Incentive compatibility constraints and dynamic programming in continuous time

Abstract 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.

[1]  A. Velasco,et al.  The Tragedy of the Commons and Economic Growth: Why Does Capital Flow from Poor to Rich Countries? , 1992, Journal of Political Economy.

[2]  Jess Benhabib,et al.  The joint exploitation of a productive asset: a game-theoretic approach , 1992 .

[3]  Patrick J. Kehoe,et al.  Sustainable plans and mutual default , 1989 .

[4]  P. Kehoe,et al.  Sustainable plans and mutual default , 1990 .

[5]  E. Dockner,et al.  Analysis of Nash equilibria in a class of capital accumulation games , 1996 .

[6]  H. Ishii,et al.  A New Formulation of State Constraint Problems for First-Order PDEs , 1996 .

[7]  Aldo Rustichini,et al.  Dynamic programming solution of incentive constrained problems , 1998 .

[8]  Aldo Rustichini,et al.  Social conflict and growth , 1996 .

[9]  A. Rustichini,et al.  Optimal Taxes without Commitment , 1997 .

[10]  G. Hardin,et al.  The Tragedy of the Commons , 1968, Green Planet Blues.

[11]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[12]  Jerzy Zabczyk,et al.  Mathematical control theory - an introduction , 1992, Systems & Control: Foundations & Applications.

[13]  Pierpaolo Soravia.  Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints , 1999, Differential and Integral Equations.

[14]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[15]  P. Ireland.  Sustainable monetary policies , 1997 .

[16]  S. Domenico.  RECURSIVE CONTRACTS , 1998 .

[17]  H. Soner.  Optimal control with state-space constraint I , 1986 .

[18]  P. Kehoe,et al.  Sustainable Plans , 1990, Journal of Political Economy.

[19]  A. Rustichini.  Lagrange multipliers in incentive-constrained problems , 1998 .

[20]  Albert Marcet,et al.  Communication, commitment, and growth , 1992 .

[21]  Prajit K. Dutta,et al.  How Different Can Strategic Models Be , 1993 .

[22]  Suresh P. Sethi,et al.  A Survey of the Maximum Principles for Optimal Control Problems with State Constraints , 1995, SIAM Rev..

[23]  Pierpaolo Soravia.  Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. I. Equations of unbounded and degenerate control problems without uniqueness , 1999 .

[24]  A. Rustichini.  Second best equilibria for games of joint exploitation of a productive asset , 1992 .

[25]  Drew Fudenberg,et al.  Capital as a commitment: Strategic investment to deter mobility , 1983 .

[26]  Piermarco Cannarsa,et al.  A boundary-value problem for Hamilton-Jacobi equations in hilbert spaces , 1991 .

[27]  Sub- and superoptimality principles of dynamic programming revisited , 1996 .

[28]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[29]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[30]  Prajit K. Dutta,et al.  The tragedy of the commons? , 1993 .

[31]  Gerhard Sorger,et al.  Existence and Properties of Equilibria for a Dynamic Game on Productive Assets , 1996 .

[32]  Knut Sydsæter,et al.  Optimal control theory with economic applications , 1987 .