alpha)\delta\rho^{2}\lambda\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)+y\left(\rho\tau_{2}^{B}-(1-\alpha)\delta\rho^{2}(1-\lambda)\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)\right)\right]. \end{align} \end_inset Then, define \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} x\coloneqq\tau_{3}^{B}-(1-\alpha)\delta\rho^{2}\lambda\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)+y\left(\rho\tau_{2}^{B}-(1-\alpha)\delta\rho^{2}(1-\lambda)\left(\rho\tau_{2}^{B}+\tau_{3}^{B}\right)\right) \end{equation} \end_inset where we observe that \begin_inset Formula \begin{equation} \eta=\frac{p^{B}}{x}, \end{equation} \end_inset and \begin_inset Formula \begin{equation} \zeta=\frac{p^{B}y}{x}. \end{equation} \end_inset Finally, we get that \end_layout \begin_layout Standard \begin_inset Formula \begin{align} (1-\alpha) & =\frac{\eta\tau_{2}^{B}}{\delta\rho\left[-p^{B}+(\eta-\zeta)\tilde{\tau}_{3}^{B}+\zeta\rho\tau_{2}^{B}\right]}\\ & =\frac{\frac{p^{B}}{x}\tau_{2}^{B}}{\delta\rho\left[-p^{B}+(\frac{p^{B}}{x}-\frac{p^{B}y}{x})\tilde{\tau}_{3}^{B}+\frac{p^{B}y}{x}\rho\tau_{2}^{B}\right]}. \end{align} \end_inset \end_layout \begin_layout Standard Since all scalars are known in the last equation, we can now solve for \begin_inset Formula $\alpha$ \end_inset . Then, we can estimate \begin_inset Formula $\eta$ \end_inset and \begin_inset Formula $\zeta$ \end_inset by substituting \begin_inset Formula $\alpha$ \end_inset in Equations ( \begin_inset CommandInset ref LatexCommand ref reference "eq:eta" plural "false" caps "false" noprefix "false" \end_inset ) and ( \begin_inset CommandInset ref LatexCommand ref reference "eq:zeta" plural "false" caps "false" noprefix "false" \end_inset ) respectively. \end_layout \begin_layout Subsection Naivete about Temptation \end_layout \begin_layout Standard \series bold Derivation of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Naivete" plural "false" caps "false" noprefix "false" \end_inset ). \series default The Euler equation \emph on predicted \emph default for period \begin_inset Formula $t$ \end_inset on the survey at the beginning of period \begin_inset Formula $t$ \end_inset is \begin_inset Formula \begin{equation} \eta x_{t}^{*}\left(s_{t},\tilde{\gamma},\boldsymbol{p}_{t}\right)+\zeta s_{t}+\xi_{t}-p_{t}+\tilde{\gamma}=(1-\alpha)\delta\rho\left[\eta\tilde{x}_{t+1}+\zeta s_{t+1}+\xi_{t+1}-p_{t+1}+\tilde{\gamma}+\tilde{\gamma}\tilde{\lambda}-\left(\zeta\tilde{x}_{t+1}+\phi\right)\right]. \end{equation} \end_inset This equation uses the assumption that consumers are aware of period \begin_inset Formula $t$ \end_inset projection bias when predicting period \begin_inset Formula $t$ \end_inset consumption on survey \begin_inset Formula $t$ \end_inset , so the only reason why the period \begin_inset Formula $t$ \end_inset survey-taker mispredicts the period \begin_inset Formula $t$ \end_inset objective function is naivete about period \begin_inset Formula $t$ \end_inset temptation. \end_layout \begin_layout Standard Habit stock evolution implies \begin_inset Formula $\tilde{s}_{t+1}=\rho(s_{t}+\tilde{x}_{t})$ \end_inset . Linear policy functions imply \begin_inset Formula $\tilde{x}_{t+1}=\tilde{\lambda}\tilde{s}_{t+1}+\tilde{\mu}_{t+1}$ \end_inset . Substituting these equations into the predicted Euler equation gives \begin_inset Formula \begin{align} \eta x_{t}^{*}\left(s_{t},\tilde{\gamma},\boldsymbol{p}_{t}\right)+\zeta s_{t}+\xi_{t}-p_{t}+\tilde{\gamma} & =(1-\alpha)\delta\rho\left[(\eta-\zeta)\left(\tilde{\lambda}\tilde{s}_{t+1}+\tilde{\mu}_{t+1}\right)+\zeta\tilde{s}_{t+1}+\xi_{t+1}-p_{t+1}+\tilde{\gamma}+\tilde{\gamma}\tilde{\lambda}-\phi\right].\\ & =(1-\alpha)\delta\rho\left[\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)\tilde{s}_{t+1}+(\eta-\zeta)\tilde{\mu}_{t+1}+\xi_{t+1}-p_{t+1}+\tilde{\gamma}+\tilde{\gamma}\tilde{\lambda}-\phi\right]\\ & =(1-\alpha)\delta\rho\left[\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)\rho(s_{t}+\tilde{x}_{t})+(\eta-\zeta)\tilde{\mu}_{t+1}+\xi_{t+1}-p_{t+1}+\tilde{\gamma}+\tilde{\gamma}\tilde{\lambda}-\phi\right]. \end{align} \end_inset \end_layout \begin_layout Standard Analogously, the actual Euler equation for period \begin_inset Formula $t$ \end_inset can be written as \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \eta x_{t}^{*}\left(s_{t},\gamma,\boldsymbol{p}_{t}\right)+\zeta s_{t}+\xi_{t}-p_{t}+\gamma=(1-\alpha)\delta\rho\left[\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)\rho(s_{t}+x_{t}^{*})+(\eta-\zeta)\tilde{\mu}_{t+1}+\xi_{t+1}-p_{t+1}+\tilde{\gamma}+\tilde{\gamma}\tilde{\lambda}-\phi\right]. \end{equation} \end_inset \end_layout \begin_layout Standard Differencing the actual and predicted Euler equations for period \begin_inset Formula $t$ \end_inset versus period \begin_inset Formula $t+1$ \end_inset for the Control group gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \eta\left(x_{t}^{C}-\tilde{x}_{t}^{C}\right)+\gamma-\tilde{\gamma}=(1-\alpha)\delta\rho\left[\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)\rho\left(x_{t}^{C}-\tilde{x}_{t}^{C}\right)\right] \end{equation} \end_inset \end_layout \begin_layout Standard Solving