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 themarginal propensity to consume and themarginal propensity to saveand, of course, \(c + s = 1\).
(a) Let \({S_n}\) be the total spending that has been generated after \(n\) transactions. Find an equation for \({S_n}\).
(b) Show that \(\mathop {\lim }\limits_{n \to \infty } {S_n} = kD\), where \(k = \frac{1}{s}\). 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.
