Linear Quadratic Approximations: An Introduction

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Abstract This is a brief introduction to dynamic programming and the method of using linear quadratic (LQ) approximations to the return function; the method is an approximation because it computes the solution to a quadratic expansion of the utility function about the steady state or the stable growth path of model economies. The main purpose of the chapter is to review the theoretical basis for the LQ approximation and to illustrate its use with a detailed example (social planning). The author demonstrates that, using the LQ approximation approach and the certainty equivalence principle, solving for the value function is a relatively easy task. The different sections of the chapter describe the standard neoclassical growth model, present a social planner problem that can be used to solve the model, give a recursive formulation of the social planner's problem, and describe an LQ approximation to this problem. Exercises are included throughout, and an appendix presents a MATLAB program to illustrate the LQ method.