for \begin_inset Formula $\gamma-\tilde{\gamma}$ \end_inset and substituting \begin_inset Formula $m^{C}=x_{t}^{C}-\tilde{x}_{t}^{C}$ \end_inset gives equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Naivete" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Subsection Temptation \end_layout \begin_layout Standard \series bold Limit effect: derivation of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gamma_Limit" plural "false" caps "false" noprefix "false" \end_inset ). \series default Consider a \begin_inset Quotes eld \end_inset zero temptation \begin_inset Quotes erd \end_inset intervention that fully eliminates both perceived and actual temptation starting in period 2, generating treatment effects \begin_inset Formula $\tau_{t}^{0}$ \end_inset . Differencing the average Euler equations for periods 2 versus 3 for the zero temptation group versus its control group gives \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \eta\left(x_{2}^{0}-x_{2}^{0C}\right)-\gamma & =(1-\alpha)\delta\rho\left[(\eta-\zeta)\left(\tilde{x}_{3}^{0}-\tilde{x}_{3}^{0C}\right)+\zeta\left(\tilde{s}_{3}^{0}-\tilde{s}_{3}^{0C}\right)-\tilde{\gamma}-\tilde{\gamma}\tilde{\lambda}\right]\\ \eta\tau_{2}^{0}-\gamma & =(1-\alpha)\delta\rho\left[(\eta-\zeta)\tilde{\tau}_{3}^{0}+\zeta\rho\tau_{2}^{0}-\tilde{\gamma}-\tilde{\gamma}\tilde{\lambda}\right] \end{align} \end_inset \end_layout \begin_layout Standard Solving for \begin_inset Formula $\gamma$ \end_inset and substituting \begin_inset Formula $\tau^{0}=\tau^{L}/\omega$ \end_inset gives equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gamma_Limit" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Standard To solve for \begin_inset Formula $\gamma$ \end_inset as a function of data and known parameters, we solve equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Naivete" plural "false" caps "false" noprefix "false" \end_inset ) for \begin_inset Formula $\tilde{\gamma}$ \end_inset , substitute into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gamma_Limit" plural "false" caps "false" noprefix "false" \end_inset ), and rearrange, giving \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \gamma=\frac{\eta\tau_{2}^{L}/\omega-(1-\alpha)\delta\rho\left(\left[(\eta-\zeta)\tilde{\tau}_{3}^{L}/\omega+\zeta\rho\tau_{2}^{L}/\omega\right]+(1+\tilde{\lambda})m_{2}^{C}\cdot\left[-\eta+(1-\alpha)\delta\rho^{2}\left((\eta-\zeta)\tilde{\lambda}+\zeta\right)\right]\right)}{1-(1-\alpha)\delta\rho(1+\tilde{\lambda})}.\label{eq:gamma_Limit_appendix} \end{equation} \end_inset \end_layout \begin_layout Standard \series bold \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold Bonus valuation: derivation of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:DeltaV(pB)" plural "false" caps "false" noprefix "false" \end_inset ). \series default When we elicited the bonus valuation on survey 2, we had not yet told participa nts whether the bonus would be in effect for period 2 or 3. The theoretical valuations for a period 2 vs. period 3 bonus are identical if we assume that consumers predict no anticipator y effect of the period 3 bonus. Otherwise, this derivation would need to account for the period 2 survey taker’s valuation of the perceived internality reduction from the anticipatory effect. Since the actual bonus was for period 3, we focus the derivation on that case and maintain the assumption of zero predicted anticipatory effect. \end_layout \begin_layout Standard From the perspective of the period 2 survey taker, the predicted period 3 continuation value (given naivete about future projection bias) as a function of predicted habit stock and period 3 price is \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \begin{aligned}V_{3}\left(\tilde{s}_{3},p_{3}\right)= & u_{3}\left(\tilde{x}_{3}^{*}\left(\tilde{s}_{3},\tilde{\gamma},\boldsymbol{p}_{3}\right);\tilde{s}_{3},p_{3}\right)+\delta V_{4}\left(\tilde{s}_{4},\cdot\right).\end{aligned} \label{eq:V3-1} \end{equation} \end_inset The change in that predicted continuation value from a marginal change in period 3 price is \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{dV_{3}(\tilde{s}_{3},p_{3})}{dp_{3}}= & \frac{\partial\tilde{u}_{3}}{\partial p_{3}}+\frac{\partial\tilde{x}_{3}}{\partial p_{3}}\left[\frac{\partial\tilde{u}_{3}}{\partial\tilde{x}_{3}}+\delta\frac{dV_{4}\left(\tilde{s}_{4},\cdot\right)}{d\tilde{s}_{4}}\frac{\partial\tilde{s}_{4}}{\partial\tilde{x}_{3}}\right].\label{eq:dVdp-full} \end{align} \end_inset \end_layout \begin_layout Standard People taking survey 2 predict that their period 3 selves will set \begin_inset Formula $x_{3}$ \end_inset to maximize that same function with an additional \begin_inset Formula $\tilde{\gamma}x_{3}$ \end_inset in period 3 flow utility: \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \begin{aligned}\tilde{x}_{3}^{*}(\tilde{s}_{3},\tilde{\gamma},\boldsymbol{p}_{3})=\arg\max_{x_{3}} & u_{3}\left(x_{3};\tilde{s}_{3},\boldsymbol{p}_{3}\right)+\tilde{\gamma}x_{3}+\delta V_{4}\left(\tilde{s}_{4},\cdot\right).