6

Price Differentiation

In this chapter we examine one of the most fundamental concepts in pricing and revenue optimization—price differentiation. Price differentiation refers to the practice of a seller charging different prices to different customers, either for exactly the same good or for slightly different versions of the same good. It can be a powerful tactic for sellers to improve profitability. It also adds a new level of complexity to pricing, often driving the need to use analytical techniques to improve the calculation and updating of prices over time. Price differentiation can also be dangerous if not managed correctly—it can lead to loss of profit through arbitrage and to negative reactions to pricing that some customers may perceive as unfair.

Tactics for price differentiation include charging different prices to different customers (or groups of customers) for exactly the same product, charging different prices for different versions or amounts of the same product, and combinations of the two. It also includes charging different prices for combinations of products than for individual products sold separately. I use the term “price differentiation” rather than the standard economic term price discrimination in part to avoid the negative connotations associated with the word “discrimination.” However, I also want to stress that price differentiation includes not only charging different prices to different customers for the same product (group pricing) but also the less controversial strategies of product versioning, regional pricing, channel pricing, and nonlinear pricing.

While “price differentiation” and “dynamic pricing” are often used interchangeably, there is an important distinction between the two concepts. Price differentiation refers to strategies for charging different prices to different customers based on differences in their willingness to pay. Price differentiation usually (but not always) means that different prices are in play for the same (or very similar products) at the same time. Dynamic pricing, on the other hand, refers to changing the price of a product over time. Dynamic pricing is often employed to balance supply and demand—as in the case of the variable-pricing approaches discussed in Chapter 7. Prices may also change over time as a result of changes in the desirability of a product over time—such as the declining appeal of a fashion item as the end of the season approaches. In this case, all customers may see the same price at the same time, but the price they see changes over time.

The distinction between price differentiation and dynamic pricing is important but not as clear-cut as it may seem. If two customers are being charged different prices for the same product at the same time, that is definitely price differentiation. When a ride-sharing service raises prices for all customers in a location during rush hour, that is dynamic pricing (see Section 7.6.4 for details). However, as we shall see, time can also be used as a segmentation dimension. Airline prices tend to go up as departure approaches in part because airlines know that late-booking customers typically have a higher willingness to pay for a ticket than early-booking customers. In this case, prices may be changing over time both as a form of price differentiation and to balance supply and demand (Chapter 8 discusses how airlines change prices over time as a form of price differentiation). Furthermore, a seller can perform price differentiation and dynamic pricing simultaneously—airlines raise and lower prices for the same flight prior to departure (dynamic pricing), but at each point in time they may be offering different prices to customers purchasing the ticket in Europe than in Japan (price differentiation).

There is both art and science to price differentiation. The art lies in finding a way to divide the market into different segments such that higher prices can be charged to the high-willingness-to-pay segments and lower prices to the low-willingness-to-pay segments. There is no one way to segment customers that applies to all possible markets. Instead, a variety of techniques can be applied in different ways, depending on the characteristics of a market, the competitive environment, and the character of the goods or services being sold. The science lies in setting and updating the prices to maximize overall return from all segments.

We start by using a simple example (based on the widget maker from Chapter 5) to illustrate the motivations behind price differentiation.

6.1 THE ECONOMICS OF PRICE DIFFERENTIATION

Let us return to the case of the widget maker in Example 5.3. The widget maker, looking for a way to increase his profitability, might start by taking a close look at his price-response function. He would realize that 10,000 customers are willing to pay some amount greater than zero for his widget, 5,000 are willing to pay $6.25 or more, 3,000 are willing to pay $8.75 or more (the contribution-maximizing price), and 1,200 are even willing to pay $11.00 or more. The marketing department might see an opportunity to improve profitability. After all, 2,000 of the customers who are currently purchasing at $8.75 would be willing to pay $10.00 or more for the product. Those customers are getting a great deal at $8.75, but from the seller’s point of view, it is at least $1.25 in additional revenue left on the table from each sale. Furthermore, 3,000 customers are willing to pay more than the production cost ($5.00) but less than the sales price of $8.75. Each of these is a potentially profitable sale lost because the price is too high. What if the seller could determine the maximum amount that each customer would be willing to pay and could charge that amount to everyone willing to pay more than the per-unit cost of $5.00? This would be pricing nirvana—the ultimate in one-to-one pricing.

Figure 6.1 Contribution opportunity from price differentiation.

The total potential opportunity for profit improvement from price differentiation is shown in Figure 6.1. With a single price, the widget maker charges $8.75 per unit and realizes a total profit of ($8.75 – $5.00) × 3,000 = $11,250. This is the area of region A in Figure 6.1. Better price differentiation would enable the seller to charge each customer who is willing to pay more than $8.75 per unit her exact willingness to pay. This would provide the additional revenue in region B in the figure. There is an additional opportunity to sell widgets to customers who are willing to pay more than $5.00, the unit production cost, but less than $8.75. This potential profit is shown as region C in the figure. The sum of the three regions is the total contribution that the widget maker would realize if he were able to charge every potential customer exactly at her willingness to pay. This is an upper bound on what could possibly be realized under any price differentiation program. Charging every customer exactly her willingness to pay is known as first-degree price discrimination in the economics literature.¹

While it is unrealistic to assume that this total potential could ever be captured, the sheer magnitude of the potential gain means there is a powerful motivation for sellers to tailor different prices to different customers according to their willingness to pay. Assume that the widget maker finds he can divide his market into two segments. One segment consists of all customers willing to pay more than $7.00 for widgets. The other segment consists of all the customers willing to pay $7.00 or less. The corresponding price-response functions are

where d₁(p₁) is the price-response function for customers with a high willingness to pay and d₂(p₂) is the price-response function for customers with a lower willingness to pay. These two price-response functions are shown in Figure 6.2. The sum of these two functions is the original price-response function. The difference is that now the seller can offer a different price to each of the two segments. We assume for this example that the widget maker can perfectly identify customers as belonging to one group or the other and can then offer each customer the appropriate price, without any opportunity for resale or arbitrage between the two groups.

Figure 6.2 Price-response functions for two segments.

TABLE 6.1

Impact of market segmentation

The widget maker can determine the optimal prices for each segment by solving for any of the optimality conditions in Section 5.2.1 twice—once for each segment. The results and a comparison to the unsegmented case are shown in Table 6.1. Dividing his customers between those with willingness to pay greater than $7.00 and those with willingness to pay less than $7.00, and charging the optimal prices to the different segments, enables the widget maker to increase his revenues by 18% and his contribution by 7%. It is interesting to note that customers also benefit. The same 3,000 customers who would buy under a single price at $8.75 still get to buy at $8.75. In addition, 800 additional customers get to purchase at the new low price of $6.00. These 800 customers are priced out of the market if the seller can only charge a single price. In this case, price differentiation is a win-win situation, since the seller is certainly better off and all of the customers are at least as well off as before. However, price differentiation is not always a boon for consumers, as we see in Section 6.5.

6.2 LIMITS TO PRICE DIFFERENTIATION

If price differentiation is such a powerful way for sellers to increase contribution, why is it that sellers do not always do it for all customers? There are three powerful real-world limits to price differentiation.

• Imperfect segmentation. The brain-scan technology required to determine the precise willingness to pay of each customer has not yet been developed. The best that can be done is to create market segments such that the average willingness to pay is different for each segment.

• Cannibalization. Under differential pricing, there is a powerful motivation for customers in high-price segments to find a way to pay the lower price. In the widget example, there is a strong motivation for high-willingness-to-pay customers who are being charged $8.75 per widget to masquerade as low-willingness-to-pay customers and pay only $6.00 per widget.

• Arbitrage. Price differentials create a strong incentive for arbitrageurs to find a way to buy the product at the low price and resell to high-willingness-to-pay customers below the market price, keeping the difference for themselves.

The presence of any one of these factors can drastically reduce or even eliminate the benefits of price differentiation.

