7
Chapter 5 shows how a seller can calculate the optimal single-period price when he has the freedom to produce (or order) as much of a good as he wants and he knows with certainty the price-response function he faces. In this case, the seller can maximize expected profit by charging the price at which his marginal cost equals his marginal revenue or, equivalently, the price at which his contribution margin ratio equals one over the elasticity. Things are a bit more complicated in the real world: sellers have the freedom to adjust prices over many periods, they are unsure how customers will respond to different prices, and they face constraints on their ability to satisfy demand. In this chapter, we focus on the influence of constrained supply on optimal prices.
When the supply of a good is constrained, the conditions for optimal prices presented in Chapter 5 do not necessarily give the price that maximizes net contribution or profit. Return for a moment to the example of the widget maker. With only a single period to consider and with perfect knowledge of his market, the optimal actions for him were to set his price at $8.75 and produce 3,000 units. In this case, he sells all 3,000 units and makes a total contribution of $11,250. This solution assumes that supply is totally flexible and demands are deterministic and he is only going to operate during a single period. What should he do if he can only manufacture 2,800 widgets? What price should he set? How does his optimal price change with the supply constraint? These are the sort of questions we address in this chapter.
We start by looking at the nature of supply constraints and the situations in which they occur. We then examine how a seller can determine the optimal price to charge when faced by a supply constraint, and I introduce and describe the important concept of opportunity cost. We extend the calculation of optimal prices to the case when a supplier has a segmented market and faces supply constraints. This leads to the tactic of variable pricing, which is used when a supplier has multiple units of constrained capacity and can change prices to balance supply and demand.
7.1 THE NATURE OF SUPPLY CONSTRAINTS
Most sellers face supply constraints at least some of the time. For example, most retailers replenish their stock of inventory at fixed intervals—often weekly. In between replenishment times, they are limited to selling their current inventory—in other words, their supply is fixed. In most cases, retailers stock enough inventory that a stockout is unlikely. A drugstore will typically have enough toothpaste, shaving cream, and aspirin in stock that it will not sell out, except in cases of an extraordinary run on a particular item—bottled water prior to a hurricane, for example. In this case, the seller may technically have a limited amount of inventory on hand, but he does not need to consider a supply constraint when setting price. There are, however, many cases in which a seller needs to consider supply constraints when setting a price.
• Service providers almost always face capacity constraints. A hotel is constrained by the number of rooms, a gas pipeline by the capacity of its pipes, and a barbershop by the number of seats (and barbers) it has available.
• Manufacturers face physical constraints on the amount they can produce during a particular period. For example, in 2018, the Mack Trucks company supplied all of Australia and New Zealand from a manufacturing facility west of Brisbane, Australia, with a capacity of 250 trucks per month. In any particular month, Mack could only deliver 250 new trucks plus whatever inventory it had at the beginning of the month.
• Retailers and wholesalers often sell goods that are not replenishable. These include fashion goods that are typically ordered once or electronic goods that are near the end of their life cycle.
• Intrinsically scarce or unique items, such as beachfront property, flawless blue diamonds, van Gogh paintings, and Stradivarius violins, command premium prices because of their scarcity. In these cases, marginal cost is not an important determinant of price because it is meaningless (as in the case of a van Gogh painting) or extremely low relative to the price that the item can command in the market (as in the case of diamonds).
The treatment of a supply constraint depends on its nature. A hard constraint is one that cannot be violated at any price. Typically, hotels and gas pipelines face hard constraints. On the other hand, air freight carriers may have the option to lease space on other carriers to carry cargo in excess of their capacity. In other industries, a seller can delay delivery. A freight carrier that can lease additional space or can delay delivery faces a soft constraint. Mack Trucks’ Australian production limit is also a soft constraint because it can (and does) ask customers to wait for delivery when demand exceeds supply. Whether a supply constraint is hard or soft will, in part, depend on timing. An airline that learns two months in advance that it will be facing very high demand for a particular flight may be able to assign a larger aircraft. However, it will probably not have any option to increase capacity if it only learns about the high demand a week before departure.
7.2 OPTIMAL PRICING WITH A SUPPLY CONSTRAINT
In Example 5.3, the optimal unconstrained price of widgets is $8.75, with corresponding demand of 3,000 units and a total contribution of $11,250. This is optimal if the seller is able to manufacture all 3,000 units—or order them from a third party. But what if he cannot supply the 3,000 units demanded? Perhaps he has a supply chain bottleneck that limits his production. Perhaps he is a reseller with a fixed monthly quota he cannot exceed. In either case, the widget maker faces a rationing problem—he needs to determine which customers will be served and which will not. He has three basic options for rationing supply:
1. Do nothing—keep the price at $8.75 and let customers buy on a first-come, first-served basis until supply is exhausted.
2. Allocate the limited supply to favored customers or allocate to customers using some other mechanism such as a lottery.
3. Raise the price until demand falls to meet supply. He could do this directly by increasing the price or via an auction.
Alternatives two and three are not mutually exclusive—the seller could use a combination of allocations and higher prices to manage a shortage. If he has segmented his market effectively, he could raise his average selling price by allocating most or all of the limited supply to higher-paying customers—this is the basic idea behind revenue management, which we discuss in Chapter 8.
Note that the issue faced by the seller with constrained supply is in part an allocation problem—that is, determining which consumers get to possess the desired product or service that he is selling. Options such as first come, first served and lotteries allocate the product to the early or the lucky, respectively. Other options are to choose recipients based on characteristics such as loyalty or volume of business. When the 2011 floods in Thailand drastically cut the global supply of DRAM chips, many sellers chose not to raise prices to the market-clearing level and allocated the limited supply to their most valued customers. Many sold-out concerts use first come, first served as an allocation rule. Both of these approaches appear to be broadly considered fair by customers—although both are also subject to the possibility of arbitrage in which customers who purchase at the lower price may resell at the higher price—a practice known as scalping in the case of concert or sporting tickets. The approach discussed in this chapter uses price as the allocation method—goods are sold to those who have the highest willingness to pay.1 Economists would argue that this is the most efficient approach to allocating scarce goods and, if arbitrage would otherwise occur, it results in exactly the same allocation at exactly the same prices, except that all of the economic rent accrues to the seller rather than scalpers or other middlemen.
Even in the absence of arbitrage, raising the price of a scarce good generally increases expected social welfare. Consider the case of a concert with a uniform general admission price of $25. There is only one seat remaining for sale, but there are two customers who would like tickets. One has willingness to pay of $30, and one has willingness to pay of $75. If the concert promoter holds the ticket price at $25 and uses first come, first served or a lottery to allocate the ticket, then there is an equal chance that either customer will get the ticket for $25, resulting in expected consumers surplus of 0.5 × ($30 – $25) + 0.5 × ($70 – $25) = $25, which, added to the producers surplus of $25, gives a total social surplus of $50. If the concert promoter raises the price of the last ticket to $40, then consumers surplus is $70 – $40 = $30, which, added to the producers surplus of $40, gives a total surplus of $70. Note that, in this case, both expected consumers surplus and producers surplus (profit) increased when the price of the ticket was raised. In many cases, however, using price as the sole mechanism to allocate a scarce good is frowned on by consumers and needs to be used with care. Chapter 14 discusses some of the issues with consumer acceptance of the practice of pricing to allocate scarce supply.
