Endogenous Growth and Convergence
1. Let output, Y, be a constant proportion of capital, K.
Y = aK
Investment, I, is a constant proportion of output:
I = bY
With depreciation at rate the net change in the capital stock is:
I - K = bY - K = baK - K
The proportionate change in the capital stock is:
(baK - K) / K = ba -
The impact of b changing from 0.1 to 0.2 clearly depends upon the size of a. If a = 0.4 and = 0.03 then a move in b from 0.1 to 0.2 raises growth from 1% (0.01) to 5% (0.05). If a = 0.5 then this shift in b will raise growth from 2% to 7%.
2. If the size of the gap between the countries is Gt at time to then the gap evolves according to G t+1 =0.98 Gt
In the next period G again falls to 98% its original level so that G t+2 =0.98 Gt+1 = (0.98)2 Gt
After 20 years the gap is (0.98)20 = 67% its original level. The gap has fallen by about one third.
After 40 years the gap is (0.98)40 = 45% its original level. The gap has fallen by about 55% of its original level.
3. Output for the economy with high investment and higher level of TFP is at B whilst low TFP economy with low investment is at A. Allowing only for differences in investment understates extent of inequality across countries.
4. Under these assumptions the MPK schedule looks like
Countries to the left of A will receive no investment as their MPK is less than the interest rate and so investment earns a negative return. These…