#### Problem 73E

73. 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 $100 c \%$ and saves $100 s \%$ 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, $c+s=1$.

(a) Let $S_{n}$ be the total spending that has been generated after $n$ transactions. Find an equation for $S_{n}^{\prime}$.

(b) Show that $\lim _{n \rightarrow \infty} S_{n}=k D$, where $k=1 / s .$ The number $k$ is called the multiplier. What is the multiplier if the marginal propensity to consume is $80 \%$ ? Note: The federal government uses this principle to justify deficit spending. Banks use this principle to justify lending a large percentage of the money that they receive in deposits.