Let us return to the case of the widget maker and assume that he has decided to use price as the mechanism to allocate scarce widgets. Then he can find the price that maximizes his contribution margin by solving the constrained-optimization problem
max d(p)(p – c),
subject to
d(p) ≤ b,
where b is the maximum supply available. Let p* be the optimal unconstrained price. If d(p*) ≤ b, the supplier does not need to do any further calculations: p* is also the optimal constrained price. If, on the other hand, he finds that d(p*) > b, then he needs to charge a higher price than p* in order to maximize contribution. This situation is illustrated in Figure 7.1. Here, the capacity constraint is a horizontal line—the supplier can only produce quantities below that line. At the optimal unconstrained price, p*, demand d(p*) exceeds capacity.
When the optimal unconstrained price generates demand that exceeds capacity, the supplier needs to calculate the runout price—that is, the price at which demand would exactly equal the supply constraint. In Figure 7.1, the runout price is the price p̄ at which the price-response function intersects the capacity constraint. In other words, p̄ is the price at which d(p̄) = b.
Example 7.1
Assume the widget seller faces the price-response function d(p) = 10,000 – 800p but can only supply a maximum of 2,000 widgets during the upcoming week. Demand at the optimal unconstrained price of $8.75 is 3,000 widgets. Therefore, the runout price can be determined by finding p̄ such that 10,000 – 800p̄ = 2,000 or p̄ = $10.00.
The general principle behind calculating the optimal price with a supply constraint is the following.
The profit-maximizing price under a supply constraint is equal to the maximum of the runout price and the unconstrained profit-maximizing price. As a consequence, the profit-maximizing price under a supply constraint is always greater than or equal to the unconstrained profit-maximizing price.
Figure 7.1 Pricing with a capacity constraint.
This principle is the same for the revenue-maximizing price, which is simply the profit-maximizing price with associated cost of 0.
The effects of different levels of capacity constraint on price and total revenue are shown in Table 7.1. For a binding capacity constraint, b < 3,000, the runout price can be found by inverting the price-response function:
Table 7.1 shows the results of reducing capacity incrementally from 3,000 units per month to 1,000 units per month. The optimal price increases steadily as the capacity decreases. As capacity becomes increasingly constrained, there is more and more pressure on the seller to increase the price.
7.3 OPPORTUNITY COST
Everything else being equal, imposing a supply constraint can only reduce contribution. An auto manufacturer experiencing a strike that takes 25% of capacity out of production for a month will likely see lower profits for that month than if all of the capacity were available. A 200-room hotel that takes 50 rooms out of service for two months to be refurbished is likely to give up some potential revenue. As shown in Table 7.1, the deeper or more binding a supply constraint cuts, the greater the hit the seller is likely to take on his contribution. Taking 75 rooms out of service for two months will lead to less overall contribution than taking 50 rooms out of service for the same period of time.
TABLE 7.1
Impact of constrained capacity on optimal price, contribution, and opportunity cost
The reduction in contribution resulting from a supply constraint is called the opportunity cost associated with that constraint. It is the difference between the maximum contribution the supplier could realize without the constraint and the maximum contribution with the constraint. Since the constrained contribution can never be greater than the unconstrained contribution, total opportunity cost will always be greater than or equal to zero.
Example 7.2
In Example 7.1, the optimal price for widgets with a supply constraint of 2,000 widgets per week is $10.00 per widget, with a total contribution of ($10.00 – $5.00) × 2,000 = $10,000. The optimal unconstrained price of widgets is $8.75, with demand of 3,000 and corresponding total contribution of ($8.75 – $5.00) × 3,000 = $11,250. The total opportunity cost of the supply constraint is $11,250 – $10,000 = $1,250.
Total opportunity cost is the maximum amount the seller would be willing to pay to eliminate his supply constraint entirely. In many cases, the more interesting question is how much the supplier would be willing to pay for one additional unit of supply or capacity. This is the marginal opportunity cost.
Example 7.3
The marginal opportunity cost for the widget seller facing a constraint of 2,000 widgets per week can be found by solving the constrained-optimization problem with a supply constraint of 2,001 and subtracting the optimal contribution with a supply constraint of 2,000. When the supply constraint is 2,001, the optimal contribution is $10,002.50. Thus, the marginal opportunity cost of the supply constraint is equal to $10,002.50 – $10,000.00 = $2.50.
Optimal contributions, prices, and marginal opportunity costs for the widget example are shown in Table 7.1. Note that as the amount of available capacity decreases (i.e., the supply constraint becomes more binding), the optimal price and the marginal opportunity cost both increase. The marginal opportunity cost is zero when the capacity constraint is not binding—which is the case for any constraint above 3,000 units. This illustrates the following general principle.
Total and marginal opportunity costs are nonzero only when there is a supply or capacity constraint that is binding at the optimal unconstrained price. Otherwise they are zero.
The opportunity cost can be an important input to other corporate decisions. From Table 7.1 we can see that the marginal opportunity cost faced by the supplier is $2.50 per widget when supply is limited to 2,000 units. This means that the widget maker would be willing to pay up to $2.50 in rent for an additional unit of capacity that allowed him to make one more widget at a cost of $5.00. Alternatively, he would be willing to pay up to $2.50 + $5.00 to buy another widget from a wholesaler. Fundamentally, the opportunity cost associated with constrained capacity is an economic measure of how much the company would be willing to pay for additional capacity. By calculating and utilizing this information in its supply decisions, companies can drive higher returns than they could from optimizing their pricing and their supply decisions in isolation.
7.4 MARKET SEGMENTATION AND SUPPLY CONSTRAINTS
Market segmentation can be a powerful tool for increasing profitability when supply is constrained. To see how, let us look at an example.2
Example 7.4
The football game between Stanford and the University of California at Berkeley (a.k.a. the Big Game) is going to be held at Stanford Stadium, which has 60,000 seats. Customers can be segmented into students (those carrying a student ID card) and the general public. We assume that the price-response functions for each of these segments is
where pg is the price charged to the general public and ps is the student price. What if Stanford can only charge a single price to all? In this case, the aggregate price-response function is d(p) = (120,000 – 3,000p)+ + (20,000 – 1,250p)+ and the optimal price will be $20.00. At this price, Stanford would sell exactly 60,000 tickets, grossing $1,200,000. Note that all ticket sales would be to the general public. The students are priced out of the market, since the highest willingness to pay of any student is 20,000/1,250 = $16.00.
Now, what if Stanford could charge different ticket prices to students versus the general public? Then finding the optimal price requires solving the constrained-optimization problem
subject to
We could, of course, solve this problem directly. However, we gain more insight by using the principle established in Chapter 6: Prices should be set for the two different segments so that the marginal revenues from both segments are equal. When supply is unconstrained, marginal revenues should all be set to the marginal cost. When supply is constrained, marginal revenues should still be equated, but they need to be set so the supply constraint is satisfied.
Example 7.5
We want to find the revenue-maximizing prices for students and the general public. The marginal opportunity cost is 2pg – 40 for the general public and 2ps – 16 for students. Equating the two marginal revenues and simplifying gives pg = ps + 12. In other words, the ticket price for the general public will be $12.00 higher than the ticket price for students. The other condition that must be satisfied is that the total demand from both students and the general public be equal to the capacity of the stadium; that is, (120,000 – 3,000pg)+ + (20,000 – 1,250ps)+ = 60,000, which, when simplified, gives 3pg + 1.25ps = 80. Solving both conditions simultaneously gives ps = $10.35 and pg = $22.35. At these prices, Stanford will sell 52,941 tickets to the general public and 7,059 seats to students, generating revenue of $1,256,471, a 4.7% increase over the best that could be achieved with a single ticket price.
