In this paper, the authors study a general concave discrete-time, infinite-horizon, optimal-control problem. They establish necessary and sufficient conditions for optimality in the weak sense of W. A. Brock and for optimality in the strong sense of D. Gale. The corresponding transversality conditions are general exhaustion properties of the limit value of the optimal state variables; these properties cover and extend the well-known results obtained in special cases.

We study different solutions to a simple one-dimensional linear quadratic game with a large number of private agents and a government. A “time-consistent” solution is defined as a solution to the Hamilton-Jacobi-Bellman equation, i.e. as a policy for which the government has noprecommitment capability. This solution is compared to a policy where the government has an “instantaneous” pre-commitment, i.e. an equilibrium in which the government has a period by period leadership. In both cases, we show how control theory should be applied to calculate the equilibrium and how to relate these equilibria to the differential game literature.

Jean Mercenier, Philippe Michel, Discrete-Time Finite Horizon Approximation of Infinite Horizon Optimization Problems with Steady-State Invariance, Econometrica, Vol. 62, No. 3 (May, 1994), pp. 635-656