Profit Maximization with Limited Capacity: Ordering a Best Seller

out, other government programs, once begun, seem to have lives of their own.

CHECK STATION 2 A firm spent $10 million to develop a product for market. In the product’s first two years, its profit was $6 million. Recently, there has been an influx of comparable products offered by competitors (imitators in the firm’s view). Now the firm is reassessing the product. If it drops the product, it can recover $2 million of its original investment by selling its production facility. If it continues to produce the product, its estimated revenues for successive two-year periods will be $5 million and $3 million and its costs will be $4 million and $2.5 million. (After four years, the profit potential of the product will be exhausted, and the plant will have zero resale value.) What is the firm’s best course of action?

The notion of opportunity cost is essential for optimal decisions when a firm’s multiple activities compete for its limited capacity. Consider the manager of a bookstore who must decide how many copies of a new best seller to order. Based on past experience, the manager believes she can accurately predict potential sales. Suppose the best seller’s estimated price equation is P 24 Q, where P is the price in dollars and Q is quantity in hundreds of copies sold per month. The bookstore buys directly from the publisher, which charges $12 per copy. Let’s consider the following three questions:

1.How many copies should the manager order, and what price should she charge? (There is plenty of unused shelf space to stock the best seller.) 2.Now suppose shelf space is severely limited and stocking the best seller will take shelf space away from other books. The manager estimates that there is a $4 profit on the sale of a book stocked. (The best seller will take up the same shelf space as the typical book.) Now what are the optimal price and order quantity? 3.After receiving the order in Question 2, the manager is disappointed to find that sales of the best seller are considerably lower than predicted.

Actual demand is P 18 2Q. The manager is now considering returning some or all of the copies to the publisher, who is obligated to refund $6 for each copy returned. How many copies should be returned (if any), and how many should be sold and at what price?

As always, we can apply marginal analysis to determine the manager’s optimal course of action, provided we use the “right” measure of costs. In

TABLE 6.1

An Optimal Book Order

The optimal number of books to order and sell depends on demand, sales costs, and opportunity costs.

Sales Forgone Final Price Revenue Cost Profit Net Profit

(a) Qs 600 $18 $10,800 $7,200 $ 0 $3,600

(b) Qs 400 20 8,000 4,800 1,600 1,600

[Qs 600] 18 10,800 7,200 2,400 1,200

(c) Qs 200 14 2,800 4,800 800 1,600

Qr 200 6 1,200 0

[Qs 400] 10 4,000 4,800 1,600 2,400

[Qr 0] 6 0 0

[Qs 0] — 0 4,800 0 2,400

[Qr 400] 6 2,400 0

Question 1, the only marginal cost associated with the best seller is the explicit $12 cost paid to the publisher. The manager maximizes profit by setting MR equal to MC. Since MR 24 2Q, we have 24 2Q 12. The result is Q 6 hundred books and P $18. This outcome is listed in Table 6.1a.

By comparison, what are the optimal order quantity and price when shelf space is limited, as in Question 2? The key point is that ordering an extra best seller will involve not only an out-of-pocket cost ($12) but also an opportunity cost ($4). The opportunity cost is the $4 profit the shelf space would earn on an already stocked book—profit that would be forgone. In short, the total cost of ordering the book is 12 4 $16. Setting MR equal to $16, we find that Q 4 hundred and P $20. Given limited shelf space, the manager orders fewer best sellers than in Question 1. Table 6.1b compares the profitability of ordering 400 versus 600 books. The cost column lists the store’s payment to the publisher ($12 per best seller). Forgone profit is measured at $4 per book.

We confirm that ordering 400 books is the more profitable option, taking into account the forgone profit on sales of other books. Indeed, the logic of marginal analysis confirms that this order quantity is optimal, that is, better than any other order size.

Finally, Question 3 asks how the manager should plan sales and pricing of the 400 best sellers already received if demand falls to P 18 2Q. The key here is to recognize that the original $12 purchase price is irrelevant; it is a sunk cost. However, opportunity costs are relevant. The opportunity cost of keeping the best seller for sale has two elements: the $4 profit that another book would earn (as in Question 2) plus the $6 refund that would come if the copy were returned. Therefore, the total opportunity cost is 6 4 $10.

Setting MR equal to MC implies 18 4Q 10, or Q 2 hundred. The store manager should keep 200 books to be sold at a price of 18 (2)(2) $14 each. She should return the remaining 200 books to obtain a $1,200 refund. As Table 6.1 indicates, this course of action will minimize her overall loss in the wake of the fall in demand. The table also shows that selling all 400 copies or returning all copies would generate greater losses.

Pricing E-books For book publishers, the penetration of e-books represents both an opportunity and a threat. On the one hand, this new platform has the potential to spur overall book sales. On the other, e-books threaten to cannibalize sales of highermargin print books. E-book sales were accelerating—growing from 2.9 percent of total sales in 2009 to 8.5 percent in 2010. Book publishers faced the key question: How should they market and price print books and e-books to maximize overall profit? Of one thing publishers were sure: Amazon, in its attempt to seize control of the e-book market, was the enemy. Amazon’s $9.99 pricing strategy for most e-books—though attractive to customers—had the effect of battering printbook sales. In self-defense, five of the largest book publishers banded together in 2010 to establish a key pricing agreement with Apple (and subsequently Google). In short order, Amazon was forced to acquiesce to the same terms. Under the so-called agency pricing arrangement, the book publisher would set the e-book price, while the online retailer would serve as an agent. Sales revenue would be split, 70 percent to the publisher and 30 percent to the internet seller. Book publishing has long been criticized for its culture of two-martini lunches—that is, cultivating authors, and publishing a plethora of titles with too little regard to the tastes of the buying public and to bottom-line profits. Bookstores, not publishers, are the points of contact with readers. Today, the catch phrase is “Profit or Perish.” So with profit in mind, let’s consider the stripped-down economics of the book business. The typical hardcover best seller is priced at about $26 retail. Of this amount the publisher keeps 50 percent (or $13) and the remaining 50 percent goes to the bookstore. Out of its share, the publisher pays a 15 percent royalty ($3.90) to the author and costs due to printing, shipping, and book returns (typically about $3.50). The economics of e-books are much simpler. The marginal cost per e-book is negligible, and as noted earlier, revenues are split 70–30 between the publisher and the online agent. Given the power to establish prices, what hardcover and e-book prices should book publishers set? Part of the answer lies in recognizing the opportunity cost associated with aggressive e-book pricing. Yes, selling additional e-books provides publishers with additional revenue. But it also means selling fewer print books and foregoing some of the associated profit from that high-margin