Equilibrium price dispersion under demand uncertainty: the roles of costly capacity and market structure


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1 RAND Journal of Economics Vol. 30, No. 4, Winter Equilibrium rice disersion under demand uncertainty: the roles of costly caacity and market structure James D. Dana, Jr.* When caacity is costly and rices are set in advance, firms facing uncertain demand will sell outut at multile rices and limit the quantity available at each rice. I show that the otimal rice strategy of a monoolist and the unique urestrategy Nash equilibria of oligoolists both exhibit intrafirm rice disersion. Moreover, as the market becomes more cometitive, rices become more disersed, a attern documented in the airline industry. While generating similar redictions, the model differs from the revenue management literature because it disregards market segmentation and fare restrictions that screen customers. 1. Introduction In markets with costly caacity and uncertain demand, firms will not necessarily set the same rice. When firms set their rices before demand is known, they are indifferent between offering units at higher rices with the exectation that these units will sell with a lower robability (when demand is high) and offering units at lower rices with the exectation that these units will sell with higher robability (when demand is high or low). Equilibrium rice disersion with homogeneous goods and erfect cometition was first described by Prescott (1975) in a model of hotel cometition and was develoed more formally by Eden (1990). Prescott s model offers an exlanation for interfirm (and intrafirm) rice disersion in industries such as airlines, automobile rentals, hotels, restaurants, theaters, and sorting events. Furthermore, it has also been the basis of recent alied research on the real effects of monetary shocks (Lucas and Woodford, 1993; Eden, 1994), wage rigidities (Weitzman, 1989), resale rice maintenance (Deneckere, Marvel, and Peck, 1996), stochastic eakload ricing (Dana, 1999b), and advance urchase discounts (Dana, 1998). 1 This article is the first to extend Prescott s model to monooly and imerfect cometition. By exanding firms strategy sets to include rice distributions, i.e., sets * Northwestern University; I would like to thank Severin Borenstein, Robert Gertner, Tom Holmes, Carsten Kowalczyk, Boaz Moselle, Alejandro Manelli, Ed Olmstead, Steve Postrel, Kathryn Sier, Olaf Staffans, and SangSeung Yi for helful comments. I would also like to thank the Editor, Kyle Bagwell, and several anonymous referees. Financial suort from the Rockefeller Grant in Economics at Dartmouth College is gratefully acknowledged. 1 See also Eden (forthcoming) for a broader discussion of alications of the model to monetary economics. 632 Coyright 1999, RAND
2 DANA / 633 of rices and quantity limits at each rice, I show that there exists a unique urestrategy equilibrium in rice distributions even when no urestrategy equilibrium exists in rices (see Bryant, 1980). In other words, the model redicts an equilibrium with intrafirm rice disersion in which each firm offers its outut at multile rices (as oosed to random rices). The oligooly equilibrium is symmetric and the market rice distribution converges to Prescott s cometitive equilibrium as the number of firms aroaches infinity. As cometition increases, the average rice level falls and the degree of rice disersion increases. This result arises for the same reasons that a monoolist tyically does not raise its rice by the entire amount of an increase in costs, while cometitive firms ass through all of their cost increases. This inverse correlation between rice disersion and market concentration has been observed in the airline industry. In articular, Borenstein and Rose (1994) showed emirically that rice disersion is greater on cityair routes that are served by a larger number of carriers. Borenstein and Rose attribute this result to rice discrimination and argue that oint using a monoolisticcometition model with certain demand. 2 However, their emirical results are also consistent with this article s theory that rice disersion is due to caacity costs (i.e., erishable assets) and demand uncertainty. 3 Furthermore, this model is consistent with other characteristics of the airline industry, characteristics that the rice discrimination theory does not address. In articular, caacity utilization rates (load factors) are higher for airlines that charge lower fares. For examle, in 1993 the major airlines whose yields (rices) were below average had an average load factor (caacity utilization) of 64.25% and a yield of cents, while the major airlines whose yields were above average had an average load factor of 59.92% and a yield of cents. 4 This is the natural rediction of a model with costly and erishable caacity, rice rigidities, and demand uncertainty and is difficult to exlain in models that do not have each of these assumtions. 5 Yield management, now commonly called revenue management, is the use of market segmentation and seatinventory control (i.e., assigning limits on the availability of seats at each fare) to maximize firm revenues (or rofits). Sohisticated revenue management systems are now used by airlines, hotels, car rental comanies, commercial shiers, and increasingly in other industries to manage demand uncertainty and to ensure that their roducts and services are available even when demand is high. Because of customer segmenting restrictions (or fencing ) such as advancedurchase discounts, Saturdaynight stayover requirements, nonrefundable urchases, and volume discounts, revenue management is usually considered to be a form of thirddegree rice discrimination. However, in a model in which market segmentation and rice discrimination are not feasible, I show that demand uncertainty and the erishable nature of the assets are alone sufficient to exlain intrafirm rice disersion and may hel exlain 2 They consider a variant of Borenstein s (1985) model where consumers have heterogeneous travel (waiting) costs in a circle location model and roduct variety (dearture times) is fixed while they vary the market structure. However, most models of oligooly rice discrimination, including Borenstein (1985), Holmes (1989), and Gale (1993), do not redict a ositive correlation between rice disersion and market structure. 3 These two theoretical exlanations are not inconsistent with one another. In fact, if the existence of rice disersion for other reasons facilitates airlines use of discriminatory restrictions, then they may be comlementary. 4 These calculations are based on data from Deartment of Transortation Form 441. Yield is defined as total oerating revenue divided by the number of assenger miles flown by aying customers. Load factor is defined as the number of assenger miles flown by aying customers divided by the number of available seat miles. 5 An alternative exlanation of this result is that airlines with greater market ower (and hence higher rices) have excess caacity in order to deter entry. RAND 1999.
