# Thread: Forming first-order difference equation

1. ## Forming first-order difference equation

Hi,

I was wondering if anyone can help with this mathematical economics problem:

The multiplier-accelerator model of growth has the following three equations

St = aYt
It+1 = b(Yt+1 - Yt)
St = It

a) Briefly interpret the above three equations and combine them as a first-order difference equation.

b) Assume that output at time zero is Y0 and solve the difference equation. Which are the determinants of output growth in the model? Explain intuitively.

It's clear from b) that the difference equation must refer to Y(t). But I can't work out how to get a single first-order difference equation that includes the three equations, that is set up to refer to Y(t).

2. ## Re: Forming first-order difference equation

I assume "Yt+1" is $Y_{t+1}$, not " $Y_t+ 1$" and that " $It+1$" is $I_{t+1}$. So the middle equation is the difference equation: $Y_{t+1}- Y_t= \frac{I_{t+1}}{b}$

Since you are given that $I_t= S_t$ and $S_t= aY_t$, it follows that $I_t= aY_t$ so that $I_{t+1}= aY_{t+1}$. Putting that into your equation, $Y_{t+1}- Y_t= \frac{a}{b}Y_{t+1}$ which is the same as $\left(1- \frac{a}{b}\right)Y_{t+1}= Y_t$. I would write that as $Y_{t+1}= \frac{b}{a- b}Y_t$ is relatively easy to solve.

3. ## Re: Forming first-order difference equation

Excellent, thanks