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Piece-rate contracts for other-regarding workers.

I. INTRODUCTION

A wide array of experimental evidence suggests that individuals care not only about their own outcomes but the outcomes of their opponents as well; that is, individuals do not necessarily have "selfish" preferences and rather exhibit "other-regarding" or "social" preferences (see Camerer 2003, for a summary). Given the pervasive nature of other-regarding preferences, accounting for them in real-world decision-making problems is important because other-regarding behaviors such as fairness, trust, and inequality aversion may significantly affect policy and contracting decisions. This paper concentrates on one prevalent type of other-regarding preferences, behavior toward inequality, and applies it to a principal-agent contracting model. Using piece-rate contracts, (1) we focus on two important questions. (2) First, how do attitudes toward inequality affect how workers respond to incentives? Second, how do attitudes toward inequality affect the incentives offered by the firm?

Analyzing the impact of attitudes toward inequality on contracting issues is reasonable because when workers are paid according to their performance, pay inequality arises naturally. Frank (1985) argues that humans are hardwired with a preference for higher status and therefore like any change that moves them up relative to their coworkers. On the other hand, experimental evidence on ultimatum games, dictator games, and the like has led researchers to assume that subjects dislike inequality (e.g., Bolton and Ockenfels 2000; Fehr and Schmidt 1999) and therefore dislike any change that makes payoffs less equal. Either way, pay inequality matters to workers, and it is important to derive the correct incentive system in the presence of these preferences.

Our paper has two major results that can best be summarized when workers have identical preferences and effort costs. First, other-regarding workers exert more effort than self-oriented ones when they are either competitive (i.e., have a preference for status) or behindness averse. This result establishes that piece rates can motivate other-regarding workers more than they motivate purely self-interested ones, and so firms can get a bigger bang for their buck when they use incentive pay with the right types of other-regarding workers. Second, if workers are identical and either inequality averse or behindness averse, the firm's profit-maximizing piece rate is lower than it would be if workers were purely self-interested. In other words, inequality and behindness aversion both lead to wage compression.

To capture the different inequality attitudes, we employ the Fehr and Schmidt (1999) model of other-regarding preferences. There are several reasons for using their model: it can fit data from a variety of experimental settings (as shown in their original paper), it has an axiomatic foundation (Neilson 2006), and it has been used successfully in other applications of other-regarding preferences (e.g., Itoh 2004). In the FehrSchmidt model, an individual receives (self-oriented) utility from his own payoff but (social) disutility from any difference between his partner/opponent's payoff and his own. In a principal-agent setting, the agents' payoffs are determined by both the pay they receive from the principal and the effort costs they bear. We modify the original Fehr-Schmidt model in two ways: first to allow for other behavioral patterns besides inequality aversion and second to make social disutility depend on pay differences, while self-oriented utility depends on pay minus effort costs.

Although there are several good reasons for assuming that workers compare pay but not effort costs, the current literature suggests that what workers compare is likely context dependent. In his book addressing wage rigidity as support for internal pay equity, Bewley (1999) states that employees care about their pay relative to that of their coworkers; he also finds that even when pay secrecy is a company policy, workers often reveal how much they make to others. Accordingly, we choose to focus our analysis on comparisons of gross wages, but in Section VII, we also explore the implications of our model when workers compare net wages. (3)

Other researchers have also examined the effects of other-regarding preferences in performance-pay settings. Battling and von Siemens (2005) derive optimal incentive contracts for risk, inequality, and behindness averse agents in a moral hazard setting. They find that behindness aversion increases the agency costs of providing incentives because agents suffer from both inequality and risk. However, their model is restricted to a setting in which agents have a binary effort choice. Our model is more general than this, as we allow for a continuum of effort choices. Rey Biel (2004) examines a two-worker joint production game with binary effort choices and finds that inequality aversion allows the firm to induce the desired effort for less pay by designing a compensation scheme that imposes inequality off of the equilibrium path. Demougin and Fluet (2006) also demonstrate how firms can use the threat of inequality to induce workers to exert effort for less pay. Englmaier and Wambach (2005), Dur and Glazer (2008), and Itoh (2004) all examine incentive contracts when agents care about inequality relative to the principal (instead of their coworkers). However, Itoh (2004) also compares agents with other-regarding preferences and finds that as agents become more inequity averse, wages become smaller and the principal's expected utility increases. Furthermore, when there are multiple symmetric other-regarding agents, as agents become more inequity averse, wages become smaller and the principal's expected utility increases. Grund and Sliwka (2005) show that inequality aversion can lead competitors to increase their effort in an appropriately designed tournament.

The paper proceeds as follows. Section II provides a brief description of the model. Other-regarding preferences are discussed in a general context in Section III, and Section IV compares the effort choices of other-regarding workers and self-oriented workers. Section V introduces the inequality premium and determines how it responds to changes in the piece rate and the parameters of the preference representation. Section VI examines the impact of other-regarding preferences on the optimal piece rate. Section VII checks the robustness of the results when workers compare net rather than gross wages, and Section VIII offers a conclusion. All proofs are collected in an Appendix.

II. DESCRIPTION AND TIMING OF THE MODEL

In this principal-agent model, the principal is risk neutral and the workers are risk neutral but other-regarding. (4) The timing is as follows. The principal offers all workers a single pay schedule (s, b) consisting of a salary component, s, and a piece-rate component, b. (5) Workers observe the contract and decide whether they want to participate or not, and if they choose to participate, they exert the level of effort that maximizes their expected utility. The principal observes the output from each worker and pays the agent according to the specified contract.