3 634 / THE RAND JOURNAL OF ECONOMICS why rice disersion increases with cometition. In the next section I describe more carefully the relationshi of the model to airline revenue management ractices and research. Another alication of the monooly model resented here is theater and stadium ticket ricing. Although in these examles firms roducts are not homogeneous, ticket sellers face the same tradeoff between selling the marginal unit at a low rice for sure and at a high rice with uncertainty. Since rices are clearly set in advance, theaters and stadiums reserve some seats at higher rices (or some lowerquality seats at higher margins) to meet the uncertain demand. The model also alies to retail ricing and inventories. Lossleader models tyically exlain low rices as a baitandswitch tactic, but instead one might think of advertised lowriced roducts as items riced at (or near) cost for which firms then have no incentive to hold seculative inventory. When these items stock out, customers are left to buy similar goods at higher rices, but only because realized demand was high and not because the retailer intended to ration customers that he could rofitably serve at the low rice. An imortant assumtion of the model is that rices are rigid; firms must recommit to a menu or schedule of rices before demand is known. This commitment may arise from advertisements or romotions, or because the firm must hysically rint tickets in advance. Alternatively, it may reflect a strategic commitment on the art of firms not to invest in technologies to make rices more resonsive to demand. Regardless of the reason, rice commitments enforce a rice system in which customers who either arrive early or arrive late when demand is low can buy at lower rices, while customers who arrive late when demand is high ay a higher rice. An individual consumer at the time of urchase may only observe the rice of each firm s leastexensive remaining unit, but ex ost consumers will have aid different rices solely because of the random order in which they were served. Comlete rigidity is much more than I need to assume. It is enough to assume that firms do not immediately change their rices when they receive a new, relevant iece of demand information. In the airline industry, useful information about final demand might be contained only in sales that are very close to dearture time after it is too late to alter rices, and the information might not be revealed until sales are finalized. Allowing firms to change their rices in resonse to changes in cometitors rices, ast sales, market research, or advertising camaigns, or to change the rices for later flights based on information about ast flights sales, would not alter the results. Note also that if firms learn no additional useful information about the demand state during the trading rocess, then the rice rigidity assumtion can be relaxed (see Eden, 1990 and forthcoming). 6 The article adds to earlier work on oligooly models with demand uncertainty. Klemerer and Meyer (1986, 1989) consider models with some market clearing. The latter is unusual because firms strategy sets consist of suly functions that restrict the relationshi between rice and quantity, but these functions are quite different from the rice distributions described here. Deneckere and Peck (1995) consider a more closely related model; however, they assume that customers can visit only one firm and do not allow intrafirm rice disersion. Firms in their model comete in both rice and availability (see also Carlton, 1978; Peters, 1984; and Dana, 1999a). Schmidt 6 This means that during the trading rocess, the conditional robability of any feasible demand state must always be equal to the ex ante robability that the state occurs divided by the ex ante robability that any of the remaining feasible states occur. RAND 1999.
4 DANA / 635 (1991) looks at a model in which demand uncertainty and heterogeneous costs (as oosed to sunk caacity costs) lead to rice disersion. Other monooly models that redict intrafirm rice disersion are Wilson (1988), Salo (1977), and Dana (1992). Wilson shows that with an increasing marginal cost function and a marginal revenue function that is not everywhere nonincreasing, a monoolist may charge at most two rices and ration the availability of the lowerriced units (his rationing rocess is the same as that considered here). Salo shows that a monoolist can oen multile sales outlets and offer different rices at each in order to rice discriminate against consumers with high search costs. Dana considers a different mechanism by which rice rigidities might cause rice disersion. When the monoolist s otimal rice changes with the realization of demand (because the elasticity decreases with realized demand), then the firm might charge more for units that sell only when demand is high. 7 Many other oligooly and cometitive models redict rice disersion as a result of randomization. While some can be reinterreted to redict intrafirm rice disersion, none redict rice disersion as the unique urestrategy equilibrium of an oligooly model. 8 Of articular interest are articles by Varian (1980), Butters (1977), Burdett and Judd (1983), and related articles by Stahl (1989) and Rosenthal (1980). Stahl and Rosenthal find that rices increase and rice disersion decreases as the market becomes more cometitive, in contrast to the result obtained here. The difference is that rice disersion in their models is driven by search and imerfect information (or similar market imerfections) rather than caacity costs and demand uncertainty. Section 2 reviews the ractice of revenue management and its relationshi to the model. Section 3 resents a simle twostate examle to illustrate the contributions of the article. Section 4 describes the basic model, and Section 5 derives the equilibrium ricing under the different market structures of erfect cometition, monooly, and oligooly. Section 6 examines the relationshi between rice disersion and market structure, and concluding remarks are offered in Section Revenue management The term revenue management is commonly used to describe most asects of airlines ricing and seatinventory control decisions; but in reality, revenue managers rimarily ractice seatinventory control. 9 Formally, revenue management describes a rocess of setting fares for each route (origin and destination air) and each set of restrictions (nonsto, timeofday, dayofweek, refundable, advance urchase, first class or coach, and Saturdaynight stayover) and limiting the number of seats available at each fare. In the language of economics, revenue management increases airlines rofits in three ways. First, it imlements eakload ricing. Second, it imlements thirddegree rice discrimination. That is, fare restrictions screen customers and segment them by their sensitivity to rice and otentially by their demand uncertainty. 10 And third, it imlements an inventory control system for coing with uncertain demand for a erishable asset. 7 In this article the elasticity of demand is assumed to deend only on rice and not the realization of the demand state. 8 However, an oligooly version of Salo s (1977) model with differentiated roducts would. 9 See Weatherford and Bodily (1992) for an overview of research on revenue management. 10 Dana (1998) shows that cometitive firms will use advanceurchase rice discounts to screen for customers with more certain demands (because they contribute less to aggregate demand uncertainty), and Courty and Li (1998) show that a monoolist may use control over refunds to screen for customers with more certain demands. RAND 1999.
