# Adverse Selection: Random Participation Constraint

The previous section has shown how a deterministic but type-dependent participa- tion constraint could perturb the standard results on the optimal rent extraction- efficiency trade-off. We now perturb the agent’s participation constraint in another direction, by allowing some randomness in the decision to participate. Instead of the agent’s reservation utility being perfectly known, let us consider a risk-neutral agent with a random participation constraint,

and

We assume that s˜ is drawn from the interval , centered at zero with a cumulative  distribution  function  G(ε).  We  denote  by  g(ε) = G‘(ε) the  density  of this random variable.

The motivation for such a stochastic specification of the reservation utility levels is that the agent might have some random opportunity cost of accepting the contract proposed by the principal and that this cost is already revealed to the agent at the time of contracting, even if the principal has no ability to screen this information.14 Alternatively, the agent may be facing a whole set of possible trad- ing opportunities outside of his relationship with a given principal. Those trading opportunities yield a random profit ε˜ to the agent. Implicit here is the idea that the principal competes with other principals who have unknown characteristics. However, this competition is still modeled as an exogenous black box.

In this model, the incentive constraints for both types remain, as usual,

and

A deterministic incentive-feasible contract is accepted by both types if and only if (3.69) and (3.70) both hold. Acceptance is now a random event. A priori, both types only accept the contract with some probability, which is G(U) for  the θ-type  and  G(U¯) for  the  θ¯-type.  To  simplify  the  analysis,  we  will  assume that  ε¯ is  small  with  respect  to .  This  assumption  will  imply  that  G(U ) = 1, and only the inefficient agent might not participate with some strictly positive probability  1 − G(U¯).  The  optimal  contract  must  solve  the  program  below:

On  top  of  the  usual  concavity  of  S(·),  assuming  that is  increasing with ε ensures the quasi-concavity of this program. Its solution is then described in the next proposition. It is indexed by a superscript R that means random partic- ipation.

Proposition 3.3: Assume random participation constraints, but also that ε¯ is  small  enough.  Then  the  optimal  contract  entails:

• The incentive constraint of the efficient agent (3.71) is binding.
• The rent    and  the  output    of  an  inefficient  agent  are  determined together as the solutions to

and

Two important remarks should be made at this point. First, since an inefficient agent trades with the principal with a probability of less than one , the principal finds it relatively more likely that he will face an efficient agent on the condition that trade is being carried on. Hence, the principal is more willing to distort the inefficient agent’s output downward to reduce the relatively high expected cost of the efficient agent’s information rent. Indeed,    defined by (3.73) is more distorted than the usual second-best distortion   obtained with an exogenously given zero participation constraint.

Second,  the  principal  chooses  a  level  of  the  inefficient  agent’s  rent    that trades off the marginal gain of inducing slightly more participation by this type against the marginal cost of this extra participation. The marginal gain of increasing the  rent  by    is  precisely  the  net  total  surplus   times  the increase in probability that the inefficient agent chooses to participate, namely .  The  marginal  cost  takes  into  account  the  fact  that  this  extra rent  has to be given to all participating agents, i.e., both the efficient one who trades with probability of one and also the inefficient one who contracts with a probability of only less than one. The cost is thus . The marginal benefit is equal to the marginal cost when (3.74) holds.

It  is  interesting  to  note  that  the  output  q¯R    converges  towards     defined in  (2.28)  as  ε¯  goes  to  zero.  In  this  case  the  random  participation  constraint almost becomes the usual deterministic participation constraint with zero reserva- tion value.

Finally,  the  generalized  monotone  hazard  rate  property,  namely  increasing  in  ε,  guarantees  that    is  strictly  positive  when  To induce a relatively more likely participation, the principal must a priori give to the inefficient agent a strictly positive rent. Lastly, the probability that the ineffi- cient type participates is strictly lower than one when , where is the second-best optimal output with a deterministic participation  constraint.

Rochet and Stole (2000) provided a complete analysis of a model with random participation constraints and a continuum of types. In such a setting, they also looked at the interesting case of competition among principals.

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