Note that the profit-maximizing price for the general public is more than double the student price. The stadium is sold out and there are thousands of people who would be willing to pay much more than the student price. Yet the profit-maximizing decision for Stanford is to sell more seats to students at a deep discount. In this case, the effect of market segmentation is that students win while the general public loses. Market segmentation reduces the price of tickets for the more price-sensitive students while increasing the price for the less price-sensitive general public.
The optimal prices assumed that the marginal costs—and ancillary revenues—of serving both students and the general public were the same. Differences in either marginal costs or ancillary revenues between the two market segments would lead to different optimal prices (see Exercise 3). The results also relied on a perfect fence between the student and the general public segments. The optimal prices and revenues would change if student sales cannibalized sales to the general public (see Exercise 1).
7.5 VARIABLE PRICING
Consider the following pricing tactics.
• Disney World charges a different price for its general admission by day and length of ticket. The average cost of a ticket per day for each starting day in January 2020 and length-of-ticket from one to seven days is shown in Table 7.2. Average daily prices range from $139 for a one-day ticket on January 1 to $62 for a seven-day ticket starting January 19.
• The San Francisco Opera charges a lower price for weeknight performances than for weekend performances. The most expensive box seats cost $175 for a Wednesday performance and $195 for a Saturday performance, while the least expensive balcony side tickets cost $25 for a Wednesday performance and $28 for a Saturday performance.
TABLE 7.2
Average daily ticket price for a visit to Disney World in January 2020 by starting date and length of visit
• Major League Baseball teams in the United States set different prices for different games, depending on factors such as weekday versus weekend, time of year, opponent, and special events. The pricing schedule for the Colorado Rockies in 2004 is shown in Table 7.3.
• The Gulf Power Company is an electric utility serving more than 428,000 residential customers in northwestern Florida. The majority of these residential customers pay a flat rate of $0.114 per kilowatt hour for electricity. However, Gulf Power offers residential customers the alternative of purchasing their electricity under its Residential Select Variable Pricing (RSVP) program. Under the RSVP program, each weekday is divided into an off-peak, a shoulder, and an on-peak period. Electricity costs $0.083 per kilowatt hour (kWh) during the off-peak period, $0.1046 during the shoulder period, and $0.189 during the on-peak period. In the summer the weekday on-peak period is from 1:00 p.m. to 6:00 p.m., while in the winter the weekday on-peak period is from 6:00 a.m. to 10:00 a.m. The summer off-peak period is from 11:00 p.m. to 6:00 a.m. and the winter off-peak period is from 11:00 p.m. to 5:00 a.m. All other hours are shoulder period. In addition to these three periods, with a half hour’s notice to RSVP customers Gulf Power can declare a critical period during which electricity costs $0.777 per kWh. Under the RSVP program, Gulf Power can declare up to 88 hours of critical period per year.3
TABLE 7.3
Home ticket prices for the Colorado Rockies baseball team, 2004 season
SOURCE: Rascher et al. 2007.
What do these four pricing tactics have in common? They are all cases in which a seller with constrained capacity adjusts prices in response to anticipated demand in order to maximize return from fixed capacity. We call this tactic variable pricing. Industries where variable pricing is employed share four characteristics:
• Demand is variable but follows a predictable pattern.
• The capacity of a seller is fixed in the short run (or is expensive to change).
• Inventory is perishable or expensive to store—otherwise buyers would learn to predict the variation in prices and stockpile when the price is low.
• The seller has the ability to set prices in response to anticipated supply and demand imbalances.
Under these conditions, sellers can use variable prices to shape demand to meet fixed capacity (or supply) by exploiting differences in customer preference. More people want to visit Disney World during the first week in January than during the third week in February. Weekday opera performances are less popular than weekend performances. The Rockies know that a game against the Yankees will generate more demand than a game against the Cincinnati Reds. The highest electricity prices correspond to the periods of highest use and the lowest prices to the periods of lowest use. Variable pricing does not qualify as pure group pricing; rather, it is a form of product differentiation because it allows customers to choose among different products based on price.
Note that variable pricing can either be static or dynamic. Under static variable pricing, prices are set once in advance and do not change as new information is obtained. The Colorado Rockies’ ticket pricing scheme shown in Table 7.3 and the Gulf Power Company RSVP rate plan are both examples of static variable pricing. Under dynamic variable pricing, prices can be changed as new information is received. For example, many sporting teams now change prices as the season advances and tickets become either more or less in demand as a result of team performance.
TABLE 7.4
Intercepts and slopes for the price-response curves in the theme park example
7.5.1 The Basics of Variable Pricing
Consider a hypothetical theme park that can serve up to 1,000 customers per day. The theme park charges a single admission price, and all rides are free after admission. During the summer, demand follows a stable and predictable pattern, with higher demand on weekends than during weekdays. We assume that demands for different days of the week are independent; that the price-response functions are linear, with intercepts and slopes, as shown in Table 7.4; and that the theme park has a marginal cost of zero per customer.
What price should the theme park charge without variable pricing? Since its marginal cost is zero, it maximizes contribution by maximizing revenue. The revenue-maximizing single price can be found by solving the nonlinear optimization problem
subject to
where p is the optimal single price, xi is attendance on each day, Di and m i are, respectively, the intercept and slope of the price-response function for day i, and C is total daily capacity (in this case 1,000). The optimal single price is $25.00 per ticket. The corresponding attendance and revenue for each day are shown in the columns labeled “Single price” in Table 7.5. With a single price, the theme park serves a total of 4,795 customers during the week, with total revenue of $119,900. What is striking in Table 7.5 is the wide variation in attendance across the week. The park is full on Friday, Saturday, and Sunday, but it operates at only 25% of capacity on Monday and 40% of capacity on Tuesday. Furthermore, potential customers are being turned away on Friday, Saturday, and Sunday—500 on Saturday alone.
These two conditions—wide variation in utilization and a large number of turndowns—are indications that contribution could be improved through variable pricing. In the case of the theme park, this means charging a different price for each day. When demands are independent—as we have assumed—the optimal daily prices can be calculated by solving independent optimization problems for each day. The resulting daily prices, attendance, and revenues are shown in the columns labeled “Variable pricing” in Table 7.5. With variable pricing, daily prices range from a low of $15.00 on Monday to a high of $38.33 on Saturday. The total number of customers served during the week rises 29% from 4,795 to 6,205, and total revenue rises 30% to $155,294. Note that variable pricing evens out utilization across days of the week and increases overall utilization. Attendance on the low days of Monday and Tuesday increases from 250 and 400 to 750 and 700, respectively. The weekly utilization of the park—its load factor—increases from 4,795/7,000 = 68.5% to 6,205/7,000 = 88.6%.
TABLE 7.5
Theme park daily prices, attendances, and revenues under constant pricing and under variable pricing
The average price paid by all customers increases only slightly under variable pricing—from $25.00 to $25.03. This is typical—variable pricing can often lead to a major increase in total revenue with only a small increase in the average price paid by customers.
