An Introduction to Economic Dynamics

A BUSINESS CYCLES II: THE ADIFFERENCE AND DIFFERENTIAL EQUATIONS

A.1 FIRST-ORDER LINEAR DIFFERENCE EQUATION

Consider a first-order linear difference equation of the form

$$x_{n\text{+1}}\text{=}A{x}_n\text{+}B,$$

where A and B are constants and n = 0,1,2,… Since the equation is linear, the general solution is of the form

$$x_n=x_n^H+x_n^P,$$

where $x_n^P$ is a particular solution and $x_n^H$ satisfies the homogeneous equation

$$x_{n+1}^H=A{x}_n^H.$$

Clearly

$$x_n^H=A{x}_{n-1}^H=A^2{x}_{n-2}^H=\dots =A^n{x}_0^H.$$

To find a particular solution we try $x_n^P=x_0^P$, i.e. a fixed-point solution independent of n. Then

$$x_0^P=A{x}_0^P+B\Rightarrow x_0^P=\frac{B}{1-A},A\ne 1.$$

Hence, we find that ...