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.

Question: Find k* the steady state per-capita capital stock, consumption per capita (c*) and output

per capita (y*).

Answer 1

We are given :

F(K_t ,N_t) = zK_t^1/3N_t^2/3

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 * 300^1/3*100^2/3

z =1 , therefor the new production function would be : F(K_t ,N_t) = K_t^1/3N_t^2/3  

Converting Production function in per capita term : y_t = k_t^1/3

1+n = (C₀ / N₀) = 0.8 * 300^1/3*100^2/3/100 = 1.15

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

Today's total savings = sF(K_t ,N_t) = 0.2 * K_t^1/3N_t^2/3

Today's total depreciation = dK_t = 0.05*K_t

Therefore, K_t+1 = 0.2 * K_t^1/3N_t^2/3 - 0.05*K_t (1)

Divide both side in (1) by N_{t ,} we get :

(Kt+1/N_t) =( 0.2 * K_t^1/3N_t^2/3 - 0.05*K_t )/ N_t

=>  (Kt+1/N_t) = 0.2 * (K_t^1/3/N_t^1/3) - 0.05*(K_t / N_t )

=> (1+n) (Kt+1/N_t+1) = 0.2 * (K_t^1/3/N_t^1/3) - 0.05*(K_t / N_t ) { as N_t+1 = (1+n)N_t

=> 1.15 k_t+1 = 0.2*k_t^1/3 - 0.05 * k_t (2)

At steady state k_t+1 = k_t

Therefore from (2) we have :

=>1.15 k_t = 0.2*k_t^1/3 - 0.05 * k_{t (3)}

Solving (3) for k_t we have : k_t^* = 0.06

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

y_t^* = (k_t^*)^1/3 = 0.39 (per -capita output at steady state)

Per -capita consumption at steady state level = 0.8 * y_t^* = 0.8 * .39 = 0.312