7.5.2 Variable Pricing with Diversion
We have seen that variable pricing can increase operating contribution when demands are independent among periods. However, variable pricing can also shift demand from peak periods to off-peak periods—a phenomenon called diversion or demand shifting. Diversion occurs when at least some customers are flexible and will shift their consumption among periods if they can save money by doing so. Diversion can be illustrated by an example from Robert Cross (1997). His local barbershop was overcrowded and turning customers away on Saturdays, while Tuesdays were very slow. Some of the Saturday customers were working people who could only come on Saturday; others were retirees and schoolchildren, who could get their hair cut any day of the week. By raising prices 20% on Saturday and reducing them by 20% on Tuesday, the shop induced some of the Saturday customers to shift to Tuesday. As a result of this simple peak-pricing action, turnaways decreased, Saturday service improved, and total revenue increased by almost 20%.
Variable pricing segments the market for a generic product (a haircut) between those with strong preferences for a particular date and those who are relatively indifferent. Since customer preferences are not evenly distributed—more people want to see an opera, attend a theme park, or get their hair cut on a weekend—and capacity is fixed, using price to shift indifferent customers to less popular dates can increase both revenue and utilization and reduce turnaways. In the case of the Colorado Rockies, the four-tiered pricing system segments the market between customers who badly want to see a popular game (such as one against the Yankees) from those who just want a night out at the ballpark. Furthermore, variable pricing is generally well accepted because it segments via self-selection: customers choose the product and price combination they prefer.
While diversion is an important element of variable pricing, it can be a two-edged sword. Raising the price for peak capacity and lowering it for off-peak capacity will shift customers from the peak to the off-peak period. This is good (at least in moderation) because the overall gain from extracting higher prices from the remaining peak customers outweighs the lower price received for off-peak customers. However, if the price differential becomes too great, we would expect more and more peak customers to defect to the off-peak period. If the barbershop drops Tuesday prices too low and raises Saturday prices too high, it might end up with a Tuesday peak and very slow Saturdays. Ultimately the loss from peak customers who shift to the cheaper off-peak period (cannibalization) can outweigh the benefits of off-peak demand induction and higher peak rates.
While diversion is easy to describe, it is difficult to model. In effect, the different alternatives are substitute products, and each pair could have a different cross-price elasticity. One approach is to presume that every customer has a separate willingness to pay for each alternative. Each customer would compare his willingness to pay for each alternative with the price and purchase the alternative that provided the highest positive surplus—assuming that such an alternative exists.
Example 7.6
Under the separate-willingness-to-pay model, each customer would have a separate willingness to pay for attendance on each day at the theme park. The willingness to pay of each customer can be represented as a vector, so a customer with willingness-to-pay vector of ($30, $18, $22, $19, $14, $18, $32) would have a $30 willingness to pay for Sunday attendance, an $18 willingness to pay for Monday attendance, and so on. We can represent the vector of prices in the same fashion: ($33.87, $15.00, $17.50, $18.01, $19.00, $27.00, $38.33). The consumers surplus vector for this customer is the difference between her willingness to pay and the price for each day, here (− $3.87, $3.00, $4.50, $0.99, − $5.00, − $9.00, − $6.33). In this case, the customer would attend on Tuesday because that provides her with the highest surplus ($4.50). For comparison, the consumers surplus for this customer under a constant price of $25.00 would be ($5.00, − $7.00, − $3.00, − $6.00, − $11.00, − $7.00, $7.00), which means she would attend on Saturday. The effect of variable pricing on this customer is to shift her attendance from Saturday to Tuesday.
While the willingness-to-pay point of view helps us understand how an individual consumer might act, it is difficult to extend to a useful model of total market price responsiveness. If we knew the distribution of all customer willingness to pay across the population, we could, in theory, use that information to derive a multidimensional demand function. One problem is that this model would require 7 own-price elasticities (1 for each day of the week) and 42 cross-price elasticities (1 for each combination of days of the week). It is unlikely that the theme park (or most other companies) would have the data available to estimate a credible model with this many parameters.
We can use a much simpler approach to incorporating diversion in the theme park example—we assume that eight customers will shift from one day to another for every $1 difference in price. Thus, if the base prices for Monday and Tuesday were $20 and $22, respectively, we would first calculate a base demand for each day using its individual linear price-response function and then shift 16 people from Tuesday to Monday. When applied across the whole week, this means that demand will shift from days with higher-than-average prices to those with lower-than-average prices. Under this model, the unconstrained demand the theme park will see on day i is given by
where di is the unconstrained demand, pi is the price, Di is the demand at zero price, and mi is the slope of the linear price-response function for day i. Demand for each day now depends not only on the price for that day but on the price for all the other days of the week. We can now formulate the optimal pricing problem as a nonlinear optimization problem:
subject to
The optimal prices, attendance, and revenue under this model are shown in Table 7.6. Base demand is the number of customers calculated using the price-response functions for each day, and net diversion is the net number of customers induced to shift to or from each day by differential pricing. A positive net diversion means that more customers shifted to that day than shifted away; a negative net diversion means the opposite. Variable pricing shifted a total of 838 customers from the peak days of Friday, Saturday, and Sunday to the off-peak days of Monday through Thursday. Furthermore, the theme park served 6,609 total customers during the week, resulting in a 94% utilization and total revenue of $156,536—a dramatic improvement over charging a single price.
TABLE 7.6
Theme park daily prices, attendances, and revenues using variable pricing and assuming demand switching
Of course, the theme park operator might not want to set prices exactly as shown in Table 7.6. For simplicity, he may want to reduce the total number of prices and only offer prices in increments of $5. In this case, it is easy to see that he would still gain significant revenue from a three-tier pricing policy of $30 on weekends, $25 on Friday, and $20 on Monday through Thursday, as opposed to setting a constant price of $25.
Variable pricing does not need to be implemented within list prices. The theme park could implement two-tier pricing by establishing a $30 list price but distributing $10-off coupons redeemable only on weekdays. Even more effectively, the theme park might establish a $35 price for weekends and Friday, a $20 price for Monday through Thursday, and distribute $5-off coupons redeemable only for Monday admissions. In this case, variable pricing has been implemented through a combination of variable list pricing and couponing.
7.6 VARIABLE PRICING IN ACTION
We now consider how variable pricing is used in four different industries—sporting events, airlines, electric power, and ride-sharing. We also look at two industries—movie theaters and restaurants—that meet all the criteria for variable pricing but in which variable pricing is little used.
7.6.1 Sporting Events and Entertainment
Some of the most obvious candidates for variable pricing are sporting events and other forms of entertainment such as concerts and theater. In most cases, sporting arenas, concert venues, and theaters have fixed capacity but demands that vary depending on the offering. A further sign of the potential for variable pricing in these industries is the prevalence of a robust secondary market for tickets. For much of history, this secondary market was driven by scalpers (called ticket touts in the UK) who would purchase tickets at list price with the expectation of selling at a premium. More recently, companies such as StubHub and RazorGator have enabled online resale of tickets. The existence of this secondary marketplace was driven in large part by the fact that concert promoters and sporting teams were unable or unwilling to adjust prices to balance supply and demand.
