Question

1. In previous problem, the given Cobb-Douglas
production function was **Q = 6 L ^{½}
K^{½}** and the cost function was given as:

** C = 3L + 12K**. For $384
of total cost, the optimum labor usage was determined to be 64, and
capital of 16.

a. If the cost function now changes to C = 3L + 18K, it implies that the total cost will become $480. Compute the new level of total cost for Q = 192. Can you explain why it is less than $480?

b. Determine the change in average total cost with the new cost function. (The original AC from problem set 3 was average cost = $2.00 per unit.)

b. If the production
function were become **Q = 6 L ^{.6}
K^{.4}**with the same cost function, would more or
less labor be used in the long run? Support your answer, but you do
not need to compute the exact final value of L. (Hint: The solution
to this question deals with the marginal productivities
MP

Answer #1

Q = 6L^{1/2}K^{1/2}

(a) New price ratio = 3/18 = 1/6

Cost is minimized when MPL/MPK = w/r = 1/6

MPL =
Q/L
= 6 x (1/2) x (K/L)^{1/2}

MPK =
Q/K
= 6 x (1/2) x (L/K)^{1/2}

MPL/MPK = K/L = 1/6

L = 6K

Substituting in production function,

6L^{1/2}K^{1/2} = 192

L^{1/2}K^{1/2} = 192/6 = 32

Squaring both sides,

L x K = 1,024

6K x K = 1,024

6K^{2} = 1,024

K^{2} = 170.67

K = 13.06

L = 6 x 13.06 = 78.36

Total cost ($) = (3 x 78.36) + (18 x 13.06) = 235.08 + 235.08 = 470.16

Cost is less than $480 because optimal usage of the more expensive input, i.e. capital, is now lower.

(b)

When Q = 192,

Average cost ($) = TC/Q = 470.16/192 = 2.45

Average cost has increased by $(2.45 - 2) = $0.45.

(c)

Q = 6L^{0.6}K^{0.4}

MPL =
Q/L
= 6 x 0.6 x (K/L)^{0.4}

MPK =
Q/K
= 6 x 0.4 x (L/K)^{0.6}

MPL/MPK = (0.6/0.4) x (K/L) = 3K/2L = w/r = 1/6

2L = 18K

L = 9K

Therefore, more labor will be used in long run.

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