Question

An economy has a Cobb–Douglas production function:

Y=K^α(LE)^1-α

The economy has a capital share of 0.35, a saving rate of 43 percent, a depreciation rate of 3.50 percent, a rate of population growth of 3.50 percent, and a rate of labor-augmenting technological change of 2.5 percent. It is in steady state.

a.) Solve for capital per effective worker (k*), output per effective worker (y*), and the marginal product of capital.

K*=?

y*=?

marginal product of capital= ?

Answer 1

Answer-

Given-

Production Function- Y = K^a (LE) ^ (1-a)

Savings rate : s = 0.43

Population growth rate: n = 0.035

Depreciation rate : d = 0.035

Labour- augmented growth rate: g = 0.025

Capital share : a = 0.35

A)

Step-1Divide the Given production function by “LE’

Y/ LE = {  K^a (LE) ^ (1-a) } / LE

let y = Y/LE = output per effective worker

  k = K/LE = capital per effective worker

Then , y = ( K/LE) ^ a

or y = k^a

The production function in terms of per effective worker-

y = k^a

Step-2 Finding the equation for steady state

At steady state ,

Investment = Break even investment

s y = (d+n+g) k

Step-3 Substituting the given values

s k^a = ( 0.035 + 0.035 + 0.025) k

0.43 k ^ (0.35) = 0.095 k

k ^ ( 1-0.35) = 0.43 /0.095

k^ 0.65 = 4.526316

k = (4.526316)^(1/0.65)

k = 10.205593

Capital per effective worker = k = 10.205593

Now plugging thevalue of k in y

  y = k^ (.35)

y = (10.205593)^ (0.35)

y = 2.254724

Output per effective worker = y= 2.254724

Step-4 Calculating Marginal Product Of capital

dY/ dK = MPK = aK ^(a-1) .( LE) ^ (1-a)

In per effective worker terms,

MPK= dy/dk = ak^ ( a-1)

or MPK = 0.35(10.205593)^ (1-0.35)

or MPK = 1.584211

MPK = 1.584211