Workers generate output by exerting effort. When worker i exerts effort [e.sub.i], his output is given by [e.sub.i] + [[epsilon].sub.i], where [[epsilon].sub.i] is a random variable caused by measurement error by the employer. Each [[epsilon].sub.i] is drawn independently from the distribution F. Worker pay is determined by s and b: if worker i is observed to generate output [e.sub.i] + [[epsilon].sub.i], his pay is [w.sub.i] = s + b([e.sub.i] + [[epsilon].sub.i]). Exerting effort is costly, and worker i's effort-cost function is given by [c.sub.i]([e.sub.i]), which is assumed to be zero at zero and strictly increasing and strictly convex. Worker i's net income from the employment relationship is defined as s + b([e.sub.i] + [[epsilon].sub.i]) - [c.sub.i]([e.sub.i], or [w.sub.i] - [c..sub.i] ([e.sub.i])

III. PREFERENCES

It is assumed that a worker's utility depends on his own payoff and a comparison of his own income with that of his coworker. Let Worker 1 be the worker whose decision we are modeling, and let Worker 2 be his coworker. The utility function of Worker 1 is given by:

(1) U([w.sub.1], [c.sub.1], ([e.sub.1]), [w.sub.2]) = E[u([w.sub.1] - [c.sub.1]([e.sub.1]))] - aE[V([w.sub.2] - [w.sub.1])].

The function u represents Worker l's own utility from his net wage and is referred to as the self-oriented utility function, the function V represents his disutility from inequality in gross wages and is referred to as the social disutility function, and a [greater than or equal to] 0 is a shift parameter that reflects the strength of the decision maker's other-regarding preferences. (6)

We assume that u is linear and strictly increasing with u(x) = x and that V takes the following form:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where v is nonnegative, strictly increasing, and strictly convex with v(0) = 0. The function V captures different types of equity attitudes according to the value of [lambda]. (7) First, note that [eta] [greater than or equal to] 0 when [w.sub.1] [less than or equal to] [w.sub.2] which means that Worker 1 is behind in the payoff comparison, while when [eta] [less than or equal to] 0, Worker 1 is ahead in the comparison. (8) Since V is subtracted from u in the specification of U, the assumptions v(0) = 0 and v' > 0 mean that the worker dislikes being behind. Also, the property v' > 0 means that the worker prefers to catch up when he is behind. If [lambda] > 0, Worker 1 also dislikes being ahead. Since, when [lambda] > 0, the worker dislikes any situation in which wages are unequal, [lambda] > 0 implies inequality aversion. In contrast, when [lambda] < 0, Worker 1 likes being ahead and prefers to move farther ahead or, equivalently, prefers that Worker 2 move farther behind. This behavioral pattern is commonly referred to as competitiveness, where the decision maker chooses to increase the difference between his payoff and his opponent's payoff (MacCrimmon and Messick 1976). This can also be thought of as status seeking, as in Frank (1985).

An increase in [lambda] represents a movement toward inequality aversion and away from competitiveness, and so [lambda] is referred to as the inequality aversion parameter since increases make the worker more inequality averse. The magnitude of [lambda] is also meaningful. When [absolute value of [lambda]] < 1, being behind by a certain amount generates more disutility than the utility gained or lost by being ahead an equal amount. Accordingly, when [absolute value of [lambda]] < 1, the individual is behindness averse. When [absolute value of [lambda]] > 1, being ahead has a larger effect than being behind. Fehr and Schmidt (1999) support a behindness averse specification, and behindness aversion is also consistent with the logic of loss aversion. Figure 1 shows the V function for different values of [lambda].

Finally, if a worker chooses not to enter the employment relationship, he exerts no effort and receives no pay, so his income is non-stochastic and zero. Moreover, if he is not in the employment relationship, he has no "coworker" in the true sense of the word and therefore has no obvious candidate for comparison. Accordingly, we assume that if the worker is not in the employment relationship, the second component of the preference function is irrelevant, and his utility is u(0) = 0.

IV. THE AGENT'S PROBLEM

According to the preferences specified in Equation (1), the self-oriented utility function u depends on net income, but the social disutility function V depends on the difference between gross incomes. For Worker 1, then, since Worker i's gross income is s + b([e.sub.i] + [[epsilon.sub.i]), the social disutility function V is a function of b([e.sub.2] - [e.sub.1] + [[epsilon].sub.2] - [[epsilon].sub.1]). Suppose that Worker 2's effort level is given at [e.sub.2]. Worker 1 chooses [e.sub.1] to maximize: (9)

(3) U([e.sub.1], [e.sub.2]) = E[s + b([e.sub.1] + [[epsilon].sub.1]-[c.sub.1]([e.sub.1])] - aE[V(b([e.sub.2] - [e.sub.1] + [[epsilon].sub.2] - [[epsilon].sub.1]))],

which is assumed throughout to be strictly concave in [e.sub.1] so that a unique maximum exists. (10)

Let z = [[epsilon].sub.2] - [[epsilon].sub.1], and let G be the distribution function for the random variable z, with density function g. Since [[epsilon].sub.1] and [[epsilon].sub.2] are drawn independently from the same distribution, z is symmetric about zero, which implies that g(z) = g(-z). From the definition of V in Equation (2), if [e.sub.2] - [e.sub.l] = k < 0, we have V(k) = [lambda]v(-k), which combined with the symmetry of g(z), yields:

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

as illustrated in Figure 2. We can then rewrite:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first integral on the right-hand side of Equation (5) is the contribution to expected social utility from being behind, and the second integral is the contribution from being ahead.

