Multiple Products

In sum, the firm should continue production because the product generates a positive contribution, thereby minimizing the firm’s loss. The firm suffers an economic loss in the short run; nevertheless, this is better than shutting down. Thus, we have the following general rule:

In the short run, the firm should continue to produce as long as price exceeds average variable cost. Assuming it does produce, the firm maximizes contribution (and minimizes any losses) by setting marginal revenue equal to marginal cost.

In the long run, all inputs and all costs are variable. (For instance, a firm that leases its plant and equipment can shed these costs if it chooses not to renew its two-year lease. The firm can also downsize its workforce over time.) In the long run, the firm should continue operating only if it expects to earn a nonnegative economic profit. A firm that suffers persistent economic losses will be forced to exit the industry.

Earlier we noted that the repair firm’s cost function is C 270 30Q .3Q2. Suppose demand is given by P 50 .2Q. What is the firm’s optimal course of action in the

short run? In the long run?

CHECK STATION 4

In the previous section, we noted that the prevalence of multiproduct firms is explained by economies of scope. The implication of such economies is that the firm can produce multiple products at a total cost that is lower than the sum of the items’ costs if they were produced separately. As we shall see, managers must be careful to pay attention to relevant costs in a multiproduct environment.

To illustrate, consider a firm that produces two products in a common facility. The firm’s total cost of production is described as

where FC denotes the total fixed costs shared by the products. The separate variable costs for the products also are included and depend directly on the output levels of each product. The firm’s total profit is

[6.6]

where R1 and R2 denote the products’ revenues. As noted earlier, each term in parentheses is the product’s contribution. The firm’s total profit is the sum of its products’ contributions minus its total fixed costs.

C FC VC1 VC2,

(R1 VC1) (R2 VC2) FC,

As we saw in the single-product case, the firm should continue producing an item only if R VC or, equivalently, P AVC. Exactly the same principle applies to the multiproduct case. Furthermore, in the long run the firm should continue in business only if the total profit in Equation 6.6 is nonnegative; otherwise, it should shut down. The firm’s output rule for multiple goods can be stated in two parts:

1.Each good should be produced if, and only if, it makes a positive contribution to the firm’s fixed costs: Ri VCi or, equivalently,

Pi AVCi. 2.In the long run, the firm should continue operations if, and only if, it makes positive economic profits.

MULTIPLE PRODUCTS: A NUMERICAL EXAMPLE Suppose a firm’s total fixed cost is $2.4 million per year. For the first good, P1 $10, AVC1 $9, and Q1 1.2 million; in turn, P2 $6.50, AVC2 $4, and Q2 .6 million. Total profit is 1.2 1.5 2.4 $.3 million per year. Each product makes a positive contribution, and total contribution exceeds total fixed cost. Thus, the firm should stay in business. What if the second good’s price is $5.50? In the short run, both goods continue to contribute to fixed costs and, therefore, should be produced. In the long run, total contribution ($2.1 million) falls short of total fixed cost, so the firm should shut down. Finally, if the second good’s price is $3.50, then P2 AVC2. The firm should halt production of the second good immediately and cease operations altogether in the long run.

Sizing up relevant costs for production decisions and measuring costs for accounting purposes are sometimes in conflict. For instance, almost all accounting systems allocate fixed costs across the firm’s multiple products. Typically, these allocations are in proportion to the products’ volumes. If a product accounts for 20 percent of a firm’s output (whether by units, labor costs, or machine-hours), it is assigned 20 percent of these costs. These allocated costs are added to the product’s direct unit costs to determine its total cost of production. According to this cost-accounting system, the product is profitable if, and only if, its total revenues exceed its total costs.

Although cost accounting systems are useful in many respects (especially for tax purposes), they can be misleading when it comes to economic decisions. For decision-making purposes, costs that are truly fixed (i.e., don’t vary) with respect to products’ volumes should not be allocated at all. This is in keeping with the earlier proposition that fixed costs do not matter. According to Equation 6.6, contribution is the only relevant measure of a good’s performance. Production should continue if contribution is positive and should cease if it is negative. Note that the good’s accounting profit (by including a fixed-cost allocation) understates its contribution. Consequently, when it comes to product decisions, accounting profit can be very misleading.

In the preceding example, a typical accounting allocation would assign the $2.4 million in fixed costs to the products in proportion to output: $1.6 million to the first product and $.8 million to the second. (The first product constitutes two-thirds of total output.) Thus, the first product’s accounting profit would be 1 (10 9)(1.2) 1.6 $.4 million. Based on this measure, the product appears to be unprofitable. What if the firm discontinues its production? The firm will no longer earn any contribution from the first good. But there will be no decline in the $2.4 million fixed cost; now the entire fixed cost will be assigned to the second item. Left producing a single good, the firm will be unable to earn a profit (its loss will amount to 2.4 1.5 $.9 million) and will be forced to shut down. Here, allocating fixed costs leads to a disastrous series of decisions. As we noted earlier, the firm’s optimal course of action is to produce both products. To repeat, assigning fixed costs to products is unnecessary (and potentially misleading). Instead, the only relevant long-run issue is whether the firm’s total contribution covers these fixed costs in the aggregate.

