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which is very similar to the structure of the conditions used to characterize the equilib- rium in Theorem 1, although the equilibrium is in general no longer symmetric. How- ever, as we shall see below, if the conditions of Corollary 1 are satisfied, we can derive a very simple expression for the relationship between the equilibrium effort levels. Before turning to this result, we first demonstrate that, in some situations, a contest with differ- ent cost functions is equivalent to a contest with different prizes (in terms of equilibrium effort choices) and hence can, according to Proposition 1, be transformed into a contest with different production functions. This is formalized in the following remark.¹⁶

Remark 1. Suppose that players have heterogeneous cost functions that take the form $c_i(e_i) = \omega_i c(e_i)$, with $\omega_i > 0$. Then, the objective of player i can be written as:

$${\int }_{\mathbb{R}}{F}_k(g_{e_k}^{-1}(g_{e_i}(x)))f_i(x)dx{V}_i-{\omega }_{i}c(e_i)={\omega }_i({\int }_{\mathbb{R}}{F}_k(g_{e_k}^{-1}(g_{e_i}(x)))f_i(x)dx\frac{V_i}{{\omega }_i}-c(e_i))\mathrm{.}\backslash int\_\{\backslash mathbb\{R\}\}\; F\_k(g\_\{e\_k\}^\{-1\}(g\_\{e\_i\}(x)))\; f\_i(x)\; dx\; V\_i\; -\; \backslash omega\_i\; c(e\_i)\; =\; \backslash omega\_i\; \backslash left(\; \backslash int\_\{\backslash mathbb\{R\}\}\; F\_k(g\_\{e\_k\}^\{-1\}(g\_\{e\_i\}(x)))\; f\_i(x)\; dx\; \backslash frac\{V\_i\}\{\backslash omega\_i\}\; -\; c(e\_i)\; \backslash right).$$

This objective is equivalent to one in which prizes are given by $\frac{V_i}{\omega_i}$, but the cost functions are the same for both players.

Proposition 1 and Remark 1 highlight how, in certain cases, contests with different prizes or different cost functions can be transformed into equivalent contests with different production technologies. We now turn to showing that, if the assumptions underlying Corollary 1 are satisfied, these contests can be reinterpreted as contests with different skill distributions, allowing us to apply the equilibrium characterization from our baseline case.

Proposition 2 below provides results for two well-known production technologies that satisfy the assumptions of Corollary 1. We first consider the Cobb-Douglas production technology. In this case, we can show that the ratio of equilibrium efforts $e_1^*/e_2^*$ depends only on the ratio of prizes and the degree of homogeneity of the cost function. We then consider the additive production technology. In this case, we obtain a similar characterization which applies to the ratio of efforts subject to an exponential transformation, $\exp(e_1^*)/\exp(e_2^*)$.

Proposition 2. Let the two prizes be given by $V_1 = sV$ and $V_2 = V$ with $s > 0$.

(i) If the production technology is given by $g(\theta,e) = \theta^\alpha e^\beta$, with $\alpha, \beta > 0$, and $c$ is homogeneous of degree $\delta > 0$, an equilibrium exists with efforts given by $e_1^* = s^{1/\delta} e_2^*$ and

¹⁶Similar transformations between prizes and cost functions are standard in the literature.