Question

A firm production is represented by the following Cobb-Douglas function: Q=K^1/4L^3/4 The rental rate, r, of capital is given by $100 and the price of labor w 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 300 units?
c. What is the minimum cost of producing 300 units?
d. What is the additional cost of increasing output to 500 units in the short run and in the long run?
e. Does this firm have economies or diseconomies of scale? Please answer based on the long run cost
calculations in parts c and d.

Answer 1

Production function is Q=K^1/4L^3/4

The rental rate = $100 and the price of labor w = $200.

a. For a given level of output, the ratio of capital to labor in order to minimize costs = w/r = 200/100 = 2

b. Find MRTS = MPL/MPK = (3/4)(K/L)^(1/4) / (1/4)(L/K)^(3/4) = 3K/L

At optimum input mix MRTS = w/r

3K/L = 2

L = 1.5K

Now Q = K^(1/4) L^(3/4)

300 = K^(1/4) (1.5K)^(3/4)

300 = 1.5^(3/4) K

K = 221.34

L = 332

Hence, capital K = 221.34 and labor = 332 should be used to produce 300 units

c. Minimum cost of producing 300 units = C = wL + rK = 200*332 + 100*221.34 = 88534

d. In short run K is fixed at 221.34. Then we have 300 = 221.34^(1/4) L^(3/4) or L = 656

Cost C = 656*200 + 221.34*100 = 153334

In long run we have

500 = K^(1/4) (1.5K)^(3/4)

500 = 1.5^(3/4) K

K = 369

L = 553.34

C = 553.34*200 + 369*100 = 147568

The additional cost of increasing output to 500 units in the short run is 64800 and in the long run is 59034

e. ATC (Q = 300) = 88534/300 = $295 and ATC (Q = 500) = $295

Since ATC is same, there are constant returns to scale.