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

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

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

Excellent, thanks