\end{aligned} \label{eq:V3+gammatildex} \end{equation} \end_inset Thus, people taking survey 2 predict that they will set \begin_inset Formula $x_{3}$ \end_inset such that \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{\partial\tilde{u}_{3}}{\partial x_{3}}+\tilde{\gamma}+\delta\frac{dV_{4}\left(\tilde{s}_{4},\cdot\right)}{d\tilde{s}_{4}}\frac{\partial\tilde{s}_{4}}{\partial\tilde{x}_{3}} & =0.\label{eq:dUdx} \end{align} \end_inset \end_layout \begin_layout Standard Substituting equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dUdx" plural "false" caps "false" noprefix "false" \end_inset ) into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dVdp-full" plural "false" caps "false" noprefix "false" \end_inset ) gives \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{dV_{3}(\tilde{s}_{3},p_{3})}{dp_{3}} & =\frac{\partial\tilde{u}_{3}}{\partial p_{3}}-\tilde{\gamma}\frac{\partial\tilde{x}_{3}}{\partial p_{3}}\\ & =-\tilde{x}_{3}(p_{3})-\tilde{\gamma}\frac{\partial\tilde{x}_{3}}{\partial p_{3}}.\label{eq:dVdp} \end{align} \end_inset \end_layout \begin_layout Standard This illustrates a temptation-adjusted envelope theorem: the effect of a marginal price change on the long-run self's utility (given perceived misoptimi zation from the long-run self's perspective) equals the mechanical effect \begin_inset Formula $\tilde{x}_{3}(p_{3})$ \end_inset adjusted by the magnitude of the perceived misoptimization \begin_inset Formula $\tilde{\gamma}\frac{\partial\tilde{x}_{3}}{\partial p_{3}}$ \end_inset . With zero perceived temptation ( \begin_inset Formula $\tilde{\gamma}=0$ \end_inset ), this reduces to the standard envelope theorem. The derivation for a period 2 bonus would be analogous, except with \begin_inset Formula $(1-\alpha)$ \end_inset multiplying the predicted period 3 continuation value in both the survey taker's objective function and the predicted period 2 objective function. \end_layout \begin_layout Standard We integrate over equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dVdp" plural "false" caps "false" noprefix "false" \end_inset ) to determine the effect of a non-marginal price increase from 0 to \begin_inset Formula $p^{B}$ \end_inset : \end_layout \begin_layout Standard \begin_inset Formula \begin{align} V_{3}\left(\tilde{s}_{3},p_{3}=p_{3}^{B}\right)-V_{3}\left(\tilde{s}_{3},p_{3}=0\right) & =\int_{p_{3}=0}^{p_{3}=p_{3}^{B}}-\tilde{x}_{3}(p_{3})-\tilde{\gamma}\frac{\partial\tilde{x}_{3}}{\partial p_{3}}\ dp_{3}\\ & =-p_{3}^{B}\cdot\left(\tilde{x}_{3}(p_{3}^{B})+\tilde{x}_{3}(0)\right)/2-\tilde{\gamma}\cdot\left(\tilde{x}_{3}(p_{3}^{B})-\tilde{x}_{3}(0)\right), \end{align} \end_inset where the second line follows from the fact that demand is linear in price, which was shown in Proposition \begin_inset CommandInset ref LatexCommand ref reference "prop:Linearity" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold Limit valuation: derivation of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:vL" plural "false" caps "false" noprefix "false" \end_inset ). \series default The period 3 survey-taker's objective function is \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} V_{3}\left(s_{3},\tilde{\gamma}_{3}\right)=u_{3}\left(x_{3}^{*}\left(s_{3},\tilde{\gamma}_{3},\boldsymbol{p}_{3}\right);s_{3},p_{3}\right)+\begin{array}{c} \alpha\sum_{r=4}^{T}\delta^{r-3}u_{r}\left(\tilde{x}_{r}^{*}\left(s_{3},\tilde{\gamma},\boldsymbol{p}_{r}\right);s_{3},p_{r}\right)\\ +(1-\alpha)\delta V_{4}\left(\tilde{s}_{4},\cdot\right) \end{array}.\label{eq:V3} \end{equation} \end_inset This equation uses the assumption that the survey taker is projection biased. \end_layout \begin_layout Standard The change in that objective function from a marginal change in perceived period 3 temptation is \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \begin{aligned}\frac{dV_{3}\left(s_{3},\tilde{\gamma}_{3}\right)}{d\tilde{\gamma}_{3}}= & \frac{\partial x_{3}^{*}\left(s_{3},\tilde{\gamma}_{3},\boldsymbol{p}_{3}\right)}{\partial\tilde{\gamma}_{3}}\left[\frac{\partial u_{3}}{\partial x_{3}}+(1-\alpha)\delta\frac{\partial V_{4}\left(\tilde{s}_{4},\cdot\right)}{\partial\tilde{s}_{4}}\frac{\partial\tilde{s}_{4}}{\partial\tilde{x}_{3}}\right].\end{aligned} \label{eq:dVdgammatilde full} \end{equation} \end_inset People taking survey 3 predict that they will set \begin_inset Formula $x_{3}^{*}$ \end_inset such that \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{\partial u_{3}}{\partial x_{3}}+\tilde{\gamma}_{3}+(1-\alpha)\delta\frac{dV_{4}\left(\tilde{s}_{4},\cdot\right)}{d\tilde{s}_{4}}\frac{\partial\tilde{s}_{4}}{\partial\tilde{x}_{3}} & =0.\label{eq:dUdx-1} \end{align} \end_inset \end_layout \begin_layout Standard Substituting the period 3 first-order condition from equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dUdx-1" plural "false" caps "false" noprefix "false" \end_inset ) into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dVdgammatilde full" plural "false" caps "false" noprefix "false" \end_inset ) gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{dV_{3}(s_{3},\tilde{\gamma}_{3})}{d\tilde{\gamma}_{3}}=-\tilde{\gamma}_{3}\frac{\partial x_{3}^{*}\left(s_{3},\tilde{\gamma}_{3},\boldsymbol{p}_{3}\right)}{\partial\tilde{\gamma}_{3}}.