NBER WORKING PAPER SERIES PRICE DISPERSION UNDER COSTLY CAPACITY AND DEMAND UNCERTAINTY. Diego Escobari Li Gan

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1 BER WORKIG PAPER SERIES PRICE DISPERSIO UDER COSTLY CAPACITY AD DEMAD UCERTAITY Diego Escobari Li Gan Woring Paer 375 htt:// ATIOAL BUREAU OF ECOOMIC RESEARCH 5 Massachusetts Avenue Cambridge, MA 238 May 27 We than the valuable comments from Volodymyr Bilotach, James Dana, Benjamin Eden, Paan Jindaon, Eugenio Miravete, Carlos Oyarzun, Claudio Piga, Steven Puller, Ximing Wu, and seminar articiants at the Deartment of Economics, Texas A&M University, the Texas Econometrics Cam, and the International Industrial Organization Conference at Savannah, Geogia. Stehanie Reynolds rovided caable research assistance in the collection of the data. Financial suort from the Private Enterrise Research Center at the Texas A&M University and the Bradley Foundation is gratefully areciated. The usual disclaimer alies. The views exressed herein are those of the author(s) and do not necessarily reflect the views of the ational Bureau of Economic Research. 27 by Diego Escobari and Li Gan. All rights reserved. Short sections of text, not to exceed two aragrahs, may be quoted without exlicit ermission rovided that full credit, including notice, is given to the source.

2 Price Disersion under Costly Caacity and Demand Uncertainty Diego Escobari and Li Gan BER Woring Paer o. 375 May 27 JEL o. C23,L ABSTRACT This aer tests the emirical imortance of the rice disersion redictions of the Prescott-Eden-Dana (PED) models. Equilibrium rice disersion is derived in a setting with costly caacity and demand uncertainty where different fares can be exlained by the different selling robabilities. The PED models redict that a lower selling robability leads to a higher rice. Moreover, this effect is larger in more cometitive marets. Desite its alications to several imortant maret henomena, there exists little emirical evidence suorting the PED models, mostly because of the difficulty of coming u with an aroriate measure of the selling robabilities. Using a unique anel of U.S. airline fares and seat inventories, we find evidence that strongly suorts both redictions of the models. After controlling for the effect of aggregate demand uncertainty on fares, we also obtain evidence of second degree rice discrimination in the form of advance-urchase discounts. Diego Escobari Deartment of Economics Texas A&M University College Station, TX escobari@tamu.edu Li Gan Deartment of Economics Texas A&M University College Station, TX and BER gan@econmail.tamu.edu

3 I. Introduction It is widely observed that rices of homogeneous goods within the same maret exhibit rice disersion. Some of the most recent evidence includes retail rices for rescrition drugs in Sorensen (2), and internet electronic equiment marets in Baye and Morgan (24). Various models, including search frictions, information asymmetries, and bounded rationality, have been roosed to exlain this henomenon. Here we see to establish the emirical imortance of the rice disersion redictions in the Prescott (975), Eden (99) and Dana s (999b) models. Prescott (975) considers an examle of hotel rooms where sellers set rices before they now the number of buyers, then the equilibrium rices will be disersed; lower-riced units will sell with higher robability, while higher-riced units will sell with lower robability. Hence, sellers face a tradeoff between rice and the robability of maing a sell. This same tradeoff is observed in Eden (99), who formalizes Prescott s model in a setting where consumers arrive sequentially, observe all offers and after buying the cheaest available offer they leave the maret. He derives an equilibrium that exhibits rice disersion even when sellers are allowed to change their rices during trade and have no monooly ower. This flexible rice version of the Prescott model, develoed in Eden (99, 25a) and Lucas and Woodford (993), is nown as the Uncertain and Sequential Trade (UST) model. Dana (999b) extends the Prescott model with rice commitments for erfect cometition, monooly, and oligooly and shows that firms offer outut at multile rices. In the oligooly equilibrium, the maret distribution of rices converges to the Prescott s distribution as the number of firms aroaches to infinity. Moreover, as cometition is greater, average rice level falls and rice disersion increases. As exlained in Eden (25b), from the ositive economics oint of view it does not matter whether rices in the Prescott s model flexible of rigid. From the oint of view of the seller and this aer, both will have the same resulting allocation. In this aer, both the flexible and the rigid version of the model are commonly referred as Prescott-Eden-Dana (PED hereafter) models. Versions of the PED model have been alied to solve a variety of economic henomena, such as wage disersion and maret segmentation (Weitzman, 989), rocyclical roductivity (Rotemberg and Summers 99), the role of inventories (Bental and Eden 993), real effect of monetary shocs (Lucas and Woodford, 993; Eden, 994), destructive cometition in retail marets (Denecere, Marvel, and Pec 997), advance urchase discounts (Dana 998), stochastic ea-load ricing (Dana 999a), gains from trade (Eden 25) and seigniorage ayments (Eden 27). Desite its wide alications, few aers test the emirical redictions of the PED models. 2

