Tools and Tricks of the Trade, Part II:
Linear Differential Equations and
This chapter shows how to solve for the perfect foresight path in optimizing, continuous-time dynamic models. The solution procedure involves the use of optimal control theory, the methods for solving linear differential equations, and a working knowledge of how to apply these techniques in general equilibrium models of realistic complexity. I start out in Section 4.1 by developing the techniques for solving linear dynamic systems. Section 4.2 discusses the rudiments of the optimal control solution to dynamic optimization problems. The example in Section 4.3 illustrates how to apply the techniques.
A dynamic economic model usually generates a system of nonlinear differential (or difference) equations. Although the nonlinear system often lacks a closed-form solution, a linear approximation of the system can be used to investigate its dynamics in the neighborhood of a stationary equilibrium. For ease of exposition, I demonstrate how this is done in the case of a system of two simultaneous differential equations. The methodology for solving higher order systems is essentially the same and will be discussed later in Section 4.1.3.
Consider the pair of nonlinear differential equations
α is some exogenous variable and ẋ1 and ẋ2 are derivatives with respect to time (ẋ1 = dx1/dt). The stationary equilibrium, or steady state of the system, (x*1, x*2), is found by setting ẋ1 = ẋ2 = 0 and solving