# Dynamic Economics: Optimization by the Lagrange Method

By Gregory C. Chow | Go to book overview

CHAPTER TWO Dynamic Optimization in Discrete Time

2.1 The Method of Lagrange Multipliers by an Example

Much of dynamic economics is concerned with maximizing a time separable objective function subject to a set of equations that describe the dynamic evolution of the state variables xt. To illustrate, consider the problem of maximizing the objective function for three periods (t = 0, 1, 2)

(2.1)

subject to the dynamic equation

xt+1 = f(xt, ut) + εt+1. (2.2)

Here, β is the discount factor; r(xt, ut) is the return for period t, which may depend on the p-component vector xt of state variables and the q-component vector ut of control variables to be set by the decision maker; f is a p-component vector function, and εt+1 is a vector of random shocks. To treat a deterministic optimization problem first, temporarily regard εt+1 as a vector of constants known to the decision maker at time t. Assume the functions r and f to be differentiable and concave. The three-period problem is to maximize the objective function (2.1) with respect to u0, u1, and u2 subject to the constraint (2.2).

It is natural and easy to solve this problem by the method of Lagrange multipliers. Introduce the p × 1 vectors λ1 and λ2 of Lagrange multipliers and form the Lagrangean expression

-19-

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