Question

In: Economics

Consider a Solow economy with the following production function F(K,N) = zK^(1/3)N^(2/3) and parameters d =...

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

Solutions

Expert Solution

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.


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