5 636 / THE RAND JOURNAL OF ECONOMICS Like most of the literature on seatinventory control, this article addresses only this last role of revenue management. Peakload ricing and rice discrimination are, after all, rimarily ricing roblems. 11 If demand were known, then rices alone could be used to accomlish these objectives and there would be no reason to use seatinventory control. I focus instead on how firms can rofit from offering multile fares and limiting the availability of seats at low fares when they cannot segment the market. In contrast to the traditional oerations literature on revenue management, I show that market segmentation is not necessary for revenue management to increase rofits. Demand uncertainty, costly caacity (erishable assets), and rice rigidities are sufficient to exlain the use of seatinventory control. This article differs from most research on revenue management (seatinventory control) in several ways. First, I simultaneously determine rices and inventory levels; the revenue management literature treats fares as exogenous. 12 Second, I exlicitly consider strategic interactions between firms, while the revenue management literature tyically considers singlefirm models. 13 Third, because I ignore market segmentation, I am allowing consumers demands to be diverted to the nexthighest fare when the lower fare stocks out; highwillingnesstoay consumers first try to buy at low rices, but when low fares are no longer available, they buy at higher rices. An acknowledged shortcoming of the revenue management literature is that it tyically treats market segments as indeendent, both in the sense that demand is not diverted and in the sense that demands are stochastically indeendent. 14 But since airlines do in fact segment markets, the advantages here of studying a seatinventory control model with endogenous ricing, strategic interaction, and diversion must be weighed against the cost of ignoring market segmentation. There are other otential roblems in alying this model to the airline industry. I assume that firms irrevocably commit themselves to distributions of rices and seat inventories for each flight before learning demand. At many (but not all) airlines, revenue managers use modern comuter technology to calculate and adjust seatinventory allocations on a daily basis. Price changes (while much less frequent) can also be made quickly. To understand the imortance of these issues for the model, we have to look beyond a theoretical discussion of revenue management. In ractice, revenue management has traditionally described several searate functions, if not searate deartments, within an airline s organization. First is the collection of historical (and contemorary) sales data used to generate forecasts. Second is fare setting. That grou determines the restrictions that assengers must meet and the rices that tickets will sell for. In ractice, fares aly to many flights, and any limits on dearture dates or times are secified as restrictions on the fare. These deartments closely monitor cometitors fares on comuter reservation systems and quickly match any of their rice changes. Third is a otentially comuterized system that determines, 11 In ractice, seatinventory control often substitutes for otimal eakload ricing. Airlines may set the same fares for flights with different forecast demands and rely on their revenue management comuters to more severely limit the inventory of low fares on the highdemand flight, accomlishing more easily (because it is automated) the goal of charging higher fares for more oular flights. 12 To date, otimizing models of both rice and seat inventory are an academic idea. For an early effort in which rices are chosen otimally given seatinventory levels and a discussion of aroaches to the roblem in which both are chosen simultaneously, see Weatherford (1997). 13 Belobaba and Wilson (1997) examine seatinventory control in a strategic setting, but like most work in the revenue management literature, they treat fares as exogenous. 14 Models of seatinventory control with diversion have recently been considered by Belobaba and Weatherford (1996) and Bodily and Weatherford (1995) and are research riorities for the next generation of revenue management systems (see Hoerstad, 1998). RAND 1999.