\label{eq:dVdgammatilde} \end{equation} \end_inset \end_layout \begin_layout Standard We integrate over equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dVdgammatilde" plural "false" caps "false" noprefix "false" \end_inset ) to determine the effect of the non-marginal temptation reduction from \begin_inset Formula $\tilde{\gamma}$ \end_inset to \begin_inset Formula $(1-\omega)\tilde{\gamma}$ \end_inset : \end_layout \begin_layout Standard \begin_inset Formula \begin{align} v^{L}=V_{3}\left(s_{3},\tilde{\gamma}_{3}=(1-\omega)\tilde{\gamma}\right)-V_{3}\left(s_{3},\tilde{\gamma}_{3}=\tilde{\gamma}\right) & =\int_{\tilde{\gamma}_{3}=\tilde{\gamma}}^{\tilde{\gamma}_{3}=(1-\omega)\tilde{\gamma}}-\tilde{\gamma}_{3}\frac{\partial x_{3}^{*}(\tilde{\gamma}_{3})}{\partial\tilde{\gamma}_{3}}\ d\tilde{\gamma}_{3}.\\ & =\left(x_{3}^{*}(\tilde{\gamma})-x_{3}^{*}(0)\right)\cdot\tilde{\gamma}\cdot\frac{1}{2}-(1-\omega)^{2}\cdot\left(\tilde{x}_{3}(\tilde{\gamma})-\tilde{x}_{3}(0)\right)\cdot\tilde{\gamma}\cdot\frac{1}{2}\\ & =\left(x_{3}^{*}(\tilde{\gamma})-x_{3}^{*}(0)\right)\cdot\tilde{\gamma}\cdot\left(1-(1-\omega)^{2}\right)\cdot\frac{1}{2}\\ & =\left(x_{3}^{*}(\tilde{\gamma})-x_{3}^{*}(0)\right)\cdot\tilde{\gamma}\cdot\frac{\omega(2-\omega)}{2}\\ & =\frac{\left(x_{3}^{*}(\tilde{\gamma})-x_{3}^{*}((1-\omega)\tilde{\gamma})\right)}{\omega}\cdot\tilde{\gamma}\cdot\frac{\omega(2-\omega)}{2}\label{eq:vL-3}\\ & =\left(x_{3}^{*}(\tilde{\gamma})-x_{3}^{*}((1-\omega)\tilde{\gamma})\right)\cdot\tilde{\gamma}\cdot\frac{(2-\omega)}{2}, \end{align} \end_inset where the second line is the area of the long-run self's perceived deadweight loss reduction trapezoid (following from linear demand) and the fifth line follows from the assumption that \begin_inset Formula $\tilde{\tau}^{L}/\omega=\tilde{\tau}^{0}$ \end_inset . \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Subsection Temptation with Multiple Goods \begin_inset CommandInset label LatexCommand label name "app:TemptationMultipleGoods" \end_inset \end_layout \begin_layout Standard We now extend our model to include a second temptation good \begin_inset Formula $y$ \end_inset , which in our experiment is FITSBY use on other devices. Habit stock now evolves according to \begin_inset Formula $s_{t+1}=\rho\left(s_{t}+x_{t}+y_{t}\right)$ \end_inset . Before period \begin_inset Formula $t$ \end_inset , consumers now consider flow utility to be \begin_inset Formula $u_{t}\left(x_{t},y_{t};s_{t},p_{t}\right)$ \end_inset . In period \begin_inset Formula $t$ \end_inset , consumers choose as if period \begin_inset Formula $t$ \end_inset flow utility is \begin_inset Formula $u_{t}\left(x_{t},y_{t};s_{t},p_{t}\right)+\gamma_{x}x_{t}+\gamma_{y}y_{t}$ \end_inset . Before period \begin_inset Formula $t$ \end_inset , consumers predict that they will choose as if period \begin_inset Formula $t$ \end_inset flow utility is \begin_inset Formula $u_{t}\left(x_{t},y_{t};s_{t},p_{t}\right)+\tilde{\gamma}_{x}x_{t}+\tilde{\gamma}_{y}y_{t}$ \end_inset . \begin_inset Formula $x$ \end_inset is still sold at price \begin_inset Formula $p_{t}$ \end_inset , while \begin_inset Formula $y_{t}$ \end_inset has zero price. The limit treatment fully eliminates perceived and actual temptation on \begin_inset Formula $x$ \end_inset . \end_layout \begin_layout Standard We derive new equations for \begin_inset Formula $\gamma$ \end_inset or \begin_inset Formula $\tilde{\gamma}$ \end_inset for the limit effect, bonus valuation, and limit valuation strategies. With all three strategies, if \begin_inset Formula $y$ \end_inset is not a temptation good ( \begin_inset Formula $\tilde{\gamma}_{y}=\gamma_{y}=0$ \end_inset ) or if \begin_inset Formula $y$ \end_inset is neither a substitute nor a complement for \begin_inset Formula $x$ \end_inset , then our original estimating equations are unaffected. \end_layout \begin_layout Standard \series bold Limit effect. \series default To derive \begin_inset Formula $\gamma$ \end_inset using the limit effect strategy, we assume full projection bias ( \begin_inset Formula $\alpha=1$ \end_inset ). We assume that the static quadratic flow utility function is now \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} u(x,y;p)=\frac{\eta_{x}}{2}x^{2}+\xi_{x}x-px+\sigma xy+\frac{\eta_{y}}{2}y^{2}+\xi_{y}y.\label{eq:QuadraticUtility_y} \end{equation} \end_inset \end_layout \begin_layout Standard Without the limit, consumers maximize \begin_inset Formula $u(x,y;p)+\gamma_{x}x+\gamma_{y}y$ \end_inset , giving \end_layout \begin_layout Standard \begin_inset Formula \begin{align} y^{*}\left(x\right) & =\frac{\sigma x+\xi_{y}+\gamma_{y}}{-\eta_{y}}\label{eq:ystar}\\ x^{*} & =\frac{\xi_{x}-p+\sigma\frac{\xi_{y}+\gamma_{y}}{-\eta_{y}}+\gamma_{x}}{-\eta_{x}+\frac{\sigma^{2}}{\eta_{y}}} \end{align} \end_inset \end_layout \begin_layout Standard Taking the expectation over individuals, the bonus effect on \begin_inset Formula $x^{*}$ \end_inset is \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \tau_{x}^{B} & =\frac{p^{B}}{-\eta_{x}+\frac{\sigma^{2}}{\eta_{y}}}\label{eq:tauBx} \end{align} \end_inset \end_layout \begin_layout Standard The limit allows consumers to set \begin_inset Formula $x_{L}$ \end_inset before period \begin_inset Formula $t$ \end_inset . When setting the limit, consumers predict that in period \begin_inset Formula $t$ \end_inset they will set \begin_inset Formula $y$ \end_inset conditional on \begin_inset Formula $x_{L}$ \end_inset to maximize \begin_inset Formula $u\left(x_{L},y;p\right)+\tilde{\gamma}_{x}x_{L}+\tilde{\gamma}_{y}y$ \end_inset , giving \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} y^{*}\left(x_{L}\right)=\frac{\sigma x_{L}+\xi_{y}+\tilde{\gamma}_{y}}{-\eta_{y}}. \end{equation} \end_inset Consumers thus set \begin_inset Formula $x_{L}$ \end_inset to maximize \begin_inset Formula $u\left(x_{L},y^{*}\left(x_{L}\right);p\right)$ \end_inset , giving \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} x_{L}=\frac{\xi_{x}-p+\xi_{y}\frac{\sigma}{-\eta_{y}}}{-\eta_{x}+\frac{\sigma^{2}}{\eta_{y}}}. \end{equation} \end_inset \end_layout \begin_layout Standard The effect of the limit on \begin_inset Formula $y$ \end_inset is \begin_inset Formula $y^{*}\left(x_{L}\right)-y^{*}\left(x^{*}\right)=\frac{\sigma x_{L}+\xi_{y}+\tilde{\gamma}_{y}}{-\eta_{y}}-\frac{\sigma x^{*}+\xi_{y}+\gamma_{y}}{-\eta_{y}}$ \end_inset . Taking the expectation over individuals, the limit effect on \begin_inset Formula $y$ \end_inset is \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \tau_{y}^{L} & =\frac{\sigma}{-\eta_{y}}\tau_{x}^{L}\label{eq:tauLy} \end{align} \end_inset \end_layout \begin_layout Standard The effect of the limit on \begin_inset Formula $x$ \end_inset is \end_layout \begin_layout Standard \begin_inset Formula \begin{align} x_{L}-x^{*} & =\frac{\xi_{x}-p+\xi_{y}\frac{\sigma}{-\eta_{y}}}{-\eta_{x}+\frac{\sigma^{2}}{\eta_{y}}}-\frac{\xi_{x}-p+\sigma\frac{\xi_{y}+\gamma_{y}}{-\eta_{y}}+\gamma_{x}}{-\eta_{x}+\frac{\sigma^{2}}{\eta_{y}}}\\ & =\frac{\frac{\sigma}{-\eta_{y}}\gamma_{y}+\gamma_{x}}{-\eta_{x}+\frac{\sigma^{2}}{\eta_{y}}}\\ & =\frac{-\gamma\left(1+\frac{\sigma}{-\eta_{y}}\right)}{-\eta_{x}+\frac{\sigma^{2}}{\eta_{y}}} \end{align} \end_inset where the third line assumes \begin_inset Formula $\gamma_{x}=\gamma_{y}=\gamma$ \end_inset . \end_layout \begin_layout Standard Taking the expectation over individuals and substituting equations ( \begin_inset CommandInset ref LatexCommand ref reference "eq:tauBx" plural "false" caps "false" noprefix "false" \end_inset ) and ( \begin_inset CommandInset ref LatexCommand ref reference "eq:tauLy" plural "false" caps "false" noprefix "false" \end_inset ) gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \tau_{x}^{L}=\frac{-\gamma\left(1+\frac{\tau_{y}^{L}}{\tau_{x}^{L}}\right)}{p^{B}/\tau_{x}^{B}}. \end{equation} \end_inset Rearranging gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \gamma=\frac{\tau_{x}^{L}\cdot\left(p^{B}/\tau_{x}^{B}\right)}{1+\frac{\tau_{y}^{L}}{\tau_{x}^{L}}}.\label{eq:gamma_Limit_y} \end{equation} \end_inset This exactly parallels equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gamma_Limit" plural "false" caps "false" noprefix "false" \end_inset ) for the \begin_inset Formula $\alpha=1$ \end_inset case, except adjusting the denominator for substitution. If \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset are substitutes, then the estimated \begin_inset Formula $\gamma$ \end_inset increases: more temptation is required to explain a given limit when the consumer knows that she can evade the limit through substitution to another temptation good. If \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset are complements, then the estimated \begin_inset Formula $\gamma$ \end_inset decreases: less temptation is needed to explain a given limit when the consumer knows that the limit will also cause reductions in another temptation good. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold Bonus valuation. \series default The derivation for the bonus valuation with substitute goods is very similar to the one-good case. The change in the period 3 continuation value function from a marginal change in \begin_inset Formula $p_{3}$ \end_inset is \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{dV_{3}(\tilde{s}_{3},p_{3})}{dp_{3}}= & \frac{\partial\tilde{u}_{3}}{\partial p_{3}}+\frac{\partial\tilde{x}_{3}}{\partial p_{3}}\left[\frac{\partial\tilde{u}_{3}}{\partial\tilde{x}_{3}}+\delta\frac{dV_{4}\left(\tilde{s}_{4},\cdot\right)}{d\tilde{s}_{4}}\frac{\partial\tilde{s}_{4}}{\partial\tilde{x}_{3}}\right]+\frac{\partial\tilde{y}_{3}}{\partial p_{3}}\left[\frac{\partial\tilde{u}_{3}}{\partial\tilde{y}_{3}}+\delta\frac{dV_{4}\left(\tilde{s}_{4},\cdot\right)}{d\tilde{s}_{4}}\frac{\partial\tilde{s}_{4}}{\partial\tilde{y}_{3}}\right].\label{eq:dVdp-full_y} \end{align} \end_inset \end_layout \begin_layout Standard People taking survey 2 predict that they will set \begin_inset Formula $x_{3}$ \end_inset and \begin_inset Formula $y_{3}$ \end_inset according to \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{\partial\tilde{u}_{3}}{\partial x_{3}}+\tilde{\gamma}_{x}+\delta\frac{dV_{4}\left(\tilde{s}_{4},\cdot\right)}{d\tilde{s}_{4}}\frac{\partial\tilde{s}_{4}}{\partial\tilde{x}_{3}} & =0\label{eq:dUdx_y}\\ \frac{\partial\tilde{u}_{3}}{\partial y_{3}}+\tilde{\gamma}_{y}+\delta\frac{dV_{4}\left(\tilde{s}_{4},\cdot\right)}{d\tilde{s}_{4}}\frac{\partial\tilde{s}_{4}}{\partial\tilde{y}_{3}} & =0.