Since each worker's income is random, it is useful to define notions of ahead and behind in expectation. If [e.sub.1] [less than or equal to] [e.sub.2], Worker 1 is said to be behind in expectation, and when [e.sub.1] [greater than or equal to] [e.sub.2], he is said to be ahead in expectation. A worker could be behind in expectation, for example, if his marginal effort costs are higher than his opponent's, and so he exerts less effort than his opponent. Since the noise terms are drawn identically and independently, if he exerts less effort his expected output is less than his coworker's. Note that by the conventions established here, when [e.sub.1] = [e.sub.2], Worker 1 is both ahead in expectation and behind in expectation.

The optimal effort level, [e.sup.*.sub.1], satisfies the first-order condition:

(6) b-[c.sub.1]'([e.sub.1])-a[partial derivative]/[partial derivative][e.sub.1]E[V(b([e.sub.2] - [e.sub.1] + z))] = 0.

[FIGURE 2 OMITTED]

From Equation (5), we get:

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The two terms in the middle line of the equation are zero because v(0) = 0.

Let [e.sup.0.sub.1] satisfy [c.sub.1]' ([e.sup.0.sub.1]) = b. Then from Equation (6), [e.sup.0.sub.1] is the optimal effort level when the worker is not concerned with his coworker's payoff, that is, when a = 0. (11) The first proposition compares the optimal effort level when inequality matters, [e.sup.*.sub.1], to the optimal effort level when it does not, [e.sup.0.sub.1].

PROPOSITION 1. An other-regarding worker exerts more effort than a self-oriented one (i. e., [e.sup.*.sub.1] > [e.sup.0.sub.1] ) when one of the following conditions holds:

(i) The worker is competitive (i.e., [lambda] < 0).

(ii) The worker is inequality averse, behindness averse, and behind in expectation at [e.sup.0.sub.1] (i.e., [lambda] [member of] [0,1) and [e.sup.0.sub.1] [less than or equal to] [e.sub.2]).

Proof in Appendix.

The proposition states that competitive workers exert more effort for a given piece rate than self-oriented ones do and that inequality-averse workers who are also behindness averse exert more effort than self-oriented ones do when they would expect to be behind at the self-oriented level of effort.

In the absence of other-regarding preferences, the worker chooses the effort level that equates the piece rate and the marginal cost of effort. In contrast, the presence of inequality affects the marginal condition, for the simple reason that by raising or lowering his effort level, the worker can change the amount of inequality. The proposition identifies inequality attitudes that lead the worker to choose more effort.

The results of Proposition 1 are straightforward to see when one assumes no measurement error, so that [[epsilon].sub.1] = [[epsilon].sub.2] = 0. Then:

(8) U([e.sub.1],[e.sub.2]) = s+[be.sub.1] - [c.sub.1]([e.sub.1]) - aV(b([e.sub.2] - [e.sub.1])).

Suppose that choosing [e.sup.0.sub.1] leaves Worker 1 behind, that is, [e.sup.0.sub.1] < [e.sub.2]. Differentiating U with respect to [e.sub.1] and evaluating at [e.sup.0.sub.1] yields abv'(b([e.sub.2]- [e.sup.0.sub.1]))>0, so the worker prefers to exert more effort than [e.sup.0.sub.1]. Now suppose that choosing [e.sup.0.sub.1] moves Worker 1 ahead of Worker 2. This time differentiation at [e.sup.0.sub.1] yields -ab[lambda]v'(b([e.sub.2]- [e.sup.0.sub.1])), which is positive if [lambda] < 0, that is, if the worker is competitive. The conditions in the proposition are stronger than these because of the measurement error. The same intuition holds, though. If the worker is competitive, increases in his effort make the expected income difference b([e.sub.2] - [e.sub.1]) smaller, which increases his utility through the function V. Accordingly, the piece rate motivates him to exert extra effort in order to move farther ahead of (or catch up to) the other worker. In much the same way, if the worker is inequality averse, he wants to narrow any income gap. He can do this by exerting more effort when he is behind or by exerting less effort when he is ahead. However, with measurement error, both events occur with positive probability. In order for the worker to exert more effort when he is behind in expectation at [e.sup.0.sub.1], the impact of being behind must outweigh the impact of being ahead, which holds if [lambda] [member of] [0,1).

A look at Equation (6) shows that [[partial derivative].sup.2]U([e.sup.0.sub.1], [e.sub.2])/[partial derivative][e.sub.1][partial derivative]a = [[partial derivative]U([e.sup.0.sub.1], [e.sub.2])/[partial derivative][e.sub.1]]/a, and so an increase in a magnifies the effect of inequality on effort. Also,

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As the worker becomes less competitive, or more inequality averse, the effect on effort diminishes. This makes sense because a competitive worker always provides more effort than [e.sup.0.sub.1], but an inequality-averse worker may exert less than [e.sup.0.sub.1]. The move from competitiveness to inequality aversion reduces the amount of effort provided.