6 DANA / 637 given demand forecasts and fares, the otimal limit on the number of seats sold at each fare and then transmits that information to a comuter reservation system. Although now more than ten years old, Belobaba (1987) reorts the results of a survey of airlines revenue management (seatinventory control) ractices in late This is the only systematic look at airline ractices of which I am aware, and more imortant, it was done only months before the rice data analyzed by Borenstein and Rose was generated. In 1985, many airlines were just beginning to store and manage the enormous amount of historical sales data generated by reservations systems. Before 1985, forecasting was so oor that some airlines were still using a common ercentage booking limit for discount seats on all their flights, regardless of the route or season. In 1985, all revenue management systems, whether comuterized or not, chose seat inventories for each fare category using static otimizing models, as if they were making a onceandforall decision. Today some comuterized revenue management systems dynamically otimize booking limits before a flight dearts using udated demand forecasts that incororate information about actual bookings. However, in 1985, static booking limits could be changed only by an exerienced revenue manager, using his or her judgment, whenever the comuter reservation system, which automatically monitored bookings, brought to his or her attention a booking limit that was about to be reached. Thus booking limits were often relaxed (resumably whenever early bookings at the full fare were below rojections) but rarely tightened in resonse to actual bookings. While some airlines combine ricing and seatinventory control in one deartment, initiatives to change rices are usually made by marketing ersonnel, often with surrisingly little information. In many cases, ricing decisions do not even make use of historical sales information. Even if rices are otimal given the firm s best forecasts of demand (as I assume in my model), ricing deartments have no systematic method for using actual bookings to revise rices for the remaining seats on that flight. In 1985, seatinventory control relied on the assumtion that rices would not change during the final six weeks before a flight dearture, the eriod in which the bulk of airline reservations are made (Belobaba, 1987). The subsequent develoment and adotion of better tools for demand forecasting and comuterized dynamic seatinventory control has no doubt changed airline cometition significantly. However, the oneshot selection of rices and quantities early in the history of revenue management does seem to closely mirror the ricing assumtion in the model resented here. It is not clear whether airlines today are currently trying to systematically make ricing decisions a function of actual bookings, but there is certainly little evidence that they did so in the mid1980s. Though the model has obvious limits, it is nevertheless consistent with stylized facts about airlines. In articular, caacity utilization rates are higher for seatinventory allocations of lowfare seats. While information about caacity utilization by fare within an airline is rorietary (because seatinventory control information is rorietary), this is a direct consequence of the algorithms used by their revenue management comuters. Also, as mentioned earlier, airlines with lower fares tend to have higher load factors. 15 In early 1985, American Airlines introduced the first modern comuterized revenue management system that combined demand forecasting and otimizing algorithms for seatinventory control. That system has been credited with allowing American Airlines to rofitably match Peole s Exress fares across the board, leading to the raid demise of that airline (which lacked any information technology for forecasting demand or racticing revenue management). In 1985, industry analysts clearly did not understand the otential of American Airlines new system. RAND 1999.
7 638 / THE RAND JOURNAL OF ECONOMICS 3. A twodemandstate examle The intuition for much of the article can be seen in the following numerical examle with two demand states. Suose that consumers reservation values for homogeneous stadium seats are uniformly distributed on [0, 100] and that either 100 or 400 consumers will demand seats (with equal robability). So the inverse demand is P L (Q L ) 100 Q L if demand is low and P H (Q H ) Q H if demand is high. Also suose the marginal cost of offering a seat in the stadium is $20. (This may be interreted as either the marginal cost of stocking inventory or as the shadow cost of caacity.) Firms are required to rint tickets with assigned rices before learning demand. Consumers arrive in random order and choose among the unsold tickets, but since the good is homogeneous, consumers will always refer the cheaest remaining ticket. First suose that stadium seats are rovided by a erfectly cometitive market (Prescott s (1975) model). Secifically, imagine that each firm offers (and rices) a small number of the stadium s seats, taking the aggregate distribution of rices and quantities as given. In the unique cometitive equilibrium, 80 tickets are offered at $20 and 180 tickets are offered at $40. In the lowdemand state, exactly 80 consumers are willing to send $20 er ticket and all 80 are served. (No remaining consumer is willing to send $40, so the exensive tickets go unsold.) In the highdemand state, 320 consumers are willing to ay $20 er ticket but only 80 of them are lucky enough to get a $20 seat. Of the remaining 260 consumers, only 180 are willing to ay $40 for a seat (this is because of the roortional rationing assumtion) and all of these consumers are served. So 80 tickets at $20 each are urchased whether demand is high or low, while 180 tickets at $40 each are urchased only if demand is high. The exected revenue on tickets at either rice is $20, which is equal to the marginal cost of roviding a seat. It may seem odd at first that the unique cometitive equilibrium has disersed rices. Why doesn t a uniform rice equilibrium exist? To see why, suose that all of the seats are offered at a uniform rice of $30 each. The zerorofit condition imlies that the ex ante robability of sale of each seat is 2 3, so in equilibrium firms offer 210 seats (because ( 1 2 )( ) ( 1 2 )( ) 2 3 ). But this cannot be an equilibrium, because a firm could increase its rofit either by offering a seat for $29 or by offering one for $50. A $29 seat would be the first seat to sell in either demand state, would be assured of selling, and would generate a rofit of $9. A $50 seat would sell in the highdemand state (there is strictly ositive residual demand because the demand at $30 is 280, which exceeds the 210 seats available) but not in the lowdemand state, so it would earn an exected rofit of $5. Similar logic establishes that no cometitive equilibrium exists at any other uniform rice either. The cometitive equilibrium is shown grahically in Figure 1. In our discussion above, the exected revenue on each seat is equal to $20, the marginal cost. Equivalently, we can think of firms as setting the rice of a ticket equal to the marginal cost of a seat divided by the robability of sale. The grah in Figure 1 deicts the cometitive equilibrium by doubling the marginal cost curve rather than halving the demand curve. To see that this is a cometitive equilibrium, notice that exected revenue is less than or equal to marginal cost at every rice (and exactly equal to marginal cost at $20 and $40 where firms suly is ositive). At rices below $20, rofit is clearly negative; at rices between $20 and $40, the exected revenue is less than $20 because the robability of sale is still onehalf; and at rices greater than $40, exected revenue is zero. RAND 1999.