\label{eq:dUdy_y} \end{align} \end_inset \end_layout \begin_layout Standard Substituting equations ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dUdx_y" plural "false" caps "false" noprefix "false" \end_inset ) and ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dUdy_y" plural "false" caps "false" noprefix "false" \end_inset ) as well as \begin_inset Formula $\frac{\partial\tilde{u}_{3}}{\partial p_{3}}=-\tilde{x}_{3}(p_{3})$ \end_inset into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:dVdp-full_y" plural "false" caps "false" noprefix "false" \end_inset ) gives \end_layout \begin_layout Standard \begin_inset Formula \begin{align} \frac{dV_{3}(\tilde{s}_{3},p_{3})}{dp_{3}} & =-\tilde{x}_{3}(p_{3})-\tilde{\gamma}_{x}\frac{\partial\tilde{x}_{3}}{\partial p_{3}}-\tilde{\gamma}_{y}\frac{\partial\tilde{y}_{3}}{\partial p_{3}}.\label{eq:dVdp_y} \end{align} \end_inset \end_layout \begin_layout Standard Integrating over a non-marginal price increase from 0 to \begin_inset Formula $p^{B}$ \end_inset assuming linear demand, also assuming \begin_inset Formula $\tilde{\gamma}_{x}=\tilde{\gamma}_{y}=\tilde{\gamma}$ \end_inset , taking the expectation over participants, and rearranging gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \tilde{\gamma}=\frac{\bar{v}^{B}-\bar{F}^{B}+p_{3}^{B}\bar{\tilde{x}}_{3}^{B+BC}}{-\left(\tilde{\tau}_{x3}^{B}+\tilde{\tau}_{y3}^{B}\right)}\label{eq:gammatildeB_y} \end{equation} \end_inset This exactly parallels equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:gammatildeB" plural "false" caps "false" noprefix "false" \end_inset ), except adjusting the denominator for substitution. The survey taker values the total temptation reduction \begin_inset Formula $-\left(\tilde{\tau}_{x3}^{B}+\tilde{\tau}_{y3}^{B}\right)$ \end_inset induced by the bonus. If \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset are substitutes, the total temptation reduction is lower, and more temptation is needed to justify a given valuation. If \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset are complements, the total temptation reduction is higher, and less temptation is needed to justify a given valuation. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Standard \series bold Limit valuation. \series default The derivation for the limit valuation with substitute goods is also similar to the one-good case. The change in the period 3 survey-taker's objective function from a marginal change in perceived period 3 temptation for good \begin_inset Formula $x$ \end_inset only is \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \begin{aligned}\frac{dV_{3}\left(s_{3},\tilde{\gamma}_{x3}\right)}{d\tilde{\gamma}_{x3}}= & \frac{\partial x_{3}^{*}}{\partial\tilde{\gamma}_{x3}}\left[\frac{\partial u_{3}}{\partial x_{3}}+(1-\alpha)\delta\frac{\partial V_{4}\left(\tilde{s}_{4},\cdot\right)}{\partial\tilde{s}_{4}}\frac{\partial\tilde{s}_{4}}{\partial\tilde{x}_{3}}\right]+\frac{\partial y_{3}^{*}}{\partial\tilde{\gamma}_{x3}}\left[\frac{\partial u_{3}}{\partial y_{3}}+(1-\alpha)\delta\frac{\partial V_{4}\left(\tilde{s}_{4},\cdot\right)}{\partial\tilde{s}_{4}}\frac{\partial\tilde{s}_{4}}{\partial\tilde{y}_{3}}\right].\end{aligned} \label{eq:dVdgammatilde full_y} \end{equation} \end_inset \end_layout \begin_layout Standard Substituting the predicted period 3 first-order conditions for \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{dV_{3}(s_{3},\tilde{\gamma}_{x3})}{d\tilde{\gamma}_{x3}}=-\tilde{\gamma}_{x3}\frac{\partial x_{3}^{*}}{\partial\tilde{\gamma}_{x3}}-\tilde{\gamma}_{y}\frac{\partial y_{3}^{*}}{\partial\tilde{\gamma}_{x3}}.\label{eq:dVdgammatilde_y} \end{equation} \end_inset \end_layout \begin_layout Standard Integrating over this from \begin_inset Formula $\tilde{\gamma}_{x}$ \end_inset to \begin_inset Formula $(1-\omega)\tilde{\gamma}_{x}$ \end_inset assuming linear demand, also assuming \begin_inset Formula $\tilde{\gamma}_{x}=\tilde{\gamma}_{y}=\tilde{\gamma}$ \end_inset , taking the expectation over participants, and rearranging gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \tilde{\gamma}=\frac{\bar{v}^{L}}{-\left(\tilde{\tau}_{3}^{L}(2-\omega)/2+\tilde{\tau}_{y3}^{L}\right)}.\label{eq:gammatildeL_y} \end{equation} \end_inset As with the bonus valuation, the survey taker values the total temptation deadweight loss reduction induced by the limit. If \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset are substitutes, the total temptation reduction is lower, and more temptation is needed to justify a given valuation. If \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset are complements, the total temptation reduction is higher, and less temptation is needed to justify a given valuation. \end_layout \begin_layout Standard \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Subsection Intercept \end_layout \begin_layout Standard \series bold Derivation of equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Intercept" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Standard