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Profit Maximization Explained

Question 1

Consider the following short-run production function (where L = variable input, Q = output):

Q = 10L – 0.5L2

Suppose that output can be sold for $10 per unit. Also assume that the firm can obtain as much of the variable input (L) as it needs at $20
per unit.

a. Determine the marginal revenue product function.
b. Determine the marginal factor cost function.
c. Determine the optimal value of L, given that the objective is to maximize profits.

Question 2

The Blair Company’s three assembly plants are located in California, Georgia, and New Jersey. Previously, the company purchased a major subassembly, which becomes part of the final product, from an outside firm. Blair has decided to manufacture the subassemblies within the company and must now consider whether to rent one centrally located facility (e.g., in Missouri, where all the subassemblies would be manufactured) or to rent three separate facilities, each located near one of the assembly plants, where each facility would manufacture only the subassemblies needed for the nearby assembly plant. A single, centrally located facility, with a production capacity of 18,000 units per year, would have fixed costs of $900,000 per year and a variable cost of $250 per unit. Three separate decentralized facilities, with production capacities of 8,000, 6,000, 4,000 units per year, would have fixed costs of $475,000, $425,000, and $400,000, respectively, and variable costs per unit of only $225 per unit, owing primarily to the reduction in shipping costs. The current production rates at the three assembly plants are 6,000, 4,500, and 3,000 units, respectively.

a. Assuming that the current production rates are maintained at the three assembly plants, which alternative should management select?
b. If demand for the final product were to increase to production capacity, which alternative would be more attractive?
c. What additional information would be useful before making a decision?

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Solution Preview

a. Determine the marginal revenue product function.

The marginal production function is found by taking the derivative of the total revenue function. Total revenue is found by multiplying output by price:

TR = P Q = 10 (10L - 0.5L^2)= 100L - 5L^2
MR = 100 - 10L

MR = dTR/dQ = 10PL -- 0.5PL^2

b. Determine the marginal factor cost function.

Cost function: TC = 20L
Marginal revenue function: MC = 20 (first derivative of ...

Solution Summary

The solution demonstrates the use of the short run production function to maximize profits in 285 words with calculations explained.