### 1. Effort and Production** **

We consider an agent who can exert a costly effort *e*. Two possible values can be taken by *e*, which we normalize as a zero effort level and a positive effort of one:

*e** *in {0, 1}. Exerting effort *e** *implies a disutility for the agent that is equal to ψ(*e) *with the normalizations ψ(0) = ψ_{0} = 0 and ψ(1) = ψ_{1} = ψ.

The agent receives a transfer *t *from the principal. We assume that his util- ity function is separable between money and effort,2 *U *= *u(**t)* − ψ(*e)*, with *u(*·) increasing and concave (*u*^{‘} *> *0, *u*^{”} *< *0) and normalized so that *u(*0) = 0. Some- times we will use the function *h *= *u*^{−1}, the inverse function of *u(*·), which is increasing and convex (*h*^{‘} *> *0, *h*^{”} *> *0).

Production is stochastic, and effort affects the production level as follows: the stochastic production level *q*˜ can only take two values , with *q*¯ − __q__* *= Δ*q > *0, and the stochastic influence of effort on production is characterized by the probabilities , with π_{1} *> π*_{0}. We will denote the difference between these two probabilities by Δp = π_{1} − π_{0}.

Note that effort improves production in the sense of *first-order stochastic dom- inance*, i.e., is decreasing with *e *for any given production *q*^{∗}. Indeed, we have , and . This property implies that any principal who has a utility function v(·) that is increasing in production prefers the stochastic distri- bution of production induced by the positive effort level *e *= 1 to that induced by the null effort level *e* = 0. Indeed, we have , which is greater than if v(·) is increasing. An increase in effort improves production in a strong sense in this model with two possible levels of performance.

### 2. Incentive Feasible Contracts** **

Mimicking what we did in chapters 2 and 3, we start by describing incentive feasible contracts in a moral hazard environment. In such an environment, the agent’s action is not directly observable by the principal. The principal can only offer a contract based on the observable and verifiable production level, i.e., a function f*t*k*q*˜li linking the agent’s compensation to the random output *q*˜. With two possible outcomes *q*¯ and * q*, the contract can be defined equivalently by a pair of transfers

*t*¯ and

*. Transfer*

__t__*t*¯ (resp.

*) is the payment received by the agent if the production*

__t__*q*¯ (resp.

*) is realized. Keeping the same notations as in chapter 2, the risk-neutral3 principal’s expected utility is now written as*

__q__if the agent makes a positive effort (*e *= 1), and

if the agent makes no effort (*e *= 0). For notational simplicity, throughout this chapter we will denote the principal’s benefits in each state of nature by and *S(** q)* =

*.*

__S__The problem of the principal is now to decide whether to induce the agent to exert effort or not and, if he chooses to do so, then to decide which incentive contract should be used.

Each level of effort that the principal wishes to induce corresponds to a set of contracts ensuring participation and incentive compatibility. In our model with two possible levels of effort, we will say that a contract is *incentive feasible *if it induces a positive effort and ensures the agent’s participation. The corresponding *moral hazard incentive constraint *is thus written as

(4.3) is the incentive constraint that imposes upon the agent to prefer to exert a positive effort. If he exerts effort, the agent faces the lottery that gives *t*¯ (resp. * t*) with probability π

_{1}(resp. 1 − π

_{1}) and not the lottery that yields

*t*¯ (resp.

*) with probability π*

__t___{0}(resp. 1 − π

_{0}). However, when he does not exert effort, the agent incurs no disutility of effort and saves an amount ψ.

Still normalizing the agent’s reservation utility at zero, the agent’s participation constraint is now written as

(4.4) is the agent’s participation constraint that ensures that if the agent exerts effort, it will yield at least his outside opportunity utility level. Note that this par- ticipation constraint is ensured at the *ex ante *stage, i.e., before the realization of the production shock.

Figure 4.1: Timing of Contracting Under Moral Hazard

**Definition 4.1: ***An incentive feasible contract satisfies the incentive and participation constraints (4.3) and (4.4).*

The timing of the contracting game under moral hazard is summarized in figure 4.1.

### 3. The Complete Information Optimal Contract** **

As a benchmark, let us first assume that the principal and a benevolent court of law can both observe effort. This variable is now *verifiable *and can thus be included into a contract enforced by the court of law. Then, if he wants to induce effort, the principal’s problem becomes

Indeed, only the agent’s participation constraint matters for the principal, because the agent can be forced to exert a positive level of effort. If the agent were not choosing this level of effort, his deviation could be perfectly detected by both the principal and the court of law. The agent could be heavily punished, and the court could commit to enforce such a punishment.

Denoting the multiplier of this participation constraint by μ and optimizing with respect to *t*¯ and __t__* *yields, respectively, the following first-order conditions:

where *t*¯∗ and __t__^{∗} are the first-best transfers.

From (4.5) and (4.6) we immediately derive that , and finally that *t*^{∗} = *t*¯* =*t*^{∗}.

With a verifiable effort, the agent obtains *full insurance *from the risk-neutral principal, and the transfer *t*^{∗} he receives is the same whatever the state of nature. Because the participation constraint is binding we also obtain the value of this transfer, which is just enough to cover the disutility of effort, namely *t*^{∗} = *h(ψ)*. This is also the expected payment made by the principal to the agent, or the *first- **best cost **C ^{FB}* of implementing the positive effort level. For the principal, inducing effort yields an expected payoff equal to

Had the principal decided to let the agent exert no effort, *e *= 0, he would make a zero payment to the agent whatever the realization of output. In this scenario, the principal would instead obtain a payoff equal to

Inducing effort is thus optimal from the principal’s point of view when *V*_{1} ≥ *V _{0}*, i.e., , or to put it differently, when

where .

The left-hand side of (4.9) captures the gain of increasing effort from *e *= 0 to *e* = 1. This gain comes from the fact that the return *S*¯, which is greater than * S*, arises more often when a positive effort is exerted. The right-hand side of (4.9) is instead the first-best cost of inducing the agent’s acceptance when he exerts a positive effort.

Denoting the benefit of inducing a strictly positive effort level by *B *= ΔπΔ*S*, the first-best outcome calls for *e*^{∗} = 1 if and only if *B** *≥ *h(ψ)*, as shown in figure 4.2.

Figure 4.2: First-Best Level of Effort

Source: Laffont Jean-Jacques, Martimort David (2002), *The Theory of Incentives: The Principal-Agent Model*, Princeton University Press.