Question

Assume real per capita GDP in West Swimsuit is $6000 while in South Homestead it is $1500. The annual growth rate in West Swimsuit is 2.33%, while in South Homestead it is 7%. What will the income of the two countries be when it is equal?

Answer 1

Current incomes are Yws0 = 6000 and Ysh0 = 1500. Growth rates are rws = 2.33% and rsh = 7%. Let they are equal after n years. We thus have

Yws0(1 + rws)^n = Ysh0(1 + rsh)^n

6000(1 + 2.33%)^n = 1500(1 + 7%)^n

4(1 + 2.33%)^n = (1 + 7%)^n

Taking natural logs

ln 4 + n * ln (1.0233) = n * ln (1.07)

1.3863 + n*0.0230 = n*0.0677

This gives n = 1.3863/0.0446 = 31.065 years

After 31 years these incomes will be

Yws31 = 6000(1 + 2.33%)^31.065 = 12271 and Ysh31 = 1500(1 + 7%)^31.065 = 12271.

Approximately 31 years will be the time.