Consider the Cobb-Douglas production function f(L,K) = L^2/3 K^1/3 . The price o

Question

Consider the Cobb-Douglas production function f(L,K) = L^2/3 K^1/3 . The price of labor is w= 2 and the price of capital is r=3 Suppose that in the short run, capital is fixed at K = 100.

1.What are the returns to scale in the short run?

2. In the long run where capital can be adjusted freely by the firm, what are the returns to scale? What is the shape of the average cost curve? the marginal cost curve? You do not need to justify your answer.

3. Derive the long run average cost of the firm using the method of your choice, and plot it in your previous graph. Now assume that in the long run, capital can only be varied in discrete amounts. To simplify, assume that capital can only take on the two values K = 100 and K = 400.

4. Derive the short run average cost function when K = 400. Plot it in your previous graph.

5. Show the new long run average cost curve on your graph. (You can draw by hand on the graph using the mouse.)

6. Comment on the shapes of the two long run average cost curves that you derived.

Explanation / Answer

A) In the short run only labor is variable and so Q becomes Q = L2/3(100)1/3. Now increase L by A times. So that:

Q = (AL)2/3(100)1/3

Q = A2/3L2/3(100)1/3

Q = A2/3Q

This implies that output increases by a lesser amount. So in the short run there are diminishing returns.

B) Returns to scale can be computed from output elasticities of labor and capital. Since alpha = 2/3 and beta = 1/3, alpha + beta = 1. So there are constant returns to scale. Confirm this by increasing L and K by A times. So that:

Q = (AL)2/3(AK)1/3

Q = A2/3L2/3A1/3K1/3

Q = AL2/3K1/3

This implies that output increases by a same amount. So in the long run there are constant returns. Long run AC envelops all the short run AC's so that both the marginal cost and average costs are U-shaped

C) The marginal product function of labor is the partial derivative of the production function with respect to labor. This implies

MPL   = dQ/dL

            = d(L2/3K1/3)/dL

            = (2/3)(K/L)1/3

Similarly, the marginal product function of capital is the partial derivative of the production function with respect to capital

MPK    = dQ/dK

            = d(L2/3K1/3)/dK

            = (1/3)(L/K)2/3

Note that rate of technical substitution RTS is the ratio of marginal products of labor and capital

At equilibrium level, RTS is equal to w/r

MPL /MPK= w/r

(2/3)(K/L)1/3 /(1/3)(L/K)2/3 = 2/3

2K/L = 2/3

L = 3K

The equation found above indicates the input mix to be used to produce profit maximizing level of output.

Place these values in production function

Q = (3K)2/3K1/3

Q = 32/3K

Q = 2.08K

K = 0.48Q

To compute the cost function, find that

C = wL + rK

Place K = 0.48Q w = 2, r = 3,

C = 2*3K + 3K = 9K

C = 9*0.48Q = C = 4.32Q

Average cost is per unit cost so AC = C/Q = 4.32.

D) The production function is given as Q = L2/3K1/3. Substitute K = 400, we get

Q = L2/3K1/3

Q = L2/3(400)1/3

L = Q3/2/20

The Cost function is given by C = wL + rK

Place L = Q3/2/20, w = 2, r = 3, and K = 400

C = 2Q3/2/20 + 3*400

C = Q3/2/10 + 1200

Short run average cost function is C/Q = Q1/2/10 + 1200/Q

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