4 This aer rovides a formal test of the PED models while heling to exlaining rice disersion in the airline industry, which is considered to have one of the most comlex ricing systems in the world. We tae advantage of a unique U.S. airlines anel disaggregated at assenger level that contains the evolution of fares and inventories of seats over a eriod of 3 days for 228 domestic flights dearting on June 22 nd, 26. The data collection resembles exerimental data which controls for most of the roduct heterogeneities observed in the industry. This reresents the erfect control for fences that segment the maret allowing our analysis to exlain the use of seat-inventory control just under demand uncertainty, costly caacity and rice commitments. Moreover, airlines reresent the erfect environment to test the rice disersion under demand uncertainty and costly caacity. First, air ticets exire at a oint in time; once the lane dearts carriers can no longer sell ticets. Second, caacity is fixed and can only be augmented at a relatively high marginal cost. Once carriers start selling ticets they are unliely to change the size of the aircraft. 2 This imlies that we can focus on the demand side uncertainty without having to worry about any uncertainty in the suly given our time frame of study. Moreover, as in the PED models, after we control for ticet restrictions that screen costumers, all airlane seats are the same and buyers have unit demands. In order to exlain rice disersion we enlarge the definition of airlane seats by an additional selling robability dimension. Once this is achieved, although rices themselves may be disersed, this disersion can be exlained by aroriately rescaling the rice of each unit by its selling robability. At the ris of over-maing this oint, consider the following examle of a erfectly cometitive maret with zero rofits. Each time a carrier sells a seat, the exected marginal revenue is set to be equal to the marginal cost. Because of demand uncertainty, airlines hold inventories of seats that are sold only some of the times. For those seats that are sold only when demand is high, fares must be set higher to comensate for the lower robability of sale. In this aer we develo a measure of the different selling robabilities. Even though uncertainty is coming from the demand side, we follow the PED models and reresent this by adjusting the marginal cost of caacity, or ex ante shadow cost, by these selling robabilities. By dividing the constant unit cost of caacity by the robability of sale, we obtain the Effective Cost of Caacity (ECC), and then we measure the imact of ECC on fares. As redicted by Prescott (975) and Eden (99), ECC should have a ositive effect on fares. Moreover, as redicted in Dana (999b), this effect should be greater in more cometitive marets. In this aer we rovide evidence suorting both redictions. On average, a ercent 2 one of the 228 flights in the samle changed the aircraft size. 3