Finally, a look at Equation (7) allows a discussion of when the presence of other-regarding preferences might lead to less effort. According to the proposition, one looks at the case in which the worker is inequality averse and [e.sup.0.sub.1] > [e.sub.2], so that choosing [e.sup.0.sub.1] places the worker ahead of his coworker in expectation. Because [e.sup.0.sub.1] > [e.sub.2], the first integral in Equation (7) is over a smaller probability mass than the second term. Consequently, when [lambda] [greater than or equal to] 1, the second term dominates the first, and the worker exerts less effort than [e.sup.0.sub.1] when he is ahead of his coworker. This can also occur when [lambda] is less than but close to 1, which has the interpretation that the decision maker is inequality averse but not very behindness averse.

When workers have identical preferences and costs, for each value of b, there is a symmetric equilibrium in which both workers exert the same effort level, [e.sub.*]. Take the first derivative of Equation (3) with respect to [e.sub.1] and substitute [e.sub.1] = [e.sub.2] = e, and that effort level, [e.sub.*], solves:

(10) b - c'([e.sub.*]) + a [partial derivative]/ [partial derivative][e.sub.1]E[V(bz)] = 0.

The derivative d[e.sub.*]/da tells how the symmetric equilibrium effort level changes when the workers become more sensitive to inequality. From standard comparative statics techniques, the sign of d[e.sub.*]/da is the same as the sign of [partial derivative]E[V(bz)]/[partial derivative][e.sub.1], (12) which from Equation (5) is:

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which is positive when [lambda] < 1. This establishes the next proposition, whose proof is omitted.

PROPOSITION 2. If workers are identical and are either competitive, behindness averse, or both (i.e., [lambda] < 1), then in symmetric equilibrium, when they become more other regarding (i. e., a increases), the), both exert more effort.

According to Proposition 2, if the workers have identical costs and preferences that exhibit either competitiveness or inequality aversion combined with behindness aversion, their other-regarding preferences provide an intrinsic incentive that works alongside the piece rate in motivating effort. Competitive workers exert additional effort because they like their incomes moving ahead of their coworkers', and they dislike it when their incomes fall behind. Inequality-averse workers dislike moving ahead and moving behind, but behindness aversion states that moving behind has the larger effect. So, the inequality-averse workers exert additional effort in order to avoid falling behind their coworkers.

This result is reminiscent of the results from promotion tournaments (e.g., Lazear and Rosen 1981). In tournaments, workers exert effort in an attempt to win a prize, and the larger the prize, the harder they work. When workers are identical, as they are in Proposition 2, they have equal chances of winning the tournament in equilibrium; yet, they still work harder when the prize increases. Essentially, if they do not work harder when the prize increases, their opponents have an incentive to exert even more effort, and the nonresponding worker's probability of winning falls. So, workers respond to higher prizes by exerting more effort to avoid falling behind. Similarly in this setting, an increase in a intensifies the "rewards" from being ahead and the "losses" from being behind, and the workers respond to this by exerting more effort in an attempt to not fall behind.

V. INEQUALITY PREMIA

When workers are not other regarding, measurement error causes their participation constraints, or individual rationality constraints, to change. Measurement error introduces income risk, and when workers are risk averse, the firm must pay a risk premium to induce them to participate in the employment relationship. In this setting with risk-neutral but other-regarding workers, different considerations occur because of inequality, which arises for two reasons. First, measurement error introduces inequality even when both workers exert the same amount of effort. Second, inequality occurs when one worker chooses more effort than the other, which may come about because of such factors as cost differences or heterogeneous preferences. When workers are other regarding, the firm must pay an inequality premium in order to satisfy the workers' participation constraints. In this section, we explore properties of the inequality premium.

With noise, but no coworker, Worker l's utility is given by E[s + b([e.sub.1] + [[epsilon].sub.1]) - [c.sub.1]([e.sub.1])] = s + [be.sub.1] - [c.sub.1]([e.sub.1]). With a coworker, utility is given by U([e.sub.1], [e.sub.2]) = s + [be.sub.1] - [c.sub.1]([e.sub.1]) - aE[V(b([e.sub.2] - [e.sub.1] + [[epsilon].sub.2] - [[epsilon].sub.1]))]. We define the inequality premium by the value [[theta].sub.I] that solves s + [be.sub.1] - [c.sub.1]([e.sub.1] - [[theta].sub.I] = U([e.sub.1], [e.sub.2]), or:

(12) [[theta].sub.I] = aE[V(b([e.sub.2] - [e.sub.1] + [[epsilon].sub.2] - [[epsilon].sub.1]))].

PROPOSITION 3. The inequality premium is increasing in an individual's degree of inequality aversion ([partial derivative][[theta].sub.I]/[partial derivative][lambda] > 0).

Proof in Appendix.

Proposition 3 says that for any type of other-regarding preferences, the inequality premium rises when the worker becomes more inequality averse (less competitive). To understand why, first note that the parameter [lambda] only comes into play when the worker is ahead. When [lambda] < 0, he likes being ahead, but when [lambda] > 0, he dislikes it. Increasing X moves the worker either closer to the range where he dislikes being ahead or farther into that range. Either way, the inequality premium rises.

PROPOSITION 4. The inequality premium is increasing in the piece rate ([partial derivative][[theta].sub.I]/[partial derivative]b > 0) when one of the following two conditions holds:

(i) The worker is inequality averse ([lambda] [greater than or equal to] 0).

(ii) The worker is competitive, behindness averse, and behind in expectation ([lambda] [member of] (-1,0) and [e.sub.1] [less than or equal to] [e.sub.2]).

Proof in Appendix.

