When the same consumer buys more than one unit of a good or service at a time, the marginal value placed on consuming additional units declines as more units are consumed. Second-degree price discrimination takes advantage of this falling mar- ginal valuation by reducing the average price as the amount purchased increases.
For this reason, second-degree price discrimination only works for products and services for which consumers will buy multiple units during a given period. As an example of this, you may have noticed that Best Buy never offers quantity discounts on refrigerators but frequently offers “two for the price of one” deals on DVDs.
In first-degree price discrimination, the firm has complete information about every consumer’s demand, which allows the firm to sort consumers according to willingness to pay and charge the maximum price for every unit sold. In third- degree price discrimination, which we will examine in the next section, the firm does not know every consumer’s demand but does know the demands by groups of consumers and charges each group a different price. Second-degree price dis- crimination differs sharply from first- and third-degree price discrimination, because the firm possesses no information prior to the sale about individual or group demands. The second-degree price discriminator knows only that people who buy small amounts will have high marginal valuations and people who buy large amounts will have low marginal valuations. Accordingly, those who buy small amounts will be less price-sensitive than those who buy large amounts.
second-degree price discrimination
When a firm offers lower prices for larger quanti- ties and lets buyers self- select the price they pay by choosing how much to buy.
I L L U S T R AT I O N 1 4 . 1 Greyhound Ditches Uniform Pricing
for Dynamic Pricing
Businesses engage in various methods of price dis- crimination because they can capture more consumer surplus and thus increase revenues and profit with- out making any changes in the underlying demand curve. The Greyhound bus company, for example, has decided to replace its uniform flat-rate pricing plan with a pricing system that was pioneered by math- ematicians at American Airlines in the 1980s. The pric- ing technique is called “dynamic pricing” because the mathematical model varies the ticket prices for a trip on a bus or airline according to how demand condi- tions change over time.
Dynamic pricing is a complicated process that funda- mentally tries to approximate first-degree price discrim- ination by varying the price of a ticket at every point in time—both points in time leading up to the point of departure and the time of departure itself. As Grey- hound CEO Tim O’Toole explains, “No longer will a trip
on Greyhound cost the same on July 17 as the day after Thanksgiving.” O’Toole should also have mentioned that ticket prices for travel on July 17 will vary over the time period leading up to the July 17th departure.
To give you an approximate idea about the sub- stantial value from dropping simple uniform pric- ing for a price discrimination model such as dynamic pricing, Bloomberg News reports that Greyhound spent
$40 million on the computers and software that are required to predict optimal time-varying ticket prices and to implement the resulting complex schedule of prices. Because uniform pricing is so much easier to ad- minister than dynamic pricing, dynamic pricing must be richly rewarding in order to justify investing in the costly technology that the complicated pricing method requires. No airline has ever regretted its decision to ditch uniform pricing, and we are sure that Greyhound will never go back to uniform pricing either.
Source: Andrea Rothman, “Greyhound Taps Airline Pricing Models to Boost Profit,” Bloomberg.com, May 20, 2013.
Under these circumstances, it makes sense to charge a higher price to smaller, less price-sensitive buyers with high marginal valuations and to charge a lower price to larger, more price-sensitive buyers with low marginal valuations. Unfortu- nately, when a customer walks into the store, a second-degree price discriminator does not know whether the buyer plans to purchase a little or a lot of product. Only when the customer makes a purchase and chooses to buy either a small amount or a large amount does the firm learn whether the buyer is, respectively, a relatively price-insensitive buyer (i.e., small quantity/high marginal valuation) or a rela- tively price-sensitive buyer (i.e., large quantity/low marginal valuation). Since the firm cannot determine prior to a sales transaction whether buyers possess high or low price sensitivities, the firm must offer all consumers the same price schedule.
Even though the menu of prices is the same for all buyers, consumers self-select into different pricing categories through their own decisions about how much to buy: small buyers pay higher prices and large buyers pay lower prices. While implementation of second-degree price discrimination requires much less informa- tion about buyers than do other types of price discrimination, this reduced amount of information restricts the firm’s ability to capture consumer surplus. As you will see, consumers get to keep some amount of consumer surplus under second-degree price discrimination, but less surplus than under uniform pricing.
Consider Home Depot’s pricing decision for its interior wall paint. Home Depot knows that some buyers are only planning to paint one or two rooms of their homes. These smaller buyers, at the margin, will highly value an additional gallon of paint because they are buying so little. And, because they are buying so little paint, they are relatively insensitive to the price of paint. Home Depot also knows that other buyers are going to paint every room in their homes and will be purchasing many gallons of paint. These larger buyers will possess relatively low marginal valuations and will be much more sensitive to paint prices than smaller buyers. Obviously Home Depot employees cannot identify small and large buyers prior to the sales transaction, so they must offer all paint buyers the same pricing schedule—one that is designed to give larger buyers lower prices.
