Question

Consider two countries, A and B.
In the last 30 years, the average annual growth rate of the real per capita GDP of country A is 1.5%
and country B is 2.0 %.
In the current year, the real per capita GDP of country A is $10,000, and the real per capita GDP of
country B is $X.
Based on the current real per capita GDP of the two countries and the average annual growth rate of
the real per capita GDP of the two countries in the last 30 years, we find that 25 years from now,
the two countries will have the same real per capita GDP.
(a) Solve for the real per capita GDP of country B in the current year, $X.
(b) According to the rule of 70, how many years will it take for country A to double its per capita real
GDP?
(c) Suppose that the production function of country A satisfies the assumption that the marginal
product of capital is positive but diminishing and that the growth of the real per capita GDP in
country A is driven purely by capital accumulation.
Then, would your answer in (b) tend to over-estimate or under-estimate the length of time that it
takes the country to double its real per capita GDP? Explain.

Answer 1

After n years, the real per capita GDP of countries A and B will be:

GDPA = 10,000*(1.015)n GDPB = X * (1.02)n

a)

If the two countries will have the same amount of real per capita GDP 25 years from now then;

GDPA = GDPB

10,000 * (1.015)25 = X * (1.02)25

Hence,

X = 10,000 * (1.015)25 / (1.02)25

X = $8,843.9598

Hence X is approximately equal to $8,844.

b)

The rule of 70 states that:

No. of years to double = 70/ Annual percentage growth rate

Hence for country A,

No. of years to double its per capita real GDP = 70/ 1.015 = 68.9655

No. of years to double the per capita GDP of country A is approximately 69 years.

c)

If the assumptions of marginal product of capital is positive and diminishing and the growth of real per capita GDP in country A is driven purely by capital accumulation are satisfied, then we can say that the annual growth rate of country A goes on diminishing. Hence we can say that the answer is part b tends to underestimate the length of time it takes for country A to double its real per capita GDP.