In the example that opens this chapter, the managers of a sports shoe company were engaged in a debate over what strategy would lead to the greatest profit. Should production of the boys’ shoes be increased? Cut back? Discontinued? The correct answers to these questions depend on a careful analysis of relevant costs. To clarify the situation, management has gathered cost information about different sales quantities. The firm’s production managers have supplied the data on direct (i.e., variable) costs. Recall that production of women’s and boys’ running shoes share $90,000 in fixed costs. The firm’s accountants allocate this cost to the two lines in proportion to numbers of pairs. The output of women’s shoes is 8,000 pairs.

Allocating Costs Revisited

Pairs Allocated Average of Shoes Price Revenue Direct Cost Cost Total Cost

1,600 $40 $64,000 $66,400 $15,000 $50.88 2,400 36 86,400 74,400 20,769 39.65 3,200 32 102,400 85,600 25,714 34.79 3,600 30 108,000 92,400 27,931 33.43 4,000 28 112,000 100,000 30,000 32.50

Thus, if the volume of boys’ shoes is 4,000 pairs, the product’s output is one-third of the total; hence, its allocation is (1/3)($90,000) $30,000. Allocations for other outputs are computed in the same way. Average total cost is the sum of direct and allocated costs divided by total output.

The firm currently is charging a price of $36 per pair and selling 2,400 pairs per week. How would management evaluate the current profitability of this strategy, and how might

it improve its profits? First, consider the wrong method of approaching these questions. Management observes that when it sells 2,400 pairs, total average cost is $39.65 per pair. This exceeds the $36 selling price. Therefore, management believes its current strategy is unprofitable. What are its other options? An obvious possibility is to increase price to a level above $39.65, say, to $40. The table shows the results of this strategy. Volume drops to 1,600 pairs, but average total cost rises to $50.88. (Because the decline in volume is much greater than the reduction in total cost, average cost rises dramatically.) Price still falls well short of average cost. A price cut will do no better. The other prices in the table tell the same story: Average total cost exceeds price in all cases. Therefore, management concludes that the boys’ running shoe cannot earn a profit and should be discontinued.

Let’s now adopt the role of economic consultant and explain why management’s current reasoning is in error. The problem lies with the allocation of the $90,000 in “shared” costs. Recall the economic “commandment”: Do not allocate fixed costs. In a multiproduct firm, contribution is the correct measure of a product’s profitability. A comparison of columns 3 and 4 in the table shows that the boys’ shoe makes a positive contribution for four of the price and output combinations. Thus, the shoe should be retained. The firm’s optimal strategy is to lower the price to P $32. The resulting sales volume is Q 3,200. Maximum contribution is $102,400 $85,600 $16,800. Beyond P $32, however, any further price reduction is counterproductive. (The additional cost of supplying these additional sales units exceeds the extra sales revenue.) Thus, the production manager would be wrong to advocate a policy of minimizing direct costs per unit of output. We can check that of the five output levels, average variable cost (AVC) is minimized at Q 4,000. (Here AVC is $100,000/4,000 $25 per pair.) Nonetheless, this volume of output delivers less contribution than Q 3,200 because the accompanying drop in price is much greater than the decline in average variable cost. To sum up, the firm’s correct strategy is to maximize the product’s contribution.

SUMMARY

Decision-Making Principles

1.Cost is an important consideration in decision making. In deciding among different courses of action, the manager need only consider the differential revenues and costs of the various alternatives. 2.The opportunity cost associated with choosing a particular decision is measured by the forgone benefits of the next-best alternative. 3.Economic profit is the difference between total revenues and total costs (i.e., explicit costs and opportunity costs). Managerial decisions should be based on economic profit, not accounting profit. 4.Costs that are fixed (or sunk) with respect to alternative courses of action are irrelevant. 5.In the short run, the firm should continue to produce as long as price exceeds average variable cost. Assuming it does produce, the firm

maximizes its profit (or minimizes its loss) by setting marginal revenue equal to marginal cost. 6.In the long run, all revenues and costs are variable. The firm should continue production if, and only if, it earns a positive economic profit. A multiproduct firm should continue operating in the long run only if total revenue exceeds total costs. There is no need to allocate shared costs to specific products.

