Nonverifiability: Risk Aversion

The previous sections have discussed how various sorts of mechanisms may help to implement the first-best allocation rule when both the principal and the agent are risk neutral. The objective of this section is to discuss the potential of those mechanisms when either the principal or the agent is risk averse.

1. Risk-Averse Agent 

When the agent is risk averse, signing no contract ex ante remains optimal if all the bargaining power is still left to the principal ex post. Indeed, ex post take-it-or- leave-it offers impose no risk on the agent, who is always maintained at the zero utility level.

An incentive contract performs badly, because there is now a trade-off between insurance and incentive compatibility. However, the Nash (and subgame-perfect) implementation performs rather well. It is straightforward to extend sections 6.3 and 6.4 to the case of a risk-averse agent to show that Nash implementation allows one to implement the first-best outcome with full insurance for the agent.14

2. Risk-Averse Principal 

Clearly, signing no contract at the ex ante stage can no longer be optimal even if the principal has all of the bargaining power ex post. Indeed, ex post take-it-or- leave-it offers impose some risk on the principal from an ex ante point of view. However,  an  incentive  contract  can  still  implement  the  first-best, as we have seen in section 2.11.2. Making the agent residual claimant for the hierarchy’s profit is again optimal in the case of nonverifiability. Of course, we still have to be sure that nonresponsiveness does not occur.

Finally, unique Nash implementation of the first-best outcome can also be obtained using a game form similar to that in figure 6.5. In our standard example, efficiency  still  requires  that  q  and  q¯*   are  produced,  such  that  S(q) = θ  and .  Providing  insurance  to  the  principal  also  requests  that  the  principal gets the same payoff in each state of nature:

Finally, the agent’s ex ante participation constraint should also be binding:

Since trade is more valuable in state θ than in state θ¯, we have .  Solving  (6.27)  and  (6.28)  therefore  yields .

Figure 6.10: Unique Nash Implementation with Risk Aversion

In  figure  6.10,  we  have  represented  the  out  of  equilibrium  contracts and ,  which  uniquely  implement  the  first-best  allocation  rule.  Proceeding as in Section 6.3, these contracts must again satisfy the following constraints:

We let the reader check that the set E (crossed area) (resp. F (dotted area)) of possible values of  satisfying the constraints (6.29) to (6.31) (resp. (6.32) to (6.34)) can be represented as in figure 6.10. In particular, since the areas E and F  now have a nonempty intersection, can be chosen equal to .  Risk  aversion  on  the  principal’s  side  tends  to  simplify  the  mechanisms that can be used to implement the first-best.

The above construction may appear somewhat pointless, because simple incentive contracts {t(q)} also achieve the first-best. However, it can be easily extended to common value environments where S(·) is also a function of θ, S(q, θ), and when incentive contracts may run into the difficulties of nonresponsiveness (see sections 2.10.2 and 2.11.1).

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

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