Question

27) A firm has a production function Q = KL, where Q is the quantity of output, K is the amount of capital and L is the amount of labor. MPL=K and MPK=L.

a) Suppose that capital is fixed at K=10 in short run. In this case, the marginal product of labor is MPL=10. Does this production function exhibit diminishing marginal returns to labor?

b) Suppose that in the short run, K is fixed at 10. The interest rate is r=4 and the wage is w=1. What is the short run total cost curve?

c) In (b), what are the functions for fixed cost, variable cost, average fixed cost, average variable cost and average cost?

d) In the long-run, capital is also variable. Does this production function exhibit increasing, decreasing or constant returns to scale?

e) What is the long run cost function? What is average cost in the long run? Hint: Use MRTS=-MPL/MPK = -w/r as we discussed in class. Note that MPL=K and MPK=L.

f) Does this cost function exhibit increasing or decreasing economies of scale?

Answer 1

Q = KL

MPL= K

MPK= L

a)

If K=10, then

Q= 10L

MPL= 10

Differentiation of MPL:

dMPL/L= 0

Also if we producer double the quantity of L, then:

Q'= 10(2L)= 2 (10L)= 2Q

The output also doubled whuch implies constant marginal return to labor.

b)

r =4

W=1

Total cost(TC)= LW+Kr

TC= L+4K

If K=10, Q= 10L which implies L= Q/10

TC= Q/10 + 40 Total cost curve equation

c)

Fixed cost is the part of toal cost that remain fixed at all levels of quantity and it also exist at Q=0. So

TC= Q/10 +40

If Q=0, then

TC= 40= fixed cost(FC)

Variable cost is the cost that varies with output.

Variable cost(VC)= TC-TFC

VC= Q/10+40-40= Q/10

Average fixed cost(AFC) = FC/Q= 40/Q

Average variable cost(AVC)= VC/Q= Q/10Q= 1/10

Average cost(AC)= AVC+AFC= (1/10)+(40/Q)

d)

In long run:

Q= KL

If producer double the quantity of both the inputs, then:

Q'= (2K) (2L)

Q'= 4(KL)= 4Q

Here Output increases by more than the double so production function exhibits Increasing return to scale.