Chapter 12.3, Problem 97AYU

Multiplier Suppose that, throughout the U.S. economy, individuals spend $90\%$ of every additional dollar that they earn. Economists would say that an individual’s marginal propensity to consume is $0.90$. For example, if Jane earns an additional dollar, she will spend $0.9(1)=\$0.90$ of it. The individual who earns $\$0.90$ (from Jane) will spend $90\%$ of it, or $\$0.81$. This process of spending continues and results in an infinite geometric series as follows:

${\mathrm{1.0.90.0.90}}^2,{0.90}^3{\mathrm{.0.90}}^4.\dots$

The sum of this infinite geometric series is called the multiplier. What is the multiplier if individuals spend $90\%$ of every additional dollar that they earn?

To determine

To calculate: The value of multiplier if individual spend $90\%$ of every additional dollar that they earn.

Answer to Problem 97AYU

Multiplier value $=10$.

Explanation of Solution

Given:

It is given that throughout the U.S. economy, individuals spend $90\%$ of every additional dollar that they earn.

It is also given that the process of spending continues and results in an infinite geometric series as follows:

$1,0.90,{0.90}^2,{0.90}^3,{0.90}^4,\cdots$

First term $a_1=1$ and the common ratio $r=\frac{a^n}{a^{n-1}}=0.90$.

Formula used:

Convergence of an infinite geometric series theorem states that If $\left|r\right|<1$ converges. Its sum is ${\displaystyle \sum }_{k=1}^{\infty }{a}_1{r}^{k-1}=\frac{a_1}{1-r}$.

Calculation:

Multiplier if individual $=\frac{1}{1-0.90}=\frac{1}{0.1}=10$.