LA CALIDAD

So far,I assumed that the principal has full commitment power in that she can commit to any incentive compatible policy. In this section I relax this assumption and consider a case where the principal’s commitment power is limited. More precisely, I assume that she cannot commit to any policy with a negative continuation value. Precisely, at the beginning of each period, she needs to have a non-negative value in expectation. She still has within period commitment power and can fulfill any specified within period randomization. In other words, the limited commitment structure I impose does not rule out the possibility of having a negative realized value for some periods. For expositional convenience, I will call the model with full commitment power as the baseline model and denote the corresponding value function by VB afterwards. Clearly, when the expected social cost of the projects is

less then their economic benefits, i.e. µ ≤ v, an optimal policy for the baseline model is also optimal in this limited commitment environment. This stems from the fact that the continuation valuation of the principal never becomes negative in this case, i.e. VB(U)≥0

for everyU ∈[0, v].

On the contrary, when the expected cost is higher then the economic benefits, i.e.

µ > v, baseline policy can not be an optimal policy. Principal’s continuation value becomes negative with positive probability, as in the outcome of whitelisting. She cannot promise the maximal utility to agent in this limited commitment environment. As a result, the optimal policy differs from the one of baseline model. From now on, the analysis will be based on the caseµ > v.

For a given history, the continuation of an optimal policy must also be optimal con- dition on the agent’s expected utility. Moreover it maintains a non-negative value for the principal by definition. Therefore, it is still possible to represent principal’s problem within a stationary form. The distinction is that the state space is endogenous and will be a proper subset of [0, v]. In describing this endogenous state space, the most prominent property one shall look for is the non-negativity of the corresponding value function for each value in- side the state space. Indeed, the state space will be the maximum subset of [0, v] that is satisfying this property. This is due to the fact that an optimal policy conditional on a specific state space is also a feasible policy under any larger state space. Hence enlarging state space without hurting the non-negativity can only improve the principal’s policy.

In this regard, it is not hard to conclude that the state space is an interval and contains the utility level 0. Then, it is sufficient to find out the maximal utility Umax that the

principal can promise to the agent without violating the limited commitment constraint. This would complete the characterization of the endogenous state space [0, Umax]. However,

one needs to be sure about the existence of such a value. To this end, I define some auxiliary objects. First of all, for a given valueW < v, it is known that the Bellman equation that is defined over [0, W] is guaranteed to have a solution since Blackwell sufficiency conditions are satisfied. LetVW be the corresponding value function arising from the solution of Bellman

equation.

Lemma 1.6. When principal has a limited commitment power, there exists a utility level

Umax which is equal to the maximum utility that the principal can promise the agent in an optimal policy. Its value is given by:

Umax= sup{W |VW(W)≥0}.

Moreover, the value function arising from an optimal policy satisfies:

The value ofUmaxstrictly decreases withµ, conditional on having an optimal policy inducing self-monitoring.

Proof. See appendix A.8

The proof of this result, first indicates that the existence ofUmax is guaranteed. Then

it is argued that VW(W) is continuous, and hence the resulting value function always takes

non-negative values over the interval [0, Umax]. The fact that the value of the principal at

the maximal promised utility is equal to 0 follows from the continuity ofVW(W). Finally,

the proof points out an important observation in order to conclude the monotonicity of

Umax with respect to µ. Incentive compatibility of an optimal policy is independent of

the prior belief. Therefore an optimal policy at the maximal utility for some prior is also incentive compatible and brings a higher value to principal for any other smaller prior.

Then by using the arguments provided in the earlier sections one can conclude the following result.

Proposition 1.3.When principal has a limited commitment power and the expected so- cial cost of a project is larger than its value, whitelisting never appears in an equilibrium. Conditional on having an informative optimal policy,

• If principal internalizes the cost of self-monitoring, then the agent gets blacklisted eventually in the optimal contract.

• If principal does not internalize the cost of self-monitoring, then the optimal policy never reaches to a stable outcome and fluctuates over time.