# 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  (u’i  > 0, u”i  < 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  h”0  > 0).  Consider  the  case where  u1(·) is  a  concave  transformation  of  u0(·),  i.e.,  u1(·) = gu0(·) 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

and

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

and

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.25   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  gf·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  u(t − ψ(e)), where  u(·) 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  c(π, x) 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

and

The principal’s problem becomes:

It is important to note that these constraints depend in a separable way on, first, the average transfer (π1t¯ + (1 − π1(t) received by the agent and, second, the risk  created  by  these  transfers  (t¯ − t).  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:

and

By inducing effort, the principal therefore gets

If  he  does  not  induce  effort,  the  principal  can  offer  t¯ = t = 0.  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 ≥ e.

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.

Remark 1: The fact that the principal can play independently both on the agent’s expected transfer to ensure his participation and on the power of incentives to induce him to exert effort is, of course, a direct consequence of the agent having CARA preferences that do not exhibit wealth effects. The agent’s average wealth level does not affect incentives.

Remark 2: The second direct consequence of this model is that the power of incentives and the decision to induce effort would be the same if the agent’s certainty equivalent from not working with the prin- cipal was w instead of zero as we have assumed above. The solution to this new problem is directly translated from the solution in the case where w  = 0, and we obtain . This translation result will be particularly useful in section 9.5.2.

Remark  3:  When r  is small enough, we have   . The model is then akin to assuming that the agent has mean-variance pref-erences  over  the  monetary  payoff  t˜ − ψ.  In  this more general case, we can solve (5.86) explicitly for ΔtSB, and we find that  (provided  that  1 − π1 − π0  >  0  and  that  ψ is  small  enough  to ensure existence of the solution)

which is approximately equal to  when r is small.

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