# Moral Hazard: Redistribution and Moral Hazard

In chapter 3, we have already seen how the conflict between incentive compat- ibility and budget balance leads to the under-provision of output in an adverse selection model. The same qualitative result still holds in a moral hazard environ- ment. Expected volume of trade may be reduced by moral hazard. To illustrate this point, we consider a simple model of redistribution and moral hazard. There is a unit mass population of agents who are all ex ante identical and have a utility function  U  = u(t) − ψ(e),  where  u(·) (u > 0, u < 0) is  defined  over  monetary gains and efeg is the disutility of effort. Each of those agents exerts an effort e in {0, 1},  and  may  be  successful  or  not  in  producing  output.  When  successful  (resp. unsuccessful), i.e., with probability lfeg (resp. 1 − π(e)), the return of this effort is q¯ (resp. q < q¯). The agents are all ex ante identical, so the government maximizes an objective function V = U , which corresponds to the utility of a representative agent.

A  redistributive  scheme  is  a  pair  of  transfers that  depend  on  whether the agent is successful or not. To be incentive feasible, such a scheme must satisfy the following budget constraint:

as well as the usual incentive compatibility constraint,

Note that (5.91) means that the budget is balanced in expectation over the whole  population  of  agents.  Indeed,  by  the  Law  of  Large  Numbers, π1  can  also be viewed as the fraction of successful agents in society.

When effort is verifiable, the government solves the following problem if it wants to implement a high level of effort:

Let us denote the multiplier of the budget constraint (5.91) by μ. The nec- essary and sufficient Kuhn and Tucker optimality conditions with respect to t¯ and t then lead to

The complete information optimal redistributive scheme calls for complete insurance and the constant transfer received by each agent in both states of nature is

i.e., it is equal to the average output. The optimal redistributive scheme amounts to a perfect insurance system. Taxation provides social insurance by transferring income from those agents who have been lucky to those who have been unlucky. Let us now consider the case where effort is nonobservable by the govern- ment. If the government wants to induce zero effort, it still relies on the complete insurance scheme  above,  and  the representative  agent  gets  an  expected utility

If the government wants to induce a high effort, it instead solves the following problem:

Denoting the respective multipliers of those two constraints by μ and λ, the first-order  conditions  for  optimality  with  respect  to  t¯ and  t  can  be  written  respec- tively as

and

Dividing  (5.95)  by   and  (5.96)  by  u‘(tSB)  and  summing,  we  obtain that μ is  strictly  positive  since . Therefore, the budget constraint  is  binding.  Similarly,  we  also  find  that 0 because  is  necessary  to  satisfy  the  incentive  compatibility  constraint (5.92)  and  u(·) is  concave.  Hence,  this  latter  constraint  is  also  binding  and  both t¯SB   and  tSB  are  obtained  as  solutions  to  the  following  nonlinear  system:

and

Under moral hazard complete redistribution is not achieved. Furthermore, it is socially optimal to induce a high effort when

Because uf·g is strictly concave and  Jensen’s inequality implies that the left-hand side of (5.99) is strictly lower than

Hence, the second-best rule (5.99) is more stringent than the first-best rule, which calls for a positive effort if and only if

A high effort is implemented less often under moral hazard because the ben-efit of doing so is lower. The reader will have recognized the similarity of this section with section 4.8.5. Indeed, the redistributive scheme analyzed above is akin to the insurance contract that would be offered by a competitive sector.

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