In this way, Home Depot customers self-select themselves into lower- or higher- price groups.
There are many ways of designing pricing schedules to offer lower prices for larger quantities. We will now examine two of the most common ones: two-part pricing and declining block pricing. In both of these types of pricing, the firm designs a common pricing structure offered to all buyers and lets buyers self-select the price they will pay by choosing the amount of product to buy.
Two-Part Pricing
A two-part pricing plan creates average prices that decline with the amount pur- chased by a consumer. This declining price is accomplished by charging both a fixed access charge for the right to purchase as many units as desired and a con- stant usage fee for each unit purchased. Thus, the total expenditure (TE) for q units of a product is the sum of the fixed access charge (A) plus the usage charge, which
two-part pricing A form of second-degree price discrimination that charges buyers a fixed access charge (A) to purchase as many units as they wish for a con- stant fee (f ) per unit.
is computed by multiplying the per-unit usage fee (f) times the number of units purchased (q)
TE = A + fq
The average price (or price per unit) is equal to the total expenditure divided by the number of units purchased
p = TE ___ q = ______ A + fq q
= A __ q + f
The final expression for price shows clearly that product price falls as the consumer buys more of the good, because the fixed access fee is spread over more units of the good. The firm sets the values for the access charge and usage fee, and then buyers self-select the prices they will pay by choosing the quantities they wish to buy. Notice that all buyers face the same pricing menu or formula. Through the process of self-selection, however, those who buy larger amounts pay lower prices than those who choose to buy smaller amounts.
Finding the profit-maximizing values of A and f can be rather complex. To keep matters manageable, we will examine two of the easier situations for designing a two-part pricing plan. First, we will show you how to choose A and f when all consumers have identical demands for the product and the firm knows ev- erything about consumer demand (i.e., knows the equation for demand). In this special case, as in the case of first-degree price discrimination, two-part pricing captures all of the consumer surplus. Second, we will show you how to extend the pricing plan to two different groups of identical buyers, and, again, the firm knows everything about consumer demand in each of the two groups (i.e., knows the equations for both demands). In this second case, a single or common two-part pricing menu applies to both groups of buyers, and, for this reason, the firm will not be able to capture the entire consumer surplus. This case allows for some dif- ference among consumer demands, and so it more closely represents real-world applications of two-part pricing than the first case.
Before continuing, we should mention that more complicated pricing schemes arise when firms offer buyers multiple two-part pricing menus instead of a single plan that applies to all buyers. By letting consumers “subscribe” to their favorite one of the multiple pricing plans, firms can take advantage of buyer self-selection of price plans to target higher average prices at less elastic consumers and lower average prices at more elastic buyers. Cellular phone companies, for example, usu- ally offer a variety of calling plans. In plan 1, buyers pay a high monthly charge for the right to purchase all the minutes they wish for a very low fee per minute (zero in some cases). The phone company also offers an alternative plan, plan 2, which allows buyers to pay a low monthly access charge (zero in some cases) coupled with a relatively high fee per minute. The phone company lets consumers choose the plan under which their bill will be computed. The choice of plans is yet another example of a self-selection process. The phone company devises values for the two plans— A1, A2, c1, and c2—so that it can tailor the two-part pricing structure for each
kind of buyer in a way that captures even more consumer surplus than using only a single two-part pricing schedule. Finding the optimal access charges and usage fees for multiple two-part pricing menus is rather complex, and, for this reason, we will not cover multiple two-part pricing plans in this text.3
All consumers are identical Let’s first consider how to design a two-part pricing plan when all consumers have identical demands, and the firm knows this demand curve precisely. We can best explain this pricing practice with an example. Suppose you are the new manager of Northvale Golf Club, which is a private club catering exclusively to retired senior citizens who play golf most every day of the week.
Northvale faces only limited competition from other golf clubs, because the nearest competing golf course is 25 miles away. The club’s membership is composed of 100 seniors, all of whom possess identical demand curves for playing rounds of golf at Northvale. Based on marketing research done by an outside consulting firm, you know that the annual (inverse) demand equation for each one of the identical senior golfers is PSR = 125 2 0.5QSR. Figure 14.3 shows this demand curve.
