n22 - Atsushi Inoue ECG 765 Mathematical Methods for...

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Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 22 The System of First-Order Difference and Differential Equations When we want to introduce interactions among variables, we will consider the system of difference and differential equations. First we will show that higher-order difference and differential equations can be written as a system of first-order difference and differential equations, respectively. Then we will analyze the system of first-order difference and differential equations. Outline A. Second-Order Difference Equations as a System of First-Order Differ- ence Equations B. The System of First-Order Difference Equations C. The System of First-Order Differential Equations A. Second-Order Difference Equations as a System of First-Order Difference Equations. Consider a second-order difference equation y t +2 = ay t +1 + by t + c. Let x t = y t +1 . Then we can express the original second-order equation by means of two first-order simultaneous equations as follows: x t +1 = ax t + by t + c, y t +1 = x t . B. The System of First-Order Difference Equations Consider " x t +1 y t +1 # = " a b c d # " x t y t # + " e f # 1
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or z t +1 = Az t + w.
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