Chapter 19, Problem 1.1P

To determine

The expected years that the per capita income of the Democratic Republic of Congo will take to surpass the per capita income of the United States of America.

Concept Introduction:

The per capita income index is an indicator of how much an individual earns in comparison to his peers in an economy during a specific period of time. This indicator helps to understand the standard of living of individuals in different countries by comparing their average per capita income with one another.

Explanation of Solution

Currently the per capita income of United States is 160 times greater than the per capita income of Congo. The per capita income growth of United States is 3% per annum in comparison to 6% per annum per capita growth of Congo. Thus, we can understand that Congo’s per capita income growth is 3% faster per annum than the United States.

To calculate the number of years that Congo requires for surpassing the per capita growth of United States:

$\begin{array}{l}{\left(l+g\right)}^t=N\\ tlog\left(l+g\right)=log\left(N\right) \\ t =\frac{log\left(N\right)}{log\left(l +g \right)} \end{array}$

Where,

g = growth rate

t = number of years required

n = per capita income

$\begin{array}{l}t=\frac{log\left(160\right)}{log\left(1.03\right)} \\ = \frac{ 5.075 }{0.0295}\\ \begin{array}{l} \hfill \\ =171.70\hfill \end{array}\end{array}$

Hence, in 171.70 years, the Democratic Republic of Congo’s per capita income will surpass the per capita income of the United States of America, if both the countries continue to grow at the current growth rate annually.