Managerial Economics Fall 1994

Professor Henderson Final Exam

Answer two of the following three questions.

1. Consider the following production function for a firm: Q = 100L - L² + 200K - 0.5K². The current price of output is $2. The current price of L is $100 per day, and the price of K is $200 per day. Assume that the firm is a price taker in all markets.

a. How much K and L should the firm employ?

b. How much output should the firm produce daily?

c. What are revenues, costs, and profit per day?

2. Santa Fe Enterprises has determined that its profit function is of the following form:

= -60 + 140Q₁ + 100Q₂ - 10Q₁² - 8Q₂² - 6Q₁Q₂, where Q₁ and Q₂ are the monthly amounts of the two products produced by the firm. Suppose that the raw materials needed to make the product are in short supply and that the firm has a contract with a supplier calling for the delivery of 200 units of the raw material per month. No other sources of the raw material are available. Also, assume that all raw materials must be used during the month and that none can be carried over to the next month. Furthermore, suppose that Q₁ requires 20 units of the raw material and Q₂ requires 40 units of the raw material to produce one unit of output.

a. Write the constrained optimization problem.

b. Using the Lagrangian method, solve the problem for the optimal output of Q₁ and Q₂.

c. What is the value of ? How do you interpret this value?

3. Martin Casey, the executive vice president of the Summit Company, must allocate the firm's three production processes to the manufacturing of either cotton or wool rugs. Rugs must go through all three processes before a finished product is available. The following characteristics are relevant for production.

	Type of Rug
Characteristics	Cotton	Wool
Minutes of process 1	6	3
Minutes of process 2	2	4
Minutes of process 3	8	8
Profit per rug	12	8

The firm has a total of 72 minutes of process 1 time available, 48 minutes of process 2 time, and 120 minutes of process 3 time that it can use per production cycle (4 hours long) for making of rugs. Be sure to label your answers carefully and clearly.

a. Solve the primal problem for minutes of each production process to allocate to cotton and wool rug production. Where is the slack?

b. How many of each type of rug should be produced to maximize profit?

c. Solve the dual for the shadow prices of the three resources.

Answer three of the following four questions.

4. A local grocery store has collected the following facts: A loaf of bread costs $0.69 from the wholesaler. It costs the store an additional $0.11 to price the bread, shelve it, and sell it. Overhead adds $0.05 more per loaf to the cost. The grocer has estimated the demand curve for bread to be the following function, Q = 220 - 120P. The current retail price of bread is $1.19 per loaf. Is this the profit-maximizing price? Explain.

5. The Radnor Company believes that the demand curve for its product is:

Total cost is given by TC = 5Q.

a. Explain why the demand curve has this sort of shape.

b. What is the firm's current price and output? its profit?

c. What price and output will maximize the firm's profit? Explain.

d. Roughly sketch the demand, marginal revenue, marginal cost, and average cost curves for this problem on a single diagram.

6. Ann McCutcheon is hired as a consultant for a firm producing precision ball bearings. This firm sells in two distinct markets, one of which is completely sealed off from the other. The demand curve for this firm's output in the one market is P₁ = 160 - 8Q₁, where P₁ is the price of the product, and Q₁ the amount sold in the first market. The demand curve for the firm's output in the second market is P₂ = 80 - 2Q₂, where P₂ is the price of the product, and Q₂ the amount sold in the second market. The firm's marginal cost curve is 5 + Q, where Q is the firm's entire output (Q₁ + Q₂). The firm asks Ms McCutcheon to suggest a pricing policy. How many units of output should the firm sell in each market? What should the price be in each market?

7. The Xerxes Company is composed of a marketing division and a production division. The marketing division packages and distributes a plastic item made by the production division. The demand curve for the finished product sold by the marketing division is given by, P_M = 200 - 3Q_M, where P_M is the price (in dollars per pound) of the finished product, and Q_M is the quantity sold (in thousands of pounds). Excluding the production cost of the basic plastic item, the marketing division's total cost function is TC_M = 100 + 15 Q_M, where TC_M is the marketing division's total cost (in thousands of dollars). The production division's total cost function is TC_P = 5 + 3Q_P + 0.4Q_P², where TC_P is the total cost of production (in thousands of dollars), and Q_P is the total quantity produced of the basic item (in thousands of pounds). There is a perfectly competitive market for the basic plastic item, the price being $20 per pound.

a. What is the optimal output for both divisions? If Q_M Q_P, describe the firm's transaction in the external market.

b. What is the optimal transfer price for the plastic item?

c. At what price should the marketing division sell its product?