Example 6.1

Assume that the widget seller has segmented his market and is charging the two prices shown in Table 6.1. If 300—that is 10%—of the high-willingness-to-pay customers are able to find a way to purchase widgets at the lower cost of $6.00, it would totally eliminate the benefits of price differentiation.

The deterioration of benefits in Example 6.1 could arise from imperfect segmentation, cannibalization, arbitrage, or a combination of any of them.

6.3 TACTICS FOR PRICE DIFFERENTIATION

The previous section establishes two properties of price differentiation.

1. Successful price differentiation allows sellers to increase profitability by charging different prices to customers with different willingness to pay.

2. Cannibalization, imperfect segmentation, or arbitrage can destroy—or even reverse—the benefits of price differentiation.

The first property means that sellers have a tremendous incentive to find creative ways to position and price their products differently to different market segments. The second means that price differentiation needs to be carefully planned and managed in order to be effective. In this section we look at some of the most common and effective approaches to price differentiation used in different markets.

6.3.1 Group Pricing

Group pricing is the tactic of offering different prices to different groups of customers for exactly the same product at the same time.² The idea is, of course, to offer a lower price to customers with a low willingness to pay and a higher price to those with a high willingness to pay. In practice, pure group pricing requires determining whether a customer belongs to an identified group and using that information to determine which price to offer. Examples of pure group pricing include

• Student discounts

• Senior citizen discounts

• Ladies’ Night specials

• Family specials

• Discounts for favored customers

• Favorable terms offered by manufacturers or wholesalers to large retailers such as Walmart

• Lower prices offered to government, educational, and nonprofit organizations by suppliers

Four criteria must hold for group pricing to be successful.

• There must be an unambiguous indicator of group membership. Examples of such indicators include a student ID card or a driver’s license that lists age. Furthermore, it must be difficult or impossible for members of one group to masquerade as members of another. Otherwise, cannibalization could easily reduce or even erase the benefits of price differentiation.

• Group membership must strongly correlate with price sensitivity. Senior citizen discounts are predicated on the knowledge that senior citizens, on average, are more price sensitive than the public in general.

• The product or service cannot be easily traded among purchasers. This is necessary to avoid arbitrage, in which some customers with access to low prices resell to customers who are quoted higher prices.

• The segmentation must be both culturally and legally accepted. Group pricing can be extremely controversial. Certain group pricing practices—setting different prices for different races, for example—are illegal in many jurisdictions. Other group pricing practices such as gender-based pricing may or may not be illegal (depending on the location) but may be extremely unpopular. Note that the social acceptability of many group pricing practices depends not only on the prices themselves but the ways in which the prices are communicated. Thus, a senior citizen discount may be acceptable, while a middle-age surcharge would likely not be acceptable. We examine these issues in depth in Chapter 14.

Taken together, these criteria are so stringent that pure group pricing is relatively rare in direct consumer sales. Group pricing is most common in services. There are a number of reasons for this. Many service suppliers can verify group membership; for example, a movie theater can check a driver’s license before applying a senior citizen discount. Most services, from health care to haircuts, are intrinsically nontransferable, so arbitrage is not an issue. Airlines check passenger identification in part as a way to prevent arbitrage. Finally, group pricing is common in consumer and small business lending where the rate for a mortgage or loan might depend on the credit rating of the customer.

Pure group pricing is also common in business-to-business sales. Chapter 13 discusses how businesses can estimate price responsiveness and develop customized prices for different business segments. Customized pricing in business-to-business sales is often combined with some form of product versioning, particularly in industries where individual orders are very complex and/or configurable.

6.3.2 Channel Pricing

Channel pricing is the practice of selling the same product for different prices through different distribution channels. Here are some examples.

• Barnes and Noble sells books for different prices online than in its stores.

• Special web-only fares for airline tickets are available through the internet but not through travel agencies.

As with other price differentiation schemes, there can be more than one reason why a seller might charge different prices through different channels. One is cost: for many companies, selling through the internet is cheaper than selling through traditional channels. In 2002, the average distribution cost for an airline ticket booked through an offline travel agency was $30.66 compared to $11.75 for one booked directly on an airline’s website (Barnes 2012). This cost difference alone would be motivation for an airline to charge less for tickets purchased on its website than through a travel agency.

However, it is also the case that customers arriving via different channels have different price sensitivities. For personal loans, it has been shown that customers inquiring through the internet are more price sensitive than those contacting a call center, who are in turn more price sensitive than those who apply for a loan at a retail branch. Furthermore, internet customers who access a consolidator website or use a shopping bot tend to be more price sensitive than those who go directly to a bank’s website. This is not surprising given the characteristics of the channels—it is generally easier and more convenient to shop and compare prices during a single internet session than by making many phone calls. Thus, differential willingness to pay is also a motivation for channel pricing.

6.3.3 Regional Pricing

Regional pricing is a common price differentiation technique. Here are some examples.

• In Latin America, McDonald’s sells hamburgers for higher prices in wealthy neighborhoods than in poorer ones (The Economist 2004).

• A round-trip New York–Tokyo ticket purchased in Japan will usually cost more for many airlines than the same ticket purchased in the United States.

• A cold 20-ounce bottle of Coke costs $1.50 at the local drugstore, $1.99 at a gas station, $2.79 at the airport newsstand, and $3.49 at a hotel gift shop (Simon and Fassnacht 2018, 210).

• In the United States and in Canada, banks often charge different rates for mortgages in different regions of the country.³

In each case, the price difference is based on the supplier’s desire to exploit differences in price sensitivity between locations. One limiting factor in regional pricing is typically arbitrage—entrepreneurs can theoretically purchase in the lower-price location, transport to the higher-price location, and resell, keeping the difference. Note that, in each of the cases listed above, arbitrage is difficult or impossible.

In addition to arbitrage, another limiting factor in regional pricing is the potential for region to be perceived as a proxy for other customer characteristics such as income levels or ethnicity. Redlining was the name given to the policies of many U.S. banks not to lend to customers in certain urban neighborhoods—neighborhoods that were typically predominantly African American. Redlining was widely condemned and was ultimately outlawed in the United States.⁴ More recently, a 2015 ProPublica study revealed that Princeton Review charged higher prices for its test preparation services in zip codes that had higher proportions of Asian Americans than other zip codes. Princeton Review admitted differential pricing by region but denied discrimination in “the literal, legal and moral meaning of the word” (Larson, Mattu, and Angwin 2015).

6.3.4 Couponing and Self-Selection

We have seen that group pricing is often both difficult and unpopular—difficult because it requires the seller to categorize customers on the basis of price sensitivity before quoting them a price, and unpopular because it often seems unfair to consumers. It is often much more convenient to differentiate prices in ways that allow customers to self-select. In a self-selection approach, both the list price and a discounted price are available to all customers, but it takes additional time, effort, or flexibility to obtain the discounted price. The idea is that those willing to make the additional effort to get the discount are generally more price sensitive than those who are not. Here are some examples.

• Retailers commonly offer discount coupons through newspapers, direct mail, and email.

• Retailers often offer mail-in rebates for purchasers of a good.

• Movie theaters charge lower prices for a weekday matinee than for a Saturday night show. (This is also an example of variable pricing, as discussed in Section 7.6.5.)

• Brand-name retailers such as Ralph Lauren, the Gap, and Liz Claiborne operate outlet stores in out-of-the-way locations, making merchandise available for a substantial discount. For example, Ralph Lauren operates several branded stores in Manhattan that sell its clothing at full price. It also operates a so-called factory store in Elizabeth, New Jersey, about 20 miles away, in which it sells some of the same items at a deep discount. Local customers are likely to be more price sensitive than Manhattan dwellers. Manhattan dwellers who are willing to make the trek to New Jersey are likely more price sensitive than those who are willing to simply purchase Ralph Lauren from a local store.

