Chapter 16, Problem

To determine

Calculate the level of output after four years if initial output equals $\$1,000$ and the economy grows at a rate of $8\%$ a year.

Explanation of Solution

Calculate the output for $4$ years with a growth rate of $8\%$ by multiplying the initial output with growth rate per year then add obtained value with growth rate.

Given,

Initial output as $8\%$ as $\$1,000$

Growth rate:

  $\begin{array}{c}8\%=\frac{8}{100}\\ =0.08\end{array}$

Now,

Multiply initial year output with growth of rate per year

Then,

Add previous year output with growth rate of current year.

That is,

Growth	Amount
Initial output	$\$1,000$
Growth rate at the end of First year   $\left(\text{Initial}\text{output}\times \text{growth}\text{rate}\text{per}\text{year}\right)$	$\begin{array}{l}1,000\times 0.08\\ =80\end{array}$
Level of output at the end of first year   $\left(\text{Initial}\text{output}+\text{Growth}\text{at}\text{the}\text{end}\text{of}\text{the}\text{year}\right)$	$\begin{array}{l}1000+80\\ =1080\end{array}$
Growth rate at the end of second year   $\left(\text{First}\text{year}\text{output}\times \text{growth}\text{rate}\text{per}\text{year}\right)$	$\begin{array}{l}1,080\times 0.08\\ =86.40\end{array}$
Level of output at the end of second year   $\left(\text{First}\text{year}\text{output}+\text{Growth}\text{at}\text{the}\text{end}\text{of}\text{the}\text{second}\text{year}\right)$	$\begin{array}{l}1080+86.4\\ =1,166.4\end{array}$
Growth rate at the end of third year   $\left(\text{Second year}\text{output}\times \text{growth}\text{rate}\text{per}\text{year}\right)$	$\begin{array}{l}1,166.4\times 0.08\\ =93.312\end{array}$
Level of output at the end of third year   $\left(\text{Second year}\text{output}+\text{Growth}\text{at}\text{the}\text{end}\text{of}\text{the}\text{third}\text{year}\right)$	$\begin{array}{l}1,166.4+93.3\\ =1,259.71\end{array}$
Growth rate at the end of fourth year   $\left(\text{Third year}\text{output}\times \text{growth}\text{rate}\text{per}\text{year}\right)$	$\begin{array}{l}1,259.71\times 0.08\\ =100.77\end{array}$
Level of output at the end of fourth year   $\left(\text{third year}\text{output}+\text{Growth}\text{at}\text{the}\text{end}\text{of}\text{the}\text{fourth}\text{year}\right)$	$\begin{array}{l}1,259.71+93.3\\ =1,360.49\end{array}$

Therefore, the level of output at the end of fourth year is $\$1,360.49$

Economics Concept Introduction

To calculate the value of growth at the end of first year:

Multiply initial year output with growth of rate per year.

That is,

  $\begin{array}{c}\text{Growth}\text{at}\text{the}\text{end}\text{of}\text{first}\text{year}=\text{Intial}\text{output}\times \text{growth}\text{rate}\text{at}\text{end}\text{of}\text{the}\text{year}\\ \text{=1000×0}\text{.08}\\ =80\end{array}$

Growth at the end of the year is $\$80$

  $\begin{array}{c}\text{Level of output at the end of first year}=1,000+80\\ =1,080\end{array}$

Similarly

Calculate the level of output for $4$ years with a growth rate of $8\%$ and add obtained value with growth rate.

Therefore, the level of output at the end of fourth year is $\$1,360.49$