5 decrease in the robability of sale would lead to a.29 ercent increase in rices. Moreover, this effect was found to be larger in more cometitive marets. The reason is straight forward, in a erfectly cometitive marer where firms have no marus; every dollar increase in the ECC will be transferred to rices. On the other hand, in less cometitive marets, art of the increase in the ECC will be absorbed by the maru. The findings in this aer can be additionally motivated as an examle of a sot maret subject to demand uncertainty and oened to advance urchases. The standard formulation of a sot marets subject to uncertain excess demand, assumes either imlicitly or exlicitly, a tatonnement rocess that restricts trade until the maret-clearing rice is found. As ointed out in Dana (999b), a sot maret subject to rice commitments should be oened to advance urchases. As we aroach the dearture date, the dynamics of fares and inventories in a flight is an examle of how the maret clearing rice is achieved without having to restrict trade in the resolution of uncertainty in the demand. Along the aer we discuss how the analysis carried out resembles a sot maret with rice commitments. By heling to exlain one of the sources of rice disersion, this aer has an imortant imlication for the airline industry as well. Borenstein and Rose (994) calculated that the exected absolute difference in fares between two assengers on a route is 36 ercent of the airline s average ticet rice. One imortant source of this rice disersion is the existence of intrafirm rice disersion due to advance-urchase discounts (APD). Substantial discounts are generally available to travelers who are willing to urchase ticets in advance. This ind of ricing ractices can romote efficiency by exansions in outut when demand is elastic or may be the only way for a firm to cover large fixed costs. Gale and Holmes (993) justify the existence of APD in a monooly model with caacity constraints and erfectly redictable demand. They show that firms using APD can divert demand from ea eriod to off-ea eriod and achieve a rofit-maximizing method of selling ticets. In a similar setting, but with demand uncertainty, Gale and Holmes (992) show that APD can romote efficiency by sreading consumers evenly across flights before timing of the ea eriod is nown. In cometitive marets, Dana (998) finds that firms may offer APD when individual and aggregate consumer demand is uncertain and firms set rices before demand in nown. The PED models that we test, exlain why carriers offer lower riced seats to earlier urchasers. 3 Our results show that one source of the rice variation found by Borenstein and Rose (994) comes from the 3 ote that the term earlier used refers to the case when assengers who buy before other assengers, rather than a temoral dimension. Travelers urchasing seats even long before dearture may not benefit from APD if most of the seats in the airlane have already been sold. 4

6 fact that carriers face caacity constraints and have to deal with uncertainty in the demand. Moreover, we find that this source of rice disersion is greater in more cometitive marets, result consistent with Borenstein and Rose (994), who also found greater rice disersion in more cometitive marets. Our findings reresent a refinement of Borenstein and Rose (994). They attribute this result to rice discrimination using a model of monoolistic-cometition with certain demand. We argue that if demand uncertainty is considered, art of this rice disersion can be exlained by carriers dealing with caacity costs and uncertain demand. The resent aer is the first emirical aer, to our nowledge, that includes uncertainty in the determination of rices in the airline industry. Desite a number of alications of the PED models, few aers test the emirical redictions of the model. Eden (2) rovides a test and finds a negative relationshi between inventories and outut. However, as ointed in the same article, this negative relationshi is not necessarily an outcome of the PED models. In fact, other models, such as the model of inventory control, would generate the same rediction. Wan (27) tests art the models using data from online boo industry. She tests the effect of stoc-out robability and search cost on rice disersion and finds evidence that higher stoc-out robabilities are associated with higher rices. The PED models requires caacity (how many boos to store or how many seats on an airlane) to be fixed in the short run. This is less liely to be true for the online boo industry than for the airline industry. In addition, Wan (27) does not test the effect of cometition on the rices. 4 The organization of this aer is as follows. Section II describes the data and its characteristics. The theoretical motivation and the emirical secification are resented in Section III; first exlaining the theoretical motivation, then showing how we model demand uncertainty with an alication. Section IV exlains the emirical results. Finally, Section V concludes the aer. II. The Data and Its Main Characteristics The main data source in this aer comes from data collected on the online travel agency Exedia.com for flights that dearted on June 22 nd, 26. It is a anel with 228 cross section observations during 35 eriods maing a total of 798 observations. Each cross section observation corresonds to a secific carrier's non-sto flight between a air of dearting and destination cities. The data across time has one observation every three days. The first was 4 Bilotach (26) mentions the otential role of the PED models in exlaining rice airline disersions, but his dataset does not allow him to formally test the model. 5

