**Economics 5315 Professor Henderson**

**Managerial Economics Fall 1994**

1. The Silverado Mining Company owns two different mines (A and B) that produce uranium ore. After the ore is mined it is separated into three different grades--high, medium, and low. The hourly production of uranium ore of the two mines are as follows. Mine A produces 0.75 tons of high-grade ore per hour, 0.25 tons of medium-grade ore per hour, and 0.50 tons of low-grade ore per hour. Mine B produces 0.25, 0.25, and 1.50 tons of high-, medium-, and low-grade ore per hour. Silverado has a contract with a uranium-processing plant to supply a minimum of 36 tons of high-grade ore, 24 tons of medium-grade ore, and 72 tons of low-grade ore per week. It costs Silverado $50 per hour to operate mine A and $40 per hour to operate mine B. The objective is to minimize the cost of fulfilling the contracts.

a. Formulate the primal problem and solve for the optimal values for the weekly operating hours for mines A and B. How many tons of each grade ore will be produced? What is the weekly cost of operating the two mines?

b. Set up the dual problem and solve for the shadow prices of
the three grades of ore.

2. There are two companies (C and D) that produce an identical product. Demand for the product is given by the following linear demand function,

where Q = Q_{C} + Q_{D}, Q_{C }is the
quantity sold by firm C, and Q_{D} is the quantity sold
by form D. Total cost functions for the two companies are

TC_{C} = 25,000 + 100Q_{C }TC_{D} =
20,000 + 125Q_{D}

Assume that the firms act independently, that is, in determining its own price and output level, each firm assumes that the other firm's output will not change.

a. Determine the long-run equilibrium outputs and selling price for each firm.

b. How would your answer change if the two firms joined together
to form a cartel? (You don't have to come up with a numerical
answer. Just explain the different behavior you would witness
and the impact on P, Q_{C}, Q_{D}, and .)

3. General Statics Corp. has two divisions--production (P) and
marketing (M). Each division manager is responsible for division
profits. The P division makes an electronic ignition system which
M packages and distributes to automobile manufacturers. The firm's
demand curve for this product is estimated to be P = 1,000 - 0.625Q,
where Q represents thousands of units. The P division's marginal
cost is MC_{P} = 200 + 0.375Q_{P}, and the M division's
is MC_{M} = 100 + 0.5Q_{M}.

a. Assuming there is no external market for the ignition systems, calculate the profit-maximizing transfer price.

b. Assuming there is a perfectly-competitive external market in which the price of ignition systems is $350 per unit, how many units should be produced? How many should be sold internally? How many should be sold externally?

c. Explain your answers.

**Answer either 4 or 5. It's your choice.
**

4. The price elasticity of demand for a textbook sold in the United
States is estimated to be -2.0, whereas the price elasticity of
demand for books sold overseas in -3.0. The U.S. market requires
hardcover books with a marginal cost of $6; the overseas market
is normally served by softcover texts, having a marginal cost
of $4.50. Calculate the profit-maximizing price in each market.
[Hint: Remember that MR = P(1 + 1/E_{P})].

5. Alchem (L) is the price leader in the polyglue market. All
ten other manufacturers [follower (F) firms] sell polyglue at
the same price as Alchem. Alchem allows the other firms to sell
as much as they wish at the established price and supplies the
remainder of the demand itself. Total demand for polyglue is
given by the following demand function, P = 20,000 - 4Q_{T},
where Q_{T} = Q_{L} + Q_{F}. Alchem's
marginal cost function for manufacturing and selling polyglue
is MC_{L} = 5,000 + 5Q_{L}. The aggregate marginal
cost function for the follower firms is MC_{F} = 2,000
+ 4Q_{F}.

a. To maximize profits, how much polyglue should Alchem produce and what price should it charge?

b. What is the total market demand for polyglue at the price established by Alchem? How much of the total demand will the follower firms supply?

6. Consolidated Sugar Company has two divisions, a farming-preprocessing
division (P) and a processing-marketing division (M). The P division
grows sugar cane and crushes it into juice, which it may sell
to the M division or externally in a perfectly competitive open
market. The M division buys cane juice, either from the P division
or externally in the open market, and then evaporates and purifies
it as processed sugar. One unit of cane juice is converted into
one unit of processed sugar.

