1) Consider an economy described by the production function: Y = F (K, L ) = K1/

Question

1) Consider an economy described by the production function: Y = F (K, L ) = K1/2L1/2 where the depreciation rate is and the population growth rate is n.

a) Find the per worker production function.

b) Find the steady-state capital stock per worker as a function of the savings (s), depreciation rate () and population growth rate (n) c) Now assume the depreciation rate is 4% a year, population growth (n) is 2% a year and the savings rate is 24%. Find the steady-state level of capital per worker and the corresponding levels of output per worker and consumption per worker.

d) If the populatin growth rate were to increase what would happen to total output (Y) in the steady state? How about output per worker (y) in the steady state?

Explanation / Answer

Part a)

Y = K1/2 L1/2

Dividing the function by L, to obtain per worker production function we get,

Y/L = K1/2 L1/2 / L

Y/L = K1/2 / L1/2

y = k1/2 ( assumingY/L = y and K/ L= k)

This is the per worker production function

Part b)

The steady state capital stock per worker refers to the stock of capital at which investments are equal to depriciation

At equilibrium however we know, Investments (I) = Savings (S = sy)

Now steady state capital stock can be written as

Investments = Depriciation + Population growth rate

sy = k + nk

Part c)

From the above formula, steady state level of capital per worker (k*)

0.24 y = 0.04 k - 0.02 k

0.24 k1/2 = 0.06 k

k* = 16

Output per worker (y) = k1/2 = 4

Consumption per worker = (1-s) y = 0.76 * 4 = 3.04

Part d)

If the population rate (n) increases, the total output declines as a result

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