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.