Chapter 6, Problem 1TY

To determine

Difference between the growth rate of GDP among countries A and B.

Explanation of Solution

GDP measures the value of all final goods and services produced by an economy in an accounting year.

It is given that the economy of countries A and B grow at an annual rate of 3 percent and 4 percent respectively. And the two countries start with a similar GDP.

The growth rate after 25 years can be calculated using the following formula-

  $\text{FV}\text{=}{\text{PV(1 + rate of growth)}}^{\text{t}}\left(\text{1}\right)$

Here,

FV represents the future value of the growth rate of GDP

PV represents the present value of GDP

t represents the total number of years

Assuming, the GDP of both countries equal to $10 at the initial stage. Plug the given values in (1) to determine the value of the GDP of both the countries after 25 years.

Country A

  $\begin{array}{c}\text{FV}\text{=}10{\left(\text{1+}\text{0}\text{.03}\right)}^{25}\\ =10{\left(1.03\right)}^{25}\\ \text{=}10\left(2.094\right)\\ \text{=}\text{20}\text{.94}\end{array}$

Thus, the future value of country A's GDP is equal to $20.094.

Country B

  $\begin{array}{c}\text{FV}\text{=}10{\left(\text{1+}\text{0}\text{.04}\right)}^{25}\\ =10{\left(1.04\right)}^{25}\\ \text{=}10\left(2.665\right)\\ \text{=}\text{26}\text{.65}\end{array}$

Thus, the future value of country B's GDP is equal to $26.65.

After 25 years, country B's economy is larger than country A by the following proportion,

  $\begin{array}{c}\left(\frac{\text{Future}\text{value}\text{of}\text{GDP}\text{of}\text{country}\text{B}}{\text{Future}\text{value}\text{of}\text{GDP}\text{of}\text{country}\text{A}}\right)\text{-1}\\ \text{= }\left(\frac{\text{26}\text{.65}}{\text{20}\text{.94}}\right)\text{-1}\\ \text{=}1.27\text{-1}\\ \text{=0}\text{.27}\end{array}$

Thus, country B's economy is approximately 27% higher than country A. The answer is not 25% because the rate at which the economy is growing is compounding.