Nuts and Bolts

1.The firm’s cost function indicates the (minimum) total cost of producing any level of output given existing production technology, input prices, and any relevant constraints. 2.In the short run, one or more of the firm’s inputs are fixed. Short-run total cost is the sum of fixed cost and variable cost. Marginal cost is the additional cost of producing an extra unit of output. In the short run, there is an inverse relationship between marginal cost and the marginal product of the variable input: MC PL/MPL. Marginal cost increases due to diminishing returns. The short-run average cost curve is

U-shaped. 3.In the long run, all inputs are variable. The firm chooses input proportions to minimize the total cost of producing any given level of output. The shape of the long-run average cost curve is determined by returns to scale. If there are constant returns to scale, long-run average cost is constant; under increasing returns, average cost decreases with output; and under decreasing returns, average cost rises. Empirical studies indicate L-shaped (or U-shaped) long-run average cost curves for many sectors and products. 4.Many firms supply multiple products. Economies of scope exist when the cost of producing multiple goods is less than the aggregate cost of producing each good separately. 5.Comparative advantage (not absolute advantage) is the source of mutually beneficial global trade. The pattern of comparative advantage between two countries depends on relative productivity, relative wages, and the exchange rate.

Questions and Problems

1.The development of a new product was much lengthier and more expensive than the company’s management anticipated. Consequently, the firm’s top accountants and financial managers argue that the firm should raise the price of the product 10 percent above its original target

to help recoup some of these costs. Does such a strategy make sense?

Explain carefully. 2.Comment on the following statement: “Average cost includes both fixed and variable costs, whereas marginal cost only includes variable costs.

Therefore, marginal cost is never greater than average cost.” 3.A company produces two main products: electronic control devices and specialty microchips. The average total cost of producing a microchip is $300; the firm then sells the chips to other high-tech manufacturers for $550. Currently, there are enough orders for microchips to keep its factory capacity fully utilized. The company also uses its own chips in the production of control devices. The average total cost (AC) of producing such a device is $500 plus the cost of two microchips. (Assume all of the $500 cost is variable and AC is constant at different output volumes.)

Each control device sells for an average price of $1,500. a.Should the company produce control devices? Is this product profitable? b.Answer part (a) assuming outside orders for microchips are insufficient to keep the firm’s production capacity fully utilized. c.Now suppose $200 of the average cost of control devices is fixed.

Assume, as in part (a), that microchip capacity is fully utilized. Should control devices be produced in the short run? Explain. 4.The last decade has witnessed an unprecedented number of megamergers in the banking industry: Bank of America’s acquisitions of Fleet

Bank, MBNA, and U.S. Trust; Bank of New York’s acquisition of Mellon

Financial; and Wells Fargo’s acquisition of Wachovia, to name several of the largest consolidations. Besides growth for its own sake, these superbanks are able to offer one-stop shopping for financial services: everything from savings accounts to home mortgages, investment accounts, insurance vehicles, and financial planning. a.In the short run, what are the potential cost advantages of these mergers? Explain. b.Is a $300 billion national bank likely to be more efficient than a $30 billion regional bank or a $3 billion state-based bank? What economic evidence is needed to determine whether there are long-run increasing returns to scale in banking? c.Do you think these mergers are predicated on economies of scope? 5.An entrepreneur plans to convert a building she owns into a video-game arcade. Her main decision is how many games to purchase for the arcade. From survey information, she projects total revenue per year as

R 10,000Q 200Q2, where Q is the number of games. The cost for each game (leasing, electricity, maintenance, and so on) is $4,000 per year. The entrepreneur will run the arcade, but instead of paying herself a salary, she will collect profits. She has received offers of $100,000 to sell

her building and a $20,000 offer to manage a rival’s arcade. She recognizes that a normal return on a risky investment such as the arcade is 20 percent. a.As a profit maximizer, how many games should she order? b.What is her economic profit? 6.Suppose the manufacturer of running shoes has collected the following quantitative information. Demand for the boys’ shoe is estimated to be

Q 9,600 200P, or, equivalently, P 48 Q/200. The shoe’s direct cost is C $60,000 .0025Q2 . a.Check that these demand and cost equations are consistent with the data presented in the “Allocating Costs Revisited” section. b.Find the firm’s profit-maximizing price and output. 7.You are a theater owner fortunate enough to book a summer box office hit into your single theater. You are now planning the length of its run.

Your share of the film’s projected box office is R 10w .25w2, where R is in thousands of dollars and w is the number of weeks that the movie runs.

The average operating cost of your theater is AC MC $5 thousand per week. a.To maximize your profit, how many weeks should the movie run? What is your profit? b.You realize that your typical movie makes an average operating profit of $1.5 thousand per week. How does this fact affect your decision in part (a), if at all? Explain briefly. c.In the last 25 years, stand-alone movie theaters have given way to cineplexes with 4 to 10 screens and megaplexes with 10 to 30 screens (yes, 30 screens!) under one roof. During the same period, total annual movie admissions have barely changed. What cost factors can explain this trend? In addition, what demand factors might also be relevant? d.The film’s producer anticipated an extended theater run (through

Labor Day) and accordingly decided to move back the DVD release of the film from Thanksgiving to January. Does the decision to delay make sense? Explain carefully. 8.Firm A makes and sells motorcycles. The total cost of each cycle is the sum of the costs of frames, assembly, and engine. The firm produces its own engines according to the cost equation:

CE 250,000 1,000Q 5Q2 .