Around the beginning of this century, sporting teams began to experiment with variable pricing. Variable pricing is particularly attractive in baseball since each team has 81 home games each season that vary widely in terms of fan appeal. The Colorado Rockies were among the earliest adopters, instituting the variable-pricing scheme shown in Table 7.3 in 2004. Over the next 15 years or so, every team in Major League Baseball followed suit, with the New York Yankees being the last to adopt variable pricing in 2017 (Best 2017). While there was some resistance from fans, variable pricing was generally viewed as successful, which has spurred uptake in other leagues including the National Football League, the National Basketball Association, the National Hockey League, and the Premier League (soccer) in England, among others.
Many sporting teams employ dynamic variable pricing, establishing an initial set of prices according to a scheme such as that shown in Table 7.3. However, prices can also be updated over time based on information gained as the season progresses—broadly speaking, if the team is doing well, prices will increase; if they are doing poorly, prices will drop. Prices may also be adjusted based on weather, the record of the opposing team, or the emergence of a popular player (for example, it was estimated that, in the 1991–1992 NBA season, Michael Jordan generated over $50 million in revenue for opposing teams through increased attendance and ancillary revenue).4
Sporting teams like variable pricing because it increases total revenue without increasing ticket prices. This is attractive in an environment where ticket price increases are likely to be condemned by fans and sportswriters as evidence of greedy owners and players gouging loyal fans. In 2003, the New York Mets adopted a variable-pricing plan under which the price was increased $10 for the 17 most popular games while lowering or holding prices unchanged for 43 games. “That allowed the Mets, who finished last in their division in 2002, to trumpet the plan as holding ticket price increases to only 4% on average” (Fatsis 2003, 82).
Broadway and other long-run theaters are also an ideal candidate for variable pricing—demand varies in predictable ways across seasons and by the day of the week, and customers purchase advance tickets. Nonetheless, theaters were relatively slow to adopt variable pricing beyond charging more for weekend night shows than matinees and weekday nights. One of the pioneers in dynamic pricing in Broadway theater was Disney’s The Lion King. As Disney vice president David Schrader described pricing for The Lion King, “First the range of prices across the theater is set according to an estimate of the demand for a particular performance, based on the 15 years of past B.O. [box office] from which to extrapolate annual trends. Second, after tickets go on sale, Disney re-examines each perf [performance] to see how demand is shaping up in reality, and then tweaks pricing accordingly” (Cox 2013, 1). Dynamic pricing was widely touted as one of the reasons that The Lion King generated the most revenue of any Broadway show in the least amount of time despite being in one of the smaller Broadway theaters.5 With the success of variable pricing at The Lion King, many other Broadway theaters, as well as performing arts groups such as the San Francisco Symphony and attractions such as the Indianapolis Zoo, have adopted more dynamic pricing. Ticketmaster, by far the largest distributor of event and sporting tickets in the United States, has adopted dynamic pricing to adjust prices based on expected supply and demand.
7.6.2 Passenger Airlines
Passenger airlines certainly meet the criteria for variable pricing—constrained capacity with demand that fluctuates over time. Southwest Airlines implements variable pricing directly by changing fares for its departures to match demand with the restricted capacity. However, the major nondiscount airlines also practice variable pricing. They do so by opening and closing the availability of various fare classes on different flights. Figure 7.2 shows average fares and load factors on a flight of a major airline from the Midwest to Florida during the winter. For operational reasons, the airline has scheduled the same aircraft for the flight each day. The airline is charging a higher average price on high-demand days (Saturday and Sunday) and lower prices on low-demand days (Tuesday, Wednesday, and Thursday). In this case, the airline is adjusting the average fare not by changing the list price but by opening and closing fare classes. When total demand is anticipated to be high, the airline will restrict availability of discount fares, causing the average fare to increase. When total demand is anticipated to be low, the airline will open availability of deep discounts, causing the average fare to drop. It is continually monitoring demand using its revenue management system to determine which classes to open and close on each flight. Chapters 8 and 9 describe in detail how this is done.
Figure 7.2 Average daily load factors for a Midwest–Florida flight on a domestic US discount carrier. Source: Courtesy of Garrett J. van Ryzin.
Discount airlines use variable pricing even more directly. If demand at Southwest Airlines begins to rise more quickly or more slowly than anticipated, Southwest may change the price. Southwest is not doing this to segment customers between high-elasticity and low-elasticity segments—rather, it is using variable pricing without segmentation to maximize the return from each flight and, to some extent, to smooth demand through demand diversion from high-fare, high-utilization flights to low-fare, low-utilization flights.
7.6.3 Electric Power
Electric power is another obvious candidate for variable pricing. Suppliers have constrained capacity, and electricity is difficult and expensive to store. Demand follows reasonably predictable patterns, and many customers have the flexibility and willingness to adjust their usage in response to prices. In fact, electric utilities have a further motivation to vary prices: There are tremendous cost differences between generation during peak and off-peak periods. For many utilities, it can be 10 times more expensive to generate power during a peak period than during an off-peak period. Adopting dynamic pricing could reduce generation costs by 8% to 20%, leading to price reductions for all customers, not just those shifting their consumption.6
Despite the obvious benefits that electric suppliers and consumers could realize from time-of-day pricing, utilities in the United States have been slow to adopt programs that would adjust the price of electricity in response to changes in demand. Some of the reasons are technical, relating to the unique characteristics of electric power transmission and distribution. However, the most powerful reasons are political and historical. Historically, electric utilities were granted service monopolies over a certain territory and guaranteed a reasonable return on their investments in generation capacity and transmission. In return, they submitted to government regulation of their prices to avoid abuses of monopoly power in pricing. While some jurisdictions are moving to less regulated regimes and seeking to encourage competition, the transition from fully regulated to a more competitive market for electric power has proven to be difficult and in some cases a political disaster. The attempt by California to deregulate its electricity market was a debacle that was one of the drivers behind the California governor recall and election of 2003.
Despite the political barriers, many electric power utilities have instituted some form of variable-pricing program. Most of these programs are aimed at industrial customers, who often have the flexibility to schedule their operations to take advantage of variable prices. In a typical program, the electricity supplier charges the customer for electricity on an hour-by-hour basis, with expected prices for the next day announced 24 hours in advance. This enables industrial customers to reschedule their operations to minimize consumption costs. Programs to charge variable prices to residential customers are much rarer—the Gulf Power RSVP program described earlier is a typical example. It is likely that, despite setbacks, the trend toward more competitive power markets in the United States will continue and that the need for variable-pricing methods will only increase.
7.6.4 Ride-Sharing
Major players in the ride-share industry, such as Lyft, Uber, and Didi Chuxing, have all adopted dynamic pricing. These ride-share operators connect prospective riders with independent drivers who are willing to provide transportation. When a prospective rider opens the app and specifies her destination, the ride-share company quotes a price that can depend on both the origin and the destination. The price for a particular origin and destination combination will change over the course of time in response to current demand and supply conditions, as we shall see.
While business models vary, the ride-share operator typically determines both the rider price, or what the rider pays, and the driver price, or what the driver will be paid for the trip. The ride-share operator keeps the difference between the rider price and the driver price—the trip margin. In most cases, the trip margin is a fixed percentage of the rider price: as of 2019, Uber was advertising that its trip margin percentage was 25%, and Lyft was advertising 20%. This means that, on an Uber trip costing the rider $20, the driver would get $15 and Uber $5.7 Ride-share operators typically set a time- and distance-based base rate for each trip. These base rates, which may vary by city, establish a minimum rider price and hence a minimum driver payment. However, most ride-sharing operators soon established dynamic pricing in which the rider price and driver price varied by time and by location.
