If real GDP per capita in a year was 50% greater than a decade earlier and the population grew by 20% over the same period, while the real GDP per capita did not experience a decli... If real GDP per capita in a year was 50% greater than a decade earlier and the population grew by 20% over the same period, while the real GDP per capita did not experience a decline in any year, what is the minimum increase in nominal GDP over the decade? Read more

Steps to Solve

1. Understanding Growth Rates We start with what is given: the real GDP per capita increased by 50%, and the population grew by 20%.

   Let’s assume the original real GDP per capita was $R$ and the original population was $P$. Then the new real GDP per capita can be expressed as: $$ R' = R \times (1 + 0.50) = 1.5R $$

2. Calculating Total New Real GDP The total real GDP after the growth can be calculated as follows: $$ \text{New Real GDP} = R' \times \text{New Population} $$ The new population is: $$ P' = P \times (1 + 0.20) = 1.2P $$ Thus, we have: $$ \text{New Real GDP} = 1.5R \times 1.2P = 1.8RP $$

3. Calculating Initial Real GDP The initial real GDP can be calculated as: $$ \text{Initial Real GDP} = R \times P $$

4. Determining the Growth Rate of Nominal GDP The nominal GDP must grow in a way that maintains the growth of real GDP per capita and accommodates population growth. The growth rate ($g$) can be represented as: $$ \text{Nominal GDP Growth Rate} = \frac{\text{New Nominal GDP} - \text{Initial Nominal GDP}}{\text{Initial Nominal GDP}} $$ Since nominal GDP must cover real GDP growth while adjusting for population: $$ \text{Nominal GDP} = \text{New Real GDP} $$

5. Setting up the Equation Now, we set the new nominal GDP as: $$ \text{New Nominal GDP} = (1 + g) \text{Initial Nominal GDP} $$ This can be equated to the calculated new real GDP: $$ 1.8RP = (1 + g)(RP) $$

6. Solving for Growth Rate Dividing both sides by (RP) (assuming (R) and (P) are not zero): $$ 1.8 = 1 + g $$ Thus, we can isolate (g): $$$ g = 1.8 - 1 = 0.8 \text{ or } 80% $$ However, to find the minimum growth rate, we take into account that real GDP per capita doesn't decrease, so we need to mitigate the effects of the population growth. This gives: $$ g = 0.8 - 0.2 = 0.6 \text{ or } 60% $$