Difference Equations

First Order Difference Equations

Following equation is the linear first order difference equation:

$$y_t=\phi y_{t-1}+w_t.$$

Recursive Solution

Consider the following system

Date	Equation
0	$y_0=\phi y_{-1}+w_0$
1	$y_1=\phi y_{0}+w_1$
2	$y_2=\phi y_{1}+w_2$
$\vdots$	$\vdots$
t	$y_t=\phi y_{t-1}+w_t$

Given the initial conditions $y_{-1}$ and $w_0$, we are equipped to determine all future values of $y$.

We can write $y_1$ as

$$\begin{align*} y_1&=\phi y_{0}+w_1\\ &=\phi (\phi y_{-1}+w_0) +w_1\\ &=\phi^2 y_{-1}+ \phi w_0 +w_1. \end{align*}$$

Similarly $y_2$ can be written as

$$\begin{align*} y_2&=\phi^3 y_{-1}+ \phi^2 w_0 +\phi w_1 + w_2, \end{align*}$$

and $y_{100}$ can be written as

$$y_{100}=\phi^{101}y_{-1}+\phi^{100}w_0 + \phi^{99}w_1 + \cdots+\phi w_{99}+w_{100}.$$

In general

$$y_{t}=\phi^{t+1}y_{-1}+\phi^{t}w_0 + \phi^{t-1}w_1 + \cdots+\phi w_{t-1}+w_{t}.$$

Another general form is

$$y_{t+j}=\phi^{j+1}y_{t-1}+\phi^{j}w_t + \phi^{j-1}w_{t+1} + \cdots+\phi w_{t+j-1}+w_{t+j}.$$

Effect of $w_t$ on $y_{t+j}$ is

$$\frac{\partial y_{t+j}}{\partial w_t}=\phi^j.$$

Points to note:

1. The above equation is also called Impulse Response Function (IRF).
2. When $|\phi|<1$, the system is stable. When $|\phi|>1$, the system is explosive.
3. If $|\phi|<1$, the cumulative effect of one unit increase in $w_t$ on $y_{t+j}$, where $j \in \{0,1,2,\cdots\}$, is

$$\sum_{j=0}^{\infty}\frac{\partial y_{t+j}}{\partial w_t}= \sum_{j=0}^{\infty}\phi ^j = \frac{1}{1-\phi}.$$

Example

Let the stream of future values be $\{y_t,y_{t+1},y_{t+2},\cdots\}$ and the interest rate be $r$. Present value (PV) of the stream at time $t$ is given by

$$PV=y_t+ \frac{y_{t+1}}{1+r}+\frac{y_{t+2}}{(1+r)^2}+\frac{y_{t+3}}{(1+r)^3}+ \cdots$$

Let $\beta$ be the discount factor, hence

$$\beta=\frac{1}{1+r}.$$

Therefore

$$\begin{align*} PV&=y_t+ \beta y_{t+1}+ \beta^2 y_{t+2}+ \beta^3 y_{t+3}+\cdots\\ &=\sum_{j=0}^{\infty}\beta^j y_{t+j}\\ \end{align*}$$

Effect of one time increase

What would be the impact of increasing $w_t$ by one unit on $PV$ while keeping $w_{t+1}, w_{t+2}, \ldots, w_{t+j}$ unchanged?

$$\begin{align*} \frac{\partial PV}{\partial w_t}&=\sum_{j=0}^{\infty}\beta^j \frac{\partial y_{t+j}}{\partial w_t}\\ &=\sum_{j=0}^{\infty}\beta^j \phi^j\\ &=\frac{1}{1-\beta \phi} \end{align*}$$

provided $|\beta \phi|<1$.

Effect of permanent change on PV

What would be the impact of a permanent change in $w$ on $PV$?

A permanent change in $w$ means that $w_t, w_{t+1}, \cdots ,$ and $w_{t+j}$ would all increase by one unit.

Total change in $PV$ can be calculated as follows

$$\begin{align*} \text{Total Change in PV}&= \frac{\partial PV}{\partial w_t} + \sum_{j=1}^{\infty}\beta^j \frac{\partial y_{t+j}}{\partial w_{t+1}} + \sum_{j=2}^{\infty}\beta^j \frac{\partial y_{t+j}}{\partial w_{t+2}}+ \cdots + \sum_{j=j}^{\infty}\beta^j \frac{\partial y_{t+j}}{\partial w_{t+j}}\\ &=\sum_{j=0}^{\infty}\beta^j \phi^{j} + \sum_{j=1}^{\infty}\beta^j \phi^{j-1} + \sum_{j=2}^{\infty}\beta^j \phi^{j-2}+\cdots+\sum_{j=j}^{\infty}\beta^j \phi^{0}\\ &=\sum_{j=0}^{\infty}\beta^j \phi^{j} + \beta\sum_{j=1}^{\infty}\beta^{j-1} \phi^{j-1} + \beta^2\sum_{j=2}^{\infty}\beta^{j-2} \phi^{j-2}+\cdots+\sum_{j=j}^{\infty}\beta^j \phi^{0}\\ \end{align*}$$

Effect of permanent change on $y_{t+j}$

The effect on $y_{t+j}$ of a permanent change in $w$ beginning in period $t$ is given by

$$\begin{align*} \frac{\partial y_{t+j}}{\partial w_{t}}+\frac{\partial y_{t+j}}{\partial w_{t+1}} + \frac{\partial y_{t+j}}{\partial w_{t+2}}+ \cdots + \frac{\partial y_{t+j}}{\partial w_{t+j}}&=\phi^j+\phi^{j-1}+\phi^{j-2}+\cdots+\phi + 1.\\ \end{align*}$$

When $|\phi|<1$, then

$$\begin{align*} \lim_{j\to \infty}\Big[\frac{\partial y_{t+j}}{\partial w_{t}}+\frac{\partial y_{t+j}}{\partial w_{t+1}} + \frac{\partial y_{t+j}}{\partial w_{t+2}}+ \cdots + \frac{\partial y_{t+j}}{\partial w_{t+j}}\Big]&=1 + \phi + \phi^2 + \cdots\\ &=\frac{1}{1-\phi}. \end{align*}$$