As we have already mentioned, in most of our discussion dealing with the case of adverse selection, we consider the case of contracts offered at the interim stage, i.e., once the agent already knows his type. However, sometimes the principal and the agent can contract at the ex ante stage, i.e., before the agent discovers his type. For instance, the contours of the firm may be designed before the agent receives any piece of information on his productivity. In this section, we characterize the optimal contract for this alternative timing under various assumptions about the risk aversion of the two players.
1. Risk Neutrality
Suppose that, instead of contracting after the agent has discovered l, the principal and the agent meet and contract ex ante, i.e., before the agent obtains infor- mation. If the agent is risk neutral, his ex ante participation constraint is now written as
This ex ante participation constraint replaces the two interim participation con- straints (2.23) and (2.24) in problem jP k. What matters now to ensure participation is that the agent’s expected information rent remains non-negative.
From (2.20), we see that the principal’s objective function is decreasing in the agent’s expected information rent. Ideally, the principal wants to impose a zero expected rent to the agent and have (2.61) be binding.
Moreover, the principal must structure the rents U and U¯ to ensure that the wedge between those two levels is such that the incentive constraints (2.21) and (2.22) remain satisfied. An example of such a rent distribution that is both incentive compatible and satisfies the ex ante participation constraint with an equality is
With such a rent distribution, the optimal contract implements the first-best out- puts without cost from the principal’s point of view as long as the first-best is monotonic as requested by the implementability condition. This may not be the case, for instance, when the nonresponsiveness property holds, as in section 2.10.2. In that case, even under ex ante contracting and risk neutrality some inefficiency still arises.
In the contract defined by (2.62), the agent is rewarded when he is efficient and punished when he turns out to be inefficient. There must be some risk in the distribution of information rents to induce information revelation, but this risk is costless for the principal because of the agent’s risk neutrality. However, to be feasible, such an ex ante contract requires a strong ability of the court of law to enforce contracts that could possibly lead to a negative payoff when a bad state of nature realizes.23
Proposition 2.4: When the agent is risk neutral and contracting takes place ex ante, the optimal incentive contract implements the first-best outcome.
Remark: The principal has in fact much more leeway in structuring the rents U and U¯ in such a way that the incentive constraints (2.21) and (2.22) hold and the ex ante participation constraint (2.61) is an equality. Consider the following contracts where t∗ = , with T ∗ being a lump-sum payment to be defined below. This contract is incentive compatible since
by definition of q∗, and
by definition of q¯*.
Note that the incentive compatibility constraints are now strict inequalities. Moreover, the fixed-fee T ∗ can be used to satisfy the agent’s ex ante participation constraint with an equality by choosing
This implementation of the first-best outcome amounts to having the principal selling the benefit of the relationship to the risk-neutral agent for a fixed up-front payment T ∗. The agent benefits from the full value of the good and trades off the value of any production against its cost just as if he was an efficiency maximizer. We will say that the agent is residual claimant for the firm’s profit.
Harris and Raviv (1979) proposed a theory of the firm as a mech- anism allocating resources at the ex ante stage. The first best allo-cation remains implementable when the firm has a strong ability to enforce contracts.
2. Risk Aversion
A Risk-Averse Agent
The previous section has shown us that the implementation of the first-best is feasible with risk neutrality. The counterpart of this implementation is that the agent is subject to a significant amount of risk. Such a risk is obviously costly if the agent is risk-averse.
Consider now a risk-averse agent with a Von Neumann–Morgenstern utility function uj·k defined on his monetary gains t − θq, such that u‘ > 0, u” < 0 and u(0) = 0. We suppose, as in section 2.11.1, that the contract between the prin-cipal and the agent is signed before the agent discovers his type.25 The incentive constraints are unchanged but the agent’s ex ante participation constraint is now written as
As usual, we guess a solution such that (2.22) is slack at the optimum, and we let the reader check this ex post. The principal’s program reduces now to
We summarize the solution in the next proposition.26
Proposition 2.5: When the agent is risk-averse and contracting takes place ex ante, the optimal menu of contracts entails:
- No output distortion for the efficient type qSB = q∗. A downward output distortion for the inefficient type q¯SB < q¯*, with
- Both (2.21) and (2.65) are the only binding constraints. The efficient (resp. inefficient) type gets a strictly positive (resp. negative) ex post information rent, U SB > 0 > U¯SB
With risk aversion, the principal can no longer costlessly structure the agent’s information rents to ensure the efficient type’s incentive compatibility constraint, contrary to section 2.11.1. Creating a wedge between U and U¯ to satisfy (2.21) makes the risk-averse agent bear some risk. To guarantee the participation of the risk-averse agent, the principal must now pay a risk premium. Reducing this pre- mium calls for a downward reduction in the inefficient type’s output so that the risk borne by the agent is lower. As expected, the agent’s risk aversion leads the principal to weaken the incentives.
