Adverse selection and pay compression.

• Author: Stewart, Jay

1. Introduction

   A fundamental result of the principal-agent literature is that pay will be linked to performance when it is difficult for the principal to monitor the agent's actions. But in the real world, we often find this link to be weak. Several authors (Lazear 1989; Holmstrom and Milgrom 1991; Garvey and Swan 1992) have shown that weaker incentives may be optimal when performance pay leads to undesired behavior on the part of workers. In Lazear's tournament model, firms weaken the link between pay and performance when workers can sabotage their coworkers' efforts. Holmstrom and Milgrom show that straight salaries are optimal when the worker's job includes some activities that cannot be easily monitored. And Garvey and Swan show that a hybrid compensation scheme that is part tournament and part managerial discretion is optimal when workers can sabotage coworkers' efforts and managers face adverse incentives.

   The common thread in these papers is that principals weaken the link between pay and performance to mitigate adverse incentives.(1) This is what Garvey and Swan call "pay compression." In this paper, I show that adverse incentives, while sufficient, are not necessary for pay compression to occur.

   I consider a labor market where workers are heterogeneous with respect to ability (measured as cost of effort, with more able workers having a lower cost of effort), jobs have a single dimension, and workers compete in contests.(2) In this market, a pooling equilibrium is common, and the equilibrium contest maximizes the utility of low ability workers. This implies that the difference between the winner's pay and the loser's pay is smaller in the equilibrium pooling contest than would be the case if firms could observe workers' abilities before contracting. Pay compression results naturally in equilibrium even though all parties involved (firms, low ability workers, and high ability workers) would opt for greater incentives in the absence of adverse selection.

2. The Model

   I begin this section by introducing the basic model of contests when there is adverse selection in the labor market and describing how workers determine their optimal levels of effort. Next I describe the pure moral hazard model, which I use as a benchmark in section 3. Finally, I describe how workers' expectations affect their choice of which contest to enter.

   Firms produce output according to a production function that is additively separable across workers and linear in effort. In particular, worker i produces output ([q.sub.i]) according to [q.sub.i] = [a.sub.i] + [Theta] + [e.sub.i], where [a.sub.i] is an action (effort) taken by the worker, [Theta] is a random shock that is common to all workers in the firm, and [e.sub.i] is an individual-specific random shock. The [Theta] term may be thought of as factors that affect the profitability of the firm, while the [e.sub.i] term represents factors that affect the individual worker's performance. The random variable [Theta] is distributed according to H([center dot]) over [[[Theta].sub.0], [[Theta].sub.1]], and the [e.sub.i] are i.i.d. according to F([center dot]) on the interval [[e.sub.0], [e.sub.1]]. Also E([e.sub.i]) = E([Theta]) = 0.(3) A firm's total output is given by Q = [[Sigma].sub.i] [q.sub.i]. Firms maximize E([Pi]) = pQ - [[Sigma].sub.i] [r.sub.i], where p is the price of output and [r.sub.i] is the income received by worker i. In addition, firms can freely enter and exit the market so that E([Pi]) = 0 in equilibrium.(4)

   The labor market is composed of two types of workers, high (H) ability and low (L) ability, who differ in their cost of effort. All workers are risk averse and maximize utility of wealth net of the cost of exerting effort. If worker i is a type J worker (J = H, L), this is given by [U.sub.J] = V([r.sub.i]) - [C.sup.J]([a.sub.i]), where a [element of] [[a.sub.0], [a.sub.1]]. Utility of wealth V([center dot]) is increasing, strictly concave, and twice differentiable, while the cost of effort [C.sub.J]([center dot]) is increasing, strictly convex, and twice differentiable. High ability workers have a lower cost of effort. Specifically, [C.sub.L]([a.sub.0]) = [C.sub.H]([a.sub.0]) = 0 and [Mathematical Expression Omitted] for all a [element of] [[a.sub.0], [a.sub.1]]. Firms cannot distinguish between high and low ability workers before hiring.

   Nalebuff and Stiglitz (1983) and Green and Stokey (1983) have shown that when the variance of [Theta] is sufficiently large, contests dominate individualistic contracts because they impose less risk on workers. Under individualistic contracts such as piece rates, a high variance of [Theta] results in high variance in compensation. But in a contest, [Theta] does not affect relative standings and hence does not affect compensation. Throughout the rest of the paper, I assume that the variance of [Theta] is large enough so that contests dominate individualistic contracts.

