# Assignment 1: Equation Paper

Decent Essays
Consider a team consisting two groups, 1 and 2, each of which contains $N$ homogeneous risk-neutral individuals. Each agent's effort is unobservable and each group's output is observable.

An individual $i$ chooses effort level $e_i\in A=[\delta,+\infty)$ where $\delta$ is positive and arbitrarily close to zero. An agent's cost function is $c(e_i)$ where $c$ is strictly convex, strictly increasing, twice continuously differentiable, and $c(0)=0$.

A group yields output from agents' effort in the team. Let group $J$'s output be denoted by $x_J$ and $x_J=f(\bm{e}_J)$ where $f$ is concave, strictly increasing, twice continuously differentiable, and $f(0)=0$, and $\bm{e}_J$ is a profile of agents' effort in team $J$. We assume that $f(\bm{e})=f(\bm{e'})$
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We interpret $r$ as the incentive power because when $r$ becomes larger, $s_J$ becomes more sensitive to the output ratio.
Note that group $J$'s share of output exceeds its own output $x_J$ when it becomes a winner, and it falls behind $x_J$ when it becomes a loser. An agent's wage is the group's share of output divided by the number of group members, i.e., $s_Jy/N$, and this is always positive. At the symmetric equilibrium, each agent splits the total output equally, in other words, he/she receives $y/2N$.

This sharing function has properties that are not necessarily for our result or needed to be modified for general cases.\footnote{Properties of generalized Tullock function is characterized by Skaperdas (1996).} First, the sharing rule only uses the output ratio and our result holds without this specification. We can substitute the generalized Tullock function with a more general function, such as
$s_J=\frac{g(x_J)}{g(x_1)+g(x_2)},$
where $g$ is differentiable, strictly increasing and $g(0)=0$. The reason why we restrict the function to the class of generalized Tullock is because we can obtain the optimal incentive power $r$