Macroeconomics question... i cant figure it out for some reason?

Question

Say that income in the EU grows an average of 2% a year and in India it grows at 6% a year. Say also that EU income is $32,000 and Indian income is $2000. In how many years will Indian income catch up with EU income? i need the equation to solve this! the answer is about 76 years but i cant get that!

Answer 1

OK you need to know about compound interest and logarithms.

To find out a country's income in x years time, with a constant growth rate, you take the initial income and multiply it by the rate of interest to the power of the number of years.

In this case, you want to know when the two values are equal and you want to know the exact time at which it happens. So create an unknown parameter x, which you will solve for.

You will have:

EU: 32000 * 1.02^x

India: 2000 * 1.06^x

Set these equal to each other:

32000 * 1.02^x = 2000 * 1.06^x

Divide through by 2000:

16 * 1.02^x = 1.06^x

Take natural logarithms and solve for x:

ln16 + x*ln1.02 = x*ln1.06

ln16= x*(ln1.06-ln1.02)

ln16= x*[ln(1.06/1.02)]

ln16/0.038 = x

x = 72.96 years, which is roughly 73 years (not sure where you got 76 from!)

Answer 2

Actually result is slightly different:

32'000 *1.02^x = 2'000 * 1.06^x

16 = (1.06/1.02)^x

Log of 16 by base of (1.06/1.02) ≈ 72.0784194

So X=72.0784194 years or = 72 years and 28.7 days.