8 DANA / 639 FIGURE 1 COMPETITIVE PRICING WITH TWO DEMAND STATES If the stadium seats are rovided by a monoolist, then 40 tickets will be offered at a rice of $60 and 90 tickets at a rice of $ To see why, imagine that the monoolist does indeed rint 40 tickets with a rice of $60. This would be the otimal strategy if demand were P L (Q L ) 100 Q L with certainty because MR 100 2Q L 20 MC, as shown in Figure 2. Taking these 40 chea seats as given, it makes sense for the monoolist to offer additional seating; for examle, one more seat with a rice of $90 would sell half the time, generating an extra exected rofit of $25. The question is, how many additional seats should the monoolist offer, and at what rice? To answer this question, it is necessary to derive the residual demand curve when demand is high. Since there are 40 tickets at $60 each for sale, when demand is high only ¼ of the 160 customers who are willing to ay $60 are lucky enough to get a chea ticket. As shown in Figure 3, the residual demand curve is RD H (P H ) (¾)(400 P H ) for rices above $ The monoolist s rule is to set (exected) marginal revenue equal to marginal cost. But since high demand occurs only half the time, the exected marginal revenue is half what it would be if the demand were certain. As in the cometitive case, Figure 3 grahically deicts the effect of demand uncertainty by doubling the firm s marginal cost rather than halving marginal revenue. The monoolist maximizes rofits by offering an additional 90 tickets at a rice of $70 er seat. 18 This is, in fact, the otimal ricing strategy for the monoolist. The monoolist does not want to ration the availability of its tickets in any other way, nor adjust the rice of the lowriced seats (which would shift the residual demand when demand is high), nor offer rice distributions with more than two rices. 19 The monoolist sets 16 Equivalently, if the monoolist has a caacity of 130 seats, then its shadow cost of caacity is $20 and its rice strategy is the same. 17 The inverse residual demand is P H (Q H ) 100 (⅓)Q H. For rices below $60, the residual demand curve corresonds to the original demand curve because consumers would choose to buy any lessexensive units first. 18 E[MR] (½)(100 (⅔)q H ) 20 MC. 19 While the results of the article are stated for continuous distributions of demand uncertainty, this distribution of rices is the limit of the otimal rice distribution when the distribution of demand uncertainty converges to the discrete distribution. RAND 1999.
9 640 / THE RAND JOURNAL OF ECONOMICS FIGURE 2 MONOPOLY PRICING WITH TWO DEMAND STATES: PRICING LOWPRICED UNITS rices as if there were two searate segments, some consumers who always show u (those who buy in the lowdemand state and an equal number who buy in the highdemand state) and some who may or may not show u (the residual demand in the highdemand state). If the monoolist could offer only a single rice, it would charge $66 and earn an exected rofit of $2,880 versus a rofit of $3,000 with the tworice strategy. This examle hels illustrate my main oints. First, nonrandom rice disersion is redicted in a simle model with uncertain market demand, costly or constrained caacity, and rice commitments. In this examle, rice disersion arises for both erfect cometition and monooly, and later in the article I show that rice disersion FIGURE 3 MONOPOLY PRICING WITH TWO DEMAND STATES: PRICING HIGHPRICED UNITS RAND 1999.
10 DANA / 641 arises for oligoolies, at least when there is sufficient demand uncertainty and highenough caacity costs. The reasoning is straightforward: firms require a higher marginal revenue to hold inventories of seats that are exected to sell with lower robability. Second, the model redicts intrafirm rice disersion. Although the cometitive equilibrium may be interreted as exhibiting either interfirm rice disersion or intrafirm rice disersion, the monooly outcome exhibits only intrafirm rice disersion, and later in the article I show that the unique oligooly outcome exhibits intrafirm rice disersion as well. Finally, the examle illustrates that the degree of rice disersion is greater in more cometitive market structures, as we can see here by comaring the monooly and cometitive rices. Demand uncertainty reduces the exected marginal revenue of a sale by a fraction equal to the robability that the good will go unsold, so setting exected marginal revenue equal to marginal cost is analogous to setting marginal revenue equal to marginal cost divided by the robability of sale. As the robability of sale dros, rices rise to reflect firms lower exected marginal revenue or equivalently higher marginal costs. But because a monoolist usually absorbs art of its cost increases, while cometitive firms do not, the monoolist rices are comressed, or less disersed, relative to the cometitive ones. The marku of rice over the breakeven rice is decreasing in the breakeven rice, so the monooly rices are less disersed than the breakeven (or cometitive) rices. 