Steady-State Solutions to Dynamic Optimization Models with Inequality Constraints

Abstract

Many problems in resource economics are inherently dynamic in the sense that time is essential to the analysis. As a consequence, dynamic optimization models have played a much larger role in resource economics than in economics in general. While general techniques are available for solving dynamic optimization problems (dynamic programming and Pontryagin's Maximum Principle), computational considerations usually limit direct application of these methods to empirical models with one or two state variables. This severely constrains the realism and accuracy of the model. In many cases, however, it is possible to determine steady-state solutions to much larger problems even if the full solution cannot be determined. Since the time paths of the optimal control and state variables typically converge to a steady-state, knowledge of the optimal steady-state provides valuable information which can be used for managing the resource.1 In a recent paper, Burt and Cummings (1977) derived conditions to be satisfied by steady-state solutions to optimization problems in resource economics. While their model is general with respect to functional forms, it does not allow inequality constraints on the state and control variables. This would seem to be a serious limitation for many problems in resource economics where there may be a priori restrictions on the range of feasible control values as well as restrictions depending on current values of the state variables. The purpose of this paper is to extend the analysis of Burt and Cummings (1977) to the case of inequality constraints involving state