# Moral Hazard: The Trade-Off Between Insurance and Efficiency

Let us now turn to the second source of inefficiency in a moral hazard context— the agent’s risk aversion. When the agent is risk-averse, the principal’s program is written as:

It is not obvious that (P) is a concave program for which the first-order Kuhn and Tucker conditions are necessary and sufficient. The reason for this possible lack of concavity is that the concave function u(·) appears on both sides of the incentive compatibility constraint (4.3). However, the following change of variables shows  that  concavity  of  the  program  is  ensured.  Let  us  define  u¯ = u(t¯)  and  u = u(t), or equivalently let t¯ = h(u¯) and t = h(u). These new variables are the levels of ex post utility obtained by the agent in both states of nature. The set of incentive feasible contracts can now be described by two linear constraints:

which replaces the incentive constraint (4.3), and also

which replaces the participation constraint (4.4).

Program (P ) can now be replaced by the new program (P’ ), which writes as

Note  that  the  principal’s  objective  function  is  now  strictly  concave  in  (u¯, u) because h(·) is strictly convex. The constraints are now linear and the interior of the constrained set is obviously nonempty, and therefore (P’ ) is a concave problem, with the Kuhn and Tucker conditions being sufficient and necessary for character- izing optimality.

### 1. Optimal Transfers

Letting λ and μ be the non-negative multipliers associated respectively with the constraints (4.24) and (4.25), the first-order conditions of this program can be expressed as

where t¯SB  and tSB  are the second-best optimal transfers. Rearranging terms, we get

The four variables are simultaneously obtained as the solutions to the system of four equations (4.24), (4.25), (4.28), and (4.29). Multiplying (4.28) by π1 and (4.29) by 1 − π1, and then adding those two modified equations, we obtain

Hence, the participation constraint (4.25) is necessarily binding.

Using (4.30) and (4.28), we also obtain

where  λ must  also  be  strictly  positive.  Indeed,  from  (4.24)  we  have   implying  that  the  right-hand  side  of  (4.31)  is  strictly positive since u < 0. Using that (4.24) and (4.25) are both binding, we can immediately  obtain  the  values  of  u(t¯SB)  and  u(tSB)  by  solving  a  system  of  two equations with two unknowns.

Note that the risk-averse agent does not receive full insurance anymore. This result must be contrasted with what we have seen under complete information in section 4.1.3. Indeed, with full insurance, the incentive compatibility constraint (4.3) can no longer be satisfied. Inducing effort requires the agent to bear some risk. Proposition 4.4 provides a summary.

Proposition 4.4: When the agent is strictly risk-averse, the optimal con- tract that induces effort makes both the agent’s participation and incentive constraints binding. This contract does not provide full insurance. More- over, second-best transfers are given by

and

It is also worth noting that the agent receives more than the complete infor- mation  transfer  when  a  high  output  is  realized,  t¯SB   > h(ψ).  When  a  low  output is realized, the agent instead receives less than the complete information transfer, tSB  < h(ψ).  A  risk  premium  must  be  paid  to  the  risk-averse  agent  to  induce  his participation  since  he  now  incurs  a  risk  by  the  fact  that  tSB  < t¯SB.  Indeed,  when (4.4) is binding we have

where the right-hand side inequality in (4.34) follows from Jensen’s inequality. The expected  payment   given  by  the  principal  is  thus  larger  than the  first-best  cost  CFB  = h(ψ),  which  is  incurred  by  the  principal  when  effort  is observable (as we have seen in section 4.1.3).

### 2. The Optimal Second-Best Effort

Let us now turn to the question of the second-best optimality of inducing a high effort, from the principal’s point of view. The second-best cost CSB of inducing effort under moral hazard is the expected payment made to the agent .  Using  (4.32)  and  (4.33),  this  cost  is  rewritten  as

The benefit of inducing effort is still B = ΔπΔS, and a positive effort e = 1 is the optimal choice of the principal whenever

Figure 4.4: Second-Best Level of Effort with Moral Hazard and Risk Aversion

With h(·) being strictly convex, Jensen’s inequality implies that the right-hand side of (4.36) is strictly greater than the first-best cost of implementing effort CFB = h(ψ). Therefore, inducing a higher effort occurs less often with moral hazard than when effort is observable. Figure 4.4 represents this phenomenon graphically.

For  B  belonging  to  the  interval  [CFB, CSB],  the  second-best  level  of  effort  is zero and is thus strictly below its first-best value. There is now an under-provision of effort because of moral hazard and risk aversion.

Proposition 4.5: With moral hazard and risk aversion, there is a trade-off between inducing effort and providing insurance to the agent. In a model with two possible levels of effort, the principal induces a positive effort from the agent less often than when effort is observable.

In order to get further insights on the dependency of the second-best cost of implementation on various parameters, we can specialize the model and assume that  , where r > 0 and .   Equivalently,    for  . For this quadratic expression of h(·), we have

where  E(·)  denotes  the  expectation  operator  and  u˜SB   is  the  random  utility  level that the agent gets in the different states of nature. More precisely, we have

taking into account that (4.25) is binding at the optimal contract .  Moreover,

and

The  agent  receives  a  premium  of (in utility terms) when production is  high  and  suffers  a  penalty (in utility terms) when production is low. This risky payoff has a variance var

Gathering everything, we finally have the following expression of CSB:

The first-best cost of implementing effort with such a utility function would instead be

Hence, the agency cost AC, which is also the principal’s loss between his first-best and second-best expected profit when he implements a positive effort, can be defined as

This agency cost increases with r , a measure of the agent’s degree of risk aversion,  with  ψ,  the  cost  of  one  unit  of  effort,  and  with ,  which  is  a measure of the informational problem for the principal. Everything else being kept equal, it becomes harder and less often optimal for the principal to induce a high effort as η increases. When π1 is close to 1/2 , η is larger. In this case, the variance of the measured performance q˜ is the greatest possible one: the observable output is a rather poor indicator of the agent’s effort. Therefore, more noisy measures of the agent’s effort will more often call for inducing a low effort at the optimum and for a fixed wage without any incentives being provided. Finally, note that y is also  larger  when Δπ  is  small,  i.e.,  when  the  difference  in  utilities necessary to incentivize the agent gets larger. More generally, the agency cost’s dependence on η shows that the informational content of the observable output plays a crucial role in the design of the optimal contract. This is a general theme of agency theory that we will cover more extensively in section 4.6.

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