# Moral Hazard: Nonseparability of the Utility Function

The separability in transfer and effort of the agent’s utility function simplifies the principal-agent theory with moral hazard by ensuring that the agent’s participation constraint is binding. However, it neglects one significant incentive effect, namely that one way to provide incentives is to make the agents richer by decreasing their marginal disutility of effort. The interaction between consumption and effort is clearly important in development economics when one wants to study agrarian contracts. For example, richer agents may be able to have a better diet, which increases their resistance. Then, the optimal compensation may entail a fixed payment whose objective is to bring the agent to a utility level where the marginal disutility of effort is lower. This type of compensation may require leaving him a rent which has a different (even though related) motivation than the limited liability rent exhibited in chapter 4.

In section 5.3.1, we show graphically how nonseparability calls for relaxing the participation constraint. Nevertheless, section 5.3.2 provides an example without wealth effect where the nonseparability is not exploited by the principal to relax the participation constraint.

### 1. Nonbinding Participation Constraint

Let us now assume that the agent has a general utility function defined over transfers and effort, namely U  = u(t, e). Contrary to the standard framework used so far, we no longer postulate a priori the separability between transfer and effort. Effort  can  still  take  either  of  two  values  e  in  {0, 1}; to simplify notations, let us denote u1(t) = u(t, 1) and u0(t) = u(t, 0). Because effort is costly, we obviously have u1(t) < u0(t) for all t. Moreover, for i in {0, 1}, ui(.) is still increasing and concave in t for all t (ui‘ > 0, ui” < 0).

For what follows, it is also interesting to denote the inverse function of u0(-) by h0(-), which is increasing and convex (h’0 > 0 and h0” > 0). Consider the case where u1(.) is a concave transformation of u0(-), i.e., u1(.) = g o u0(-) with g(-) increasing and concave. Intuitively, this means that exerting effort makes the agent more averse to monetary lotteries.

In this framework, incentive and participation constraints are written respec- tively as Extending the methodology of chapter 4, we now introduce the following change  of  variables: .  With  these  new  variables,  the incentive and participation constraints (5.76) and (5.77) are written respectively as The risk-neutral principal’s problem is thus written as  Figure 5.9: Solution in the Case of Separability

The  fact  that  g(·) is  concave  ensures  that  the  constrained  set  C  of  incentive feasible  contracts is  a  nonempty  convex  set.   Since  the  principal’s  objec- tive function is strictly concave, the first-order Kuhn and Tucker conditions will again be necessary and sufficient to characterize the solution to this problem.

Instead of proposing a general resolution to this problem, we restrict our- selves to a graphical description of the possible features of the solution for general functions u1(·) and u0(·) to satisfy the above properties. As a benchmark, it is use- ful to represent graphically the usual case of separability where, in fact, we have u1(t) = u0(t) − ψ for all t. In this case, we have immediately g(u) = u − ψ for all u,  and  u1(·) is  simply  a  linear  transformation  of  u0(·).

In figure 5.9, we have represented the set C of incentive feasible contracts in the case of a separable utility function. It is a dieder turned downward and lying strictly above the 45◦ line. The principal’s indifference curve VSB is inversely U-shaped. It is graphically obvious that the optimal contract must therefore be on the extremal point ASB of the dieder. We easily recover our analytical result of section 4.4. The risk-averse agent receives less than full insurance at the opti- mum, and the agent’s participation constraint is also binding. Figure 5.10: Solution in the Case of Nonseparability

Let us now turn to the case of nonseparability, which is the focus of this section. Figure 5.10 represents the set C of incentive feasible contracts and the possible second-best optimal contract A.

With a nonseparability, the binding incentive constraint (5.78) defines a locus of contracts that is no longer a straight line but a strictly convex curve in the plan Similarly, the binding participation constraint (5.79) also defines a convex locus. The set C of incentive feasible contracts is again strictly convex, with an extremal point A still obtained when both constraints are binding. However, the strict convexity of C now leaves some scope for the optimal contract to be at point BSB, where only the incentive constraint is binding. In this case, the best way to solve the incentive problem is to give up a strictly positive ex ante rent to the agent. This case is more likely to take place when the IC constraint defines a very  convex  curve,  i.e.,  when  g(·) is  very  concave.  This  occurs  when  the  agent  is much more risk averse when he exerts a positive effort than when he does not. In this case, offering a risky lottery to induce effort and keep the agent’s expected utility relatively low is costly for the principal. The principal prefers to raise the agent’s expected utility to move toward areas where a risky lottery is much less costly.

Remark: Before closing this section, let us notice that we have already presented in chapter 4 a simple example of a contracting environment where the agent’s participation constraint is slack at the optimum, that is when the risk-neutral agent is protected by limited liability.

### 2. A Specific Model with No Wealth Effect

Sometimes, even without any separability between transfer and effort in the agent’s utility function, the agent receives zero ex ante rent. For example, suppose that the agent’s cost of effort is counted in monetary terms. The agent’s utility function is no  longer  separable  between  income  and  effort,  and  it  is  written  as  uft − efegg, where  uf·g is  again  increasing  and  concave.  With  our  usual  notations,  the  moral hazard incentive constraint is now written as The participation constraint is now where u(0) is the agent’s reservation utility that he obtains when he refuses the contract.

Let us also assume that the agent has a constant risk aversion, namely the constant absolute risk aversion (CARA) utility function u(x) = − exp(−rx). When facing a binary lottery yielding wealths a and b with respective probabilities π and 1 − π,  this  agent  obtains  a  certainty  equivalent  of  income,  which  is  defined  as where is a risk premium.

One  can  check  that  cfli xg is  increasing  with  x  for  all  x ≥ 0.

Using this formulation based on certainty equivalents, we can now rewrite (5.80) and (5.81) as The principal’s problem becomes: It is important to note that these constraints depend in a separable way on, first, the average transfer received by the agent and, second, the risk  created  by  these  transfers .  More  precisely,  the  principal  can  ensure that the participation constraint (5.84) is binding by reducing the agent’s average transfer without perturbing the power of the incentive contract, i.e., still keeping the incentive constraint (5.83) satisfied. Indeed, this latter constraint can also be written as where is the incentive power of the contract. We leave it to the reader to check that this incentive constraint must be binding. The second-best power of incentives ΔtSB is thus the unique positive solution to Given that the participation constraint (5.84) is also binding, the optimal second-best transfers are defined by: By inducing effort, the principal therefore gets If  he  does  not  induce  effort,  the  principal  can  offer .  The  principal thereby gets an expected payoff . Hence, the principal prefers to induce effort when . Under complete information, the principal induces a first-best effort by offering a constant wage t¯ = t = ψ, and the optimal effort is positive when ΔπΔS ≥ ψ.

Therefore, as in a model with separability between consumption and effort, the principal induces a positive effort less often than in the first-best world, because is  strictly  positive  when ΔtSB > 0.

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