The cost of frames and assembly is $2,000 per cycle. Monthly demand for cycles is given by the inverse demand equation P 10,000 30Q. a.What is the MC of producing an additional engine? What is the MC of producing an additional cycle? Find the firm’s profit-maximizing quantity and price.

b.Now suppose the firm has the chance to buy an unlimited number of engines from another company at a price of $1,400 per engine. Will this option affect the number of cycles it plans to produce? Its price?

Will the firm continue to produce engines itself? If so, how many? 9.A firm’s long-run total cost function is

a.What is the shape of the long-run average cost curve? b.Find the output that minimizes average cost. c.The firm faces the fixed market price of $140 per unit. At this price, can the firm survive in the long run? Explain. 10.A firm uses a single plant with costs C 160 16Q .1Q2 and faces the price equation P 96 .4Q. a.Find the firm’s profit-maximizing price and quantity. What is its profit? b.The firm’s production manager claims that the firm’s average cost of production is minimized at an output of 40 units. Furthermore, she claims that 40 units is the firm’s profit-maximizing level of output. Explain whether these claims are correct. c.Could the firm increase its profit by using a second plant (with costs identical to the first) to produce the output in part (a)? Explain. 11.As noted in Problem 5 of Chapter 3, General Motors (GM) produces light trucks in its Michigan factories. Currently, its Michigan production is 50,000 trucks per month, and its marginal cost is $20,000 per truck.

With regional demand given by: P 30,000 0.1Q, GM sets a price of $25,000 per truck. a.Confirm that setting Q 50,000 and P $25,000 is profit maximizing. b.General Motors produces the engines that power its light trucks and finds that it has some unused production capacity, enough capacity to build an additional 10,000 engines per year. A manufacturer of sports utility vehicles (SUVs) has offered to purchase as many as 25,000 engines from GM at a price of $10,000 per engine. GM’s contribution is estimated to be about $2,000 per engine sold (based on a marginal cost of $8,000 per engine). Should GM devote some of its engine capacity to produce engines to sell to the SUV manufacturer? Does this outside opportunity change GM’s optimal output of light vehicles in part (a)? c.GM also assembles light trucks in a West Coast facility, which is currently manufacturing 40,000 units per month. Because it produces multiple vehicle types at this mega-plant, the firm’s standard practice is to allocate $160 million of factorywide fixed costs to light trucks. Based on this allocation, the California production manager reports that the average total cost per light truck is $22,000 per unit. Given this report, what conclusion (if any) can you draw concerning the

C 360 40Q 10Q2 .

marginal cost per truck? If West Coast demand is similar to demand in Michigan, could the West Coast factory profit by changing its output from 40,000 units? 12.a.Firm K is a leading maker of water-proof outerwear. During the winter months, demand for its main line of water-proof coats is given by:

P 800 .15Q,

where P denotes price in dollars and Q is quantity of units sold per month. The firm produces coats in a single plant (which it leases by the year). The total monthly cost of producing these coats is estimated to be:

C 175,000 300Q .1Q2 .

(Leasing the plant accounts for almost all of the $175,000 fixed cost.)

Find the firm’s profit-maximizing output and price. If the firm’s other outerwear products generate $50,000 in contribution, what is the firm’s total monthly profit? b.From time to time corporate customers place “special” orders for customized versions of Firm K’s coat. Because they command premium prices, corporate orders generate an average contribution of $200 per coat. Firm K tends to receive these orders at short notice usually during the winter when its factory is operating with little unused capacity. Firm K has just received an unexpected corporate order for 200 coats but has unused capacity to produce only 100. What would you recommend? In general, can you suggest ways to free up capacity in the winter? c.Because of rival firms’ successes in developing and selling comparable (sometimes superior) coats and outerwear, Firm K’s winter demand permanently falls to P 600 .2Q. What is the firm’s optimal operating policy during the next three winter months? When its plant lease expires in June? 13.A manufacturing firm produces output using a single plant. The relevant cost function is C 500 5Q2. The firm’s demand curve is P 600 5Q. a.Find the level of output at which average cost is minimized. Hint: Set AC equal to MC. What is the minimum level of average cost? b.Find the firm’s profit-maximizing output and price. Find its profit. c.Suppose the firm has in place a second plant identical to the first. Argue that the firm should divide production equally between the plants. Check that the firm maximizes profit at total output Q* such that

MR(Q*) MC1(Q*/2) MC2(Q*/2).

*Starred problems are more challenging.