The common thread among these examples is that the seller has chosen a mechanism that allows customers to self-select, depending on the value they place on time or flexibility. Any customer can obtain an item at a discount if she is willing to make an additional effort. Research has confirmed that users of coupons are more price sensitive than nonusers of coupons (Narasimhan 1984). Locating outlet stores away from large cities segments the market between those who are willing to spend additional time in transit—for example, driving from Manhattan to Elizabeth, New Jersey, and back—and those who are not. Peak-load, day-of-week, and time-of-day pricing segments the market between those who have the flexibility to change their plans in order to save some money and those who are not willing to do so. Since these mechanisms are based on self-selection, they are far more acceptable to most consumers than mechanisms in which the seller unilaterally selects customers to receive discounts.

6.3.5 Product Versioning

When pure group pricing is not feasible, companies use other strategies to differentiate prices. The most notable of these is designing or developing products (either virtual or real) that may have only minor differences but enable the seller to exploit differences in price sensitivity among customer segments. This can involve developing an inferior variant and/or a superior variant of an existing product. We look at examples of both strategies as well as their logical extension into the creation of a product line.

Inferior goods. Consider the following cases.

• Well-known international brand names such as ExxonMobil and Shell sell excess gasoline in bulk at low prices to so-called off-brand independent dealers who resell it under their own brands.

• A well-known premium wine producer sells some of its production under a different label at about half the price.

• Brand-name vegetable canners sell their products under their own brand but also sell to retailers who sell the product to consumers as a house brand or a generic brand.

While the specifics of each of these cases differ, the motivation on the part of the seller is the same—a desire to sell a product cheaply to customers with lower willingness to pay without cannibalizing sales of the full-price product. This is achieved in each case by creating an inferior version of the standard product.

A particularly extreme example of inferior goods is the category of so-called damaged goods. This is a term coined to refer to the situation in which a manufacturer or supplier creates an inferior good by damaging, degrading, or disabling a standard good (Deneckere and McAfee 1996). Since this process usually starts with the standard good, it may actually cost the supplier more to create the inferior good, which it then sells at a lower price. One example is the 486SX processor developed and sold by Intel Corporation.

The 486SX processor of Intel Corporation was initially produced in a curious way. Intel began with a fully functioning 486DX processor, then disabled the math coprocessor, to produce a chip that is strictly inferior to the 486DX but more expensive to produce. Nevertheless, in 1991, the 486DX sold for $588, and the 486SX for $333, a little over half the price of the chip that is less expensive to produce. (Deneckere and McAfee 1996, 149; emphasis in original)

Another example is the Tesla. According to an article in the Economist, “The cheaper models in Tesla’s line-up have parts of their batteries disabled by the car’s software in order to limit their range” (2019, 3). When Hurricane Dorian was heading toward America’s East Coast in August 2019, Tesla, with the “tap of a keyboard in Palo Alto,” was able to temporarily remove the artificial restrictions and “give drivers access to the full power of their batteries” (3). The implication is that the range restriction for the less expensive models was imposed simply to create an inferior product that could be sold at a lower price.

Complex application software packages such as supply chain software or enterprise resource planning (ERP) software are often sold at different prices, depending on the number of features purchased by a customer—the more features purchased, the more expensive the license. In many cases, the software is configured for a particular customer by starting with the complete package and then disabling the features that the customer did not purchase.

The concept of damaging a good to create an inferior good to sell at a lower price may initially seem somewhat bizarre. However, it is really only a special case of the more general category of inferior goods. There can be a tremendous gain from offering an inferior good at a lower price, even if the supposedly inferior product is more expensive to produce. Starting from a standard product and then paying to have it damaged is only a special case of this more general principle.

Superior goods. Spendrups is the largest brewery in Sweden. Traditionally Spendrups brewed medium- or low-priced lagers aimed at the mass market. In the 1980s, it created Spendrups Old Gold, which it advertised as a premium beer and sold in a special, highly distinctive bottle. Although Old Gold did not generally fare better than Spendrups’s other brands in comparative taste tests, Spendrups was able to establish Old Gold as a premium brand and maintain a price 25% to 50% higher than that of its other brands. This is the obvious complement to the inferior-good strategy: creating a superior good to extract a higher price from less price-sensitive customers. When sales of Old Gold began to flag, it was relaunched in a distinctive 10-sided bottle in 2010 to maintain the premium image.

Another example of the superior-good strategy was employed by Proctor-Silex. In 1985, Proctor-Silex priced a top-of-the-line iron at $54.95, while its next best model was priced at $49.95. The only difference between the two was that the top model had a small light indicating when the iron is ready to use. The difference in manufacturing cost between the two models was only $1.00, yet Proctor-Silex was able to maintain a $5.00 price difference because, as a Proctor-Silex marketing manager put it, “There is a segment of the market that wants to buy the best, despite the cost” (quoted in Birnbaum 1986, 175). By creating a superior product, Proctor-Silex enabled the less price-sensitive segment of the market to self-select and extracted an additional $4.00 in contribution margin from each high-end customer. In some ways, a superior-good strategy is safer than an inferior-good strategy because it does not threaten cannibalization of existing sales. Of course, it presumes an ability to create and establish a product that the market perceives as truly superior to the existing product and presumes that there is a customer segment willing to pay a premium for the superior product.

Product lines. Establishing a product line is the natural extension of creating inferior or superior products. A product line is a series of similar products serving the same general market but sold at different prices. For our purposes, we consider vertical product lines, where almost all customers would agree that a higher-priced product is superior to a lower-priced one. This applies, for example, to a hotel that charges more for an ocean-view room than a parking-lot-view room—almost all customers would prefer the ocean view. It also applies to personal computers offered by Apple, where each product in the line has higher performance (faster CPU, more memory, etc.) than the product just below it in the line. This can be contrasted to horizontal product lines, where different customers prefer different products within the line, even at the same price. Coca-Cola’s offering of Classic Coke, Diet Coke, Cherry Coke, Diet Cherry Coke, Coke Zero, and so on is a horizontal product line because no single product is unambiguously higher quality or more desirable than another.

An example of a vertical product line is shown in Table 6.2, which gives prices for four versions of the QuickBooks financial software offered by Intuit. The versions are the Simple Start version, which costs $25.00 per month; the Essentials version, which costs $40.00 per month; the Plus version, which costs $70.00 per month; and the Advanced version, which costs $150.00 per month.⁵

In all likelihood, there is only a tiny difference (if any) in the marginal cost of producing and delivering the four different versions of QuickBooks software. Yet the Advanced package is priced six times higher than the Simple Start version. The rationale for the product line and the broad difference is market segmentation—very small businesses that need less functionality and have fewer users are presumably more price sensitive than larger businesses with more users. The establishment of a product line allows Intuit to segment its market via self-selection on the part of its customers.

TABLE 6.2

Online prices offered by Intuit for the QuickBooks product family, May 2020

TABLE 6.3

Rates offered by Budget on Expedia for a one-day midweek rental, January 2020

Another example of product-line pricing is illustrated in Table 6.3, which shows the rates on display through the online travel agency Expedia for a Budget Rental Car one-day rental from the Seattle airport on a midweek day in January 2020. Budget lists 10 different products, with prices ranging from $68.00 per day for an economy rental to $123 per day for a premium SUV. By providing a menu of alternatives at different prices, Budget allows customers to self-select: The economy-minded will choose the economy or compact cars, while those who are willing to pay for greater comfort may opt for the full-size or luxury. Furthermore, those traveling with large families may want or need to rent a larger car or an SUV. It is important to stress that the range of rates offered by Budget for these different products to rental car companies is not driven by cost differences. Taking into account the revenue from resale, life-cycle costs do not vary much among different car types. More important, there is little or no difference in the daily incremental cost to Budget for renting out an economy car or a premium SUV. The spread in daily rates is driven almost entirely by Budget’s desire to segment its market.

The pricing menus offered by Budget and Intuit are openly communicated and available to all comers.⁶ Consumers get to choose among the alternatives, and the concept of paying more to get more is widely accepted. This makes product-line pricing more acceptable to most customers than group pricing.