20 Another useful benchmark for comarison is the marketclearing rice. If rices cleared the market, cometitive firms would offer 240 seats and sell them for $0 when demand was low and $40 when demand was high. The monoolist would offer 120 seats and sell them for $50 in the lowdemand state and $70 in the highdemand state. Note that in this examle, because the uncertainty about demand is so large, the monoolist does not sell all its caacity in the lowdemand state, but this is not imortant for the inferences being drawn from the examle. Clearly rices are less volatile for a monoolist than for cometitive firms. The reason is the same. A monoolist facing linear demand absorbs art of the increase in its cost (in this case, the shadow cost of caacity is increasing from $0 to $40). But while rices here are volatile, there is no rice disersion when rices clear the market; in each demand state all consumers ay the same rice. Before turning to the continuousdemand uncertainty model, it is useful to see why a urestrategy equilibrium does not exist for the case of imerfect cometition in this discrete examle. Imagine a duooly in which each firm is endowed with 65 seats, so the industry caacity is 130 seats, the same as the monoolist s equilibrium choice of caacity. This is clearly not the caacity choice that duoolists would choose in equilibrium, but it illustrates why no urestrategy equilibrium exists. One might exect that the equilibrium of the duooly game is where each firm offers 20 tickets at $60 and 45 tickets at $70, the same distribution as a monoolist. But this is not an equilibrium. Either firm could lower the rice of all 20 of its tickets at $60 and of 20 of its 45 tickets at $70 to a new rice of $60. At that rice it would cature all the sales in the lowdemand state and still sell all its remaining 25 tickets at $70 in the highdemand state. This increases the firm s rofits dramatically because the exected revenue on a $60 seat is much higher than on a $70 seat ($60 versus $35 because the 20 Here demand is linear, so a monoolist does not ass through all its cost increases. In general, a monoolist will absorb art of a cost increase as long as the elasticity of demand increases sufficiently fast with rice. Later in the article, I assume that the demand curve has a finite rice intercet that is sufficient to guarantee that the uer bound of rices is the same for all market structures; this also imlies that the elasticity of demand increases with rice (in a neighborhood of the intercet). RAND 1999.
11 642 / THE RAND JOURNAL OF ECONOMICS lowerriced tickets are sold twice as often). The rice cut increases rofits at the exense of the other firm (consumers are better off) by causing the cometitor s $60 seats to be sold only in the highdemand state. It is not hard to show that no urestrategy equilibrium of this game exists. 4. The model In this section I describe the basic model. Demand is arameterized by rice,, and a random state variable, e, drawn from a continuously differentiable cumulative robability distribution, F(e), with strictly ositive and bounded robability density, f (e), on a suort [e, e ]. Continuity of F(e) is necessary only for the oligooly model. For the cometition and monooly models I also briefly discuss the concetually simler case where e has a discrete distribution with ossible values {e 1, e 2,...,e m } and associated robabilities { f 1, f 2,..., f m }. For simlicity, I consider the case in which demand is multilicatively searable, e.g., D(, e) g(e)d( ), which without loss of generality can be written D(, e) ed( ). 21 I also assume the inverse demand function has a finite intercet, inf{ D( ) 0}, which is clearly the same for all e. Without this assumtion the model would redict infinite rices. I assume that D() is twice continuously differentiable for all [0, ). Furthermore, there are no income effects, and for all [0, ), D () 0 and D D 2(D ) 2 0, where the derivatives at are defined aroriately using limits. The second condition is identical to the standard assumtion that the marginal revenue function is strictly decreasing in outut. The elasticity of demand, () D ()/D(), is indeendent of e because demand is multilicatively searable. Let m (c) denote the monooly rice for a firm with known demand, ed(), and constant marginal cost, c, and note that both m (c) and the marginal revenue function are also indeendent of e. I assume that demand uncertainty has f ull suort, by which I mean that demand is zero in the lowest demand state, or D(c, e) 0. For multilicatively searable demand this clearly imlies e 0. This assumtion can be relaxed for the monooly or erfect cometition cases. But in the oligooly case this assumtion, like the assumtion that F(e) is continuous, eliminates mass oints in the firstorder conditions and is necessary for the existence of a urestrategy equilibrium. Production exhibits constant returns to scale. Firms have two costs: a strictly ositive unit cost (or shadow cost) of caacity,, incurred on all units, and a unit marginal cost of roduction, c, incurred only on the units that are sold. I also assume c. Firms strategies are distributions of rices and quantities. 22 The most general way to reresent the set of such distributions would be the set of nonnegative measures on the ositive real numbers, where the measure over the entire range of rices would equal the firm s total outut or caacity. I have considered only those distributions that can be reresented by a Lebesgue integrable density, q i ( ), that is strictly ositive on some finite domain, [, ], and zero everywhere else, and a finite number of mass oints. To simlify the exosition, however, I maintain throughout the assumtion that 21 The class of multilicatively searable demand functions is a secial case of a more general set of demand functions, D(, e), D: 2, which are continuous, nonincreasing in (strictly decreasing when D(, e) 0), and nondecreasing in e (strictly increasing when D(, e) 0); however, no closedform solution exists for the general class of demand functions (see Dana, 1996). 22 The assumtion that rices and quantities do not deend on the state of demand, either directly or indirectly through the attern of sales, is imlicit. Other forward contracts, such as riority service ricing contracts considered by Harris and Raviv (1981), Chao and Wilson (1987), Wilson (1989), and Sulber (1992), are also ruled out. RAND 1999.