For the ride-sharing operators, the need to balance supply and demand motivated the adoption of peak-load pricing. In particular, in certain times and locations, the demand for rides at the base price exceeds the supply of available drivers—sometimes by a large margin. Demand for rides can vastly outstrip available supply at the end of the workday in business districts in London, Manhattan, and San Francisco; at peak airport arrival times; and when sporting events or concerts let out. When demand exceeds supply, a ride-sharing operator faces the problem of allocating constrained supply. If the operator does not raise the price, many customers would see long arrival times or that no cars are available. As a result, some customers would drop out and supply would be allocated to the customers who were most willing to wait. Alternatively, the ride-share operators could raise the price and allocate the scarce supply using prices. This is the approach that ride-share operators generally adopted.
Of the dynamic pricing schemes in use by ride-sharing companies, Uber’s has been the most widely studied. Uber divided each city it serves into hexagons about 0.2 mile on a side, with each hexagon covering an area of about 0.1 square mile. In the initial application of “surge pricing,” the rider fare for every trip originating in a hexagon was subject to a multiplier greater than or equal to 1. If the trip multiplier was greater than 1, then the hexagon was “surging” and the rider fare would be set to the trip multiplier times the base price. So, for example, if the base fare is equal to $12 for a trip and the surge multiplier were 2.5, the rider price would be 2.5 × $12 = $30 plus any fees and tolls. If the trip multiplier were 1, then the hexagon was not surging and the rider would pay the base price plus fees and tolls.
Under this scheme, the problem for Uber is what multiplier to apply at each time at each geographical location and how to update it as conditions change. The multiplier can be changed at relatively short intervals—typically every one to three minutes. We assume for the moment that the same multiplier will be applied to every ride originating in the same region during a time interval. The ride-sharing service typically knows the number of people in the area who have the app open—we denote that potential demand by D. Not everyone who opens the app will book a ride, and we can expect that the fraction of open apps that will book will depend on the multiplier applied, with a higher multiplier leading to a smaller fraction of potential demand booking.
The service also knows the number of open cars in the area, S. The ride-sharing service needs to determine the multiplier p (S, D) to set as a function of the current supply and demand situation. If D ≤ S, then p (S, D) = 1. In this case, there is sufficient supply to serve demand even if every open app converted to a booking, so there is no need to raise prices. Because the base price is a minimum, we cannot set p (S, D) < 1. If D > S, then we are in a situation of excess demand and we want to tamp down demand, so we need to set p (S, D) > 1.
The multiplier function p (S, D) needs to satisfy three conditions.
• D ≤ S → p (S, D) = 1. If supply is more than adequate to meet demand, the multiplier is equal to one.
• ∂p (S, D)/∂D ≥ 0. The multiplier is increasing in demand.
• ∂p (S, D)/∂S ≤ 0. The multiplier is decreasing in supply.
Let’s dig a little deeper into the calculation of the multiplier. First, we assume that demand responds instantly to the multiplier. This is realistic; riders tend to respond to fares at the moment they open the app, and if they see a fare that is too high, they may decide to use another mode of transportation or postpone their trip. This will be reflected immediately in the booking conversion rate—the fraction of people who book after viewing a fare on their app will decrease as the multiplier increases. We can model this situation as
where D is the number of people who open the app, 0 ≤ ρ(p) ≤ 1 is the fraction of people who open the app who will book a ride given multiplier p, and d(p) is the number of rides that will be booked. We assume that ρ'(p) < 0 so that the number of bookings will decrease as the multiplier increases.
This representation is identical to the formulation in Equation 3.1 with the difference that p represents the multiplier instead of the price. This reflects the fact that, from a customer’s point of view, the buying experience is much the same: with the ride-hailing app, the customer signals potential interest in purchasing by opening the app. The customer does not decide whether to book until seeing the price, and a higher fraction of customers will book if the price (multiplier) is low than if it is high. It also means that any of the functional forms listed for ρ(p) in Table 3.3, such as linear, logit, and probit, can be used in Equation 7.4 depending on the assumption about the underlying distribution of willingness to pay among riders.
When demand exceeds supply (D > S) and supply is fixed, then the ride-sharing operator faces a variable-pricing problem and the optimal (profit-maximizing) multiplier p* is the one that balances supply and demand. This means that p* must be such that Dρ(p*) = S, or, equivalently,
We can solve for p* by substituting the appropriate function ρ(p) into Equation 7.5 and solving.
Example 7.7
There are 100 empty cars available in a region at a time when D = 300. Assume that the price-response function is linear with ρ(p) = (1 – .33p)+. From Equation 7.5, the multiplier that balances supply and demand is given by 
which gives p* = 2.
For each city, the multipliers for a ride-sharing service like Uber or DiDi are updated for all of the regions within a city every few minutes based on changing supply and demand conditions.
One question about dynamic pricing of this sort is whether it is good for customers. Note that, if a region is in a state of excess demand and prices are not increased, service quality will decrease: riders will see increased time to arrival for their ride or, in the extreme, that no cars are available. While an excess of demand may seem like a good situation for drivers, in actuality it is not. The increased arrival times mean that drivers are being sent from distant regions to satisfy the demand and are thus spending excessive time driving to pickups—time for which they do not get paid. Furthermore, the price has been kept low, so the result is lower driver payments per hour. Riders get good prices, but they need to accept long arrival times, and many will drop out because of excessive arrival times or unavailability of cars. Juan Castillo, Dan Knoepfle, and Glen Weyl (2017) call this a wild-goose chase equilibrium, in which drivers spend their time chasing riders. They show that balancing supply and demand using price can lead to an equilibrium that provides more social welfare (as measured by producers plus consumers surplus) than the wild-goose chase equilibrium. Effective variable pricing can make both riders and drivers better off.
This discussion has assumed that supply is fixed—that is, while riders respond to changing prices, drivers do not. This is not entirely true in the short run: drivers do tend to move toward areas of higher surge. It is also likely that, in the longer run, drivers anticipate the time and locations of likely surge (e.g., downtown rush hour) and plan their driving accordingly. However, in the short-run time frame in which drivers make decisions (minutes), supply in a region is largely fixed, meaning that the effect of changing price on immediate supply is relatively small. The impact of incorporating the supply-induction effects of prices into ride-sharing pricing would be to reduce the multipliers somewhat.8
7.6.5 Movie Theaters and Restaurants
It is interesting to examine two industries in which the variable-pricing dog has not barked—that is, variable and dynamic pricing has not been widely adopted despite an apparent ideal fit. The first is movie theaters. While movie theaters employ many standard promotional approaches such as senior and student discounts and matinee discounts, they generally do not vary pricing according to the popularity or anticipated popularity of a movie; nor do they change it over time as demand begins to dwindle. The reason they do not is a bit of a mystery. Movie demand has a predictable element to it—demand is higher in the summer and on holidays than other times of the year, and demand is higher on the weekends than on weekday nights. Furthermore, demand tends to decline following the release of a new movie in a fairly consistent and predictable fashion. While there can be unexpected hits and duds, relative demand for different movies is reasonably predictable: the next James Bond movie will certainly outdraw even the most popular documentary released in the same year. Finally, movie theaters meet one of the most fundamental criteria for variable pricing: capacity is fixed and immediately perishable.
