Question

Consider a small island nation. Assume the economy is following the Solow Growth Model. Let K = $100 Billion dollars and L =100 million people. The production function is Y = K3/10 L 7/10. Let savings rate = 10% and depreciation rate = 5%.

1. Foreign Investment: Imagine the country in Question 2 did not suffer an earthquake (ignore Question #4). Instead, many foreign companies invested in the country. They added $100 Billion dollars to the capital.

a. What is the steady-state value of capital per capita, k*? (show your work)

b. What is steady-state value of output per capita, y*? (show your work)

Answer 1

a)

n = population growth rate

n = 0

d = depreciation rate  

d = 5% = 0.05

s = saving rate

s = 10% = 0.1

Y = K^3/10L^7/10

= K^0.3L^0.3

Y/L = K^0.3L^0.7/L

= K^0.3/L^{1- 0.7}  

= K^0.3/L^0.3  

= (K/L)^0.3

y = (k)^0.3 where y = Y/L output per worker

k = K/L capital per worker

Steady state condition

sy - (n + d)k = 0

sy = (n + d)k

s(k)^0.3 = (n+ d)k

(0.1)(k^0.3) = (0 + 0.05)k

0.1k^0.3 = 0.05k

0.1/0.05 = k^{1- 0.3}   

2 = k^0.7  

k = (2)^1/0.7

   = 2.69

thus, steady state value of capital per worker is 2.69

b)

y = (2.69)^0.3  

= 1.34  

thus, steady state value of output per worker is 1.34