Find Q*. Explain why total output is greater than in part (b). d.In the long run, the firm can produce using as many or as few plants as it wishes (each with the preceding cost function). In this case, what kind of returns to scale hold? What are the firm’s optimal output and price in the long run? How many plants will the firm use to produce the good? Hint: Refer to the value of minimum AC you found in part (a). 14.A firm produces digital watches on a single production line serviced* during one daily shift. The total output of watches depends directly on the number of labor-hours employed on the line. Maximum capacity of the line is 120,000 watches per month; this output requires 60,000 hours of labor per month. Total fixed costs come to $600,000 per month, the wage rate averages $8 per hour, and other variable costs (e.g., materials) average $6 per watch. The marketing department’s estimate of demand is P 28 Q/20,000, where P denotes price in dollars and Q is monthly demand. a.How many additional watches can be produced by an extra hour of labor? What is the marginal cost of an additional watch? As a profit maximizer, what price and output should the firm set? Is production capacity fully utilized? What contribution does this product line provide? b.The firm can increase capacity up to 100 percent by scheduling a night shift. The wage rate at night averages $12 per hour. Answer the questions in part (a) in light of this additional option. c.Suppose that demand for the firm’s watches falls permanently to P 20 Q/20,000. In view of this fall in demand, what output should the firm produce in the short run? In the long run? Explain. Discussion Question Explain why the cost structure associated with many kinds of information goods and services might imply a market supplied by a small number of large firms. (At the same time, some Internet businesses such as grocery home deliveries have continually suffered steep losses regardless of scale. Explain why.) Could lower transaction costs in e-commerce ever make it easier for small suppliers to compete? As noted in Chapter 3, network externalities are often an important aspect of demand for information goods and services. (The benefits to customers of using software, participating in electronic markets, or using instant messaging increase with the number of other users.) How might network externalities affect firm operating strategies (pricing, output, and advertising) and firm size?

Spreadsheet Problems

S1.A firm’s production function is given by the equation

Q 12L.5K.5 ,

where L, K, and Q are measured in thousands of units. Input prices are 36 per labor unit and 16 per capital unit. a.Create a spreadsheet (based on the example shown) to model this production setting. (You may have already completed this step if you answered Problem S2 of Chapter 5. An algebraic analysis of this setting appears in this chapter’s Special Appendix.) b.To explore the shape of short-run average cost, hold the amount of capital fixed at K 9 thousand and vary the amount of labor from 1 thousand to 2.5 thousand to 4 thousand to 5.5 thousand to 7.5 thousand to 9 thousand units. What is the resulting behavior of

SAC? Use the spreadsheet optimizer to find the amount of labor corresponding to minimum SAC. What is the value of SACmin? c.In your spreadsheet, set L 9 thousand (keeping K 9 thousand) and note the resulting output and total cost. Now suppose that the firm is free to produce this same level of output by adjusting both labor and capital in the long run. Use the optimizer to determine the firm’s optimal inputs and LACmin. (Remember to include an output constraint for cell I3.) d.Confirm that the production function exhibits constant returns to scale and constant long-run average costs. For instance, recalculate the answer for part (c) after doubling both inputs. e.Finally, suppose the firm’s inverse demand curve is given by P 9 Q/72.

With capital fixed at K 9 in the short run, use the optimizer to determine the firm’s optimal labor usage and maximum profit. Then

1 2 3

4 A B C D E F G H I J

COST ANALYSIS

Output 36 Price 8.50

5 Labor 1.00 Capital 9.00 6 MPL 18.00 MPK 2.00 MR 8.00 7 Revenue 306 8 MRPL 144.00 MRPK 16.00 9 MCL 36.00 MCK 16.00 Cost 180 10 Avg. Cost 5.00

11 12

Profit 126

13

find the optimal amounts of both inputs in the long run. Explain the large differences in inputs, output, and profit between the short run and the long run. S2.A multinational firm produces microchips at a home facility and at a foreign subsidiary according to the respective cost functions:

The firm sells chips in the home market and the foreign market where the inverse demand curves are

respectively. Here D denotes the quantity sold in each market, and Q denotes the quantity produced in each facility. Chips can be costlessly shipped between markets so that DH need not equal QH (nor DF

CH 120QH and CF 50QF .5QF 2 .

PH 300 DH and PF 250 .5DF,

A B C D E F G H

1 2 WORLDWIDE CHIP DECISIONS

3

4 Set Sales & Output Quantities (000s)

5

6 Home Abroad Total

7

8 Sales 10 10 20 Extra

9

10 Output 10 10 20

11

12 Price 290 245

13

14 Revenues 2,900 2,450 5,350 15 MR 280 240 —

16

17 Costs 1,200 550 1,750 18 MC 120 60 —

19

20 Profit 1,700 1,900 3,600

21

0 Output

45 Price Gap

equal QF). However, total production must match total sales:

QH QF DH DF. a.Create a spreadsheet (based on the accompanying example) to model the firm’s worldwide operations. Find the firm’s profit-maximizing outputs, sales quantities, and prices. Are chips shipped overseas? (Hint: The key to maximizing profit is to find sales and output quantities such that MRH MRF MCH MCF. Also note that MCH is constant. When using your spreadsheet’s optimizer, be sure to include the constraint that cell F9—extra output—must equal zero. That is, total sales must exactly equal total output.) b.Answer the questions in part (a) under an “antidumping” constraint; that is, the company must charge the same price in both markets. (Hint: Include the additional constraint that cell F12, the price gap, must equal zero.) S3.The accompanying spreadsheet depicts the pricing options (cells C10 and C18) for a best seller that is released in hardback and as an e-book.

