Question

1. Given the production function Y = zK^1/3N^2/3, complete the below table by including the values corresponding to the Golden Rule of Capital Accumulation capital per worker, output per worker, and consumption per worker.   Assume that d = 0.075 across all countries.

	z	n	k	Y	c
United States	1.00	0.01
Argentina	0.45	0.014
Thailand	0.23	0.015
Cameroon	0.05	0.028

Answer 1

Per capita production function:

Y/N = z x (K/N)^1/3  [dividing both sides by N]

y = z x k^1/3

At Golden Rule level, MPK = d + n

From aggregate production function:

MPK = Y/K = (1/3) x z x (N/K)^2/3 = (z/3) / k^2/3 = 0.075 + n

Per capita consumption: c = y - [(d + n) x k]

(A) US

(1/3) / k^2/3 = 0.075 + 0.01

(1/3) / k^2/3 = 0.085

k^2/3 = (1/3) / 0.085 = 3.92

k = (3.92)^(3/2) = 7.77

y = 1 x (7.77)^(1/3) = 1.98

c = 1.98 - (0.085 x 7.77) = 1.98 - 0.66 = 1.32

(B) Argentina

(0.45/3) / k^2/3 = 0.075 + 0.014

0.15 / k^2/3 = 0.089

k^2/3 = 0.15 / 0.089 = 1.69

k = (1.69)^(3/2) = 2.19

y = 0.45 x (2.19)^(1/3) = 0.58

c = 0.58 - (0.089 x 2.19) = 0.58 - 0.19 = 0.39

(C) Thailand

(0.23/3) / k^2/3 = 0.075 + 0.015

(0.23/3) / k^2/3 = 0.09

k^2/3 = (0.23/3) / 0.09 = 0.85

k = (0.85)^(3/2) = 0.79

y = 0.23 x (0.79)^(1/3) = 0.21

c = 0.21 - (0.09 x 0.79) = 0.21 - 0.07 = 0.14

(D) Cameroon

(0.05/3) / k^2/3 = 0.075 + 0.028

(0.05/3) / k^2/3 = 0.103

k^2/3 = (0.05/3) / 0.103 = 0.16

k = (0.16)^(3/2) = 0.07

y = 0.05 x (0.07)^(1/3) = 0.02

c = 0.02 - (0.103 x 0.07) = 0.02 - 0.007 = 0.01