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 *Y*_{0 }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).

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.

Re: Forming first-order difference equation