During the last years, there has been a revived interest in the theory of dynamic contracting. However, although most of the research incorporates some form of limited commitment/enforcement, little has been done in terms of extending the notion of commitment per se. In particular, there is no reason to believe that the outside option is constant across the history of observables. For example, it is unrealistic to treat the reservation utility of a CEO as fixed regardless of the situation in his/her firm, industry, or the economy as a whole. The dependence could come through many channels externalities, different types of agents, a certain structure of beliefs, but more importantly, it can significantly influence the nature of the relationship and the form of the optimal contract. Moreover, extending the notion of commitment can bring some important insights into various contractual problems. For example, in order to address the wide use of broad based stock option plans, Oyer (2004) builds a simple 2 period model where adjusting compensation is costly and employeels outside opportunities are correlated with the firms performance.
In this sense, what remains to be done is to generalize the notion of commitment by defining the outside options on the history observed in a dynamic contractual setting. The current paper considers a moral hazard problem in an infinitely repeated principal agent interaction while allowing the reservation utilities of both par ties to vary across the history of observables. More precisely, to keep the model tractable, the reservation utilities are assumed to depend on some finite truncation of the publicly observed history. The rest of the model is standard in the sense that the principal wants to implement some sequence of actions which stochastically affect a variable of his/her interest, but suffers from the fact that the actions are unobservable. For this purpose, the optimal contract needs to provide the proper incentives for the agent to exercise the sequence of actions suggested by the principal. The incentives, however, are restricted by the in ability of the parties to commit to a long term relationship. It is here where the dynamics of the reservation utilities enters the relationship by reshaping the set of possible self#enforcing, incentive compatible contracts.
In order to be able to characterize the optimal contract in such a setting, I construct a reduced stationary representation of the model in line with the dynamic insurance literature. The representation benefits from Green (1987) the notion of temporary incentive compatibility, Spear and Srivastava (1987) the recursive formulation of the problem with the agentls expected discounted utility taken as the state variable, and Phelan (1995) the recursive structure with limited commitment, but is closest to Wang (1997) as far as the recursive form is concerned. Unlike Wang (1997), however, I formally introduce limited commitment on both sides and provide a rigorous treatment of its effect on the structure of the reduced computable version of the model. A parallel research by Aseff (2004) uses a similar general formulation, but via a transformation due to Grossman and Hart (1983) constructs a dual, cost minimizing recursive form closer to Phelan (1995) in order to solve for the optimal contract. Such a procedure, however, exogenously imposes the optimality of a certain action on every possible contingency.
After existence is proved, the general form of the model is reduced to a more tractable, recursive form where the state is given by the agentls (promised) expected discounted utility. On a different dimension, the state space includes the set of truncated initial price histories in order to account for their influence on the reservation utilities. This recursive formulation does not rely on the first order approach and is not based on Lagrange multipliers [cf. Marcet and Marimon, (1998)]. In fact, all I need is continuity of the momentary utilities. I first consider an auxiliary version where the participation of the principal is not guaranteed. The solution of this problem can be computed through standard dynamic programming methods once the state space is determined. Following the approach of Abreu, Pearce and Stacchetti (1990), the state space is shown to be the fixed point of a set operator and can be obtained through successive iteration on this operator until convergence. Given the solution of the auxiliary problem, I resort to a procedure outlined by Rustichini (1998) in order to solve for the optimal incentive compatible, two side participation guaranteed supercontract. This is achieved by severely punishing the principal for any violation of his/her participation constraint. The procedure allows of recovering the subspace of agents expected discounted utilities supportable by a self enforcing incentive compatible contract.
Regarding the numerical computation, one point deserves special attention. In computing the endogenous state space we are iterating on sets and therefore need to represent them efficiently. For the class of infinitely repeated games with perfect monitoring, Judd, Yeltekin and Conklin (2003) are able to construct inner and outer convex polytope approximations based on the convexification of the equilibrium value set through a public randomization device. The algorithm I use may be of independent interest since it does not rely on the convexity of the underlying set. The main idea is to discretize the guess for the equilibrium set element wise, extract small open balls around the grid points unfeasible with respect to the (non updated) guess and use the remaining set, i.e. the guess less the extracted intervals, as a new guess for the equilibrium set. The procedure stops if the structure of the representations of two successive guesses coincides and the suitably defined difference between the representations is less than some prespecified tolerance level.
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