Question

2. A firm has the production function y = 4LK. The marginal products are given by MPL = 4K and MPK = 4L. (a) Provide an expression for the long run total cost function. (b) Now suppose that wi = WK = 25. Write out the expression for the long run total cost curve, and plot it on a graph. (C) With wi = wK = 25, derive the long run average cost curve, and plot it on a graph. Does this firm experience increasing or decreasing average cost, as the output increases? Explain. (d) Suppose that wi = wk increase to 30. How will the cost minimizing L and K to produce a certain level of output change? Explain. How will the long run average cost curves change?

Answer 1

a)

In long run, all inputs are variable. Profit maximization requires that

MPL/MPK=wL/wK

4K/4L=25/25

K/L=1

K=L

Given y=4LK

Set K=L

y=4L2

L=y0.5/2

K=L=y0.5/2

Long run total cost is given as

LRTC=wL*L+wK*K

LRTC=wL*(y0.5/2)+wK*(y0.5/2)=0.5*(wL+wK)*y0.5

b)

Set wL=wK=25

LRTC=0.5*(25+25)*y0.5

LRTC=25y0.5

80.00 Total Cost, $ 02 8 10 Output, y

c)

LRATC=LRTC/y=(25y0.5)/y=25/y0.5

30.00 LRATC, $ 10.00 5.00 0.00 02 8 10 Output, y

We observe that ARATC is slopping downward. Firm faces decreasing average cost.

d)

In long run, all inputs are variable. Profit maximization requires that

MPL/MPK=wL/wK

4K/4L=30/30

K/L=1

K=L

Given y=4LK

Set K=L

y=4L2

L=y0.5/2

K=L=y0.5/2

We observe that cost minimizing L and K remains the same.

LRTC will be given by

LRTC=0.5*(30+30)*y0.5=30*y0.5

We observe that LRTC will increase at every output level.