Question

When money is spent on goods and services, those who receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call this chain reaction the multiplier effect. In a hypothetical isolated community, the local government begins the process by spending D dollars. Suppose that each recipient of spent money spends 100c% and saves 100s% of the money that he or she receives. The values c and s are called the marginal propensity to consume and the marginal propensity to save and, of course, course, c + s = 1. Let Sn be the total spending that has been generated after n transactions.
Required:
Find an equation for Sn.

Answer 1

Solution :

If $S_n$ is the total spending generated after n transactions, then $a_n$ is the spending generated during the n-th transaction. Since the government spends D dollars, then $a_1$ = D.

Then the second person will receive D dollars and spend $D_c$ dollars. Therefore, $a_2=D_c$. The next person will receive $D_c$ dollars, which means they are spending $a_3 = (D_c)c = Dc^2$ dollars. Therefore,

$a_1 = D$

$a_2=D_c$

$a_3 = Dc^2$

$a_4 = (Dc^2)c= Dc^3$

....

$a_n=Dc^{n-1}$

∴ $S_n=a_1+a_2+a_3+....+a_n$

  $S_n = D+Dc+Dc^2+Dc^3+...+Dc^{n-1}$

  $S_n= D(1+c+c^2+...+c^{n-1})$

  $S-n=D . \frac{1-c^n}{1-c}$