For service companies like Budget, creation of a product line has an important side benefit—it creates opportunities for upgrading. A rental car company or a cruise line has the ability to oversell lower-quality car types and upgrade customers into higher-quality types. Not only does this provide the company with greater flexibility to manage its inventory, but being upgraded is usually viewed favorably by customers. An important advantage of pricing differentiation by establishing a product line is that consumers generally perceive it as fair.

6.3.6 Time-Based Differentiation

Time-based differentiation is a common form of product versioning. Here are some examples.

• Target.com offers Standard Shipping of 3 –5 business days free for items of $35 or more. For selected items and locations, 2-Day Shipping and Express Shipping of 1-Day are offered at an additional cost.

• Passenger airlines offer discount rates to customers who book a week or more prior to departure. Hotels offer higher rates to customers who “walk up” to obtain a room for that evening.

• Software- and hardware-support contracts charge more for two-hour response than for two-day response.

• Fashion goods cost more during the beginning of the season and are marked down toward the end of the season.

In each of these cases, companies have created differentiated products that allow customers to self-select. In the case of Target, customers who are willing to wait for delivery can pay less. For passenger airlines, customers who have the flexibility to book earlier can pay less. For the passenger airlines, it costs no more to accommodate a passenger who books later than one who books earlier. It may be that the higher price charged by Target for early delivery exactly matches the incremental cost. But it is likely that Target is also using time of delivery as a segmentation variable, relying on the willingness of some of its customers to pay a premium to have the product in their hands sooner, just as the airline is using time of booking to segment customers.

Time-based differentiation plays an important role at passenger airlines, hotels, and rental car companies, in which time of booking and other factors, such as willingness to accept a Saturday night stay-over, are used as indicators of whether a potential customer is traveling for leisure purposes or business purposes. Those traveling for leisure purposes are presumed to be more price sensitive than those traveling for business. This segmentation is the foundation of revenue management in those industries, and Chapter 8 discusses it in detail. The willingness of some customers to wait to purchase fashion goods at a discount is the basis of markdown management, which is treated in Chapter 12.

6.3.7 Product Versioning or Group Pricing?

We have treated product versioning and group pricing as separate strategies for price differentiation. In reality, no clear line separates the two approaches, and many price differentiation strategies contain elements of both. For example, consider the classic airline example (which we explore in detail in Chapter 8) of a round-trip ticket from San Francisco to Chicago costing $250 if purchased a week in advance and including a Saturday stay-over versus $750 if purchased at the last minute without restrictions. Is this group pricing or product versioning? Disgruntled customers might argue that it is simply group pricing, since different customers are paying different amounts for exactly the same service—namely, a round-trip coach seat from San Francisco to Chicago. The airline would reply that the two types of tickets are distinct products and that the added cost of the full-fare ticket is fully justified by the flexibility of being able to purchase late and return without staying over a Saturday night.

The reality is, of course, that airline pricing—like many successful examples of price differentiation—includes elements of both group pricing and product versioning. The airlines consciously created restricted discount fares as an inferior product. They did so, however, as a way to enable them to offer different fares to different customer groups: lower fares to leisure travelers, who are more price sensitive but more flexible; and higher fares to business travelers, who are less price sensitive but also less flexible. Viewed one way, we could say that the airlines created an inferior product as the most efficient and least controversial way to institute group pricing.

The airline example illustrates an important point. Pure group pricing is difficult to pull off in most consumer markets. There are few cases in which consumers can unambiguously be identified as belonging to a particular group and prices can be targeted accordingly with no fear of arbitrage or customer backlash. Airlines have no reliable objective marker to tell them whether a particular customer is flying for business or for pleasure. In the absence of such a marker, they rely on (among other factors) the imperfect criterion of whether a customer can book early. This works well enough, but it is imperfect. For example, plenty of highly price-sensitive leisure customers would be happy to book late. And some business customers are able to book early and take advantage of lower fares. Furthermore, the airlines have lots of empty seats they would like to fill with these customers, even at a very low price. But allowing customers to book at the last minute at a low price would damage the ability of the airlines to segment customers between leisure and business.

6.4 CALCULATING DIFFERENTIATED PRICES

If market segments are completely independent (i.e., there is no cannibalization) and the seller faces no capacity constraints, then calculating differentiated prices is quite simple—the seller simply finds the contribution-optimizing price for each segment. This is the right approach whether the underlying differentiation is based on channel, geography, or pure group pricing.

Example 6.2

An electronics distributor sells portable MP3 players through its exclusive retail outlets and on its website. The unit cost for each MP3 player is $200. The additional cost per sale through the internet is $35, including shipping, but $70 through the retail stores. Through price testing, the distributor has discovered that price elasticity is 2.5 for internet customers versus 2.2 for retail customers. Using Equation 5.8, the distributor calculates the contribution-maximizing price as (2.5/1.5) × ($200 + $35) = $392 for internet sales and (2.2/1.2) × ($200 + $70) = $495 for retail sales.

6.4.1 Optimal Pricing with Arbitrage

Unfortunately for sellers, perfect price differentiation is usually impossible. Cannibalization is likely whenever customers cannot be perfectly segmented according to willingness to pay. Arbitrage is likely whenever a third party can purchase the product at a low price and resell it at a high price. Regional pricing is subject to arbitrage whenever a product can be purchased in a low-price region and transported cheaply to be resold at a higher price elsewhere. For this reason, global companies often set price bands for various markets to avoid resales from low-price countries that would cannibalize sales in higher-price countries.

Example 6.3

A computer chip manufacturer finds that the contribution-maximizing prices for his chips are $2.54 in the United States and $2.43 in Brazil. However, if it costs $0.08 per unit to ship chips from Brazil to the United States, the potential for arbitrage between the two countries means he will not be able to charge those prices. He needs to set prices for the two countries that do not vary by more than $0.08.

Regional pricing with arbitrage can be formulated as a constrained optimization problem. Assume that a manufacturer is selling a common product to n different locations. These could be cities or regions, but for purposes of discussion we will consider them to be countries. The delivered cost (including transportation) in country i is c_i. The cost to an arbitrageur to transport the product from country i to country j is a_ij.⁷ To avoid arbitrage, the supplier needs to set prices such that p_j ≤ p_i + a_ij for each i and j. Otherwise, the supplier faces the possibility that an arbitrageur will purchase the product in country i, transport it to country j, and sell it for a price p̂_j. If p̂_j > p_i + a_ij, then the arbitrageur can undercut the supplier in country j while still turning a profit. It is clear that a contribution-maximizing supplier would usually like to avoid this situation.

Let p_i be the price in country i and d_i(p_i) be the price-response function faced by the supplier in country i. Then the problem of optimizing international prices under the possibility of arbitrage can be formulated as

subject to

This problem will not, in general, be a linear program. However, if the price-response function for each country is continuous and downward sloping, it will have a single optimum and will be easy to solve using standard optimization approaches. In many cases, global companies do not solve the full optimization problem but rather determine prices for major markets (e.g., North America, Western Europe, Japan) and then use price bands to determine prices for smaller markets that are close enough to major market prices to eliminate the potential for arbitrage through transshipment.

6.4.2 Optimal Pricing with Cannibalization

Cannibalization presents a somewhat different problem than arbitrage. To analyze differentiated pricing with cannibalization, let us return to the case in which we divided widget customers into those with willingness to pay more than $7.00 and those with willingness to pay less than $7.00. The corresponding price-response functions, shown in Figure 6.2, assume perfect differentiation—the company can perfectly distinguish between those customers willing to pay more than $7.00 and those willing to pay less than $7.00 and can charge the optimal price to each group. What if some of the higher-willingness-to-pay customers find a way to buy at the lower price? Let α be the cannibalization fraction—the fraction of higher-willingness-to-pay customers who find a way to buy at the lower price. α = 0 corresponds to the case of perfect differentiation, in which none of the high-willingness-to-pay customers buy at the lower price, while α = 1 corresponds to no differentiation, in which all of the customers buy at the lower price. Values of α between 0 and 1 represent different segmentation efficiencies. Given a value of α, the price-response functions for each of the two segments are given by

The second term in the equation for d₂(p₂) reflects the fact that 4,400 total customers have a willingness to pay greater than $7.00 and that a fraction α of them will find a way to buy at the lower price, p₂.

