محدودیت سازگاری با انگیزه و برنامه ریزی پویا پیوسته در زمان

This paper is devoted to the study of infinite horizon continuous time optimal control problems with incentive compatibility constraints that arise in many economic problems, for instance in defining the second best Pareto optimum for the joint exploitation of a common resource, as in Benhabib and Radner [Benhabib, J., Radner, R., 1992. The joint exploitation of a productive asset: a game theoretic approach. Economic Theory, 2: 155–190]. An incentive compatibility constraint is a constraint on the continuation of the payoff function at every time. We prove that the dynamic programming principle holds, the value function is a viscosity solution of the associated Hamilton–Jacobi–Bellman (HJB) equation, and that it is the minimal supersolution satisfying certain boundary conditions. When the incentive compatibility constraint only depends on the present value of the state variable, we prove existence of optimal strategies, and we show that the problem is equivalent to a state constraints problem in an endogenous state region which depends on the data of the problem. Some economic examples are analyzed.

In this paper, we study optimal control problems in continuous time/infinite horizon with incentive compatibility constraints. We consider a classical infinite horizon optimal control problem with a constraint on the continuation value of the plan at each time t≥0. The continuation value at each time t is constrained from below by a function of the state and of the control at time t. The constraint can be interpreted from an economic point of view as an outside option/incentive compatibility constraint. This type of constraint gives rise to two different problems: optimal stopping problems and optimal control problems with incentive compatibility constraints. In the first case, we have a positive perspective: we want to study the optimal behavior of an economic agent in a dynamic setting allowing him/her at any time in the future to stop the process and to exercise an outside option which gives him/her a reward, which is a function of the state (termination payoff). In this setting, the set of admissible controls includes those violating the incentive compatibility constraint. The agent is allowed to get the termination payoff. In the second case, we have a normative perspective. The point of view is one of a social planner who wants to characterize optimal contracts or second best solutions to dynamic problems under incentive compatibility constraints for the agents of the economy. The set of admissible controls does not include those violating the incentive compatibility constraint. The social planner looks for an optimal policy among the policies, which do not include the termination of the process. The goal is the definition of a social contract taking into account the fact that the agents can decide in the future to go out of the contract. Such an event is prevented by including the incentive compatibility constraint. This type of problems is analyzed in this paper. Many economic problems can be put in this setting. They belong to two main classes of models of the so-called second best literature. The first type of problems comes from the study of differential games such as the exploitation of an exhaustible common resource. The second type of problems arises in analyzing policy making without full precommitment (time consistency problems). In Section 2, we will fully describe a second best problem arising in the analysis of a differential game for the exploitation of a common exhaustible resource. It is well recognized that in differential games such as the exploitation of a common resource. Benhabib and Radner, 1992, Dockner and Sorger, 1996, Tornell and Velasco, 1992, Dutta and Sundaram, 1993a and Dutta and Sundaram, 1993b, capital accumulation (Fudenberg and Tirole, 1983), pollution, voluntary provision of a public good (Dockner et al., 1996), the outcome of the noncooperative interaction obtained as a subgame perfect equilibrium or as trigger strategy equilibrium may be Pareto inefficient, i.e. the reward for the agent is smaller than the one obtained by a representative agent under perfect competition (the so-called tragedy of commons). This result leads to the problem of designing contracts which are efficient among the subgame perfect equilibria (see Rustichini, 1992 and Benhabib and Rustichini, 1996) and to the problem of designing contracts yielding at every time in the future a utility level higher than the one obtained according to a specific subgame perfect equilibrium. This type of problems can be formalized as the maximization of the utility of the representative agent under the constraint that at every time in the future, the continuation value of the consumption plan is greater than the utility obtained from the strategy of a subgame perfect equilibrium. Typically, the constraint is given by the value function of a control problem without constraints. The second class of models comes from optimal taxation problems in an intertemporal setting without full precommitment or in great generality from the analysis of an economy where there is a private sector and the government. The problem is the definition of an optimal plan without full commitment at time zero (for instance, a tax plan made up of taxes by the government and saving decisions by the private sector) in such a way that the private sector and the government do not have an incentive to deviate at each time in the future (decisions are taken sequentially without commitment), (see Chari and Kehoe, 1990, Marcet and Marimon, 1992, Marcet and Marimon, 1996, Benhabib and Rustichini, 1997 and Ireland, 1997) for some interesting examples. The incentive compatibility constraint is endogenous to the model, it is usually given by a function only of the state and not of the control (in many cases, it is a value function of the associated unconstrained problem). The outside option can also be interpreted in some cases as a policy variable (fixed costs, royalties, taxes) or as an exogenous opportunity (a different investment opportunity). This type of constraints has been recently analyzed in discrete time in Marcet and Marimon, 1996, Rustichini, 1998a and Rustichini, 1998b. In the first two papers, the problem has been studied by means of Lagrange multipliers, in the third one, through dynamic programming. Our paper provides a dynamic programming solution to the problem in the continuous time case. Some similarities between our characterization of the value function of the constrained