Re-arranging steady state consumption from equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS" plural "false" caps "false" noprefix "false" \end_inset ) gives \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \begin{aligned}(1-\alpha)\delta\rho(\phi-\xi)+\xi-\left(1-(1-\alpha)\delta\rho\right)p+(1-\alpha)\delta\rho\left[(\zeta-\eta)m_{ss}-\left(1+\tilde{\lambda}\right)\tilde{\gamma}\right]+\gamma=\\ x_{ss}\left[-\eta-(1-\alpha)\delta\rho(\zeta-\eta)-\zeta\frac{\rho-(1-\alpha)\delta\rho^{2}}{1-\rho}\right]. \end{aligned} \end{equation} \end_inset \end_layout \begin_layout Standard Solving for the intercept and substituting \begin_inset Formula $x_{i1}=x_{ss}$ \end_inset gives equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:Intercept" plural "false" caps "false" noprefix "false" \end_inset ). \end_layout \begin_layout Section Counterfactual Simulations Appendix \begin_inset CommandInset label LatexCommand label name "app:EstimationResults_Counterfactuals" \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects of Temptation and Habit Formation on FITSBY Use \begin_inset CommandInset label LatexCommand label name "tab:CounterfactualsTable" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (2) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Restricted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Unrestricted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout FITSBY use (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout ( \begin_inset Formula $\tau_{2}^{B}=0$ \end_inset , \begin_inset Formula $\alpha=1$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout ( \begin_inset Formula $\alpha=\hat{\alpha}$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Baseline \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssLeffectres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssLeffect$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssLeffectresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssLeffectboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout No naivete \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssnaiveteres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssnaivete$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssnaiveteresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xssnaiveteboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout No temptation \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsstemptationres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsstemptation$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsstemptationresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsstemptationboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout No habit formation \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsshabitres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsshabit$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsshabitresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsshabitboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout No temptation or habit formation \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsshabittemptationres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsshabittemptation$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsshabittemptationresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\xsshabittemptationboot$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table presents point estimates and bootstrapped 95 percent confidenc e intervals for predicted steady-state FITSBY use with different parameter assumptions, using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS counterfactual" plural "false" caps "false" noprefix "false" \end_inset ). The numbers are as plotted in Figure \begin_inset CommandInset ref LatexCommand ref reference "fig:Counterfactuals" plural "false" caps "false" noprefix "false" \end_inset . \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Effects of Temptation on FITSBY Use Under Alternative Assumptions \begin_inset CommandInset label LatexCommand label name "tab:AltAssumptionsTable" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (1) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout (2) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Restricted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Unrestricted \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Effect of temptation on FITSBY use (minutes/day) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout ( \begin_inset Formula $\tau_{2}^{B}=0$ \end_inset , \begin_inset Formula $\alpha=1$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout ( \begin_inset Formula $\alpha=\hat{\alpha}$ \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Limit effect \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcap$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Bonus valuation \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxtildeBres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxtildeB$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxtildeBresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxtildeBboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Limit