To find the values of p₁ and p₂ that maximize total contribution, we need to solve the optimization problem

subject to p₂ ≤ $7.00. The prices that maximize total contribution in this case are

When α = 0 (no cannibalization), As α increases, cannibalization increases, and increases as well. This corresponds to intuition—the more our low-price product is cannibalizing a high-price product, the higher we need to price the low-price product to maximize total contribution. When α = 0.36 (i.e., 36% of the high-willingness-to-pay customers find a way to pay the lower price), the optimal low price has risen to $7.00. At this point, none of the low-willingness-to-pay customers are buying any more (since their willingness to pay is less than $7.00) and the only low-price customers are cannibalized high-willingness-to-pay customers.

Figure 6.3 shows optimal total contribution as a function of α. As α increases, total contribution decreases as more and more high-willingness-to-pay customers are cannibalized by the lower price. At a value of α of about 13%, total contribution is $11,250, the same total contribution that could be achieved with no segmentation. In this simple example, if more than 13% of our high-willingness-to-pay customers will be cannibalized, we are better off offering only a single price than trying to differentiate and offer different prices to the different segments.

Figure 6.3 Total contribution as a function of the cannibalization fraction α.

While the threshold of 13% is specific to this example, the lesson is general: Even fairly low rates of cannibalization can destroy the benefits of price differentiation. This means that when establishing customer segments and determining the optimal prices, it is critical that we understand the potential for cannibalization and price accordingly within each segment. Otherwise, we can end up in a situation where we are worse off than if we had simply established a single price to the entire market.

6.5 PRICE DIFFERENTIATION AND CONSUMER WELFARE

We have seen that, correctly executed, price differentiation can be good for sellers. But is it good for customers? This is not merely an academic question. Some forms of price differentiation can be unpopular among customers—particularly group pricing, which, as we see in Chapter 14, is often viewed by customers as unfair. This can lead to loss of customers as well as regulatory scrutiny. If a seller can show that a particular differential pricing approach actually makes consumers better off as a whole (or at least not worse off) in comparison with a single price, the seller will at least have an argument to use with regulators.

To look at who gains and who loses from price differentiation, we draw on some concepts from social welfare theory, as illustrated in Figure 6.4. The left-hand diagram in this figure illustrates the case of a seller with unit cost c who is facing a linear price-response function and can set only a single price. We assume that he sets the optimal price p* with corresponding sales of d(p*). The seller is not capacity constrained, and so sales equal demand. The shaded area in the figure corresponds to the seller’s profit, which is given by (p* – c)d(p*). In consumer welfare theory, this amount is called producers surplus.

Figure 6.4 Elements of social surplus for a profit-maximizing seller setting a single price (left) and two prices with perfect differentiation (right).

On the customer side, the analog of producers surplus is consumers surplus. For an individual customer who purchases the product, her resulting surplus is equal to the difference between her willingness to pay and the price. The idea is that a consumer who was willing to pay up to $54 for an item but purchased it at $45 has enjoyed a surplus of $9—money she would have been willing to pay for this item but now has available to purchase something else (or to save). The consumers surplus associated with a particular pricing scheme is the sum of the individual surpluses of consumers who purchase the item. The surplus associated with a customer who does not purchase is zero. In the left-hand graph in Figure 6.4, the consumers surplus associated with a single price is equal to the area of the triangle labeled B and the producers surplus (profit) is equal to the area of square A.

The social surplus associated with the pricing scheme is the consumers surplus plus the producers surplus—that is, the area of triangle B plus square A. Social welfare theory in its simplest form posits that total economic welfare is maximized when social surplus is maximized.⁸

There is one other area of interest, represented by the triangle labeled C in the left-hand graph in Figure 6.4. A customer whose willingness to pay is between c and p* will not purchase at the price of p* and hence realizes a consumers surplus of zero. However, this customer would be willing to purchase at some price greater than c, and a profit-maximizing seller would be willing to sell to them, but because the seller is restricted to a single price, these transactions do not take place. The area of triangle C represents surplus lost because the seller can only charge the single price p*—it is called deadweight loss.

It is instructive to consider what happens to social surplus if the seller decided to sell at a price different from p*. If he raises his price above p*, then producers surplus will be reduced, consumers surplus will be reduced, and the deadweight loss will increase, which means that overall social welfare is decreased. On the other hand, if he reduces his price below p*, producers surplus will be reduced, but consumers surplus will increase and dead-weight loss will decrease. Since deadweight loss decreases, social welfare is increased. In the extreme case, if he sets p = c, consumers surplus and social welfare would be maximized, but producers surplus (and deadweight loss) would both be zero. Other than regulation, the most likely motivation for a producer to price at cost would be competition—and, in fact, perfect competition maximizes consumers surplus and social welfare by eliminating producers surplus.

What happens if the seller is able to offer two prices? The situation is shown in the right-hand graph in Figure 6.4. Here, we assume that the seller is able to perfectly segment the market according to willingness to pay and there is no cannibalization or arbitrage. Dead-weight loss in this case is the area of triangle C and consumers surplus is the sum of the areas of triangles B₁ and B₂. Producers surplus is the area of the shaded region (not labeled in the figure). We know that producers surplus (profit) increases with two prices, and it is clear from the figure that deadweight loss has decreased. It can be shown in this case that consumers surplus has decreased and that overall social surplus has increased. Thus, in this case, price differentiation has increased social surplus but at the cost of converting some consumers surplus to producers surplus. The ideal for the profit-maximizing producer would be to charge each customer a price exactly equal to her willingness to pay (first-degree price discrimination), in which the producer would capture all of the available surplus—A + B + C in the left-hand figure. Perfect competition and first-degree price discrimination by a monopolist both maximize social surplus, although in the first case the surplus is entirely captured by customers and in the second it is entirely captured by the seller as profit.

Does this mean that price differentiation always reduces consumers surplus? The answer is no: it turns out that price differentiation can make consumers better off. Consider the case where the widget maker is able to divide his market into two segments, with the two price-response curves shown in Figure 6.2. As shown in Table 6.1, the seller’s optimal policy in this case is to charge $8.75 to the high-willingness-to-pay segment and $6.00 to the low-willingness-to-pay segment, with a concurrent increase in contribution of $800 relative to the best he can do when charging a single price. The total consumers surplus under this new policy is the old surplus, denoted by region C₁ in Figure 6.5, plus the new surplus, C₂, enjoyed by the customers paying $7.00. The seller’s profit is the sum of regions A₁ and A₂. Comparison with Figure 6.1 shows that this two-price policy has resulted in both more profit for the seller and higher total consumers surplus than the single-price policy. In other words, in this case price differentiation is a win-win for customers and sellers.

The case in Figure 6.5 is notable because some customers are better off and no customer is worse off than with a single price. The customers who paid $8.75 with the single price still pay $8.75, so they are unaffected. Customers who pay $7.00 under the two-price policy are better off since they would not have purchased with a single price. And the seller is better off since he makes more money.

So depending on how it is applied, price differentiation can either increase or decrease consumers surplus. There is one general rule: If a price differentiation scheme increases profits for a seller but does not result in additional production, it must reduce total consumers surplus. This rule is most relevant when capacity is constrained—if a theater is already selling out at a single price and it goes to a two-price system that increases revenue, then it has done so by making customers, on the average, worse off. Again, this does not mean that no customers are better off under the new scheme: it simply means that the total consumers surplus is less under the new scheme than under the old. If, on the other hand, a theater that is not full finds that it can increase total revenue by a price differentiation tactic, then the effect of the differentiation on total consumers surplus will depend on the situation. A study of the effects of pricing and consumer behavior for an actual Broadway show concluded that price differentiation increased both attendance and profit by about 5% while having a negligible effect on consumers surplus (Leslie 2004).

