Question

Suppose an economy is characterized by the following production function: Y = 3K^0.1L^ 0.9 Assume the saving rate is 10%, the depreciation rate is 3% and the population growth rate is 2%. a) Find the per worker production function, steady-state capital per worker and output per worker. b) Find steady-state consumption per worker, investment per worker, and saving per worker. c) Briefly explain what would happen to steady-state values of output per worker, capital per worker, and consumption per worker if the saving rate increased from 10% to 15%.

Answer 1

a)we are given production function as 3K^0.1L^0.9 = Y

dividing Y by L we get

Y/L= 3(K/L) ^0.1 which is y= 3k^0.1

per worker production function is given by y= 3k^0.1

steady state is given by

sf(k)= (n+dep)k

we are given s= 0.1 or 10%

depreciation rate is 3% and the population growth rate is 2%

putting values in the solow model staedy state we get

0.1(3k^0.1) = (0.03+0.02)k

0.3k^0.1 = 0.05k

0.3/ 0.05= k^0.9

so k = 6 ^1/0.9 = 7.32

y= 3k^0.1 = 3 (7.32)^0.1= 3.660

so output per worker is y= 3.66 , capital per worker is 7.32 and production function at steady state is y=3k^0.1

b)investment is given by

I= sY

per worker is

I/L= sY/ L

i=sy

=> i= 0.1(3.66)= 0.366

consumption is given by C=(1-s)Y

per worker is given by C/L= (1-s) Y/L

c= 0.9 y

=> c= 0.9 (3.66) = 3.294

S=sY

so saving per worker is same as investemnet per worker which is 0.366

c)if s changes to 0.15%, then steady state is

sf(k) = (n+dep)k

0.15(3k^0.1) = (0.05)k

0.45k^0.1= 0.05k

9=k^0.9

k= 11.48 which is capital per worker

so output per worker will be y = 3k^0.1= 3.829

consumption per worker is

per worker is given by C/L= (1-s) Y/L

c= 0.85 y

c= 0.85*3.829 = 3.254 which is consumption per worker