12 DANA / 643 i there are no mass oints. 23 Let q( ) q i ( ) denote the market rice distribution, equal to the sum of the individual firms rice distributions, and let Q( ) Q i i ( ) denote the cumulative market rice distribution where Q i ( ) # q i ( ) d. In the roositions below a rice distribution is said to be unique when the cumulative market rice distribution is the same for all equilibrium values of q( ). Because rices do not adjust to clear the market, I must secify how goods will be rationed in equilibrium. I imagine that outut is sold sequentially in order of rice, with the cheaest units going to the consumers who buy first, so the rationing rule determines the order in which consumers demands are met. Note that the rationing rule describes how aggregate demand is rationed and does not deend on individual firms rice distributions; to describe the rationing rocess it is sufficient to consider Prescott s model where infinitesimal firms each charge a single, heterogeneous rice. The two most commonly used rationing rules are roortional rationing and arallel rationing. 24 I assume roortional rationing that is consistent with homogeneous consumers with elastic demands queuing in random order and urchasing according to their demand at the available rices. It is also consistent with heterogeneous consumers with unit demands queuing in random order and urchasing as long as their valuation exceeds the rice of the available units. The residual demand is the quantity demanded at rice given that all the available units with rices less than have already been sold. Under roortional rationing, the residual demand curve for in state e, given D(, e) 0, is max{rd(, e;, q), 0}, where D(r, e) D(, e) RD(, e;, q) D(, e) q(r) dr. (1) Residual demand is further defined to be zero when D(, e) 0. The intuition for equation (1) is easiest to see in the case of a discrete rice distribution. Suose there are Q units sold at a rice. If Q D(, e), then there is excess suly in state e, and the residual demand is zero at any rice. However, if Q D(, e), then the residual demand at any rice is D(, e) QD(, e)/d(, e), or equivalently D(, e)(1 Q/D(, e)). The robability that a customer who is willing to ay actually receives one of the Q available units is Q/D(, e), and the fraction of the customers willing to ay at least whose demand is unsatisfied at rice is 1 Q/D(, e). Note that this robability is indeendent of, so 1 Q/D(, e) is also the fraction of the customers willing to ay at least whose demand is unsatisfied. Equation (1) defines the analogous residual demand function for more general rice distributions, q( ). The fraction of the demand that can be satisfied by a single unit sold at rice r is 1/D(r, e). Since we are interested in the residual demand at rice, D(, e)/d(r, e) is the fraction of the unit (or units) riced at r that are sold to consumers who would have been willing to ay the rice. Note that because roortional rationing does not change the relative roortion of consumers with different demands, this roortion can be calculated using the original demand curve instead of the residual demand curve (at rice given that all of the units below r have already been sold). 23 The more general roofs are in an earlier version of the article, available from the author. 24 The terms roortional and arallel rationing take their names from the effect that each rule has on the shae of the residual demand curve (see Beckman, 1965; Davidson and Deneckere, 1986). Parallel rationing is sometimes called efficient rationing because it maximizes consumer surlus, but arallel rationing is actually very inefficient in this context. Because rices are disersed, the rationing rule affects which customers are rationed not only when there is excess demand but also when there is excess suly. RAND 1999.
13 644 / THE RAND JOURNAL OF ECONOMICS The residual demand function determines the number of units sold in each state. Suose that in state e all the units riced below have already been sold. If RD(, e;, q) 0, then there are still consumers who are willing to ay or more for the good. I define the marketclearing rice, (e, q), as the rice of the highestriced unit that sells in state e, although markets do not clear in the traditional sense. So RD((e, q), e; (e, q), q) 0 imlicitly defines the marketclearing rice (e, q) as a function of e given the rice distribution q. From (1), using the fact that demand is multilicatively searable, we have [ ] q(r) RD(, e;, q) ed( ) 1 dr. (2) ed(r) So, RD(, e;, q) 0 imlies that either ed( ) 0or1 # (q(r)/[ed(r)]) dr 0. Clearly, ed( ) 0 only if or e 0. Hence, for all e 0, either the market clearing rice (e, q) is defined imlicitly by 1 # (q(r)/[ed(r)]) dr 0 or all the available units sell and there is excess demand at. I define (, q) as the minimum demand state in which all the units offered at rices and below are sold, also using RD(, e;, q) 0. So (, q) is uniquely defined for all [, ]by q(r) (, q) dr (3) D(r) and is continuous and weakly increasing. Equation (3) can also be used to define (e, q) for all e (, q). Without loss of generality,, so I define (e, q) for all e (, q), since any unit riced at or less would be sold in those states. So (e, q) is uniquely defined for all e in [0, e ] and is strictly increasing on [0, (, q)]. Note that if q( ) 0 on some subset [ 1, 2 ]of(, ), then (, q) is constant on [ 1, 2 ] and (e, q) is discontinuous at ( 1, q). However, we shall see that in equilibrium q( ) is strictly ositive on (, ). 5. Equilibrium ricing Perfect cometition. Imagine that cometitive firms choose how much to sell and at what rice (or rices), and that consumers, arriving in random order, costlessly observe the lowest rices still available from each firm and buy from the firm (or firms) offering the lowest rices. Since the market does not clear in the traditional sense, it is necessary to begin with a definition of a cometitive equilibrium in this context. Definition. A market rice distribution q*( ), with an associated robability of sale function y*( ), is a cometitive equilibrium if (1) there exist functions q i such that (i) i, q i maximizes firm i s rofit given y*( ) and (ii) q* q i i, and (2) y*( ) is equal to the robability that any unit of outut offered at a rice will sell given the market rice distribution q*( ). In the standard Walrasian model of erfect cometition, we assume that each firm chooses outut to maximize its rofit, taking the equilibrium rice as given. Here, since there are many rices and quantities, I have generalized the Walrasian assumtion. I assume that firms believe they are unable to affect the terms of trade facing other firms, RAND 1999.
