Question

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?

(A-D HAS BEEN ANSWERED)

e) What is the long run cost function? What is average cost in the long run?

A.The long run total cost function is C = 4Q^1/2 and the average cost is AC = 4Q^–1/2.

B.The long run total cost is C = L + 4K and the average cost is AC = (L+4K)/Q

C.The long run total cost function is C = 4L + K and the average cost is AC = 4 + K

D.The long run total cost function is C = 4√Q and the average cost is AC = 4/Q.

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

A. This cost function exhibits economies of scale

B. This cost function exhibits diseconomies of scale

C. This cost function exhibits neither economies nor diseconomies of scale

D. This production function exhibits economies of scale

Answer 1

a)

short run K = 10

Q = KL  

Q = 10L  

dQ/dL = MP_L   = 10  

MP_L   = 10  

dMP_L/dL = 0 therefore production function does not exhibit diminishing marginal returns to labor

b)  

Q = 10L

L = Q/10

w = 1

r = 4

Short run total cost function

STC = wL + rK  

= Q/10 + 4(10)

= Q/10 + 40

c)

FC = 40  

VC = Q/10  

AFC = FC/Q  

= 40/Q  

AVC = VC/Q  

= 1/Q(Q/10) = 1/10

ATC = STC/Q  

= 1/Q(Q/10 + 40)  

= 1/10 + 40/Q

d)  

Long run

Q(K,L) = KL

Q(tK,tL) = (tK)(tL)  

= t²(KL)  

= t²Q(K,L)  

Therefore production function exhibits increasing returns to scale

e)

Q = KL

MP_L = K

MP_K = L  

MRTS_L,K   = MP_L/MP_K  

= K/L  

w = 1  

r = 4

cost minimizing condition

MRTS_L,K = w/r  

K/L = 1/4  

L = 4K  

Put L = 4K in production function

Q = KL

Q = K(4K)

Q = 4K²

K² = Q/4

K = (Q/4)^1/2

K = Q^1/2/2

L = 4K

L = 4(Q^1/2/2 )  

L = 2Q^1/2  

Long run cost function C = wL + rK

C = (1)(2Q^1/2) + 4(Q^1/2/2 )

C = 2Q^1/2 + 2Q^1/2  

C = 4Q^1/2  

AC = C/Q  

AC = 4Q^1/2 /Q

AC = 4Q^{1/2 - 1}

AC = 4Q^{- 1/2}    

f)

AC = 4Q^{- 1/2}    

dAC/dQ = 4(-1/2)Q^{- 1/2 -1}

= - 2Q^{- 3/2} < 0  

Since dAC/dQ < 0 which means cost function exhibits economies of scale