Question

Per capita income is160 times greater in the U.S than in the Congo. per capita grows an average of 3% per year in the U.S. and 6 % per year in the Congo. how many years would it take before per capita in the Congo exceeds that of the United States?

(To simplify the math, suppose at the outset per capita income is $160,000 in the richer country and $1,000 in the poorer country.) I need to know how to figure it out.

Answer 1

`n=log(160)/log(1.06/1.03)~~176.77` years

Explanation:

Suppose that the Congo has per capita income of `$1`, while the US has `160` times greater of `$160` (to simplify calculation, any other values would also do).

Congo's per capita income grows at `6%` per year. Thus, next year it would be `$1*1.06=$1.06`, and the year after that would be `$1*1.06*1.06=$1*1.06^2`. After `n` years, the per capita income would rise to `$1*1.06^n`.

Similarly, the US per capita income would rise to `$160*1.03^n` after `n` years.

Congo's per capita income would exceed that of the US when

`1.06^n=160*1.03^n`

`160=1.06^n/1.03^n=(1.06/1.03)^n`

Take the logarithm of both sides:
`log(160)=log((1.06/1.03)^n)=nlog(1.06/1.03)`

where we use the fact that `log(a^b)=blog(a)` on the right-hand side.

Thus,
`n=log(160)/log(1.06/1.03)~~176.77` years.