The M division has the following demand function, P_{M}
= 24 - Q_{M}. Its cost function (excluding the cost of
cane juice) is C_{M} = 8 + 2Q_{M}. The P division's
total cost function for cane juice is C_{P} = 10 + 2Q_{P}
+ Q_{P}^{2}. Assume that the open market price
for cane juice is $14 per unit.

a. What is the profit-maximizing price and output level for the P division?

b. What is the profit-maximizing price and output level of the M division?

c. How much cane juice should the P division sell internally to the M division and how much externally on the open market?

d. How much cane juice should the M division buy internally from the P division and how much externally on the open market?

e. What is the minimum price the P division would be willing to sell cane juice to the M division?

f. What is the maximum price the M division would be willing to pay the M division for cane juice?

g. To maximize profits, what is the optimal transfer price for cane juice from division P to division M?

**Economics 5415 Professor Henderson**

**Managerial Economics Fall 1994**

1. Fiori Pasta Company produces high-quality pasta products. It has estimated its demand curve for its spaghetti to be P = 39.898 - 0.03757Q, where Q represents thousands of cartons (each containing five dozen packages of spaghetti) demanded per year by its wholesale customers. Its cost of producing this spaghetti has been estimated as

C = 2,500 + 12Q + 0.01538Q^{2}. Fiori is having a management
meeting to consider its pricing strategy. Its current price for
the spaghetti is $27.50 per carton. The president, Don Paulo
Fiori, wants to maximize sales volume subject to earning a target
profit of $500,000 per year. The vice president of sales, Tony
Baroni Fiori, wants to maximize sales revenue, since his bonus
relates only to sales revenue. The vice president of production,
Gina Elena Fiori, wants to maximize profits so they can afford
to install the latest high-technology manufacturing equipment.
You have been hired to give an impartial analysis of the problem
facing Fiori Pasta. Prepare a report carefully exploring all
three options.

2. Assume that two companies (A and B) are duopolists who produce
identical products. Demand for the products is given by P = 200
- Q, where Q = Q_{A} + Q_{B}. Total costs for
the two firms are C_{A} = 1,500 + 55Q_{A} + Q_{A}^{2}
and C_{B} = 1,200 + 20 Q_{B} + 2Q_{B}^{2}.
Assume that the firms act independently (that is, each firm assumes
the other firm's output will not change).

a. Determine the long-run equilibrium output, price, and for each firm.

b. Assume the firms form a cartel and act like a monopolist. Determine the optimal output and price for each firm. What is the total for the cartel? What is the marginal cost for each firm at their respective optimal output levels?

c. Explain the underlying theory you used to derive your answers.

3. DeSoto Engine, a division of International Motors, produces automobile engines. It sells these engines to the automobile assembly division within the corporation. A dispute has arisen between the managers of the DeSoto division and the assembly division concerning the appropriate transfer price for intracompany sales of engines. The current transfer price of $385 was arrived at by taking the standard cost of the engine ($350) and adding a 10 percent markup for profit, based on an estimated volume of 450,000 engines per year. The manager of DeSoto argues that the transfer price should be raised because the division's average profit margin on other products is 18 percent. The manager of the assembly division argues that the price should be lowered because an assembly division manager at a rival firm claims that (under similar circumstances) engines only cost her division $325. As the corporation's chief economist, you have been asked to settle this intracompany pricing problem. You estimate the following demand and cost information.

where P_{M} is the selling price per automobile and Q_{M}
is the number of vehicles sold. Total cost for the assembly division
(excluding the cost of the engine) is

where C_{M} is total cost. The DeSoto division's total
cost function is

where Q_{P} is the number of engines produced and C_{P}
is the cost.

Assume that the DeSoto division cannot sell any excess engines to outside buyers and that the assembly division cannot purchase engines from outside suppliers.

a. Determine the profit maximizing output for each division.

b. Determine the optimal transfer price for engines.

c. Calculate total revenue, total cost, and total profit for each
division.

In performing your research, you discover that there is a competitive market for engines overseas and a well-known German auto manufacturer has offered to buy DeSoto engines (up to a maximum of 700,000 per year) for $425.

d. Determine the profit maximizing output for each division.

e. Determine the optimal transfer price the intracompany sale of engines.

f. How many engines should DeSoto sell internally to the assembly division and how many should it sell externally to the German manufacturer?

g. Calculate total revenue, total cost, and total profit for each division.

h. What is your recommendation to the Board of Directors?