14 DANA / 645 that is, the set of ossible rices and the associated robabilities of sale. This is equivalent to assuming that each firm chooses its rice distribution believing that it has no effect on the aggregate rice distribution. In a cometitive equilibrium, the rofit from offering a single unit must be nonositive at each rice. Otherwise, because of the constantreturnstoscale assumtion, any firm could increase its rofits by offering additional units at that rice. The robability of sale for a unit at rice given the market rice distribution q( ) is y(, q) 1 F((, q)), so a firm that offers units at rice will earn an exected rofit ( c)y(, q) er unit of caacity. In a cometitive equilibrium, ( c)y(, q) 0 for all [, ], and ( c)y(, q) 0 for all [, ], so 25 RAND c (4) 1 F((, q)) for all [, ], and c /[1 F((, q))] for all [, ]. 26 Any rice that is offered in equilibrium must be equal to marginal cost lus the unit cost of caacity divided by the robability that a unit offered at that rice will be sold. The latter term can be interreted as an effective cost of caacity, since it is equal to the revenue (net of marginal cost) that the firm must earn if the good is sold in order to cover the unit caacity cost it incurs whether or not the good is sold. Of course, this definition of cost is somewhat ad hoc, since it imlies that cost is endogenous and deends on the rice chosen by the firm. An alternative interretation of the zerorofit condition is simly that the exected net revenue er unit of caacity, ( c)y(, q), must equal the marginal cost of caacity for all rices. In the cometitive equilibrium, as in other models of rice disersion (such as Butters (1977) and Burdett and Judd (1983)), firms are indifferent between setting a higher rice with a smaller robability of sale and setting a lower rice with a higher robability of sale. If instead of having a direct cost of caacity,, firms had a binding industry caacity constraint of K, then the cometitive equilibrium would still be characterized by (4); however, in this case would denote the shadow cost of caacity and be endogenously determined. If the caacity constraint is equal to the level of caacity that would have been chosen if the direct cost of caacity were, then the shadow cost of caacity is, and the equilibrium rice distributions are the same. This imlies that the equilibrium rice distribution is the same whether firms choose quantity and rice sequentially or simultaneously. Proosition 1 (Prescott). If caacity is either costly or constrained, the cometitive equilibrium rice distribution is strictly disersed and is uniquely defined by on the suort [c, ]. ( c) q ( ) D( ) 2 c (5) ff1 1 c [ ] 25 This clearly demonstrates that the cometitive equilibrium cannot contain mass oints; the condition cannot hold for all [, ]ifv(, q) is discontinuous at any oint k [, ]. 26 Equation (4) is the firstorder condition obtained from a erfectly cometitive firm s rofit maximization. To see this, consider the oligoolist s rofit function, equation (11) below, with (e, q) and (, q) fixed.
15 646 / THE RAND JOURNAL OF ECONOMICS Proof. See the Aendix. Similarly, if caacity is either costly or constrained, and there are m discrete demand states, the cometitive equilibrium rice distribution is strictly disersed and is uniquely defined by and RAND c i m f j ji i1 q c,j ] j1 i j [ qc,i ed( i i )1. ed( ) Proosition 1 generalizes Prescott (1975) (and Eden, 1990) by considering a continuum of demand states. Note also that many different equilibrium rice distributions for individual firms yield the same aggregate equilibrium rice distribution. It is natural to suose that each firm sets a single rice, as Prescott did, in which case firms secialize in different rices and in serving different market niches. In some of the examles suggested earlier there does aear to be some rice secialization, e.g., discount hotels and discount car rentals, although those firms roducts are often differentiated in ways other than rice. However, Proosition 1 is equally consistent with the interretation that firms choose symmetric rice distributions, q c ( )/n. This distinction is imortant because I find that in the oligooly model, only a symmetric equilibrium exists. That is, imerfect cometition imlies intrafirm rice disersion and not secialization in rices. Three other observations are worth noting. First, there is excess demand when the demand state is between (, q) and e, so stockouts do occur. If not, then there would be some unit that was sold only in state e, but the rofit on that unit, ( c)(1 F( e )), is necessarily negative because the unit would be sold with robability zero. Second, the cometitive equilibrium rice distribution is not continuous in at 0. As 0 the suort of the rice distribution is actually increasing, even though the mean of the distribution is converging to c. At 0, the suort of the distribution collases to the oint c, and the equilibrium is the standard cometitive equilibrium with a uniform rice, c. Third, the cometitive equilibrium is not Pareto efficient. There are clearly deadweight losses that arise in each state of nature. This is manifested in two ways. First, goods may not be allocated to the consumers who value them most (because of the roortional rationing rule), and second, caacity may not be fully utilized even when there are consumers whose willingness to ay exceeds marginal cost. Of course, it may not be relevant to comare the equilibrium here to one that might arise if comlete forward contracts existed, since any ricing olicy imlemented through regulation would undoubtedly be subject to most of the same rice rigidities and rationing as in the unregulated marketlace.
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