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

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 . (1-c^n)/(1-c)`