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.*

- Consider a
**short-run**situation where K is fixed at 12,000 machine-hours.- What is the short-run production function for this firm?
- 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.
- 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)?
- 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

Q^{4}=12000L

**L=Q ^{4}/12000**

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

TC(Q)=12000*1200+8(Q^{4}/12000)

**TC(Q)=14400000+(1/1500)Q ^{4}**

iv)

For Q=600

**L=Q ^{4}/12000=600^{4}/12000=10,800,000
labor hours**

**TC(600)=14400000+(1/1500)600 ^{4}=$100,800,000**

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