Question

2) Suppose that the production function is given as: q=L^0.75 K^0.25
a) What is the formula for the average product of labor, AP? in terms of L and K?

b) What is the formula for the marginal product of labor, MP? in terms of L and K ?

c) If at the current production, cost minimizing level of capital, ?=16, and the market price of labor (?) is three times the rental price of capital (?), what is the cost minimizing level of labor (?) used in the production? 3) Consider

Answer 1

2) The production function is given as,

q = L^0.75 K^0.25

a) The average product of labor will be simply equal to the output q divided by labor L.

Average product of labor will be equal to,

APL = q/L

APL = (L^0.75 K^0.25)/L

APL = L^0.75 - 1 K^0.25

APL = L^-0.25 K^0.25

APL =(K/L)^0.25

So the average product of labor in terms of capital and labor is equal to, (K/L)^0.25

b) The marginal product of labor is equal to the partial derivative of production function with respect to labor L

MPL = q/L

MPL = (L^0.75 K^0.25)/L

MPL = K^0.25×(0.75 × L^0.75 - 1)

MPL = 0.75K^0.25 L^-0.25

MPL = 0.75 (K/L)^0.25

So the marginal product of labor in terms of labor and capital is equal to, 0.75(K/L)^0.25.

C) We are given that the market price of wage w is three times the rental rate r of capital. So

w = 3r

And we are also given that K = 16

And the cost minimization combination of capital and labor is chooses such that,

MPL/MPL = w/r

MPK = q/K

MPK = (K^0.25 L^0.75)/L

MPK = L^0.75 ( 0.25 × K^0.25 - 1)

MPK = 0.25 (L^0.72 K^-0.75)

MPK = 0.25 (L/K)^0.75

Putting in the values we get

0.75 × K^0.25/L^0.25 × 1/0.25 × K^0.75/L^0.75

0.75/0.25 × K/L = w/r

Now putting w = 3r and K = 16

0.75/0.25 × 16/L = 3r/r

3 × 16/L = 3

L = 16

So the cost minimizing combination of labor and capital is L= 16 and K =16.

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