Question

What is the total effect on the economy of a government tax rebate of $\$1,000$ to each household in order to stimulate the economy if each household will spend$90\%$ of the rebate in goods and services?

Short answer

Total effect on the economy is equal to$\$5000$

Step-by-step solution

Step 1. Given information

Rebate is equal to$\$1,000$and each household will spend $90\%$of the rebate in goods and services.

Step 2. Find the total effect on the economy.

Every time money goes into the economy, $90\%$of it is spent and is then in the economy to be spent. Again, $90\%$of this money is spent in the economy again. This situation continues and so leads us to an infinite geometric series.

role="math" localid="1645374137180" $1000+1000\left(0.8\right)+1000{\left(0.8\right)}^2+1000{\left(0.8\right)}^3+...$

Let $a,r$denote the first term and the common ratio respectively.

$\begin{array}{rcl}a& =& 1000\\ r& =& \frac{1000(0.8)}{1000}\\ & =& 0.8\end{array}$

$S_n=\frac{a}{1-r}$

$\begin{array}{rcl}1000+1000\left(0.8\right)+1000{\left(0.8\right)}^2+1000{\left(0.8\right)}^3+...& =& \frac{1000}{1-0.8}\\ & =& 5000\end{array}$

The total effect of the $\$1,000$ received by each household will be a$\$5,000$ growth in the economy