Chapter 20, Problem 1P

To determine

(a)

To compute:

The years taken by a country to double its output.

Answer to Problem 1P

The years taken by a country to double its output is shown in the table below:

0.5 percent	140 years
1 percent	140 years
1.4 percent	140 years
2 percent	140 years
2.8 percent	140 years
3.5 percent	140 years
7 percent	140 years

Explanation of Solution

The calculation for years taken to double the annual growth rate:

By applying Rule $70$,

The number of years is calculated by dividing $70$ with the growth rate.

1. If the growth rate is 0.5%, then $140$ years will be required to double the output. $\left(\text{70}÷\text{0}\text{.5}=\text{140 years}\right)$

2. If the growth rate is $1\%$, then $70$ years will be required to double the output. $\left(\text{70}÷\text{1}=\text{70 years}\right)$

3. If the growth rate is $1.4\%$, then $50$ years will be required to double the output. $\left(\text{70}÷\text{1}\text{.4}=\text{50 years}\right)$

4. If the growth rate is $2\%$, then $35$ years will be required to double the output. $\left(\text{70}÷\text{2}=\text{35 years}\right)$

5. If the growth rate is $2.8\%$, then $25$ years will be required to double the output. $\left(\text{70}÷\text{2}\text{.8}=\text{25 years}\right)$

6. If the growth rate is $3.5\%$, then $20$ years will be required to double the output. $\left(\text{70}÷\text{3}\text{.5}=\text{20 years}\right)$

7. If the growth rate is $7\%$, then $10$ years will be required to double the output. $\left(\text{70}÷\text{7}=\text{10 years}\right)$

Economics Concept Introduction

Annual growth rate:

Annual growth rate refers to the rate at which the value of an investment changes over a particular timeperiod.

Ruleof 70:

The years needed by a country to double its output at a given growth rate is calculated by the rule of 70. It is calculated or estimated by dividing 70 with the given growth rate.

To determine

(b)

To compute:

The value of real GDP in $50$ years at the given annual growth rate.

Answer to Problem 1P

The real GDP in 50 years when the annual growth rate is 1.4% is $200.40, for annual growth rate of 2.8% is $397.79 and for annual growth rate of 7% is $2,945.70.

Explanation of Solution

The country's real GDP is $100 billion.

Calculation for real GDP is shown below for different growth rates:

Using the compound interest formula,

Real GDP in $50\text{years}$ when the annual growth rate is $1.4\text{percent}$ :

  $\begin{array}{c}\text{Real GDP in 50 years}=\text{ Present GDP}\times {\left(\text{1}+\text{0}\text{.014 }\right)}^{50}\\ =\text{100}{\left(\text{1}+\text{0}\text{.014}\right)}^{50}\\ =\text{\$200}\text{.40}\end{array}$

Real GDP in $50\text{years}$ when the annual growth rate is $2.8\text{percent}$ :

  $\begin{array}{c}\text{Real GDP in 50 years}=\text{Present GDP}\times {\left(\text{1}+\text{0}\text{.028 }\right)}^{50}\\ =1\text{00}{\left(\text{1}+\text{0}\text{.028}\right)}^{50}\\ =\text{\$397}\text{.79}\end{array}$

Real GDP in $50\text{years}$ when the annual growth rate is $7\text{percent}$ :

  $\begin{array}{c}\text{Real GDP in 50 years}=\text{Present GDP}\times {\left(\text{1}+\text{0}\text{.07 }\right)}^{50}\\ =\text{100}{\left(\text{1}+\text{0}\text{.07}\right)}^{50}\\ =\text{\$2,945}\text{.70}\end{array}$

Economics Concept Introduction

Annual growth rate:

Annual growth rate refers to the rate at which the value of an investment changes over a particular timeperiod.

Real GDP:

Real gross domestic product is defined as a measure of economic output with respect to changes in prices.It is a macro-economic measure.