Microeconomics homework question: Suppose that a firm uses three inputs to produ

Question

Microeconomics homework question:

Suppose that a firm uses three inputs to produce its output: Capital K, Labor L, and Materials M. The Firm's production function is given by q=K0.5L0.25M0.25. For this production function, the marginal products for labor and materials, respectively, are MPL = 0.25K0.5L-0.75M0.25 and MPM = (0.25)K0.5L0.25M-0.75. the prices of capital, labor and materials are r = $15, w=$2 and m=$1. Suppose that capital is fixed at K = 36in the short run.

A) Find the quantities of labor and materials that the firm will use to minimize the cost of producing an output level q in the short run. Do not assign the output level q a specific numerical value.

B) Making use of your results in part (a), derive the short-run total cost function, the short-run variable cost function, and the short-run fixed cost function for this firm.

C) Making use of your results in part (b), derive the shrot-run average cost function and the short-run variable cost function

Explanation / Answer

The cost minimization problem can be solved using Lagrangian method.

The cost structure of the firm is given by

C = rK + wL + mM

C = 15*36 + 2L + M

C = 540 + 2L + M

Setting Lagrangian would imply:

Minimize C = 540 + 2L + M - (K0.5L0.25M0.25)

Finding the partial derivatives and setting them equal to zero gives

2 - MPL = 0

1 - MPM = 0

K0.5L0.25M0.25 = 0

The first two equations give

MPL = 2MPM

0.25K0.5L-0.75M0.25 = 2*0.25K0.5L0.25M-0.75

L = 2M

Substitute this value in the production function to get

L = q2/50 and M = q2/25

These are the optimum values of L and M that minimizes the cost.

b) Total cost function now becomes

C = rK + wL + mM

C = 540 + q2/25 + q2/25

C = 540 + 2q2/25

Fixed cost function is FC = 540

Variable cost function = 2q2/25

c)

Short-run average cost function is C/q = 540/q + 2q/25

Related Questions

Navigate

• Browse (All)
• Browse M
• Subjects