Question

Suppose that per capita GDP of India and USA are growing at annual average rates of 8.8% and 1.8%, respectively. Further, consider that in 2019-20, per capita GDP of USA was USD 41099 and per capita GDP of India was USD 1570. Assuming that the two countries continue to grow at the above rates, India's per capita GDP will be equal to the per capita GDP of USA in _____ years (round off to 2 decimal places).

Answer 1

Correct Answer: 45

To determine when India's per capita GDP will equal that of the USA, given their respective growth rates, we use the formula for exponential growth: GDP = GDP₀(1 + r)^t, where GDP₀ is the initial GDP, r is the growth rate, and t is time in years.
Given:

• India's initial GDP (GDP_0,Ind): USD 1570
• India's growth rate (r_Ind): 8.8% = 0.088
• USA's initial GDP (GDP_0,USA): USD 41099
• USA's growth rate (r_USA): 1.8% = 0.018

We need to find t such that:
1570(1 + 0.088)^t = 41099(1 + 0.018)^t
Taking the natural logarithm on both sides:
ln[1570(1.088)^t] = ln[41099(1.018)^t]
Which simplifies to:
ln(1570) + tln(1.088) = ln(41099) + tln(1.018)
Rearranging terms:
tln(1.088) - tln(1.018) = ln(41099) - ln(1570)
Factoring t out:
t(ln(1.088) - ln(1.018)) = ln(41099) - ln(1570)
Solving for t:
t = [ln(41099) - ln(1570)] / [ln(1.088) - ln(1.018)]
Calculating each part:

• ln(1570) ≈ 7.360
• ln(41099) ≈ 10.626
• ln(1.088) ≈ 0.0844
• ln(1.018) ≈ 0.0178

Now, substituting these values:
t = (10.626 - 7.360) / (0.0844 - 0.0178) ≈ 49.727 years
This result, approximately 49.73 years.