valuation \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxtildeLres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxtildeL$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxtildeLresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxtildeLboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Limit effect, multiple-good model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapmultipleres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapmultipleresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Bonus valuation, multiple-good model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaBmultipleres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaBmultiple$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaBmultipleresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaBmultipleboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Limit valuation, multiple-good model \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxLmultipleres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxLmultiple$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxLmultipleresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxLmultipleboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Limit effect, \begin_inset Formula $\omega=\hat{\omega}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapomegares$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapomega$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapomegaresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapomegaboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Limit valuation, \begin_inset Formula $\omega=\hat{\omega}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxLomegares$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxLomega$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxLomegaresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxLomegaboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Heterogeneous limit effect \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxspecres$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxspec$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxspecresboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxspecboot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Limit effect, weighted sample \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapresbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapbalancedmedian$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapresbalanced$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\deltaxcapbalanced$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Plain Layout \begin_inset VSpace defskip \end_inset \end_layout \begin_layout Plain Layout \size small Notes: This table presents point estimates and bootstrapped 95 percent confidenc e intervals for the effects of temptation on average steady-state FITSBY use, using equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS counterfactual" plural "false" caps "false" noprefix "false" \end_inset ). The first nine estimates are for the nine temptation estimation strategies presented in Table \begin_inset CommandInset ref LatexCommand ref reference "tab:Alternative_gamma" plural "false" caps "false" noprefix "false" \end_inset . The tenth estimate is for the limit effect strategy after reweighting the sample to be more representative of U.S. adults. Appendix Tables \begin_inset CommandInset ref LatexCommand ref reference "tab:SampleDemographicsWeighted" plural "false" caps "false" noprefix "false" \end_inset – \begin_inset CommandInset ref LatexCommand ref reference "tab:StructuralEstimatesWeighted" plural "false" caps "false" noprefix "false" \end_inset present the demographics, moments, and parameter estimates in the weighted sample. Average baseline FITSBY use is \begin_inset Formula $\avguse$ \end_inset and \begin_inset Formula $\avgusebalancedmedian$ \end_inset minutes per day for the unweighted and weighted samples, respectively. We do not have a limit effect estimate for the unrestricted multiple-good model. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \begin_inset Caption Standard \begin_layout Plain Layout \series bold Distribution of Effects of Temptation on FITSBY Use \series default \begin_inset CommandInset label LatexCommand label name "fig:IndividualTemptation" \end_inset \end_layout \end_inset \end_layout \begin_layout Plain Layout \align center \begin_inset Graphics filename ../input/treatment_effects/hist_individual_temptation_effects.pdf scale 90 \end_inset \end_layout \begin_layout Plain Layout \size small Notes: Using the heterogeneous limit effect strategy, we estimate temptation \begin_inset Formula $\hat{\gamma}_{i}$ \end_inset for each Limit group participant, which we then insert into equation ( \begin_inset CommandInset ref LatexCommand ref reference "eq:xSS counterfactual" plural "false" caps "false" noprefix "false" \end_inset ) to predict the individual-specific effect of temptation on steady-state FITSBY use. This figure presents the distribution of effects across participants, winsorize d at 300 minutes per day. \end_layout \end_inset \end_layout \end_body \end_document