problem and the one provided for the discrete time case in Rustichini, (1998a) can be noted (see Remark 4.4). The optimal control problem is a state constraints problem with infinite horizon, a discounted objective function and an additional constraint on the continuation value for the plan. Such a constraint concerns the future of the trajectory and this “backward–forward” structure gives rise to nonstandard technical problems. Even simple one-dimensional problems (like the ones in Section 6) require a considerable amount of technical work. In fact, this type of problems has not been addressed in the present optimal control literature in continuous time: the classical tools of optimal control theory (like dynamic programming and maximum principle) are not available here. In this paper, we first analyze a constraint described by a function of both the control and of the state and then we restrict our attention to the case of a function only depending on the state. We prove that the dynamic programming principle holds. This allows us to write the Hamilton–Jacobi–Bellman equation (HJB in the rest of the paper) associated with our problem and to prove, under some additional assumptions, that the value function of the constrained problem is a solution in the viscosity sense of the HJB equation. We cannot obtain uniqueness of solutions of the HJB equation in general, but we are able to characterize the value function as the minimal viscosity supersolution (satisfying suitable boundary conditions) of the equation. Restricting our attention to the case where the incentive constraint only depends on the state variable, we prove a result about existence (and uniqueness) of optimal strategies and then we prove that the above problem is equivalent to a state constraints problem in a region E which is implicitly determined by the data of the problem. Then, the issues of determining the region E and its topological properties are discussed. All of this allows us to adapt known results and techniques on state constraints problems so that we can study, in some cases, the properties of the optimal trajectories. In Section 6, two simple examples are fully developed in order to make clear the main points of our second best analysis. The examples are characterized by a linear state equation, concave objective function, and an incentive compatibility constraint defined by a constant. The first example is the optimal saving problem, the second is the firm's capital accumulation problem with adjustment costs. In the first example, if the interest rate is larger than the discount rate, then both the value function and the optimal policy of the constrained problem coincide with those of the unconstrained problem, provided that the initial stock of capital is large enough. If the opposite condition holds, then the constrained value function is smaller than the unconstrained value function and the second best optimal control induces a smaller rate of consumption than the first best policy. The minimal stock of capital allowing existence of the second best policy is larger than in the first case. In the second example, as the constant describing the constraint goes up, we observe four different parameter regions. For a small constant (first region), the constraint is not binding and therefore, the unconstrained solution coincides with the constrained solution, for a higher constant (second region), the unconstrained solution is equal to the constrained solution but the initial stock of capital should be large enough to have a solution. As the constant is furthermore increased (third region), the investment rate is higher than the rate obtained in the first best case and an initial stock of capital larger than in the previous case is needed to have a solution. Finally, when the constant is beyond a certain level (fourth region), the problem becomes ill posed for every initial stock of capital. Summing up in the above two examples, we have that an incentive compatibility constraint has two effects: it restricts the state region for which a solution exists and it induces a higher rate of investment. The optimal policy foresees a stationary level of the state variable when the incentive constraint becomes binding. The paper is organized as follows. In Section 2, we present a motivating example and then we introduce the mathematical framework of the problem. The basic theory addressing the main technical problems is developed in 3, 4 and 5. 3 and 4 are concerned with the dynamic programming principle and the HJB equation, respectively, in a general framework. To improve the readability of these sections, we relegated the proofs of some technical results to the Appendix. Section 5 deals with the case when the incentive constraint depends only on the state (which occurs in main economic examples). Here, we prove the equivalence theorem (Section 5.1) that allows to reduce the problem to a state constraints problem in an endogenous region E, and then (in Section 5.2), we discuss the properties of E. Moreover in 5.3 and 5.4, we prove the upper-semicontinuity of the value function and the existence of optimal strategies. In Section 6, we analyze two examples.

In this paper, we have analyzed dynamic incentive compatibility constrained problems in continuous time. The incentive constraint is a constraint on the continuation value of the payoff function. More precisely, at every time the residual payoff is supposed to be greater than or equal to a certain function of the state and/or of the control. We have characterized the value function associated with the constrained problem by proving that the dynamic programming principle holds and that it is a viscosity solution of the HJB equation. Restricting our attention to an incentive compatibility constraint, which only depends on the value of the state, we have shown the equivalence of the constrained problem with a state-constrained problem in an endogenous region. This equivalence is useful to define the optimal strategy by means of the Pontryagin maximum principle. Two simple economic problems have been analyzed where the incentive compatibility constraint is given by a positive constant. We have shown that the constrained problem coincides with the unconstrained problem only in some cases. In general, as the constraint becomes more binding, we have three effects: the state region allowing existence for the constrained problem shrinks, the rate of capital accumulation becomes higher than the first best rate, and the value function becomes smaller than the one obtained in the unconstrained problem.