Optimal Growth Models with Discounted Return

Abstract

In this chapter, we provide a unified treatment of a class of optimal growth models by using dynamic programming methods. In the economies we consider in this chapter, a social planner maximizes a discounted sum of utilities which depend on the current and past period states subject to a feasibility constraint. We show that this problem can be brought down to a sequence of static problems by using the value function of the problem and the associated Bellman equation. The Bellman equation allows us to state that (i)the value function is continuous with respect to the initial data and to the discount factor, (ii) the optimal trajectory of state variables can be described as a dynamical system (which may be multi-valued) We first give two examples of optimal growth models. Example 1 Consider a two-sector economy. At date t, sector 1 produces consumption good ct by using a capital stock k t which is produced in sector 2. At date t, sector 2 produces capital stock kt+1 which will be used in period t + 1 by the two sectors. To produce kt+1, sector 2 needs a quantity k t of capital good. The social planner solves at date 0 the following problem: