Question

1. Suppose China’s GDP is growing by 7% a year and its population grows by 1% a year. Also suppose that US GDP grows by 3% a year and its population grows by 1% a year, and that in 2019 US GDP is 1.5 times China’s while US population is one quarter of China’s. If these growth rates continue                                    
   1. By what year will China’s GDP double? Its population? Its per capita GDP?
   2. In what year will China’s GDP equal US GDP?
   3. In what year will China’s GDP per capita equal US GDP per capita?

Answer 1

(a) Let doubling period be N years.

(i) If initial GDP be G:

G x (1.07)^N = 2 x G

(1.07)^N = 2

N x ln 1.07 = ln 2

N x 0.0677 = 0.6931

N = 10.24 years

(ii) If initial population be G:

G x (1.01)^N = 2 x G

(1.01)^N = 2

N x ln 1.01 = ln 2

N x 0.01 = 0.6931

N = 69.66 years

(iii) Per capita GDP growth rate = GDP growth rate - Population growth rate = 7% - 1% = 6%

If initial per capita GDP be G:

G x (1.06)^N = 2 x G

(1.06)^N = 2

N x ln 1.06 = ln 2

N x 0.0583 = 0.6931

N = 11.89 years

(b)

Let China's current GDP be G, then US current GDP = 1.5 x G

Let the year of equality be N. Then

1.5 x G x (1.03)^N = G x (1.07)^N

(1.07/1.03)^N = 1.5

(1.0388)^N = 1.5

N x ln 1.0388 = ln 1.5

N x 0.0381 = 0.4055

N = 10.64 years

(c)

US Per capita GDP growth rate = US GDP growth rate - US Population growth rate = 3% - 1% = 2%

Let China's current GDP be G and current population be P. Then China's Per capita GDP = (G / P)

US current GDP = 1.5 x G

US current population = P/4 = 0.25 x P

US Per capita GDP = (1.5 x G) / (0.25 x P) = 6 x (G / P)

Let the year of equality be N. Then

6 x (G / P) x (1.02)^N = G x (1.06)^N

(1.06/1.02)^N = 6

(1.0392)^N = 6

N x ln 1.0392 = ln 6

N x 0.0385 = 1.7918

N = 46.60 years