Question

# 3. A firm employs labor and capital by paying \$40 per unit of labor employed and...

3. A firm employs labor and capital by paying \$40 per unit of labor employed and \$200 per hour to rent a unit of capital. The production function is given by: Q = 120L – 2L2 + 240K – 3K2, where Q is total output.

a. At what level of labor (L) and capital (K) will output no longer increase (the maximum number to labor and capital that a firm would ever employ)?

b. Determine the firm’s optimal combination or ratio of capital (K) and labor (L)?

c. Given your answer in part b, and if the company employs 21 worker units (L), how many units of capital (K) will the employ, and how many units will be produced (Q)?

d. Using your answer in b, state the production function with K as the only variable (substitute out your Ls).

a)

A firm will employ L and K until their marginal product equals zero.

Q = 120L - 2L2 + 240K - 3K2

Marginal product of Labour = dQ/dL = 120 - 4L

120 - 4L = 0

L = 30

Marginal Product of capital = dQ/dL = 240-6K

240-6K = 0

K = 40

b)

For optimality, the firm equates the marginal rate technical substitution(MRTS) with the price ratio.

MRTS = Marginal product of Labour / Marginal Product of capital = 120 - 4L / 240-6K

Hence 120 - 4L / 240-6K = 40/200

120 - 4L / 240-6K = 1/5

600-20L = 240-6K

10L - 3K = 180

c)

If L = 21

10L - 3K = 180

210 - 3K = 180

3K = 30

K = 10

Q = 120L - 2L2 + 240K - 3K2

Q = 120*21 - 2*212 + 240*10 - 3*102 = 2520 - 882 + 2400 - 300 = 3738

d)

10L - 3K = 180

10 L = 180 + 3K

L = 18 + 3/10K

Put the value of L in production function we get,

Q = 120L - 2L2 + 240K - 3K2

Q = 120(18+3/10K) - 2(18+3/10K)2+ 240K - 3K2

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