7 gathered 3 days rior dearture, the second days and so on until 7, 4, and day(s) rior dearture, maing the 35 observations in time er flight. As in Stavins (2), the date of the flight is a Thursday to avoid the effect that weeend travel could have. The carriers considered are American, Alasa, Continental, Delta, United and US Airways. The number of flights er carrier was chosen to mae sure the share of each of these carriers on the dataset is close to its share on the US airlines' maret. For each flight at each time eriod, this dataset gives us the cheaest available economy class fare and the number of seats that have been sold u to that eriod. To calculate the sold out robabilities, the analysis uses a second dataset collected also from Exedia.com. Most airlines and online travel agencies do not dislay sold-out flights on their websites. The reason, according to Roman Blahosi, soesman of orthwestern, is that they do not want to disaoint travelers. Keeing the online dislay simle may also be a motive, and according to Dan Toore, soesman of Travelocity.com, showing sold-out flights alongside available flights could be confusing. 5 Regardless of the reason, this fact allows us to get the information about the sold out robability in each of the routes. We initially mae a census of all the available nonsto flights in each of the 8 routes used in the first dataset for seven days from February 2 nd to February 8 th in 27. The total number of flights is 5,88. The collection is done coule of wees before the beginning of February when we exect that no flights have yet been sold out, hence Exedia.com should show them all. Then, for each of these seven days of the wee we chec Exedia.com once again late at night the day before dearture to see whether each of the flights has still ticets available. If the flight is no longer there, we assume that it has already sold all its ticets. This rocedure ermits us to calculate the sold out robabilities for each of the routes. We interret this sold out robability as a lower bound because i) February is not necessarily a high demand eriod, and ii) because there may still be some ticets sold the day of the flight that did not enter the comutation. A second imortant source of data is the T- data from the Bureau of Transortation Statistics. From the T- we obtain a anel containing the yearly average load factors at dearture for the same routes as in the main dataset over the eriod 99 to 25. This heled us to calculate the exected number of ticets sold in each route. Moreover, this T- gave us the number of enlanements at each endoint airort to construct some of the instruments. 2. Fares, Inventories and Ticet Characteristics 5 Both quotes are from David Grossman, Gone today, here tomorrow, USA Today, August 26. 6

8 A tyical flight in the samle loos lie the American Airlines Flight 323 from Atlanta, GA (ATL) to Dallas-Forth Worth, TX (DFW) deicted in Figure. The best way to loo at the evolution of seat inventories, in a way that is comarable between flights, is to loo at the load factor, defined as the ratio of seats sold at each oint in time rior dearture to total seats in the aircraft. 6 Load factor will go from zero when the lane is emty to one when it is full. In Figure, the load factor for this flight increases from.2, 3 days rior dearture to.88 with one day left to deart. The increase is not necessarily monotonic as can be observed when moving from 34 to 3 days rior dearture. This is because some ticets may have been reserved and never bought or maybe bought and cancelled later. In this flight fares initially loo fairly stable between $4 and $44, but they have a shar increase during the last two wees before dearture and ea its maximum at $279 the last day. FIGURE [somewhere here] Figure 2 deicts the average fares for the 228 flights in the samle for each of the days rior to dearture. The most imortant characteristic is how fares trend uwards from an average of $258, 3 days rior dearture to an average of $473, the last day rior dearture. This means that average fares almost doubled during the eriod of study. FIGURE 2 [somewhere here] Figure 3 shows the nonarametric regression of daily sales (as ercentage of total caacity) on days rior dearture using 7752 observation over the 228 flights. The bandwidth of.4 days is obtained by least squares cross-validation. The figure suggests that as the flight date aroaches, more seats get sold. The majority of the seats are being sold during the last month and there seems to be a dro in sales during the last few days close to dearture. FIGURE 3 [somewhere here] 6 Airline's literature defines load factor only once the lane has dearted and as the ercentage of seats filled with aying assengers. It is calculated by dividing revenue-assenger miles by available seat miles. Here the load factor is defined at each oint in time as the flight date aroaches. Escobari (25) also uses the ratio of seats sold to total seats at the ticet level to obtain some evidence of ea-load ricing. 7

