Question

Suppose a firm’s long-run production function is given by Q=K^0.25 L^0.25 ,where K is measured in machine-hours per year and L is measured in hours of labor per year. The cost of capital (rental rate denoted by r) is $1200 per machine-hour and the cost of labor (wage rate denoted by w) is $12 per hour.

Hint: if you don’t calculate the exponential terms (or keep all the decimals when you do), you will end up with nice numbers on all parts below.

1. Consider a short-runsituation where K is fixed at 12,000 machine-hours.
   1. What is the short-run production function for this firm?
   2. What is an expression for the marginal product of labor (MPL), dQ/dL for this firm? At what levels of output is there a diminishing MPL? Explain.
   3. What is an equation for L(Q) the firm’s labor demand given some desired quantity of output Q? What is an equation for the firm’s total cost as a function of desired output, TC(Q)?
   4. Suppose the firm’s desired level of output is Q = 600. What is L(600) and TC(600)?

Answer 1

i)

Given

Q=K^0.25L^0.25

Put K=12000

Q=12000^0.25L^0.25

Short run production function is given by

Q=(12000^0.25)*L^0.25

ii)

Marginal Product of labor is given by

MPL=dQ/dL=(12000^0.25)*0.25*L^-0.75

MPL=dQ/dL=(12000^0.25)*0.25*L^-0.75

MPL=2.616588L^-0.75

We can see that as L increases, Q increases.

As L increases, MPL decreases.

So, we can say that MPL is decreasing for all increasing level of output.

MPL is decreasing for Q>0

iii)

We have derived in part i that

Q=(12000^0.25)*L^0.25 or

Q⁴=12000L

L=Q⁴/12000

Total Cost=TC(Q)=rK+L*w

TC(Q)=12000*1200+8(Q⁴/12000)

TC(Q)=14400000+(1/1500)Q⁴

iv)

For Q=600

L=Q⁴/12000=600⁴/12000=10,800,000 labor hours

TC(600)=14400000+(1/1500)600⁴=$100,800,000