Proposition 4 states that the inequality premium rises as the piece rate rises when either the worker is inequality averse or when he is competitive, behindness averse, and behind in expectation. An inequality-averse worker dislikes being either ahead or behind and, since v is increasing, he dislikes it more the further ahead or behind he is. Increasing the piece rate scales up the inequality, making the worker worse off and raising the inequality premium. A competitive worker dislikes being behind but likes being ahead. Behindness aversion means that the effect when behind outweighs the effect when ahead, and the assumption that he is behind in expectation places greater probability mass on the event that he is behind. An increase in the piece rate scales up the inequality, and since the effect of being behind outweighs the effect of being ahead, the inequality premium rises.

VI. THE OPTIMAL PIECE RATE

The purpose of this section is to address two issues concerning the optimal piece rate set by the firm. First, is the choice of piece rate impacted by the inequality attitudes of the workers? Second, are there circumstances under which the workers' other-regarding preferences lead to wage compression?

If p is the price of output, the optimal piece rate chosen by the firm maximizes expected profit, [pi], given by:

(13) [pi] = p[e.sub.*] - (s + b[e.sub.*]),

subject to the agent's incentive compatibility constraint (given by Equation (6), which defines [e.sub.*]) and the individual rationality constraint, which says that accepting employment earns him at least a certain equivalent income of zero: (13)

(14) s + b[e.sub.*] -c([e.sub.*]) - [[theta].sub.I](b,s,a,[lambda]) [greater than or equal to] 0.

Suppose that the workers are identical, in terms of both preferences and costs, so that they both choose the same effort level. (14) The expression for per capita expected profit, after substituting in the constraint, is given by:

(15) [pi] = p[e.sub.*]-c([e.sub.*]) - [[theta].sub.I] (b,s,a,[lambda]).

Differentiating Equation (15) with respect to b yields the first-order condition given by:

(16) [partial derivative][pi]/[partial derivative]b = (p - c'([e.sub.*]))d[e.sub.*]/db - [partial derivative][[theta].sub.I]/[partial derivative]b = 0.

This determines the optimal piece rate [b.sub.*]. (15)

If workers are not other regarding, the first-order condition reduces to (p - c'(e)) [partial derivative][e.sub.*]/[partial derivative]b = 0, and the optimal piece rate is the one that induces the amount of effort that equates the price and marginal effort cost. When workers are other regarding, the optimal piece rate could be either higher or lower than the one that is optimal when workers are inequality neutral. Equation (16) suggests two counteracting forces that govern the choice of the optimal piece rate when workers become more other regarding. One is an inequality premium effect, which puts downward pressure on piece rates, and it is captured by the change in the term [partial derivative][[theta].sub.I]/[partial derivative]b as a increases. As the workers become more other regarding, the firm has an incentive to reduce the piece rate, thereby reducing the inequality premium and hence the total compensation it must provide the worker. The second is an incentive effect, which may put upward pressure on piece rates, and it is captured by the change in the term (p c'(e))[partial derivative] [e.sub.*]/[partial derivative]b as a increases. As workers become more other regarding, they may become more responsive to incentives, prompting the firm to increase the piece rate to take advantage of this increased responsiveness. Neither of these effects exists when workers are only risk averse and not other regarding.

The impact on the optimal piece rate depends on which of these two effects dominates. The next proposition establishes that the inequality premium effect dominates in an interesting class of problems. Suppose that both workers are identical, and let [b.sup.0] be the piece rate for which c'([e.sup.0]([b.sup.0])) = p, so that [b.sup.0] is the optimal piece rate when workers are not other regarding. Letting [b.sup.*] denote the optimal piece rate when workers are other regarding (as defined by Equation (16)), we get the following result.

PROPOSITION 5. Suppose that workers are identical, inequality averse, and/or behindness averse ([lambda] > -1), and not too other regarding (0 < a < A for some A >0). Then [b.sup.*] < [b.sup.0].

Proof in Appendix.

To understand Proposition 5, first assume that de*/db > 0, so that workers respond to incentives. Proposition 4 states that if the workers are inequality averse, behindness averse, or both, [partial derivative][[theta].sub.I]/[partial derivative]b > 0. Equation (16), the first-order condition governing the optimal piece rate, can then only hold if p > c' ([e.sup.*]). Recalling that c' ([e.sup.0]) = p and that marginal costs are increasing, it follows that [e.sup.*] < [e.sup.0], and de*/db > 0 then implies that [b.sup.*] < [b.sup.0]. Proposition 5 establishes that other-regarding preferences can contribute to wage compression, and this may help explain why, as Frank (1984) discusses, the degree of wage compression observed in firms is more than that predicted by risk aversion alone. (16)

The condition that workers are not too other regarding guarantees that d[e.sup.*]/db > 0. When a = 0, the workers are not other regarding and their effort choices are governed by the condition c' ([e.sup.0]) = b. Consequently, de*/db > 0 when a = 0. By continuity, there exists some A > 0 such that d[e.sup.*]/db > 0 whenever a [member of] (0,A).

VII. COST COMPARISONS IN THE SOCIAL DISUTILITY FUNCTION

In this section, we explore the robustness of the results to a change in the specification of preferences. In particular, we change the worker's preferences so that he cares not just about pay differences but also about effort cost differences.