9 It is imortant to now that inventories evolve not just as a result of sales at the one-way, non-sto flight we are considering. Seats for each city airs in the samle can be sold as art of a larger tri or as art of a round tri with an extremely large amount of ossible otions. Along this aer we will be looing at the carriers otimal ricing decision for the one-way, non-sto flight of June 22 nd and this will have its own dynamics. This detail is imlicit in these tyes of datasets that loo at non transaction data lie Stavins (2), McAfee and Velde (26), Chen (26). The fares used in this aer are the cheaest fare available at each oint in time for a seat in economy class. The cheaest economy class fare at each oint in time rior dearture is just the search results found by Exedia.com for any other online travel agency or carrier's website when searching for the fare of a given flight. 7 It is worth ointing out that every time a carrier changes its rices, it also changes some characteristics associated with this fare. 8 The ey oint here is that these ticet characteristics that change along with fares are irrelevant for the travelers, and if buying online, it is sometimes imossible for the buyer to change these characteristics. Carriers change these irrelevant ticets characteristics to justify the changes in fares. They do not want to charge two different fares for exactly the same roduct just because the transactions 7 Different tyes of fares sometimes available are the ones travel agencies directly negotiate with airline artners. One examle is Clearance Fares and FlexSaver offered by Hotwire.com. These fares come with substantial discounts but imose additional restrictions and involve higher uncertainty. They do not allow changes or refunds and do not allow the traveler to ic the flight times or airline at the moment of booing. Additionally, the traveler cannot earn frequent flyer miles and the fare aid does not guarantee a secific arrival time. Delays can be greater than a day. 8 To show how fares can be exlained with irrelevant ticet characteristics, let's loo again at the fares of American Airlines Flight 323 deicted in Figure. In this examle, when the rice decreased from $34 to $4 between 3 (March th ) and (March 4 th ) days rior to dearture, the ticet characteristics changed from a - to a 4-days-in-advance-urchase-requirement, it changed the first-day-of-travelrequirement from February th to March 4 th, and some blacout dates where included along with changes in day-and-time-of-the-flight restrictions. one of these restrictions have a real imact on the urchase decision or the effective quality of the ticet unless the traveler nows these characteristics and carries out a detailed analysis evaluating the ossibility of canceling the flight later on. If the ticet is bought either 3 or days rior the flight day, having a - or a 4-days-in-advance-urchaserequirement is irrelevant. If the assenger has already decided to fly on June 22 nd and is buying the ticet either on March th or March 4 th, the first-day of-travel-requirement of February th or March 4 th are irrelevant as well. Blacouts and day-and-time-of-the-flight restrictions are only imortant if the traveler decides to change the day of the flight and the new date haens to be exactly in one of the blacout dates. Changing dates will be anyway subject to further restrictions on the ticets available in the new date, and a enalty of 5 lus the differences in fares. The fact is that really few assengers actually now these restrictions even exist since you cannot modify them online and are not rinted out in the ticet or the e- ticet. This examle also shows that even if the ticet is bought with more that 2 days in advance, it does not necessarily mean it gets the discount of a 2-days-in-advance-urchase-requirement. The same goes along with other restrictions; even if the traveler is willing to accet any blacout or urchase a nonrefundable ticet, if only refundable ticets are available, she may well end u buying it, sometimes without nowing the extra benefits. Stavins (2), McAfee and te Velde (26), and Chen (26) also loo at these tye of fare changes, but do not mention this oint. 8

