Question

Consider a Solow economy with the following production function

F(K,N) = zK^(1/3)N^(2/3)

and parameters d = 0.05, s = 0.2, N0 = 100 and z = 1.0. Suppose K = 300 in period 0 and the

unit period is one year. In contrast to the standard Solow model, we assume that the population

growth rate n is no longer exogenous but rather endogenous and determined by

(1 + n) = N’/N = g(C/N) = (C/N)^3 as it is the case in the Malthusian model.

1) Determine the dynamics for the per worker capital (k). This is the first question in the problem

Answer 1

Hi

The answer of the following question are as follows :

The given parameters are as follows :

F(Kt ,Nt) = zKt1/3Nt2/3

d (depreciation rate) = 5% (or 0.05)

s (saving rate) = 20% (or 0.2)

N0 (initial labor stock) = 100

K0 (initial capital stock) = 300 C0 = (1-s)* F(K0,N0) = 0.8 * 3001/3*1002/3

z =1 , therefor the new production function would be : F(Kt ,Nt) = Kt1/3Nt2/3

Now Converting the Production function in per capita term : yt = kt1/3

1+n = (C0 / N0) = 0.8 * 3001/3*1002/3/100 = 1.15

The next year capital stock (Kt+1) = Today's total savings - Today's total depreciation

So Today's total savings = sF(Kt ,Nt) = 0.2 * Kt1/3Nt2/3

Today's total depreciation = dKt = 0.05*Kt

Therefore, Kt+1 = 0.2 * Kt1/3Nt2/3 - 0.05*Kt (1)

Let's Divide both side in (1) by Nt , now we get :

(Kt+1/Nt) =( 0.2 * Kt1/3Nt2/3 - 0.05*Kt )/ Nt

=> (Kt+1/Nt) = 0.2 * (Kt1/3/Nt1/3) - 0.05*(Kt / Nt )

=> (1+n) (Kt+1/Nt+1) = 0.2 * (Kt1/3/Nt1/3) - 0.05*(Kt / Nt ) { as Nt+1 = (1+n)Nt

=> 1.15 kt+1 = 0.2*kt1/3 - 0.05 * kt (2)

At steady state kt+1 = kt

Therefore from (2) we have :

=>1.15 kt = 0.2*kt1/3 - 0.05 * kt (3)

By Solving (3) for kt we have : kt* = 0.06

Now Substituting kt* = 0.06 in the production function in per-capita form, we have :

yt* = (kt*)1/3 = 0.39 (per -capita output at steady state)per -capita consumption at steady state level = 0.8 * yt* = 0.8 * .39 = 0.312.

I hope I have served the purpose well

Thanks.