Question

1. Explain the graph in the Solow Model (exogenous growth model) that relates capital intensity and output per worker.

Answer 1

In the exogenous growth model, we have capital intensity k which is the ratio of capital(K) and labor(L)

Output per worker is y=Y/L

We know, savings in the economy is sY

Also, Y=F(K,L)

We know,

Lt=(1+n)Lt-1 where Lt is the labor at time t and n is the population growth rate.

Also, we know, Kt+1=St+(1-d)Kt-1 where d is the depriciation rate of capital and St is the savings at time t

We know St=sF(Kt,Lt) where F is the production function and s is the savings rate.

Now, we know kt=Kt/Lt, St=sALtf(kt)

Thus, Kt+1=sALt+(1-d)Kt

Now, kt+1=Kt+1/Lt+1=[sALtf(kt)+(1-d)Kt]/(1+n)Lt=[sAf(kt)+(1-d)kt]/(1+n)

So, we have derived (1+n)kt+1=sAf(kt)+(1-d)kt

Here, the orange line is (1+n)kt+1=sAf(kt)+(1-d)kt

Now, to have a steady state, the depletion and accumulation of capital per capita should be the same.

So, to keep per capita capital constant, the amount of investment must be (n+d)*kt

Thus, we have derived at the figure above.