F I G U R E 14.3 Inverse Demand Curve for Each of 100 Identical Senior Golfers:
PSR = 125 2 0.5QSR
DSR: PSR = 125 2 0.5QSR
Price and cost (dollars per round)
c
b
a d
u
e
f g
10
0 MRSR 250
SMC = AVC
230
Quantity (rounds of golf per year) 67.50
125
115
3An excellent mathematical treatment of dual two-part pricing plans can be found in Dennis W. Carlton and Jeffrey M. Perloff, Modern Industrial Organization, 4th ed. (Pearson/Addison Wesley, 2005), pp. 344–349.
Like most golf clubs, Northvale Golf Club incurs both fixed and variable costs, and the fixed costs are much larger than the variable costs. The fixed costs of main- taining the golf course are outsourced to a company that specializes in golf course turf growth and maintenance. The turf company charges Northvale $800,000 annually. Other fixed costs, such as leasing 100 golf carts and other fixed “over- head” costs add another $200,000 to annual fixed costs. Thus, Northvale spends a total of $1 million annually on fixed costs, no matter how many rounds of golf are played each year. For each round of golf, variable costs include the cost of charg- ing the golf cart’s battery, a small amount of wear and tear on the course attribut- able to each round played, and a small amount of “administrative” labor expense.
The average variable cost per round is constant and equal to $10 per round of golf.
Because average variable cost is constant, marginal cost equals average variable cost, as shown in Figure 14.3 (SMC = AVC = $10).
The owner recently fired Northvale’s previous manager for making losses.
The fired manager practiced uniform pricing by charging a price of $67.50 (i.e., a
“green fee”) for every round of golf. As shown in the figure, each one of the 100 identical senior members chooses to play 115 rounds of golf annually (point u) when facing a uniform price of $67.50 per round. Under the uniform pricing plan, annual total revenue is $776,250, which is the total amount spent on green fees by 100 golfers each spending $7,762.50 (= $67.50 × 115) annually. With 11,500 rounds played annually (115 rounds by each member), total variable cost is $115,000 (= $10 × 11,500). Under this uniform pricing plan, Northvale’s previous manager incurred annual losses of −$338,750 (= $776,250 − $115,000 − $1,000,000). You naturally want to keep your new job, so you need to find a way to increase profit at the golf club. Increased advertising is unlikely to strongly stimulate demand—
advertising can’t create new golfers or motivate seniors to play much more than they already play—so you decide that your best hope for quickly increasing profit is to find a better pricing strategy.
After examining the demand and marginal revenue curves in Figure 14.3, you are able to see that the previous manager did, in fact, correctly implement uniform pric- ing at Northvale: the profit-maximizing uniform price is indeed $67.50 per round.
However, having recently taken a course in managerial economics, you know that uniform pricing leaves a great deal of consumer surplus in your members’ pockets.
Because you know precisely the demand for every member, you realize that you can successfully practice perfect price discrimination. Unfortunately, first-degree price discrimination requires haggling over the green fee for every round of golf sold.
Nonetheless, if you can stand all this haggling, you can collect the maximum fee golfers are willing to pay for each round played. Because the job pays you well, you decide to undertake the haggling to implement perfect price discrimination.
When you haggle over green fees to obtain the highest fee for every round played, you are aware that Northvale’s marginal revenue curve (MRSR) coincides with its demand curve (DSR). You find it optimal to sell additional rounds until every player buys 230 rounds per year at point e (where MRSR = SMC). By cap- turing the highest green fee for each round played, you can collect from each golfer $15,525 [= 230 × ($125 + $10)/2], which is the area of the shaded trapezoid
0cef in Figure 14.3. With 100 identical golfers, annual total revenue is $1,552,500 (= $15,525 × 100). As shown in the figure, each of the 100 golfers plays 230 rounds annually for a total of 23,000 rounds annually. Because average variable cost is $10 per round, total variable cost is $230,000 (= $10 × 23,000). By practic- ing perfect price discrimination, you have increased Northvale’s annual profit to
$322,500 (= $1,552,500 2 $230,000 2 $1,000,000).
Although you expect the owner will be happy to see Northvale earning posi- tive economic profit instead of losing money, you wish there was a way to avoid haggling with senior citizens over green fees for every one of the 23,000 rounds played. After some careful thinking, you realize that exactly the same amount of profit can be earned by optimally designing a two-part pricing plan. Furthermore, this two-part pricing plan will capture all consumer surplus without any haggling whatsoever over green fees! You set up a meeting with the owner of Northvale Golf Club to present your new pricing plan for approval.
