Question

An economy starts off with a GDP per capita of$\$5,000$. How large will the GDP per capita be if it grows at an annual rate of $2\%$for $20$years? $2\%$for $40$years?

$4\%$for $40$years? $6\%$for $40$years?

Short answer

Per capita GDP if the economy grows at an annual rate of $2\%$for $20$years will be$\$7429.74$.

Per capita GDP if the economy grows at an annual rate of $2\%$for $40$years will be $\$11,040.20$

Per capita GDP if the economy grows at an annual rate of $4\%$for $40$years will be $24,005.103.

Step-by-step solution

Step 1. Define GDP.

The Gross Domestic Product (GDP) can be defined as the value of all the final goods and services produced in a geogrphical boundary during a given period.

Step 2. Find value of GDP if it grows at an annual rate of 2% for 20 years.

Per capita GDP if the annual growth rate of an economy remains at $2\%$for $20$years $={\left(1+\frac{2}{100}\right)}^{20}\times \$5000$

$\begin{array}{rcl}& =& {\left(\frac{100+2}{100}\right)}^{20}\times \$5000\\ & =& {\left(\frac{102}{100}\right)}^{20}\times \$5000\\ & =& (1.02{)}^{20}\times \$5000\\ & =& 1.485\times \$5000\\ & =& \$7429.74\end{array}$

Step 3. Find value of GDP if it grows at an annual rate of 2% for 40 years.

Per capita GDP if the annual growth rate of an economy remains at $2\%$for $40$years$={\left(1+\frac{2}{100}\right)}^{40}\times \$5000$

$$\begin{gathered} ={\left(\frac{100+2}{100}\right)}^{40}\times \$5000 \\ ={\left(\frac{102}{100}\right)}^{40}\times \$5000 \\ =(1.02{)}^{40}\times \$5000 \\ =2.208\times \$5000 \\ =\$11,040.20 \end{gathered}$$

Hence, the Per capita GDP if the economy grows at an annual rate of $2\%$for $40$years will be $\$11,040.20$.

Step 4. Find value of GDP if it grows at an annual rate of 6% for 40 years.

Per capita GDP if the annual growth rate of an economy remains at $4\%$for $40$years$={\left(1+\frac{4}{100}\right)}^{40}\times \$5000$

$$\begin{gathered} ={\left(\frac{100+4}{100}\right)}^{40}\times \$5000 \\ ={\left(\frac{104}{100}\right)}^{40}\times \$5000 \\ =(1.04{)}^{40}\times \$5000 \\ =4.801\times \$5000 \\ =\$24,005.103 \end{gathered}$$