Shadow prices in infinite-horizon optimal control problems with dominating discounts

Abstract

We consider a nonlinear optimal control problem, in which an integrated discounted utility index is maximized over an infinite time interval. The problem statement is motivated by various optimization problems arising in economics. Assuming that the discount parameter dominates the growth rates in the state variables and in the gradient of the current utility, we develop a version of the Pontryagin maximum principle providing a complete set of necessary optimality conditions and also suggesting an analytic expression for the values of the adjoint variables often viewed as shadow prices in the economic literature. We illustrate the proposed methodology by applying it to the problem of optimal capital accumulation for a stylized model of an enterprise.