Figure 6.5 Profit and consumer surplus from the two-price case.

Note that social surplus is maximized in two extreme cases. If the producer could use a Vulcan mind meld to determine the exact willingness to pay of every customer and he was able to charge every customer exactly her willingness to pay, he would sell his product to every customer with willingness to pay greater than unit cost at that customer’s willingness to pay. In this case, the seller would grab every penny of potential surplus. There would be no deadweight loss, but there would be no consumers surplus either. At the other extreme, the seller would sell the product at unit cost. In this case, every customer with willingness to pay greater than cost would purchase. There would be no profit and no deadweight loss, but each consumer would enjoy a high level of consumers surplus. Of course, in the static world represented by a single price-response curve, there is no motivation for the seller to engage in this largesse. In the real world, sellers are pushed toward selling at (or near) cost by the presence or the threat of competition or, possibly, regulation.

6.6 NONLINEAR PRICING

All of the approaches to pricing discussed so far can be characterized as linear. This means that the price of a bundle including multiple items is equal to a fixed price per item times the number of items purchased. Thus, if a gallon of milk costs $3.50 and a loaf of bread costs $2.50, under linear pricing the cost to purchase two gallons of milk and four loaves of bread would be 2 × $3.50 + 4 × $2.50 = $17.00. Under linear pricing, the cost to the consumer of buying twice as much of an item is simply double the cost of buying a single item, and the cost of buying two different bundles of items is the sum of the prices of each bundle.

While linear pricing is simple and intuitive, a seller can often generate additional revenue or profit through nonlinear pricing—that is, charging a price for a bundle of goods that is different (usually lower) than the sum of the prices of the individual items in the bundle. Familiar examples include volume discounts, “10% off if you purchase at least five rolls of toilet paper”; BoGos (“buy one, get one”) such as “buy two books and get the third one for 50%”; and bundling discounts, “buy a grill for full price and get 1/2 off a bag of charcoal.”

Note that nonlinear pricing schemes are a form of price differentiation: under a nonlinear pricing scheme, customers who purchase more of an item or purchase an item in combination with other items will typically pay less per unit than other customers who purchase a single unit of the item in question. Yet, unlike group pricing, nonlinear pricing is generally noncontroversial since the same pricing is typically available to all customers. In this sense, it is similar to product differentiation because each customer is typically offered the option to buy a single item at full price or multiple items with a discount.

In this section we consider two of the most common approaches to nonlinear pricing: volume discounts and bundling.

6.6.1 Volume Discounts

Volume discounting—buy more to save more—is a time-honored form of nonlinear pricing. Some examples follow.

• A six-pack of beer purchased from the supermarket costs less than six times the cost of a single bottle, and a case (12 bottles) usually costs less than two six-packs.

• Larger boxes of laundry detergent cost less per ounce than smaller boxes.

• Verizon offers shared data plans for households, including “small” (2 GB) at $35 per month, “medium” (4 GB) at $50 per month, and “large” (8 GB) at $70 per month (Verizon, n.d.).

Volume discounting is as prevalent, if not more prevalent, in wholesale and business-to-business selling. Table 6.4 shows three typical price schedules including volume discounts for three different products offered online: Duracell batteries on Walmart.com, Holiday Signs, and Pines wheatgrass products. Each seller offers significantly lower unit prices to customers who are buying more units. And the price per unit decreases as the number of units in the order increases.

TABLE 6.4

Price schedules for Duracell batteries, holiday signs, and Pines wheatgrass products

SOURCE: Battery prices are for Duracell Optimum 1.5V AAA battery, quoted on https://www.walmart.com (August 2019); holiday sign pricing from Holiday Signs website, at https://holidaysignsdirect.com (August 2019); wheatgrass products pricing from Pines website, at https://wheatgrass.com (August 2019).

There are at least three reasons why companies offer volume discounts:

• Transaction or order costs. If some costs associated with fulfilling an order are independent of size, then the order cost per unit will decrease as the size of the order increases. In this case, it makes sense for the seller to charge a lower price per unit for large orders.

Example 6.4

A software seller has a fixed cost of $1,000 per installation and a variable cost of $40 per user. Customer 1 has 10 users and customer 2 has 100 users. The total cost for him to serve customer 1 is $1,400, or $140/user, compared with a total cost of $5,000, or only $50/user, for customer 2. Assume that both customers have a valuation of $120/user. Then, if the seller can only charge a single price, he will charge $120 per copy. At this rate, both customers will buy, but he will lose $200 on the first customer. If he offers a volume discount such that the price is greater than $140/user for 10 users or fewer and $120/user for more than 10 users, the first customer will not purchase and he will make an additional $200 in profit.

• Decreasing marginal utility. In many cases, the marginal utility to a customer decreases as the number of units purchased increases. A hot and thirsty customer walking into a convenience store places more value on the first can of cold soft drink than she does on the second. Decreasing marginal utility is also often seen in business-to-business sales because the incremental value of additional units of office equipment or supplies tends to decline past some point.

• Price sensitivity increases with volume. Customers purchasing larger amounts are often more price sensitive than customers purchasing smaller amounts. This is particularly true in business-to-business settings: corporate purchasers who buy large volumes are likely to be following a formal procurement process in which they carefully compare alternatives and will negotiate intensely to gain a lower price. A plumbing contractor who spends millions of dollars on galvanized pipe annually is much more likely to spend time and effort finding a good deal than a homeowner who is buying a single length of pipe at the hardware store. Volume discounting is one way that a seller can differentiate among large-volume, highly price-sensitive customers and smaller-volume, less price-sensitive customers. In this case, volume discounting is a form of product-based customer segmentation.

The discounts shown in Table 6.4 are non-incremental because the discount applies to the entire order. This can lead to negative incremental prices. For example, according to the second column in Table 6.4, a customer ordering 50 signs would pay $1.25 per unit for a total cost of $62.50, whereas if he ordered 51 signs, he would pay $1.20 per unit for a total cost of $61.20. The incremental price of the 51st sign is − $1.30. Negative incremental prices are arguably bad things in that they can encourage overconsumption and lost potential revenue. In incremental discounting, additional discounts are applied only to additional units—not to the entire order. A common example of incremental pricing in consumer markets is “Buy one, get the second at half price.”

Volume discounts are often applied based on total volume of business over some period, rather than on size of a single order. This is often accomplished in consumer markets by frequent-buyer schemes such as the airlines’ frequent-flyer programs. My local coffee shop keeps track of my purchases and gives me every 10th cup free. The stated purpose of such programs is to reward loyal customers, but they also serve the purpose of lowering the real price seen by frequent purchasers, who are likely to be more price sensitive. This practice is extremely common in business-to-business settings, in which discounts are often applied based on sales volume in a quarter or other period. Electronics distributors resell electronics goods to retailers, government, and educational institutions. A manufacturer (such as Hewlett-Packard) selling through the distributor will offer a progressive end-of-quarter rebate based on how much of its product the distributor has sold during the quarter. For example, Acme Electronics might offer a 0.5% rebate if the distributor sells between $1 million and $1.499 million worth of Acme products during the quarter, a 0.75% rebate if the distributor sells between $1.5 million and $1.999 million, and a 1% rebate if the distributor sells over $2 million. In freight transportation, a carrier will often offer a higher discount if a shipper commits to a higher volume of business over some future period.

Combining different terms and discount structures can lead to an extremely wide variety of volume discount schemes. All of these schemes are designed to exploit the decreasing marginal value of purchasers and/or the higher price sensitivity of large purchasers. In the case of decreasing marginal values, a seller could, in theory, capture the maximum contribution by pricing each additional unit at the willingness to pay of the customer for that unit. Of course, this is impossible because the seller is selling to a mixture of customers with different willingness-to-pay values and the exact willingness to pay of each customer is unknown. However, given overall demand curves for purchases of different sizes, a seller can derive a volume discounting scheme that will increase contribution relative to charging a single price.