The demand curve for the hardback version is described by the equation: Q 80 2.4P 2PE, where PE denotes the e-book price. In turn, the demand curve for the e-book is given by: QE 80 4PE. (Both quantities are denominated in thousands of units.) Note that lowering the e-book price by $1.00 increases e-book sales by 4 thousand units but also reduces hardbook sales by 2 thousand books. In short, each additional e-book sold replaces .5 hardback sales. The profit cells are calculated based on the economic facts noted earlier in the chapter. (1) Revenues for the e-book are split 70–30 between book publisher and online seller. The marginal cost of producing and delivering additional e-books is essentially zero. (2)

Revenues for the print book are split 50–50 between the book publisher and the book retailer. The publisher incurs $3.50 per hardback in production and related costs. (3) For both book types, the publisher pays a 15 percent author royalty based on total retail revenue. a.Re-create the spreadsheet shown. If e-books did not exist (set PE $20 so that QE is 0), what is the publisher’s profit-maximizing hardback price? b.Before 2010 when Amazon was free to set the e-book price, what price should Amazon have set? In response, what is the publisher’s profitmaximizing price for the hardback? c.Alternatively, if the publisher sets both book prices, what are the optimal prices? Why does the publisher prefer a higher e-book price than the online seller? d.Suppose that each e-book sale replaces one hardback sale. This cannibalization rate is described by the hardback demand curve: Q 40 2.4P 4PE. Using this demand curve, re-answer the questions in parts (b) and (c). Confirm that this worsens the pricing

A B C D E F G H I

1 2 Pricing Print Books and E-books 3 4 5 Book Publisher Q 80 2.4P 2Pe 6 7 Print Books E-Books Total

8

9 10 11 12 Price Quantity Profit Profit Profit

$25.00 60.00 $315 $0 $315

Combined Profit

13 E Book Seller Qe 80 4Pe 14 15 E-Books $315

16 17 18 19 Price Quantity Revenue Profit

$20.00 0.00 $0 $0

conflict between publisher and online seller. In this case, which would the publisher prefer: (1) the time when e-books didn’t exist, or (2) the young e-book market when Amazon set prices?

Suggested References

The following articles examine the existence of scale and scope economies in a variety of settings. Clark, J. “The Economies of Scale and Scope at Depository Financial Institutions: A Review of the Literature.” Economic Review (Federal Reserve Bank of Kansas City, 1988): 16–33. Koshal, R. K., and M. Koshal. “Do Liberal Arts Colleges Exhibit Economies of Scale and Scope?” Education Economics 8 (December 2000): 209–220. Silk, A. J., and E. R. Berndt. “Scale and Scope Effects on Advertising Agency Costs.” Marketing Science (Winter 1993): 53–72. Sung, N., and M. Gort. “Economies of Scale and Natural Monopoly in the U.S. Local Telephone Industry.” Review of Economics and Statistics 82 (November 2000): 694–697. Wholey, D., et al. “Scale and Scope Economies among Health Maintenance Organizations.” Journal of Health Economics 15 (December 1996): 657–684. The following article reviews research estimating cost functions. J. Panzar. “Determinants of Firm and Industry Structure,” in R. Schmalensee and R. Willig (Eds.), Handbook of Industrial Organization. Amsterdam: North-Holland, 1989, pp. 3–59.

1. The full cost to the airline of a grounded plane includes explicit costs— repair costs, overnight hangar costs, and the like. It also includes an opportunity cost: the lost profit on any canceled flights. 2. The past profits and development costs are irrelevant. If the firm drops the product, it recovers $2 million. If the firm continues the product, its additional profit is 5 3 4 2.5 $1.5 million. Thus, the firm should drop the product. 3. The related electronics products would exhibit economies of scope for several reasons. First, they have many technological elements in common (expertise in copier technology carries over to facsimile machines, for example). They also have some common components (microchips).

Second, customers see the products as complementary. Thus, brandname allegiance gained in computers could carry over to telephone equipment. Third, there are likely economies in joint advertising, promotion, and distribution. (Toshiba’s sales force can pursue sales on any or all of its products.) 4. The repair firm’s marginal revenue is MR 50 .4Q, and its marginal cost is MC 30 .6Q. Setting MR equal to MC, we find Q* 20. From the price equation, P* 50 (.2)(20) 46. In turn, profit is 920 990 70. The firm incurs a loss in the short run, but this is preferable to shutting down ( 270). It is earning a maximum contribution toward overhead. In the long run, the firm should shut down unless conditions improve.

