Chapter 4 - Mathematics for dynamic economic models

Abstract

Mathematics for economic dynamic models is a chapter of key importance for the whole book, and it concludes the section dedicated to the fundamental mathematics for the economic analysis. Within the dynamic analysis the dimension of time is added, and the dependent variables are analyzed considering how they evolve over time and whether they converge to a stationary value (the equilibrium value) or not, as time increases. Both the differential equations and the difference equations are used, depending if we work in a continuous or discrete framework, together with the systems. This chapter is a prerequisite to understand the dynamic optimization section, to which this chapter is strictly related. Here the problem is to find the general time path solution, while in the dynamic optimization the objective is also to understand whether the time path optimizes a given performance measure (i.e., the functional) or not. The chapter will begin showing two important methods for solving numerically, in Excel, the differential equations (i.e., Euler and Runge-Kutta), and it will continue showing some economic models that use the ordinary, first-order differential equations, which represent the main type of equations used in the book. Then, the chapter will continue with the difference equations and the way the phase diagrams are applied in Excel. An important model of microeconomics, the cobweb model and a basic Keynesian model are taken as applications of the difference equations. The chapter will also cover the systems of linear differential equations, which are solved resorting both to the eigenvalue problem and to the Euler's method. The phase diagrams for the systems of differential equations will be also shown. Some economic applications (e.g., the modified model for the Walrasian price adjustment) of the systems of differential equations will be developed as well at the end of the chapter.