First Order Difference Equations « Complex Math Simple Life

First Order Difference Equations

Posted in difference, equation by beauangel on the February 19, 2007

Standard Form

$y_t = a\: y_{t-1} + b$ where a and b are constants.

The solution of the first order difference equation of the form above is given by the following formula:

$y_t = \left( y_o - \frac{b}{1-a} \right) a^t + \frac{b}{1-a}$ if $a \neq 1$

$y_t = y_0 + b\:t$ if $a = 1$ where $y_0$ is the value of y when $t=0$ .

One of the economic models that employs First Order Difference Equations is the Cobweb model.

In the model, $p_{t-1}$ determines $q_t$ , which in turn determines $p_t$ , according to the rules

$q_t = q^S(p_{t-1}), \quad p_t = p^D(q_t)$

where $q^S$ is the supply function and $p^D$ the inverse demand function.