How should we think about setting volume discounts? Here we consider a simple case of a single customer (or a population of identical customers) with decreasing marginal utility for an item and show how volume discounting can be used to extract additional revenue and profitability. (Warning: While the example is conceptually simple, the algebra can be a bit tedious.)

Consider a single customer (or a population of identical customers) with utility given by where x is the number of units purchased and A > 0 is a parameter. This utility function is shown in Figure 6.6. We assume for convenience that the customer can purchase fractional units. Note that the customer’s marginal utility is given by so her marginal utility is decreasing for all x > 0, which implies that her price sensitivity increases with the number of units she purchases.

Figure 6.6 Utility of a product to a customer as a function of units purchased.

Now, assume that the product is offered at a single price p > 0. Then, for any purchase amount d, the total utility to the customer after the purchase is u(d) – pd. We can determine the purchase volume that will maximize her utility by solving the first-order condition u'(d) = p. In this case, which after rearranging gives

Equation 6.1 is the customer’s individual price-response function; that is, for any price p that the seller offers, it shows how many units she will purchase. Note that d(p) is continuous and decreasing in price, consistent with the general requirement for price-response functions.

Assume that the seller has a constant unit cost c > 0. With a single price, a profit-maximizing seller will seek to set the price that maximizes π(p) = (p – c)d(p). The profit-maximizing price can be found from the standard first-order condition π'(p) = 0. In this case, π(p) = (p – c)A²/4p², which means that the first-order condition is

Equation 6.2 has a single finite root at p* = 2c, which means that the profit-maximizing seller will set a price equal to two times his cost. His corresponding sales and profit can be computed by substituting p* into Equation 6.1: the resulting sales are d(p*) = A²/16c², and the corresponding profit is π(p*) = A²/16c.

We now consider the case in which the seller offers two prices with a single volume discount tier. In this case, the seller sets a triplet x₁, p₁, p₂, with p₁ > p₂ > c. If the customer purchases x ≤ x₁ units, she will be charged p₁x. If she purchases x > x₁ units, she will be charged p₁x₁ + p₂(x – x₁). The question facing the seller is what values of x₁, p₁, and p₂ to set to maximize profit.

At first it would seem that the seller has three variables to compute. However, a little reflection should convince you that, for optimality, it must be the case that —that is, the sales threshold for the discount must be equal to the demand at the undiscounted price. If then the seller could increase revenue by increasing and selling the additional units at p₁ rather than the seller is not selling anything at the discount price and he can decrease without decreasing sales or revenue. Thus, we can formulate the seller’s problem in terms of just p₁ and p₂. The corresponding optimization problem faced by the seller is

subject to p₁ > p₂ > c. The first term of π (p₁, p₂) is the profit realized from selling the units at price p₁ and the second term is the profit realized from selling units at price p₂. We can rearrange terms in the objective function to write

Ignoring the constraints for the moment and differentiating by p₁ gives the first-order necessary condition:

So we must have p₁ = 2p₂.

We can now differentiate π (p₁, p₂) with respect to p₂, which gives us

or

Substituting p₁ = 2p₂ and rearranging terms gives

Again, the root p₂ = 0 is clearly a minimum, so the maximum must be at p₂ = 8c/5, which implies that we must have p₁ = 16c/5. Substituting back into the objective function in Equation 6.3 and calculating the amounts purchased gives us the results shown in Table 6.5.

The total profit across the two periods is

That is, the optimal profit from the volume discount is 56.25% greater than with the single price. This represents a substantial increase, which suggests why volume discounts—often with many discount points—are popular, especially in business-to-business settings.

Results from both the single-price and the volume discount case are shown in Table 6.5. Two things are notable in the volume discount case. First, the first price is higher and the second price is lower than the optimal single price. This is typical. However, it means that, to be effective, volume discounting requires that a large fraction of the customers want to purchase in volume—if only a small number of customers are volume purchasers, the loss from increasing the initial price to enable the volume discount may outweigh the benefits. Second, total sales are significantly higher with two prices than with a single price.

TABLE 6.5

Results with single price and with volume discount price

Finally, it is important to remember that, like any market segmentation scheme, volume discounts provide opportunities for arbitrage. Referring to the price schedules in Table 6.4, a 4-pack of Duracell batteries costs 4 × $1.49 = $5.96. An enterprising entrepreneur could purchase a 24-pack of Duracell batteries for 24 × $0.68 = $16.32, repackage them as six 4-packs, and sell them for $5.00 apiece, realizing a profit of $30.00 − $16.32 = $13.68, minus any repackaging costs. Many internet resellers make a living by finding and exploiting such opportunities for arbitrage. To avoid such arbitrage (not to mention out-and-out piracy), software companies often require users to register to receive future support and maintenance. In other cases, bulk and volume purchasers are required to sign a contract that prohibits them from reselling for profit. However, in many cases, the potential for arbitrage limits the potential for volume discounting.

6.6.2 Bundling

A bundle is a collection of products, services, or attributes sold as a package in which the price of a bundle is less than the sum of the individual parts. This means that bundling is a form of nonlinear pricing. Bundling may be motivated by the fact that the cost of providing a bundle is less than the cost of delivering the individual components—for example, the shipping cost of 20 items delivered together may be less than 20 times the shipping costs of the individual items. However, as we see in this section, it is also a way to extract additional sales by pricing in a way that increases sales in different market segments.

Bundling can be used in a simple way for complementary products—say, hot dogs and hot-dog buns. Consumers may be divided into three groups—those who have hot dogs at home and need buns, those who have buns at home and need hot dogs, and those who have neither and need both to enjoy the full hot-dog experience. If the seller prices buns at $6 per pack and hot dogs at $10 per pack, it may be worthwhile for him to offer a bundle of hot dogs and buns at $15 per pack to sell more hot dogs and buns to the third group without cannibalizing sales to the first two. The reasoning and analysis in this case is similar to those of volume discounts.

A more interesting use of bundling is with noncomplementary products. Assume that Microsoft is offering Excel and Word in a business marketplace that consists of four segments: bankers, traders, lawyers, and consultants. The willingness to pay of each of these segments for each of the two products is shown in Table 6.6. For simplicity, we assume that the size of each market segment is equal to 20,000. What is the optimal price if Microsoft offers each product separately? With some calculation, it is easy to see that the optimal prices are $400 for Excel and $400 for Word. At these prices, the bankers and the traders will buy Excel with revenue of $400 × 40,000 = $16MM (million), and the lawyers and consultants will buy Word with revenue of $400 × 40,000 = $16MM for a total of $32MM.

TABLE 6.6

Willingness to pay for different segments in the bundling example

Now, what if Microsoft bundled Excel and Word together? The willingness to pay of each segment for the bundle is shown in the last row of Table 6.6. One option is for Microsoft to only offer the bundle. In this case, the revenue-maximizing price would be $500, and all of the segments would purchase for a total revenue of $500 × 80,000 = $40MM. This represents a 25% increase over the unbundled pricing. However, Microsoft could also offer the products individually in addition to the bundle. In this case, the optimal policy would be to price the bundle at $600 and both Word and Excel at $450. Bankers and consultants would purchase the bundle, traders would purchase Excel, and lawyers would purchase Word. The total revenue would be $600 × 40,000 + $450 × 40,000 = $42MM, a 31.25% increase over the best unbundled pricing.

When does bundling of noncomplementary products work? The most important condition is that customer segments differ in their valuation of the individual items—if each segment valued Excel the same, there would be no opportunity to gain revenue by bundling. In addition, there must be segments who have flipped preferences. That is, there must be one segment whose preference among products (as measured by willingness to pay) is different than at least one other segment. In the example, bankers and traders have a higher willingness to pay for Excel than Word, while the opposite is true for lawyers and consultants. The fact that there are segments whose preferences are anti-correlated in this fashion is a precondition for bundling of noncomplementary products to be effective.