Figure 7.3 Predicted hourly load at Gramercy Tavern restaurant in Manhattan on Wednesday, December 18, 2019 (left), and Saturday, December 22, 2019 (right).
Source: Google Maps.
There has been some movement toward variable pricing at movie theaters in the US industry—most cinemas across the country slash prices on Tuesday nights. As a result, Tuesday has gone from being the lowest-grossing weeknight to the highest grossing (except for Friday). While this would indicate that there is an opportunity to increase revenue and fill theaters using variable pricing, in fact, “no U.S. chain has tested variable pricing in a serious way,” notes one industry observer (McClintock 2019). There has been some debate about why the industry has been so reluctant when the gains are potentially enormous. The leading theory is that independent theaters and theater chains feel implicitly pressured by distributors who want to keep in-theater prices high for their movies so that the value perception of the movie is not degraded by a low price. It is difficult to confirm or disprove this theory, but, despite the apparent fit, adoption of variable pricing is very limited in US theaters, and there are few indications that the situation is likely to change soon.
Restaurants and bars would seem to be another candidate for variable dynamic pricing. Demand varies rather predictably by time of day, day of week, and season. For example, Figure 7.3 shows Google Maps’ hourly forecast for demand at the New York City restaurant Gramercy Tavern for Wednesday, December 18, and Saturday, December 22, 2019. These forecasts, which were derived entirely from history, show predictable differences in seating demand by both day and hour. The restaurant itself has access to reservations data, which it could use to generate even more accurate forecasts. Finally, of course, restaurants have fixed and perishable capacity.
Despite checking all of the boxes for variable pricing, restaurants and bars do very little of it aside from happy hours, the usual student and senior citizen discounts, and in some cases Groupons. Even extremely popular restaurants typically allocate capacity using scarcity—reservations need to be made well in advance and/or people need to wait in line at popular times—rather than adjusting price over time to balance supply and demand. However, restaurants may present less of a mystery than movie theaters with regard to their reluctance to adjust prices: the use of printed menus makes real-time price adjustment difficult, as does the nature of the dining experience—it would be disconcerting to have prices change between the time you decided on your order and the time you actually ordered.
That said, a development that may lead to higher penetration of variable and dynamic pricing in the restaurant industry is the rise of delivery services such as Uber Eats and Grubhub in the United States and Deliveroo in Europe, which provide apps that allow diners to order food to be delivered. The popularity of these delivery services has led to the development of so-called virtual restaurants and ghost kitchens that exist solely to serve the delivery market.9 It is likely that such establishments selling primarily (or only) through online delivery apps will adopt variable and dynamic pricing—after all, the delivery services already use dynamic pricing for their fees.
7.7 SUMMARY
• Supply constraints are commonplace in many industries. They can arise as a result of limited inventory or because capacity itself is physically constrained. Whenever there is a significant chance that demand might exceed available supply, the supply constraint needs to be explicitly incorporated in determining the optimal price.
• With a single market segment, the optimal constrained price is the maximum of the runout price and the optimal unconstrained price. The optimal price with a supply constraint is always at least as high as the optimal unconstrained price.
• With multiple market segments, the optimal price can be found by solving a constrained-optimization problem. At the optimal price, the marginal revenue will be the same for all segments, but it will not be equal to marginal cost if the constraint is binding.
• A supply constraint has an associated total opportunity cost, defined as the additional operating profit the business could achieve if the supply constraint were eliminated. The marginal opportunity cost associated with a supply constraint is defined as the additional operating profit the seller would realize if one additional unit of supply were available. Both the total and the marginal opportunity costs associated with a supply constraint are zero if the constraint is not binding.
• Variable prices can be used when capacity is constrained and demand changes over time in a predictable fashion. Variable pricing can increase profitability when demands are independent by adjusting demand to meet supply. It is even more effective when some customers are flexible, because it can shift demand from peak to off-peak periods.
• Optimal variable prices can be found by solving a constrained-optimization problem. The marginal values associated with the supply constraints are the opportunity costs of additional capacity.
• Some industries in which variable pricing is used are sporting events, concerts, passenger airlines, electric power, and ride-sharing. In some of these cases, the variable prices are set once in advance. Other industries, notably ride-sharing platforms and electric power, use dynamic variable pricing in which prices are updated as more recent information about supply and demand is received. This comes close to revenue management, the topic of the next few chapters. Currently, movie theaters and restaurants do not extensively use variable pricing despite apparent suitability.
The pricing and revenue optimization problems addressed in later chapters build on the basic techniques of pricing with constrained supply. Capacity allocation (Chapter 9) is based on opening and closing availabilities to different market segments in order to maximize return from fixed and perishable capacity when willingness to pay increases over time. Network management (Chapter 10) extends this concept to the case when customers are buying multiple resources. Overbooking (Chapter 11) deals with the question of how many total units to offer for sale when capacity is fixed. Markdown management (Chapter 12) addresses the case of setting and updating prices for a fixed stock of perishable goods over time. In each of these problems, much of the complexity of the pricing problem comes from a constraint on supply.
7.8 FURTHER READING
A seminal paper on dynamic pricing with a capacity constraint is Gallego and van Ryzin 1994, which shows that, under very weak conditions, a seller with constrained perishable capacity who is pricing dynamically should set his price to either the profit-maximizing price or the runout price—the price at which he would exactly sell all of his inventory.
Regarding specific industries, variable pricing for baseball is discussed in Rascher et al. 2007, which estimates the potential value for every major league team. The San Francisco Giants were apparently the first to apply dynamic pricing in 2009; see Shapiro and Drayer 2012. For a specific application to the Los Angeles Dodgers, see Parris, Drayer, and Shapiro 2012. An early examination of the potential for dynamic pricing for theater can be found in Leslie 2004. There is a vast literature on dynamic pricing for retail electric power: a brief overview of the issues involved in implementing dynamic pricing in the United States—particularly the barriers that have hindered adoption and the prospects for the future—can be found in Wilson 2012. A more extensive discussion can be found in the papers in Reneses, Rodriguez, and Perez-Arriaga 2013. The reasons why dynamic pricing has not been widely adopted by movie theaters are discussed at length in Orbach 2004, Orbach and Einav 2007, and Phillips 2012c.
While restaurants do not widely use variable pricing (with exceptionally rare exceptions), many have adopted various other revenue management–like techniques to maximize revenue per available seat hour (RevPASH). For an overview of these techniques, see Kimes 2004.
7.9 EXERCISES
1. Let us return to the Stanford Stadium pricing problem in Section 7.4, assuming a capacity of 60,000 seats and the price-response functions for students and for the general public as given in Equations 7.1 and 7.2. Assume that 5% of the general public will masquerade as students (perhaps using borrowed ID cards) to save money. Assuming Stanford knows that, what are the optimal prices for student tickets and general public tickets it should set in this case? What is the total revenue, and how does it compare to the case without cannibalization? What does this say about the amount that Stanford would be willing to pay for such devices as photo ID cards in order to eliminate cannibalization?
2. An earthquake damages Stanford Stadium so that only 53,000 seats are available for the Big Game. What is the optimal single price and the total revenue? What are the optimal separate prices to charge for students and the general public, and the corresponding total revenue? What is the opportunity cost per seat for the 7,000 unavailable seats in both cases?
