# Mixed Models: Moral Hazard Followed by Adverse Selection

Sometimes an agent undertakes an initial nonverifiable investment or performs an effort before producing any output for the principal. For instance, the agent can choose a costly technology that affects the distribution of his marginal cost of production. At the time of choosing whether to incur the nonverifiable investment or not, the agent is still uninformed on what will be the realization of his efficiency parameter ex post. If this efficiency parameter is privately known, we are now in a framework where moral hazard takes place before adverse selection.

### 1. The Model

We assume that the risk-neutral agent can change the stochastic nature of the production process by exerting a costly effort e, which belongs to {0, 1}. The disutility of effort is, as usual, normalized so that Ψ(0) = 0 and Ψ(1) = Ψ. When exerting effort e, the agent induces a distribution of the productivity parameter θ on 0 = {Q, Q}. With probability r(e) (resp. 1 — r(e)), the agent will be efficient (resp. inefficient); for the sake of simplicity, we denote r(1) = rq, r(0) = r0, and Δv = v1 — v0. To express the fact that exerting effort is valuable, we assume that effort increases the probability that the agent is efficient, i.e., Δv > 0.

If the efficiency parameter is Q, the agent who produces an output q and receives a transfer t from the principal gets a utility U = t — θq — ψ(e). The principal has the usual utility function V = S(q) — t.

Through the contract he offers to the agent, the principal wants to control both the agent’s effort and the agent’s incentives to tell the truth on the state of nature that is realized ex post. The timing of contracting is described in figure 7.6.

In this mixed environment, the contract {t(q)} must not only induce effort if the principal finds it sufficiently valuable, but it must also induce information reve­lation. Applying the revelation principle, there is no loss of generality in restricting the principal to offer a direct revelation mechanism Through this contract, the agent will be induced to reveal his private information on the state of nature θ.

Figure 7.6: Timing of Contracting with Moral Hazard Followed by Adverse Selection

Of course, this contract is signed before the realization of the state of nature. Therefore, we are in a case of ex ante contracting similar to, albeit more complex than, those analyzed in section 2.11 and chapter 4.

### 2. No Limited Liability

To explain the new issues that arise with this type of mixed model, we start by analyzing the case of risk neutrality. We already know that the agent’s risk neutrality calls for no allocative distortion either under pure moral hazard or under pure adverse selection. With pure adverse selection, as well as with pure moral hazard, the first-best outcome can be implemented by letting the agent be a residual claimant for the hierarchy’s profit. One may wonder whether adding those two informational problems leads to any significant new problem.

Let us start by describing the first-best outcome. The first-best outputs equalize the  marginal  benefit  and  the  marginal  cost  of  production,  so  that  S(q) = θ and .  Denoting  the  first-best  productive  surplus  in  each  state  of  nature  by, respectively, , we  find  that  inducing  and effort is socially optimal whenever:

We will assume that this last condition holds in what follows.

Let us now look at the case of moral hazard and adverse selection. Making the agent residual claimant still helps in this framework. Consider the following transfers , where the constant T will be fixed in the following.

First,  we  claim  that  the  contract  induces  information  reve-lation  by  both  types.  Indeed,  by  the  definition  of  q,  and  by  the  definition  of  q¯*.  Second,  the  contract also induces effort. The agent’s expected payoff from exerting effort is .  It  is  greater  than  his  expected  payoff  from  not  exerting effort, which is  when (7.67) holds. Finally, the principal fixes the lump-sum payment T to reap all ex ante gains from trade with the agent, namely .  Hence,  we  can  state  proposition  7.3.

Proposition 7.3: When moral hazard takes place before adverse selection and the agent is risk neutral, the first-best outcome can still be achieved by making the agent residual claimant for the hierarchy’s profit.

Note also that the contract, that makes the agent residual claimant for the hierarchy’s  profit  also  ensures  the  principal  against  any  risk,  because  S(q) − t = S(q¯*) − t¯*  = T.  This  contract  also  works  perfectly  well  if  the  principal  is  risk averse.

### 3. Limited Liability and Output Inefficiency

Introducing the agent’s risk aversion or protecting the risk-neutral agent with lim- ited liability makes the implementation of the first-best outcome obtained above no longer possible. For clarification, let us assume that the agent is still risk neutral but is now protected by limited liability. Assuming that he has no asset to start with, the limited liability constraints in both states of nature are written as

Moreover, inducing information revelation at date t = 4 through a direct rev-elation mechanism requires that we satisfy the following adverse selection incentive compatibility constraints:

In  our  mixed  environment,  the  rents  U  and  U¯ must  also  serve  to  induce effort. To induce effort as under complete information, the following moral hazard incentive constraint must now be satisfied:

Finally, the agent accepts the contract at the ex ante stage when his ex ante participation constraint is satisfied

Still focusing on the case where it is always worth inducing the agent’s effort, the principal’s problem is

Depending on the respective importance of the moral hazard and adverse selection problems, the optimal contract may exhibit different properties. Some of the possible regimes of this optimal contract are summarized in the next proposition.

Proposition 7.4: With moral hazard followed by adverse selection, and with a risk-neutral agent protected by limited liability, the optimal contract has the following features:

• For (7.69) and (7-70) are  both binding.  Outputs  are given  by  qSB  = q  when  θ  realizes  and    when θ¯ realizes with

• For , (7.69), (7.70), and (7.72) are all binding. The first-best output q is still requested when θ realizes. Instead, pro- duction  is  distorted  downward  below  the  first-best  when θ¯ realizes.  We have   with

•  For   (7.69) and (7.72) are both binding. In both states of nature, the first-best outputs are implemented.

To understand how those different regimes emerge, first note that solving the pure adverse selection problem requires the creation of a differential between the  rents  U  and  U¯.  This  rent  differential  may  be  enough  to  induce  effort  when the corresponding disutility  is small. In this  case, moral hazard  has no impact and second-best distortions are completely driven by adverse selection. As effort becomes more costly, the pure adverse selection rent may no longer be enough to induce effort. When θ¯ realizes, the output must be distorted upward, with respect to the pure adverse selection distortion, to provide enough rent so that, ex ante, the agent wants to perform an effort. Finally, still increasing the disutility of effort, it is no longer worthwhile to make an output distortion. The principal prefers to maintain allocative efficiency in both states of nature and to sufficiently reward the agent in order to induce his effort. The design of the contract is then driven purely by moral hazard.

Remark: The reader will have recognized the similarity of the analysis above with that made in section 3.3 when we analyzed type-dependent reservation values in pure adverse selection models. Indeed, one can view the design of incentives in this mixed model as a two-step pro- cedure. The first step consists of the principal offering a reward when r is realized, that is large enough to induce effort provision at the ex ante stage. By doing so, the principal is committed to solving the moral hazard problem. Then, the second step consists of solving the adverse selection problem, that takes place ex post, and inducing infor- mation revelation. At this ex post stage, the principal may or may not be constrained by his previous commitment when he wants to extract the agent’s private information on the state of nature which has been realized.

To conclude, we stress that the main impact of the initial stage of moral hazard may be that it reduces allocative distortions and calls for greater information rents with respect to the case of pure adverse selection.

Proposition 7.5: Mixed models with moral hazard followed by adverse selection tend to be characterized by less allocative distortions and greater information rents than those arising in models with pure adverse selection.

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