Assume GM produces cars following the production function q = K^3/4 L^1/4. This

Question

Assume GM produces cars following the production function q = K^3/4 L^1/4. This quarter GM has promised stockholders that it will increase production to 240 cars.

a.) If GM wants to minimize the total cost of producing 240 cars, what is the optimal number of capital and labor it will hire if the cost of capital is $4 and the cost of labor is $1?

b.) What is the total cost that minimizes the production of 240 cars?

c.) In what proportions should GM use capital and labor in order to minimize the cost of a car regardless of the amount of cars being produced? In other words, how much of GM's budget should be spent on capital and how much should be spent of labor? Prove.

d.) Assume GM has its level of capital fixed at 16 in the short run. Find the total cost as a function of output. Identify GM's fixed and variable costs.

e.) How much more will it cost GM to produce 1 additional car (what is the marginal cost of a car) when capital is fixed at 16? Use your answer in d) to show.

Explanation / Answer

q = K3/4L1/4

Total cost: C = wL + rK = L + 4K

(a) Optimal input combination occurs when MPL / MPK = w / r = 1 / 4

MPL = dq / dL = (1/4) x (K / L)3/4

MPK = dq / dK = (3/4) x (L / K)1/4

So, MPL / MPK = (1/3) x (K / L) = 1 / 4

3L = 4K, So

L = 4K / 3

Substituting in production function,

(K)3/4 x (4K / 3)1/4 = 240

(K)3/4 x (K)1/4 x (4/3)1/4 = 240

K x 1.07 = 240

K = 223.35

L = 4K / 3 = 4 x 223.35 / 3 = 297.79

(b) C ($) = 297.79 + (4 x 223.35) = 297.79 + 893.4 = 1,191.19

(c) Proportion of capital and labor: 3L = 4K

L / K = 4 / 3

(d) K = 16

q = (16)3/4 x L1/4 = 8 x L1/4

L1/4 = q / 8

L = q4 / 4,096

K = 16

C = L + 4K

C = (q4 / 4,096) + (4 x 16) = (q4 / 4,096) + 64

Fixed cost = 64 (which is independent of q)

Variable cost = q4 / 4,096

(e)

Marginal cost, MC = dC / dq

MC = (4 x q3) / 4,096 = q3 / 1,024

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