Assume in Equation (1) that the worker compares his own pay minus effort costs to his coworker's pay less effort costs. Worker 1 chooses [e.sub.1] to maximize:

(17) U([e.sub.1],[e.sub.2]) =E[s + b([e.sub.1] + [[epsilon].sub.1])-[c.sub.1]([e.sub.1])] - aE[V(b([e.sub.2] -[e.sub.1] + [[epsilon].sub.2] - [epsilon].sub.1]) - ([c.sub.2] ([e.sub.2]) - [c.sub.1]([e.sub.1]))].

The key difference between this case and the benchmark case of earlier sections is that the worker's first-order condition for maximization is changed. Differentiate Equation (17) with respect to [e.sub.1] to get:

(18) [partial derivative]U/[partial derivative][e.sub.1] = (b - [c.sub.1]'([e.sub.1]))(1 + aE[V']),

and let [e.sup.**.sub.1] solve the first-order condition in Equation (18). Note that Equation (18) is equal to zero when [c'.sub.1] ([e.sub.1]) = b, which is exactly the definition of [e.sup.0.sub.1], the optimal effort for a self-oriented worker. This proves the following proposition.

PROPOSITION 6. Assume that workers compare net income as in Equation (17). Then, other-regarding workers exert the same level of effort as self-oriented ones (i.e., [e.sup.**.sub.1] = [e.sup.0.sub.1]).

When workers compare effort costs as well as pay levels, piece rates do not provide any extra incentives over the self-oriented case. Thus, Proposition 1 is not robust to this change in the preference specification. It is informative to explore why not. In the benchmark case, at [e.sup.0.sub.1], a marginal increase in effort leads to no additional self-oriented utility, but it does lead to a marginal increase in expected income, and so it leads to a marginal change in inequality. The worker likes this marginal change in inequality when he is competitive because the increased effort makes the income comparison more favorable in expectation. He also likes the marginal change when he is inequality averse, behindness averse, and behind in expectation because the additional income reduces the amount of inequality and places more probability mass on the event where he is ahead. When effort costs enter the social comparison, though, the marginal increase in effort does not change the social comparison because it changes income and effort costs in exactly the same way, and this is regardless of whether the worker is competitive or inequality averse.

Even though the counterpart of Proposition 1 does not hold in this setting, the inequality premium exhibits properties similar to those in the benchmark setting and leads to the same wage compression result. To that end, let [b.sup.**] denote the optimal piece rate for the firm when workers compare net income.

PROPOSITION 7. Assume that workers are identical, compare net income as in Equation (17), and are inequality averse, and/or behindhess averse ([lambda] > -1). Then, the inequality premium is increasing in the piece rate and in the degree of inequality aversion ([partial derivative][[theta].sub.I]/[partial derivative]b > 0 and [partial derivative][[theta].sub.I]/[partial derivative][lambda] > 0), and the optimal piece rate is lower than it would be for self-oriented workers (b** < [b.sup.0]).

Proof in Appendix..

The result of Proposition 7, that [partial derivative][[theta].sub.I]/[partial derivative]b > O, means that the inequality premium increases when the piece rate increases, and is consistent with Proposition 3. The second result states that the inequality premium increases when the workers become more inequality averse, as measured by [lambda], and is consistent with Proposition 4. The final result states that the firm's profit-maximizing piece rate is less than the one it would set if workers were inequality neutral, that is, when a = 0. This last result is consistent with Proposition 5, and so the primary finding that other-regarding attitudes can lead to reduced incentives when workers are inequality averse, behindness averse, or both extends to the case in which workers compare net incomes rather than gross pay.

VIII. CONCLUSIONS

This paper demonstrates two major results. First, if workers are identical, care about wage differences, and are behindness averse, they work harder than they would if they were self-oriented. Second, the effect of other-regarding behavior on the optimal piece rate is governed by two effects: other-regarding workers exerting more effort puts upward pressure on the piece rate as workers become more other regarding (the incentive effect), but other-regarding workers must also be compensated for facing inequality and this puts downward pressure on the optimal piece rate (the inequality premium effect). When workers are behindness averse, the inequality premium effect dominates, and so the existence of workers with other-regarding attitudes leads the firm to reduce the incentive pay in its compensation scheme, which is wage compression.

When workers compare net rather than gross wages, the results change somewhat. In contrast to the results in Section IV, we find that piece rates do not provide any extra incentives over the self-oriented case. On the other hand, in this setting, the inequality premium exhibits similar properties to those in the setting where workers compare gross wages, and hence, the wage compression result still holds.

Behindness aversion provides a link between the two main results. Workers provide extra effort if they are competitive, or if they are both inequality averse and behindness averse. The inequality premium rises with the piece rate and wage compression results if workers are inequality averse, or if they are both competitive and behindness averse. So, when workers are behindness averse, both results hold, and it does not matter if they are also competitive or inequality averse. In this sense, and in this setting, behindness aversion is more important than the other behavioral patterns. In spite of this, behindness aversion has received little attention in the literature, and it deserves more investigation.

doi: 10.1111/j.1465-7295.2008.00129.x

APPENDIX

Proof of Proposition l

By Equation (7) and the definition of [e.sup.0.sub.1],

(A1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since v' > 0, the first term is positive. If (i) holds, so that [lambda] < 0, the second term is positive, and [partial derivative]U([e.sup.0.sub.1],[e.sub.2])/[partial derivative][e.sub.1] >0. If (ii) holds, so that [lambda] [member of] [0,1) and [e.sup.0.sub.1] [less than or equal to] [e.sub.2],

(A2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first inequality holds because by replacing el [e.sub.2] with 0, the first integral is taken over a smaller probability mass, while the second integral is taken over a larger probability mass. For the second inequality, [lambda] is replaced by 1, which is larger than any value that it can take. Under both Conditions (i) and (ii) then, [partial derivative]U([e.sup.0.sub.1] [e.sub.2])/[partial derivative][e.sub.1] > 0, and, since [partial derivative]U([e.sup.*.sub.1], [e.sub.2])/[partial derivative][e.sub.1] = 0, it follows that [e.sup.*.sub.1] > [e.sup.0.sub.1].