   In a contest, two workers, i and j, compete against each other. The worker who produces the most wins, so that worker i wins if [q.sub.i] [greater than] [q.sub.j]. This implies [a.sub.i] + [Theta] + [e.sub.i] [greater than] [a.sub.j] + [Theta] + [e.sub.j], which reduces to [e.sub.j] - [e.sub.i] [less than] [a.sub.i] - [a.sub.j]. If ([e.sub.j] - [e.sub.i]) is distributed according to G([center dot]), then Prob(worker i wins) = G([a.sub.i] - [a.sub.j]). And since [e.sub.i] and [e.sub.j] are i.i.d, the density G[prime]([center dot]) = g([center dot]) is symmetric around ([e.sub.j] - [e.sub.i]) = 0. I also assume that the density g([center dot]) has a single mode and that it reaches its strict global maximum at ([e.sub.j] - [e.sub.i]) = 0.(5)

   Letting (Y + x) and (Y - x) denote payments to the winner and loser, a contest [Tau] is defined by [Tau] = (x, y).(6) Note that total prize money in the contest is 2Y, and x [greater than] 0 is the amount paid to the winner over and above Y. Hereafter, I refer to x as the prize.(7)

   Now consider a situation in which both high and low ability workers compete in the same contest. Let [Lambda] [element of] [0, 1] be the proportion of H workers in the labor market and [a.sub.h.] and [a.sub.L] denote the optimal level of effort of H and L workers. (For the remainder of the paper, I will index workers by their types.) Then the expected utility of a type J worker in a contest becomes

   [Mathematical Expression Omitted],

   where the first term in brackets is the expected payoff if the worker faces a high ability opponent and the second bracketed term is the payoff when facing a low ability opponent. The last term is the cost of effort. This equation reduces to

   [Mathematical Expression Omitted], (1)

   where [Delta](x, Y) = V(Y + x) - V(Y - x) and the term in square brackets is the probability of winning. Note that if [Tau] earns zero expected profit, then Y = p[[Lambda]E([q.sub.H]) + (1 - [Lambda])E[q.sub.L])] = p[[Lambda][a.sub.H] + (1 - [Lambda])[a.sub.L]]. (8)

   Differentiating Equation 1 with respect to [a.sub.j] yields the reaction function for a type J worker,(9)

   [Delta](x, Y)[[Lambda]g([a.sub.j] - [a.sub.H]) + (1 -[Lambda])g([a.sub.J] - [a.sub.L])] = [C[prime].sub.J]([a.sub.J]). (2)

   Symmetry implies that all workers of the same type (H or L) choose the same action, so Equation 2 becomes

   [Delta](x, Y)[[Lambda]g(0) + (1 - [Lambda])g([a.sub.H] - [a.sub.L])] = [C[prime].sub.H]([a.sub.H]) (3H)

   [Delta](x, Y)[[Lambda]g([a.sub.L] - [a.sub.H]) + (1 - [Lambda])g(0)] = [C[prime].sub.L]([a.sub.L]). (3L)

   Equations 3H and 3L state that workers equate marginal benefit and marginal cost given the optimal levels of effort of H and L workers. The [Delta](x, Y) term is the marginal gain to winning the contest. The g(0) term is the marginal increase in the probability of winning if both workers are the same type, while the g([a.sub.H] - [a.sub.L]) term is the marginal increase in probability of winning for an H worker competing against an L worker. The optimal levels of effort are denoted as [a.sub.H]([Tau]; [Lambda]) and [a.sub.L]([Tau]; [Lambda]).(10)

   There are two special cases worth noting. If [Lambda] = 1, then all workers are high ability workers. Similarly, if [Lambda] = 0, then all workers are low ability. In these cases, Equation 3 reduces to

   [Delta](x, Y)g(0) = [C[prime].sub.J]([a.sub.J]), for J = H, L. (4)

   Since there is no adverse selection, the equilibrium contest solves the following program:

   [Mathematical Expression Omitted]. (5)

   The solution to Equation 5 is the pure moral hazard contest for a type J worker and is denoted as [Mathematical Expression Omitted].(11)

   If firms observed workers' abilities before contracting, they could offer the pure moral hazard contest for each type. But since workers' abilities are not observed, firms compete for workers in a screening market similar to the one described by Rothschild and Stiglitz (1976).(12)

   Workers and firms play the game in three stages. First, each firm offers a contest and agrees to hire the first 2n workers who accept, where n is the number of contests the firm is running. I will refer to the set of all contests offered as the market tournament T. In equilibrium, the market tournament either separates workers, [T.sup.s] = {[[Tau].sup.H], [[Tau].sup.L]}, or pools workers, [T.sup.P] = {[[Tau].sup.P]}. Second, each worker determines the utility maximizing level of effort in each contest and enters the contest that provides the highest level of utility. Finally, workers compete against each other in their respective contests, and firms award prizes to the winners and losers.

   Since a worker's expected payoff in a contest depends on his opponent's ability, his choice of a contest depends on his expectations about the decisions of other workers. Workers' expectations come into play when evaluating the profitability of possible defections from the Nash equilibrium...