APPENDIX TO CHAPTER 6 CHECK STATION ANSWERS

274

APPENDIX TO CHAPTER 6

Transfer Pricing

In the body of this chapter, we have focused on the production and sale of a firm’s products to outside buyers. Although this is the most common case, products are also sold among divisions within large firms. For example, major automobile companies consist of many divisions. The division that produces parts will transfer its output to separate assembly divisions responsible for automobiles, trucks, and vans. In turn, assembled vehicles are transferred to different sales divisions and finally to dealers. In the same way, within a major chemical manufacturer, one division may produce a basic chemical that is used as an input in the production of specialty chemicals and plastics, each housed in separate divisions.

The price the selling division charges to the buying division within the company is called the transfer price. The firm’s objective is to set the transfer price such that the buying and selling divisions take actions that maximize the firm’s total profit. Accomplishing this task requires an accurate cost assessment. To illustrate the issues at stake, consider a large firm that sells a variety of office electronics products, such as telephones, printers, desktop computers, and copiers. One division specializes in the production of microchips that serve as the “electronic brains” for many of the firm’s final products, including copiers, laser printers, and facsimile machines. For the time being, we assume there is no outside market for the firm’s specialty chips; they are produced only for use

in the firm’s final products. (We relax this assumption in the following section.) What transfer price per unit should the chip division charge the copier (or any other) division?

The answer to this question is that the firm should set the internal transfer price for the microchip exactly equal to its marginal cost of production. Figure 6A.1 summarizes the demand and cost conditions associated with the production of copiers. The key point about the figure is understanding the “full” marginal cost of producing copiers. Managers of the copier division are well aware of the direct costs they incur in assembly. This marginal cost is shown as the upward-sloping MCA curve in the figure. In addition, we must consider the marginal cost of producing the chips that are used in the copier. In Figure 6A.1, this marginal cost (labeled MCM) is superimposed on the MCA curve. The total marginal cost of producing copiers is the sum of the chip cost and the assembly cost. In the figure, this total marginal cost curve is denoted by MCT MCA MCM and is drawn as the vertical sum of the two curves. (Note that MCM is depicted as slightly upward sloping; that is, the gap between MCA and MCT steadily increases as output rises.)

The firm maximizes the total profit it earns on copiers by setting the quantity such that marginal revenue equals total marginal cost, MR MCT. In Figure 6A.1, the optimal quantity occurs at Q*, and the associated optimal selling price for the copier is P*. What transfer price for chips will lead to this outcome? The appropriate transfer price should be set at PT MCM. By paying this transfer price, the copier division incurs an additional cost of MCA PT MCA MCM for each extra copier it produces. By taking into account the true “full” cost of producing additional output, the copier division automatically maximizes the firm’s total profit.

A MARKET FOR CHIPS If there is an external market in which microchips can be bought and sold, the profit-maximizing analysis must be modified. In this case, the firm should set the internal transfer price for the microchip equal to the prevailing market price. The reasoning is straightforward. Let P denote the prevailing market price. Obviously, the chip division cannot charge the copier division a transfer price that exceeds P ; the copier division would simply opt to buy chips from the outside market. Nor would the chip division be satisfied with a transfer price below P ; it would prefer to produce and sell chips exclusively for the outside market. Consequently, PT P is the only price at which internal transfers will occur.

Here is another way to arrive at this conclusion. The correct price to impute to internally produced chips should reflect the firm’s true opportunity cost. Each chip that goes into the “guts” of the firm’s copiers is a chip that could have been sold on the outside market at price P . Since it is this market price that the firm gives up, the internal transfer price should be set accordingly.

FIGURE 6A.1

Transfer Pricing

In the absence of an external market, the optimal transfer price for an intermediate product equals the item’s marginal cost of production. Optimal output of the firm’s final product occurs at Q*, where MR equals MCT . Here MCT includes the intermediate good’s transfer price. Dollars per Unit of Output

P

Marginal revenue

PT=MCM MCM MCT Demand

MCA

Q

Output

TRANSFER PRICING: A NUMERICAL EXAMPLE Let the demand for copiers be given by P 4,000 3Q, where Q is the number of copiers demanded per week and P is the price in dollars. The total cost of assembling copiers (excluding the cost of microchips) is given by CA 360,000 1,000Q. The cost of producing microchips is CM 40,000 200QM .5QM 2, where QM is the quantity of chips. Suppose each copier uses one microchip. The total cost of producing copiers is CT CA CM 400,000 1,200Q .5Q2. In turn, the marginal cost of copiers is MCT dCT/dQ 1,200 Q. Equivalently, MCT MCA MCM 1,000 (200 Q) 1,200 Q. Setting MR MCT implies 4,000 6Q 1,200 Q. Thus, Q* 400 and P* 4,000 (3)(400) $2,800. At a production rate of 400 microchips per week, marginal cost is

MCM 200 400 $600. Thus, in the absence of an external market for microchips, the appropriate transfer price is PT MCM $600. At an output of 400 chips, the average cost per chip is ACM CM/Q $500. Thus, by selling its output to the copier division at PT 600, the chip division earns an internal profit of ($600 $500)(400) $40,000 per week. The copier division’s average total cost is ACA CA/Q $600 $2,500 per copier. At P* 2,800, the division makes a profit of $300 per copier, implying a total profit of $120,000 per week. The combined profit of the divisions is 40,000 120,000 $160,000.