Proof of Proposition 3

From Equations (12) and (5), we have:

(A3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof of Proposition 4

From Equations (12) and (5), we have:

(A4) [partial derivative][[theta].sub.I]/[partial derivative]b : a([partial derivative]E[V(b([e.sub.2] [e.sub.1] + [[epsilon].sub.2] - [[epsilon].sub.1]))]/[partial derivative]b).

From Equation (5),

(A5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Differentiating again yields:

(A6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

First suppose (i) holds. Inequality aversion implies that [lambda] > 0, and therefore, the last line in Equation (A5) and the last line in Equation (A6) are the sums of two integrals of positive-valued functions, implying that [partial derivative][[theta].sub.I]/[partial derivative]b > 0 and [[partial derivative].sup.2][[theta].sub.I]/[partial derivative][b.sup.2] > 0.

Now suppose (ii) holds. Because [lambda] [member of] (- 1,0) and [e.sub.1] < [e.sub.2],

(A7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consequently, [partial derivative][[theta].sub.I],/[partial derivative]b > 0. A similar argument establishes that [partial derivative]O:/[partial derivative]b- > O.

Proof of Proposition 5

From Equation (16), the optimal piece rate b* is determined by

(A8) (p-c'(e*))de*/db = [partial derivative][[theta].sub.I]/partial derivative]b.

When [lambda] > 1, by Proposition 4, [partial derivative][[theta].sub.I]/[partial derivative]b > 0. When a = 0, e* solves c'(e*) - b, so de*/db = 1/c" > 0. By continuity, there exists an A > 0 such that when a [member of] (0,A), de*/db > 0. From Equation (A8), p - c'(e*) = ([partial derivative][[theta].sub.I]/[partial derivative]b)/(de*/db) > O, and so p c'(e*) > 0. Letting [e.sup.0] denote the effort level that makes c'([e.sup.0]) = p, strict convexity of the effort-cost function implies that c'(e*) < c'([e.sup.0]) and therefore e* < [e.sup.0]. The result follows from de*/db > 0.

Proof of Proposition 7

Since workers are identical, [e.sub.1] = [e.sub.2], implying that:

(A9) V(b([e.sub.2] - [e.sub.1] + z) ([c.sub.2]([e.sub.2]) -[c.sub.1]([e.sub.1]))) = V(bz).

From Equations (A9) and (A4),

(A10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, [partial derivative][[theta],sub.I]/[partial derivative] b > 0 follows from [delta] > -1. Also,

(A11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, from Equation (16), we have p - c'([e.sup.**]) = ([partial derivative][[theta].sub.I]/[partial derivative]b)/(d[e.sup.**]/db), and by Proposition 6, de**/db = d[e.sup.0]/db = 1/c" > 0. Consequently, [e.sup.**] < [e.sup.0], and since d[e.sup.**]/db > 0, it follows that [b.sup.**] < [b.sup.0].

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MacCrimmon, K. R., and D. M. Messick. "A Framework for Social Motives." Behavioral Science, 21, 1976, 86-100.

Main, B. G. M., C. A. O'Reilly III, and J. Wade. "Top Executive Pay: Tournament or Teamwork?" Journal of Labor Economies, 11, 1993, 606-28.

Messick, D. M., and K. P. Sentis. "Fairness and Preference." Journal of Experimental Social Psychology, 15, 1979, 418-34.

--. "Fairness, Preference, and Fairness Biases, " in Equity Theory: Psychological and Sociological Perspectives, edited by D. M. Messick and K. S. Cook. New York: Praeger, 1983, 61-94.

Moene, K. O., and M. Wallerstein. "Pay Inequality." Journal of Labor Economics, 15, 1997, 403-30.

Neilson, W. S. "Axiomatic Reference Dependence in Behavior Toward Others and Toward Risk." Economic Theory, 28, 2006, 681-92.

Rey Biel, P. "Inequity Aversion and Team Incentives." Working Paper, University College London, 2004.

Tversky, A., and D. Kahneman. "Advances in Prospect Theory: Cumulative Representation of Uncertainty." Journal of Risk and Uncertainty,, 5, 1992, 297-323.

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WILLIAM S. NEILSON and JILL STOWE *

* W.S.N. thanks the Private Enterprise Research Center for financial support.

Neilson: Professor, Department of Economics, University of Tennessee, Knoxville, TN 37996-0550. Phone 1-865-974-1691, Fax 1-865-974-4601, E-mail wneilson@utk.edu

Stowe: Assistant Professor, Fuqua School of Business, Duke University, Durham, NC 27708-0120. Phone 1-919-660-7746, Fax 1-919-681-6246, E-mail stowe@duke.edu

(1.) We look specifically at linear wage contracts, primarily because linear wage contracts, although generally suboptimal, have proven to be an extremely useful vehicle for exploring the effects of risk attitudes on incentive pay (e.g., Gibbons 1997). However, Holmstrom and Milgrom (1987) show that in a natural class of dynamic moral hazard problems, linear contracts are actually optimal.