Example 6.5

A company offering two products to three customer segments is considering whether to offer a bundled product. Each segment consists of 10,000 customers. Using consumer research, the company establishes that the willingness to pay for each product in each segment is as shown in Table 6.7. Looking at the results, the company can see that the preference order of each segment is the same and can conclude immediately that there will be no gain from offering a bundle. This can be verified by noting that the optimal individual prices for the two products are p₁ = $50 and p₂ = $80, generating revenue of $180,000. If the company offered a bundle, the optimal bundle price would be $165, generating revenue of $165,000. This supports the conclusion that there is not an opportunity to increase revenue by bundling in this case.

TABLE 6.7

Willingness to pay for different segments in Example 6.5

It should be noted that the problem of optimal bundling—that is, determining the bundle or set of bundles to offer that maximizes revenue (or profit)—becomes extremely difficult when the number of products becomes large. In the case of two products, there are three options—no bundle, bundle only, or bundle plus individual sales. In the case of three products, there are six options: no bundle; three two-product bundles (A, B), (B, C), and (A, C); and a three-product bundle (A, B,C), which could be offered with or without individual sales options. In fact, the number of possible bundles of n items is given by 2ⁿ + 1 – n, which grows quite quickly—for 20 items, there are 1,048,557 possible bundles that would need to be evaluated.⁹

Optimal bundling of multiple products is a topic of continuing research. However, for practical purposes, the most important insight is that bundling of noncomplementary products is most effective when there are segments whose preferences are anti-correlated—that is, some segments prefer one product while others prefer a different product. In this case, a bundling strategy can often be used to increase sales and profitability. It should also be noted that, like volume discounts, bundles can be subject to arbitrage—if a seller is pricing a bundle below the sum of its constituent prices, an enterprising arbitrageur could purchase a bundle and sell the individual components below their individual prices. For this reason, bundling is often used for items for which arbitrage is difficult, like software or services.

6.7 SUMMARY

• Price differentiation is the tactic of charging different customer segments different prices for the same (or nearly the same) product or service. It can be a powerful way for a seller to increase contribution.

• The effectiveness of price differentiation can be limited by imperfect segmentation, cannibalization, or arbitrage. A poorly conceived or executed price differentiation scheme can result in lower contribution than a single-price scheme.

• There are a wide variety of price discrimination schemes, ranging from group pricing through channel and regional pricing to volume discounting and product versioning. The best approach or combination of approaches depends on the market.

• Product-versioning tactics are better accepted by customers and less susceptible to cannibalization and arbitrage than group pricing. Both inferior and superior products can be used to segment customers.

• The methods for calculating differentiated prices when the supplier does not face capacity or supply restrictions are extensions of the methods introduced in Chapter 5 for a supplier pricing a single product. The possibilities of cannibalization and arbitrage need to be specifically incorporated in order for the prices to be correct.

• The effects of price differentiation on consumer welfare (as measured by consumers surplus) can be either positive or negative, depending on the market and on the differentiation scheme. If a price differentiation scheme does not increase supply, it cannot increase total consumers surplus.

• Nonlinear pricing refers to a wide variety of pricing schemes in which the price for a number of goods is not equal to the sum of the prices of the individual goods. Common examples include volume discounts, “buy one and get one free,” and bundling discounts. Applied correctly, nonlinear pricing can lead to increased sales and profitability, although arbitrage and cannibalization can limit the benefits.

We return to the theme of price differentiation again and again. In particular, price differentiation is the basis of revenue management (Chapter 8), markdown management (Chapter 12), and customized pricing (Chapter 13).

6.8 FURTHER READING

More discussion on strategies for price differentiation can be found in Bodea and Ferguson 2012, Simon and Fassnacht 2018, and Nagle and Müller 2018.

This chapter provides only a high-level treatment of consumers surplus. Fuller treatment can be found in most microeconomics texts, such as Varian 1992 and Nicholson 2002.

There is an extensive literature in both marketing and economics on nonlinear pricing; the classic reference is Robert Wilson’s book Nonlinear Pricing (1993). A useful short summary of the topic can be found in Oren 2012, and an overview of tactics for optimal bundling is in Venkatesh and Mahajan 2009.

6.9 EXERCISES

1. Arbitrage. A supplier is selling hammers in two cities, Pleasantville and Happy Valley. It costs him $5.00 per hammer delivered in each city. Let p₁ be the price of hammers in Pleasantville and p₂ be the price of hammers in Happy Valley. The price-response curves in each city are

Pleasantville: d₁(p₁) = 10,000 – 800p₁

Happy Valley: d₂(p₂) = 8,000 – 500p₂

a. Assuming the supplier can charge any prices he likes, what prices should he charge for hammers in Pleasantville and Happy Valley to maximize total contribution? What are the corresponding demands and total contributions?

b. An enterprising arbitrageur discovers a way to transport hammers from Pleasantville to Happy Valley for $0.50 each. He begins buying hammers in Pleasantville and shipping them to Happy Valley to sell. Assuming the supplier does not change his prices from those given in part a, what will be the optimal price for the arbitrageur to sell hammers in Happy Valley? How many will he sell? What will his total contribution be? (Assume that Happy Valley customers will buy hammers from the cheapest vendor.) What will happen to the total sales and contribution for the supplier? (Remember that he is now selling to the arbitrageur too.)

c. The supplier decides to eliminate the arbitrage opportunity by ensuring that his selling price in Happy Valley is no more than $0.50 above the selling price in Pleasantville (and vice versa). What is his new selling price in each city? What are his corresponding sales and total contribution?

d. From among the Pleasantville customers, the Happy Valley customers, and the seller, who wins and who loses from the threat of arbitrage?

2. Assume that it costs Budget $20 per day for every car it rents out, regardless of model. What is the implied price elasticity of customers for each of the first five car types listed in Table 6.3?

3. For the single-customer demand function specified by Equation 6.1, what is the corresponding hazard rate and price elasticity? What do they imply about the price calculated in Equation 6.2?

NOTES

1. Definitions of first-, second-, and third-degree price discrimination were established by the economist Arthur Pigou in 1920, and the terms are still widely used although they are not exhaustive. First-degree price-discrimination refers to the case in which the seller charges each customer exactly her willingness to pay. Second-degree price discrimination refers to volume discounting. Third-degree price discrimination corresponds to what we call group pricing—charging different prices to different customers based on their membership in a group (e.g., senior citizen discounts).

2. I use the term group pricing in the sense of Shapiro and Varian 1992. This should be distinguished from use of the term to offer lower rates to groups of customers in industries such as the passenger airlines, hotels, and cruise lines, which is a form of volume discounting.

3. See Phillips 2018 for patterns of regional loan pricing in North America.

4. The classic study of the history of redlining and statistical studies of its effect on minority communities is The Color of Credit, by Stephen Ross and John Yinger (2002).

5. See the Intuit QuickBooks home page, at http://quickbooks.intuit.com.

6. This does not, of course, mean that Budget and Intuit are not also offering even deeper hidden discounts to favored customers.

7. In many cases, this cost will be symmetrical—that is, a_ij = a_ji. However, this is not always the case. For example, one country may charge a higher import duty on the product.

8. This simple calculation of social surplus is based on the assumption that society values a dollar of producers surplus (e.g., operating profit) the same as a dollar of consumers surplus. There are at least two rationales for this assumption. The first is that corporate profits ultimately flow back to consumers in the form of bonuses or stock dividends. The second rationale is that if people wanted a different distributional outcome—say, more consumers surplus and less producers surplus—the tax rates could be adjusted to tax business more and consumers less. Under this view, producers plus consumers surplus establishes the size of the pie, and the first goal of economic policy should be to make the pie as big as possible. Needless to say, this view has been heavily criticized for, among other things, neglecting the extent to which businesses can influence the political process in their favor.

9. The situation is even more complex if the seller is able to offer multiple bundles; for example, a seller with products A, B, and C could offer (A, B), (A, C), and (A, B,C) as well as only B in isolation. In this case, the number of possible combinations of bundles is truly astronomical—on the order of 2²ⁿ, which becomes unmanageably large for even relatively small values of n. In practice, sellers do not combine bundles because of the added complexity of communication to customers.