Question

33 II) A firm’s production is represented by the following Cobb-Douglas function: ? = ?^1/3 ?^2/3. The rental rate, r, of capital is given by $100 and the price of labor is $200.

a. For a given level of output, what should be the ratio of capital to labor in order to minimize costs?

b. How much capital and labor should be used to produce those 300 units?

c. What is the minimum cost of producing 300 units?

d. What is the short run and long run cost of increasing output to 500 units?

e. Does this production function exhibit increasing, decreasing, or constant returns to scale? Please answer based on the cost calculations in parts c and d.

I need help with D

Answer #1

(D)

Q = K^{1/3}L^{2/3}

When Q = 300, w = $200 and r = $100,

L = 300, K = 300 (using data from previous parts).

(i) When Q = 500, in short run, capital is typically fixed and labor is typically variable. So K is fixed at 300 and L varies.

Production function = K^{1/3}L^{2/3} = 500

(300)^{1/3}L^{2/3} = 500

L^{2/3} = 500 / [(300)^{1/3}]

L = {500 / [(300)^{1/3}]}^{(3/2)} =
[(500)^{3/2}] / [(300)^{1/2}] = 11,180.34 / 17.32 =
645.50

Short run total cost ($) = wL + rK = (200 x 645.5) + (100 x 300) = 129,100 + 30,000 = 159,100

(ii) In long run, both L and K will vary.

MPL/MPK = (2K/L) = w/r = 2

2L = 2K

L = K

Substituting in production function,

(L)^{1/3}L^{2/3} = 500

L = 500

K = 500

Long run total cost ($) = wL + rK = (200 x 500) + (100 x 500) = 100,000 + 50,000 = 150,000

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