(2.) In the Frank Hahn Lecture, Fehr and Fischbacher (2002) state that "neither the effects nor the determinants of material incentives can be adequately understood if one neglects social preferences and that the interaction between material incentives and social preferences is likely to have important effects on the optimality of different types of contracts and property rights" (p. C2).

(3.) In contrast to Bewley's findings, researchers in the social psychology literature suggest that individuals use comparisons that are biased in a self-serving manner. For example, when an individual works harder than his coworker, he compares net pay, whereas when the coworker works harder, he compares gross pay (Walster, Walster, Berscheid 1978). Messick and Sentis (1979, 1983) and Cook and Yamagishi (1983) suggest that individuals who exert less effort view equal wages as being fair, whereas those who exert more effort are likely to view wages proportional to effort as being fair.

(4.) All the results of this paper still hold if the workers are allowed to be risk averse. Risk neutrality is assumed for reasons of transparency.

(5.) If workers have different effort-cost functions, the optimal piece-rate scheme would entail different piece rates and different salaries for the two workers. Many real-world firms, however, offer a single piece-rate scheme to all their workers, and the situation described here fits those firms. Of course, if the salary is insufficient to meet one worker's participation constraint, that worker exerts no effort.

(6.) The functional form in Equation (1) corresponds to the Fehr and Schmidt (1999) model of inequality aversion if all components of income (and not just gross wages) matter for comparison, if both [w.sub.1] - [c.sub.1]([e.sub.1]) and [w.sub.2]-- [c.sub.2]([e.sub.2]) are nonstochastic, and if V is piecewise linear.

(7.) The function V is reminiscent of the value function used in cumulative prospect theory (Tversky and Kahneman 1992). In that setting, individual choice under risk, the value function is specified to capture the reflection effect (choice patterns over gains are the opposite of choice patterns over losses) and loss aversion (losses have a larger effect than gains), and [lambda] would be interpreted as a loss aversion parameter. The function also captures one of the features of the Fehr-Schmidt model: the function used to govern positive payoff differences is a multiple of the function used to govern negative payoff differences.

(8.) Ahead and behind are both weak comparisons here, implying that when q = 0, Worker 1 is both ahead and behind.

(9.) We have simplified notation, writing U([e.sub.1], [e.sub.2]) instead of the more cumbersome U(s + b([e.sub.1] + [[epsilon].sub.1]), [c.sub.1]([e.sub.1]), s + b([e.sub.2] + [[epsilon].sub.2])).

(10.) Compute [[partial derivative].sup.2]U/[[partial derivative].[e.sup.2.sub.1] = - [c".sub.1] - a[b.sup.2]EV". By assumption, [c".sub.1] [greater than or equal to] 0. U is strictly concave in [e.sub.1] if V" is not too negative, that is, if V is not too concave.

(11.) [e.sup.0.sub.1] is also the effort exerted by a risk-averse but non-other-regarding worker.

(12.) The first-order condition has [U.sub.1]([e.sub.1], [e.sub.2]) = O. Implicitly differentiating with respect to a, holding [e.sub.2] constant, yields [U.sub.11][de.sub.1]/da + [partial derivative][U.sub.1]/ [partial derivative]a. Since [U.sub.11] < 0 by assumption, [de.sub.1]/da and [partial derivative][U.sub.1]/ [partial derivative]a have the same sign. Furthermore, as already argued, [[partial derivative].sub.[]/[partial derivative] has the same sign as [U.sub.1].

(13.) In most of the incentive pay literature, analyses look at maximizing total surplus rather than expected profit. In this setup, maximizing expected profit generates the same result as maximizing total surplus. The total surplus (or total certainty equivalent) has two components: the firm's certainty equivalent (or certain equivalent income), which is expected revenue minus expected worker compensation, and the worker's certainty equivalent (or certain equivalent wealth), which is his expected wage minus the cost of both exerting effort and bearing risk and inequality.

(14.) This assumption is made because if workers had different costs, for example, the firm would want to set different piece rates for the two workers. Since we are interested in the inequality generated when all workers receive the same piece rate, the assumption of identical workers is natural. If workers had different piece rates, this would provide another potential dimension for comparison for the social disutility function.

(15.) Second-order conditions require that (p - c'(e*)) ([[partial derivative].sup.2]e*/[partial derivative][b.sup.2]) - c'(e*)[([partial derivative]e*/[partial derivative]).sup.2] - [[partial derivative].sup.2][[theta].sub.I]/[partial derivative][b.sup.2] < 0, which always holds if we assume [[partial derivative].sup.2][[theta].sub.I]/[partial derivative][b.sup.2] > 0 and sgn(p - c'(e*)) = - sgn([[partial derivative].sup.2]e*/[partial derivative][b.sup.2]).

(16.) Wage compression has been addressed in a variety of other contexts, including centralized and decentralized bargaining and tournaments (see Moene and Wallerstein 1997; Main, O'Reilly, and Wade 1993; Lazear 1989; Kandel and Lazear 1992; among others). The practice of wage compression is often supported for the reason that it should reduce disharmony among coworkers; however, it is not clear that wage compression is always beneficial because better workers may feel slighted (Lazear 1989). In addition, it is commonly posited that wage compression is a result of risk aversion. Importantly, however, the degree of wage compression observed in firms cannot be explained by risk aversion alone (Frank 1984).
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