Elsevier

Games and Economic Behavior

Volume 126, March 2021, Pages 374-386
Games and Economic Behavior

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Moral hazard with limited liability: Random-variable formulation and optimal contract structures

https://doi.org/10.1016/j.geb.2021.01.002Get rights and content

Abstract

This paper studies the optimal contract for a risk-neutral agency with limited liability. We introduce a novel formulation of the model, in which the contract design problem reduces to a problem of constructing the distribution function of a random variable. This formulation directly balances the principal's tradeoff between incentivizing the agent to exert proper effort and minimizing the cost of the agent's compensation. We show that the optimal contract may involve one or two tiers of performance-based bonuses. We obtain new sufficient conditions for the optimality of bonus contracts and provide new insights into the choice of contract parameters.

Introduction

This paper studies a standard moral hazard problem with a risk-neutral agent under the auxiliary assumption that payment should be nondecreasing in the agent's performance. We derive a set of conditions under which the optimal solution takes the form of a single-step function. The contract in such a form is called a single-bonus contract—in addition to a fixed minimum payment, the agent receives a lump-sum bonus if the task performance exceeds a pre-specified threshold. The conditions we obtain are satisfied by several commonly-used distributions. When these conditions do not hold, we show that the optimal contract is likely to take the form of a two-step function, which is referred to as a two-tier bonus contract. Below, we present the standard model and describe our assumptions.

Consider a principal (she) who hires an agent (he) to perform a task. The agent chooses an effort a0 that affects the performance of the task. We assume that the task performance can be measured by its financial value for the principal and that it is a unidimensional random variable, denoted by X with support [0,).1 X has a distribution function G(x,a), which is continuous and differentiable in x and a, and a density function g(x,a) for any given x and a. The principal observes the performance realization x but cannot observe the agent's effort a. Thus, she compensates the agent according to the task performance. The compensation contract is denoted as a function V:[0,)R, which specifies the payment V(x) to the agent for any realized performance x. This paper studies the optimal effort-implementing problem: Suppose the principal aims to induce a specific effort level ao from the agent (and thus a given expected value of the task), how to design the contract V to incentivize the agent while minimizing the expected payment to him?2

Denote the cost function of the agent's effort by C(). Provided with contract V(), the risk-neutral agent will pick the target effort a=ao only if it maximizes his expected utility:U(a):=0V(x)g(x,a)dxC(a). The condition that aoargmaxa0U(a) is referred to as the Incentive compatibility (IC) constraint. The agent has a reservation value U00, and he only accepts a contract that yields an expected utility U(ao)U0. This constraint is referred to as individual rationality (IR). The agent also has a limited liability (LL) for managing the task, meaning that the payment to the agent is nonnegative, that is, V(x)0 for all x. We restrict the payment scheme V() to be a nondecreasing function in the agent's performance metric x, meaning that the agent will never be penalized for delivering a higher performance. The practical motivation is to prevent the agent from sabotaging the project. This monotonic payment (MP) constraint on V has also been imposed by several previous papers, and we will compare ours with them in Section 4. Thus, for inducing a given effort level ao from the agent, the formulation of the principal's contract design problem isminV()0V(x)g(x,ao)dxs.t. {(IC)aoargmaxa0U(a)(IR)U(ao)U0(LL)V(x)0,x[0,)(MP)V is nondecreasing. We rule out ao=0 and focus on solving (P1) for implementing a positive effort ao>0.3

The technical difficulty in solving (P1) comes from the (IC) constraint: written as a constrained optimization problem, (IC) includes a large (in fact, infinite) number of inequalities (i.e., U(ao)U(a) for any aao). How to handle the (IC) constraint is of central interest in the literature, which will be reviewed later. A well-known method is the first order approach (FOA), which replaces the (IC) constraint by its first-order condition, i.e., aV(x)g(x,a)dxC(a)=0. We contribute to the literature by offering a different way of handling the (IC) constraint and solving the problem without using FOA.

Following a notational convention in the literature, we denote by G¯(x,a)=1G(x,a) the residual distribution function, and use a subscript of the function to denote partial derivative with respect to that variable. For instance, G¯a=G¯/a. Throughout the paper, if not otherwise specified, we use “increasing, decreasing, etc.” in a weak (i.e., non-strict) sense. The following assumption regulates the properties of the distribution function G, the cost function C, and the payment function V.

Assumption 1 Regularity

  • (i)

    G¯a(x,a)>0 for any interior x and a. That is, the agent's effort positively affects the task performance in the sense of first-order stochastic dominance (FOSD).

  • (ii)

    g(,) is twice differentiable in x and a, and gxa(,) is continuous in x and twice differentiable in a.

  • (iii)

    C() is twice differentiable with C(0)=0 and C()>0.

  • (iv)

    V() is right continuous.

  • (v)

    limxV(x)G¯(x,a)=0 for any aao.

Assumption 1(ii) is satisfied by several commonly-used distributions. The second part of it is mainly used in Online Appendix B for verifying that our method can be applied under these distributions. Assumption 1(iv) means that, if V() jumps at performance level x, (i.e., a lump-sum reward), the agent can obtain the reward as long as the performance level is reached. Assumption 1(v), while not imposing an upper bound on the payment function V, limits how fast it increases as x goes to infinity. This assumption does not rule out any known optimal contract forms in the literature. As shown in the next section, this assumption allows us to disregard the expected payment from the limiting case.4

Section snippets

The random-variable formulation

In this section, we present a transformation of the moral hazard model. It begins with a change of integration for the objective function in (P1): By Assumption 1(v), we have0V(x)g(x,ao)dx=limxV(x)G¯(x,ao)+V(0)G¯(0,ao)+0G¯(x,ao)dV(x)=V(0)+0G¯(x,ao)dV(x),where the difference of the reward function at performance level x is defined as5dV(x):={V(x)dxif V is differentiable at xlimyxV(y)limyxV(y)if V is not

Form of optimal contract

In this section, we characterize the structure of the optimal solution. It is possible that problem (P3) (equivalently, (P2)) does not have an optimal solution, despite the existence of feasible solutions.6

Related literature

In this section, we compare our solution approach and findings with related literature. To establish analytical results on the optimal contract structure, all existing works invoke certain assumptions on the effort-performance dependence and/or impose certain restrictions on the decision space. We categorize the existing papers into two groups and summarize them in Table 2. Three mathematical concepts and the related acronyms are defined below Table 2: Convex Distribution Function Condition

Acknowledgements

The authors thank David C. Parkes (editor), the advisory editor, and the three anonymous referees for their valuable comments and suggestions. The first author received support from the National Natural Science Foundation of China [Grants 71501121].

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