For the constant absolute risk aversion utility function (2.66) leads to a closed-form expression for output:
Also, the efficient agent’s ex post utility writes as
and the inefficient agent’s ex post utility is
Incentives (and outputs) decrease with risk aversion. If risk aversion goes to zero (r → 0), q¯SB converges towards the first-best value q¯*. Indeed, we know from Section 2.11.1 that, with risk neutrality and an ex ante participation constraint, the optimal contract induces an efficient outcome. Moreover, the utility levels of both types converge toward those described in (2.62).
When the agent becomes infinitely risk averse, everything happens as if he had an ex post individual rationality constraint for the worst state of the world given by (2.24). In the limit, the inefficient agent’s output q¯SB and the utility levels U SB and U¯SB all converge toward the same solution as in proposition 2.1. So, the model of section 2.1 can also be interpreted as a model with ex ante contracting but with an infinitely risk-averse agent at the zero utility level.
Salanié (1990) analyzed the case of a continuum of types. Pooling for the least efficient types always occurs when risk aversion is enough. Laffont and Rochet (1998) showed a similar phenomenon with interim participation constraints when a regulator (the principal) max- imizes ex ante social welfare and deals with a risk-averse regulated firm (the agent).
A Risk-Averse Principal
Consider now a risk-averse principal with a Von Neumann–Morgenstern utility function v(·) defined on his monetary gains from trade S(q) − t such that v’ > 0, v” < 0 and v(0) = 0. Again, the contract between the principal and the risk-neutral agent is signed before the agent knows his type.
In this context, the first-best contract obviously calls for the first-best output q∗ and q¯* being produced. It also calls for the principal to be fully insured between both states of nature and for the agent’s ex ante participation constraint to be binding. This leads us to the following two conditions that must be satisfied by the agent’s rents U ∗ and U¯*:
Solving this system of two equations with two unknowns (U ∗, U¯*) yields
Note that the first-best profile of information rents satisfies both types’ incen-tive compatibility constraints since
(from the definition of q∗) and
(from the definition of q¯*). Hence, the profile of rents (U ∗, U¯*) is incentive com- patible and the first-best allocation is easily implemented in this framework. We can thus generalize proposition 2.4 as follows:
Proposition 2.6: When the principal is risk-averse over the monetary gains S(q) − t, the agent is risk-neutral, and contracting takes place ex ante, the optimal incentive contract implements the first-best outcome.
It is interesting to note that U ∗ and U¯* obtained in (2.72) and (2.73) are also the levels of rent obtained in (2.63) and (2.64). Indeed, the lump-sum payment , which allows the principal to make the risk-neutral agent residual claimant for the hierarchy’s profit, also provides full insurance to the principal. By making the risk-neutral agent the residual claimant for the value of trade, ex ante contracting allows the risk-averse principal to get full insurance and implement the first-best outcome despite the informational problem.
Of course this result does not hold anymore if the agent’s interim participation constraints must be satisfied. In this case, we still guess a solution such that (2.22) is slack at the optimum. The principal’s program now reduces to
Inserting the values of U and U¯ that were obtained from the binding con- straints in (2.21) and (2.24) into the principal’s objective function and optimizing with respect to outputs leads to qSB = q∗, i.e., no distortion for the efficient type, just as in the case of risk neutrality and a downward distortion of the inefficient type’s output q¯SB < q¯* given by
where are the principal’s payoffs in both states of nature. We can check that V¯SB < V SB since from the definition of q∗. In particular, we observe that the distortion in the right-hand side of (2.76) is always lower than , its value with a risk-neutral principal. The intuition is straightforward. By increasing q¯ above its value with risk neutrality, the risk-averse principal reduces the difference between V SB and V¯SB. This gives the principal some insurance and increases his ex ante payoff.
For example, if , (2.76) becomes
If r = 0, we get back the distortion obtained in section 2.6 with a risk-neutral principal and interim participation constraints for the agent. Since V¯SB < V SB, we observe that the first-best is implemented when r goes to infinity. In the limit, the infinitely risk-averse principal is only interested in the inefficient state of nature for which he wants to maximize the surplus, since there is no rent for the inefficient agent. Moreover, giving a rent to the efficient agent is now without cost for the principal.
Risk aversion on the side of the principal is quite natural in some contexts. A local regulator with a limited budget or a specialized bank dealing with relatively correlated projects may be insufficiently diversified to become completely risk neutral. See Lewis and Sappington (1995) for an application to the regulation of public utilities.
Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.