Now, suppose an external market for chips exists and a chip’s current market price is PM $900. For each additional chip produced and sold, the chip division’s marginal revenue equals $900, the current market price. Setting MR MC implies 900 200 QM, where QM denotes the quantity of microchips. The solution is QM 700. Next consider the copier division. The price it pays for chips is now PT PM $900. Thus, its marginal cost (inclusive of the price of chips) is MCT 1,000 900 $1,900. Setting MR MC implies 4,000 6Q $1,900. Thus, Q* 350 and P* $2,950. To sum up, the chip division’s total weekly output is 700 chips. Half of this output (350 chips) is transferred to the copier division; the other half is sold on the open market.

Questions and Problems

1.a.A senior manager argues that the chip division’s main purpose is to serve the firm’s final-product divisions. Accordingly, these services should be offered free of charge; that is, the transfer price for chips should be PT 0. Explain carefully what is wrong with this argument. b.Suppose the chip division treats the copier division as it would an outside buyer and marks up the transfer price above marginal cost.

Explain what is wrong with this strategy. 2.In the numerical example, suppose the firm can purchase chips on the open market at a price of $300. What production decisions should the divisions make in this case?

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SPECIAL APPENDIX TO CHAPTER 6

Short-Run and Long-Run Costs

This appendix takes a closer quantitative look at the cost setting of Spreadsheet Problem S1 and its illustration in Figure 6.3. We start with the following economic facts. Let the firm’s production function be given by Q 12L.5K.5 122L # 2K [6A.1]

where L and K are in thousands of units. The prices of labor and capital are PL 36 per unit and PK 16 per unit, respectively.

SHORT-RUN COSTS We begin by deriving expressions for the firm’s SAC and SMC. To do this, we fix the amount of capital at some level; call this K . With capital fixed, we solve Equation 6A.1 for L: L Q2/(144K ). [6A.2]

Total cost is

C 16K 36L 16K Q2/(4K )

after substituting for L. In turn, short-run average cost is

SAC C/Q 16K /Q Q/(4K ) [6A.3]

and SMC dC/dQ Q/(2K ). It is easy to check that SAC is U-shaped. For instance, by setting K equal to 9 thousand square feet, we obtain the SAC function:

This is the equation of the first SAC curve graphed in Figure 6.3. By setting K 18 thousand and K 27 thousand, we trace out the other SAC curves in the figure.

LONG-RUN COSTS We can now confirm that the firm’s LAC and LMC curves are constant, as shown in Figure 6.3. One way of doing so is to note that the level of LAC is given by the minimum point of each SAC curve. Returning to the SAC expression in Equation 6A.3, we can show that the point of minimum average cost occurs at output

[6A.4] To see this, remember that the SMC curve intersects the SAC curve at its minimum point. Equating the preceding expressions for SAC and SMC, we find 16K /Q Q/4K Q/2K . The solution is Q2 64(K )2, or Q 8K . After substituting Q 8K into Equation 6A.3, we find the firm’s minimum average cost to be min SAC 16K /8K 8K /4K 4/unit. In turn, substituting Q 8K into Equation 6A.2 implies

[6.A5] This equation specifies the necessary amount of labor to be used in conjunction with a plant of size K .

In summary, Equation 6A.3 describes the short-run average cost of producing Q units of output using K units of fixed capital, whereas Equation 6A.2 specifies the requisite amount of labor. For instance, the short-run average cost of producing 54 thousand units of output in a 9-thousand-square-foot plant is SAC 144/54 54/36 $4.17. The necessary amount of labor is L (54)2/[(144)(9)] 2.25 thousand labor-hours. In turn, the LAC of producing Q units is $4 and is achieved using the amounts of capital and labor given by Equations 6A.4 and 6A.5. Thus, to produce 54 thousand units in the long run, the firm should use a 54,000/8 6.75-thousand-square-foot plant and (4/9)(6.75) 3 thousand labor-hours.

SAC 144/Q Q/36.

Q 8K .

L (4/9)K .

Questions and Problems

1.a.Using Equation 6A.3, find the short-run average cost of producing 108 thousand units in a 9-thousand-square-foot plant. Determine the necessary quantity of labor. b.Would the same output be less expensive to produce using an 18-thousand-square-foot plant?

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