Question

What is the total effect on the economy of a government tax rebate of $\$500$ to each household in order to stimulate the economy if each household will spend$85\%$ of the rebate in goods and services?

Short answer

The total effect on the economy is equal to$\$3333.33$.

Step-by-step solution

Step 1. Given information

Rebate is equal to $\$500$and each household will spend $85\%$of the rebate in goods and services.

Step 2. Find the total effect on the economy.

Every time money goes into the economy, $85\%$of it is spent and is then in the economy to be spent. Again, $85\%$of this money is spent in the economy again. This situation continues and so leads us to an infinite geometric series.

$500+500\left(0.85\right)+500{\left(0.85\right)}^2+500{\left(0.85\right)}^3+...$

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

$\begin{array}{rcl}{S}_n& =& \frac{a}{1-r}\\ 500+500\left(0.85\right)+500{\left(0.85\right)}^2+500{\left(0.85\right)}^3+...& =& \frac{500}{1-0.85}\\ & =& \$3333.33\end{array}$