Chapter 8.2, Problem 51E

a.

To determine

To find: an equation for $S_n$ . Where $S_n$ be the total spending that has been generated after n transaction.

Answer to Problem 51E

  $S_n=\frac{D(1-c^n)}{1-c}$ .

Explanation of Solution

Given information: 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 value c and s are called the marginal propensity to consume and the marginal propensity to save and of course c + s =1.

Calculation:

The local government spends D dollars.

People, who receive those D dollars, will spend $Dc$ dollars.

People, who receive those Dc dollars, will spend $D{c}^2$ dollars, and so on …..

Total spending after n transaction is $\begin{array}{l}{S}_n=D+Dc+D{c}^2+\mathrm{....}D{c}^{n-1}\\ S_n=D(1+c+c^2+\mathrm{....}{c}^{n-1})\\ S_n=\frac{D(1-c^n)}{1-c}.\end{array}$

b.

To determine

To show: that $\underset{n\to \infty }{\lim }{S}_n=kD$ ,where $k=\frac{1}{s}$ . The number k is called the multiplier and what is the multiplier if the marginal propensity to consume is 80%.

Answer to Problem 51E

Multiplier $k=5.$

Explanation of Solution

Given information: 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 value c and s are called the marginal propensity to consume and the marginal propensity to save and of course c + s =1.

Calculation:

  $\begin{array}{l}{S}_n=\frac{D(1-c^n)}{1-c}\\ \underset{n\to \infty }{\lim }{S}_n=\underset{n\to \infty }{\lim }\frac{D(1-c^n)}{1-c}=\frac{D(1-0)}{1-c}\\ \text{Note}\text{}\text{that}{c}^n=0\text{because}0<c<1\\ c+s=1\Rightarrow s=1-c.\\ \underset{n\to \infty }{\lim }{S}_n=\frac{D}{1-c}=\frac{D}{s}=kD\end{array}$

Marginal propensity to consume is c = 0.8

s= 1- c =1-0.8=0.2.

Multiplier is $k=\frac{1}{s}=\frac{1}{0.2}=5$ .