Question

If the U.S. production function is Cobb–Douglas with capital share 0.3, output growth is 3 percent per year, depreciation is 4 percent per year, and the Golden Rule steady-state capital–output ratio is 4.29, to reach the Golden Rule steady state, the saving rate must be:

A) 17.5 percent. B) 25 percent. C) 30 percent. D) 42.9 percent.

Answer 1

Answer

The correct answer is (C) 30 percent.

It is given that It is a cobb douglas production function and Capital share = 0.3 i.e. a = 0.3.

Cobb Douglas function is given by :

Y = AK^aL^{1- a} = AK^0.3L^{1- 0.3} = AK^0.3L^0.7

=> Y/L = (AK^0.3L^0.7)/L = A(K/L)^0.3

=> y = Ak^0.3

Steady state occurs when Change in k = sy - dk = 0

where s = saving rate and d = depreciation rate = 4% = 0.04

=> sy = dk -------------------(1) steady state condition

Consumption per worker(c) = y - i where i = investment per worker = sy

=> c = sy

Golden rule level of capital is that level of steady state capital per worker(k) at which c is maximized.

Maximized : c = y - i = y - sy

From (1) we have sy = dk and as y = Ak^0.3

=> Maximized : c = y - sy = y - dk =  Ak^0.3 - dk

Maximized : c

First order condition :

dc/dk = 0 => 0.3Ak^-0.7 - d = 0

As, d = 0.04 and Golden rule level of capital per worker(k) = 4.29

=> 0.3A*4.29^-0.7 - 0.04 = 0

=> A = 0.04/[0.3(4.29^-0.7)] = 0.37

So, sy = dk => s(0.37k^0.3) = dk and k = 4.29

=> s = [0.04*4.29]/(0.37*4.29^0.3) = 0.30(approx)

So, saving rate = 0.30 = 30%.

Thus To reach the Golden Rule steady state, the saving rate must be 30%

Hence, the correct answer is (C) 30 percent.