You begin your meeting with the owner by explaining that, under your two- part pricing play, members will be allowed to play as many rounds of golf as they wish by paying a green fee (f) of $10 for each round played. However, to enjoy the privilege of paying such a “low” green fee, club members must also pay a “high”
annual club membership charge (A) of $13,225 per year. Upon hearing your plan, the club owner threatens to fire you on the spot. Surely, he argues, nobody will pay an annual membership charge of $13,225 to play golf at Northvale. Fortu- nately, you are able to convince him that the pricing plan will work. On the back of an envelope, you sketch Figure 14.3 (good thing you learned how to draw graphs in economics!). Using this diagram, you carefully explain the concept of consumer surplus, and show the owner that when green fees are $10 per round, each golfer at Northvale enjoys annual consumer surplus of $13,225 (= 0.5 × 230 × $115)—
the area of triangle ace. Thus seniors are just willing to pay the $13,225 annual membership charge to gain the privilege of paying a low green fee. This convinces the owner, and you get permission to implement the two-part plan.
Under your two-part pricing plan, annual total revenue is $1,552,500, which is the sum of total annual membership charges of $1,322,500 (= $13,225 × 100) and total green fees of $230,000 (= $10 × 230 × 100). It follows that annual profit will be $322,500 (= $1,552,500 2 $230,000 2 $1,000,000). Notice that this optimally designed two-part pricing plan generates the same total profit as perfect price dis- crimination. The optimal design involves setting the usage fee equal to marginal cost (f * = SMC), and setting the access charge equal to one golfer’s consumer surplus (A* = CSi).
We must emphasize here that, even though profit is exactly the same as under first-degree price discrimination, the two-part pricing plan is nonetheless a second-degree form of price discrimination. Instead of haggling over 23,000 green fees, every golfer faces the same pricing schedule: each golfer pays an annual membership charge of $13,225 and a “haggle-free” green fee of $10 for every round played. Under two-part pricing, as we explained previously, every golfer self-selects the average price he or she will pay for a round of golf by choosing the number of rounds to play. In this particular example, all golfers are assumed to
be identical, so they all choose to play 230 rounds per year. This makes the aver- age price per round $67.50 [= ($13,225 + $10 × 230)/230]. We can now summarize this discussion with the following principle.
Principle When all consumers have identical demands for a product (and demand is precisely known), a manager can capture the entire consumer surplus through two-part pricing by setting the usage fee equal to marginal cost (f * = MC ) and setting the access charge equal to one of the identical buyers’ consumer surplus (A* = CSi).
Two groups of identical consumers Two-part pricing can also be utilized to price-discriminate when there are two or more groups of buyers and the buyers within each group possess identical demands (and the demands are known to the firm). While the procedure for finding the optimal values for f and A is a bit more complicated in this case, we can best illustrate the procedure by continuing our example of Northvale Golf Club, where you are the new club manager. Now you wish to find f * and A* when there are two groups of golfers, instead of one group of identical golfers.
Northvale’s owner decides not to fire you because your decision to replace uniform green fees with a two-part pricing plan turned the club into a profitable enterprise. But now the owner expects you to further improve profits. One of the Northvale community residents, who is not retired and plays golf mostly on weekends, complains to you that the annual membership fee ($13,225) is too high for her to consider playing at Northvale Golf Club. Like the many other weekend golfers in the Northvale community, she drives some distance to play at a rival club. You now realize that Northvale has completely ignored a potential group of golfers who would play there if a more attractive pricing plan could be offered. To determine the profitability of trying to serve this group of golfers, you hire a mar- keting research firm to estimate the demand for golf at Northvale by nonretired, weekend players. The marketing experts estimate there are 100 such “weekend”
golfers, all of whom possess identical demand curves. The marketing consultant determines that the (inverse) annual demand equation for each weekender is PWK = 120 2 QWK, which is shown as DWK in Panel B of Figure 14.4.
You now can see why no weekenders are playing at Northvale. With green fees set at $10 per round, each weekender would choose to play 110 rounds of golf annually. At this rate of play, the most a weekend golfer would be willing to pay for an annual membership is $6,050 (= 0.5 × 110 × $110)—the area below DWK above SMC up to 110 units. Because the current membership charge ($13,225) exceeds a weekend golfer’s total consumer surplus, no weekend golfer joins the club.
To capture all consumer surplus from both groups, seniors and weekenders must pay different membership charges. Setting different membership charges will not work, however, because every golfer will claim to belong to the group receiving the lower membership charge. (You can’t force applicants to truthfully reveal whether they are retired seniors or primarily weekend players.) For this reason, the membership charge must be the same for both groups. Not only must
Now try Technical Problem 2.