3. Ancillary Revenue. Stanford Stadium has been repaired so that it again seats 60,000 people. Now assume that, on average, each member of the general public will consume $20 worth of concessions, resulting in a $10 contribution margin, while each student only consumes $10, resulting in a $5 contribution margin. Assuming no cannibalization, what are the prices for students and the general public that maximize total contribution margin (including, of course, ticket revenue)?
4. A barber charges $12 per haircut and works Saturday through Thursday. He can perform up to 20 haircuts a day. He currently performs an average of 12 haircuts per day during the weekdays (Monday through Thursday). On Saturdays and Sundays, he does 20 haircuts per day and turns 10 potential customers away each day. These customers all go to the competition. The barber is considering raising his prices on weekends. He estimates that for every $1 he raises his price, he will lose an additional 10% of his customer base (including his turnaways). He estimates that 20% of his remaining weekend customers would move to a weekday to save $1, 40% would move to a weekday to save $2, and 60% would move to a weekday to save $3. Assuming he needs to price in increments of $1, should he charge a differential weekend price? If so, what should the weekend price be? (Assume he continues to charge $12 on weekdays.) How much revenue (if any) would he gain from his policy?
5. The optimal variable prices for theme park admission in Table 7.5 are based on the assumption that admission fees are the only source of revenue for the park. However, the owner determines that visitors to the theme park spend an average of $12 per person on concessions, generating an average concession margin of $5 per person. Under the same assumptions about capacity and demand, what is the single-admission price the theme park should charge to maximize total weekly margin (admission price plus concession margin)? What are the individual daily prices he should charge under a variable-pricing policy, assuming independent daily demands? What is the impact on total weekly admissions? What is the impact on total weekly margin from explicitly including concessions in the optimization relative to optimizing prices on the basis of admission revenue alone?
6. Assume that the theme park owner invested in expanding his park so that he could accommodate up to 1,500 customers each day.
a. What single price would maximize his total revenue, assuming he faces independent demands for each day and that the price-response functions per day are as specified in Table 7.4? What is his corresponding attendance and revenue per day?
b. Under the same assumptions, what variable prices should he charge for each day to maximize expected revenue, and what are the corresponding attendance and revenue per day?
7. The theme park owner performs some market research and determines that his customers can best be represented by the following model.
i. Base customer demand for each day of the week is linear and is specified by the parameters in Table 7.4.
ii. Weekend (Saturday and Sunday) customers will switch to the other weekend day at the rate of 10 customers for every $1 difference in price. They will not switch to weekdays at any price.
iii. Weekday customers will switch to any other day (including a weekend day) at the rate of 8 customers for every $1 difference in price.
What are the optimal daily prices the theme park should charge?
8. The popular a cappella choral group Here Comes Treble is coming to Columbia University for a concert. The concert will be in Lerner Auditorium, which has C seats. Customers can be segmented into students holding a student ID card and the general public. The price-response function for the general public is dg(pg) = (2,400 – 60pg)+, and the price-response function for students is ds(ps) = (400 – 25ps)+. (Note: If you cannot do part a, you can assume that C = 1,500 for the following parts.)
a. Columbia plans to sell general admission and student tickets. To find the revenue-maximizing ticket prices, Columbia decides to solve a revenue-maximization problem with the capacity constraint 12pg + 5ps ≥ 320, together with the demand nonnegativity constraints ps ≤ 16 and pg ≤ 40. Given this information, what is the capacity of the auditorium?
b. Given the capacity of the auditorium that you calculated in part a, what is the revenue-maximizing single price?
c. Columbia decides to go through with the plan of charging different prices for general admission and student tickets. Given the capacity of the auditorium you calculated in part a, what are the revenue-maximizing ticket prices for students and the general public?
d. Assume that Columbia reschedules the concert for a larger auditorium with 3,000 seats. What are the revenue-maximizing prices for students and the general public?
9. The popular ride-sharing app Lyber has determined that the general form of rider response to its price multiplier p is given by the logit formula
Here, D is the total number of people in an area who have opened the Lyber app, p is the multiplier and base price, d(p) is the number of people in the area who will order a Lyber if the multiplier is p, and a and b are parameters.
a. Let S be the number of available drivers in the area, and assume that S < D. What is the formula for the multiplier that would exactly balance supply and demand in terms of a, b, S, and D?
b. Let D = 1000, S = 500, a = 10, and b = 5. What value of p would balance supply and demand?
NOTES
1. Note that a seller does not need to be a purist in allocating scarce supply. He can, for example, raise price to less than the market-clearing price and combine that with one of the other allocation approaches. This would allow him to realize some additional profit while still favoring preferred customers.
2. This example is modified from one originally developed by Jeremy Bulow of the Stanford Business School.
3. These rates were in effect on December 12, 2019, according to the Gulf Power rate sheet found at that time through https://www.gulfpower.com.
4. The impact of Michael Jordan on revenues is from Fortune magazine, based on work originally by Jerry Hausman and Gregory Leonard, who concluded that “superstars are more important than league balance for revenues” (1997, 586). Overall, Fortune estimated that Jordan’s impact on home gate receipts was $165.5 million and on road games, $30.5 million. This was an era before dynamic pricing—it is likely that the impact would have been substantially greater with optimal dynamic pricing in which games featuring Jordan would have been priced higher than other games (Johnson 1998).
5. Through the 2017–2018 Broadway season, The Lion King had grossed $1.54 billion over its run, compared to $1.25 billion for second-place Wicked. Of course, The Lion King’s record-setting financial performance is due to more than just variable pricing; it also benefits from an excellent product that appeals to all ages, association with the Disney brand, strong marketing support, and a great location close to Times Square.
6. For the potential benefits of dynamic pricing for electric power, see Imelda et al. 2018.
7. Ride-share pricing is somewhat more complicated because of various fees and incentives. For example, as of 2018 in the United States, both Lyft and Uber charge a $1 booking fee for each trip, which is not shared with the driver. Various services—including both Lyft and Uber—enable riders to tip their drivers. In addition, many operators provide various driver subsidies that can increase driver earnings—for example, an operator may pay a driver an extra $50 for completing at least 20 trips in a week. For this reason, the total compensation for a driver over a period will not necessarily be equal to a fixed fraction of the prices paid by the riders she carried. For base rider pricing, see O’Connell 2018.
8. Ride-sharing is a relatively new industry, and the ride-sharing companies continue to explore new pricing models. For example, in early 2019 Uber began to experiment with “additive surge,” in which prices are modified not by using multiplicative factors but by using “adders.” Under a multiplicative surge of 2.0, a ride with a base fare of $5.00 would have a fare of $10.00 and a ride with a base fare of $15.00 would have a dynamic fare of $30.00. With a $6.00 additive surge, the ride with a base fare of $5.00 would have a fare of $11.00 and the $15.00 base fare ride would have a dynamic fare of $21.00. Garg and Nazerzadeh 2019 provides an argument that additive surge is superior to multiplicative surge, and Uber 2019 gives a description of how it operates in practice.
9. A virtual restaurant is one whose brand appears only online, although it may be tied to a differently named physical restaurant. A ghost kitchen is an establishment that exists only to serve the delivery market—there is no associated dining establishment. Confusingly, ghost kitchens are sometimes referred to as virtual restaurants.