10 occurred at different oints in time, even if these differences in the roduct do not have any imact on the urchase decision. In the emirical test we control for the ticet restrictions that do have an imact on the quality of the ticet. Again, a similar assumtion has been imlicitly made in McAfee and Velde (26) and Chen (26) and just loo at the variations in fares without eeing trac of the corresonding variation in irrelevant ticet characteristics. Stavins (2) omits most of these irrelevant ticet characteristics but includes dummy variables for some advance urchase restrictions. These dummy variables may exlain changes in fare, but they do not reflect the underlying force behind why carriers offer advance urchase discounts in the first lace. As we argue in this aer, once the relevant ticet characteristics are controlled for, the ey underlying force is seats inventories. 2.2 Reresentative Fare A tyical concern among eole who search to buy ticets online is to now whether or not the fare aid in one lace is effectively the cheaest. The concern for us is to now if the fares found in Exedia.com reresent the actual fares offered by the carrier. We want to mae sure that the fact that we collected the fare online does not restrict the analysis to just online fares. The fares reorted on different sites are sometimes different. One source of discreancy comes from the fact that different online travel agencies have different algorithms to reort the fares found in the Comuter Reservation Systems (CRS). This lays a roll when searching comlex itineraries that may involve international flights. In our dataset this discreancy does not arise since we are already restricting the search for a secific flight number on a secific dearture date. A second imortant source of differences comes from variation across urchasing time and seat availability at urchase, the subject matter of this aer. The third imortant source of variation arises because different fees and commissions differ across travel agencies. Exedia.com charges a lum sum booing fee of $5 for every one-way ticet, Travelocity.com charges $5 as well, while Hotwire.com charges $6. Other websites lie Priceline.com, CheaTicets.com or Orbitz.com allow fees to be a function of the base airfare, the carrier or the destination. For examle, fees at Orbitz.com range from $4.99 to $.99. Bric-andmortar travel agencies charge even higher fees that can go u to $5. Buying on the hone also imoses additional different fees i.e. CheaTicets.com charges $25 while Travelocity.com charges $5.95 for over the hone booings. Requesting a rinted ticet will also imose additional variation. Even the carriers themselves charge different rices for exactly the same ticet. For examle US Airways charges no fees if urchased through its website, but charges a 9

11 ticets. 9 Currently, there are four Comuter Reservation Systems which store and retrieve travel $5 fee for ticets urchased through the airline's reservation centers and $ for ticets issued at the airort or at the city ticet offices. Moreover, the baseline fare may still be different deending on which Comuter Reservation System (CRS) the travel agency uses to boo its information used by all travel agents. These are Amadeus, Galileo, Sabre and Worldsan. Airlines ay an average booing fee er segment of $4.25 when using a CRS, while travel agencies usually obtain CRS at no cost or receive certain ayments in exchange for agreeing to use the system. According to the 25 Reort from American Society of Travel Agents (ASTA), the bric-and-mortar travel agencies have resonded by booing art of their sales using the carriers websites and not the CRS. The main source of information of Exedia.com is the Worldsan, but as well as Orbitz.com, they have established direct connection with airlines' internal reservation systems to byass Worldsan and avoid the CRS fees. While it is difficult to evaluate rice differences for exactly the same ticet offered offline, for online marets the information is readily comarable. Chen (26) using a dataset gathered online in 22 obtained that for quotes found in multile online sites the differences in rices are on the order of.3 to 2.2 ercent. Even though not mentioned in her aer, these rice differences can be traced down just by comaring the different fees charged at each site. Currently, carriers lie American, Alasa and United offer a romise that travelers will always find the cheaest fare in its own websites. If the traveler finds a cheaer fare (with more that a $5 difference), they offer aying bac the difference lus additional bonus frequent flyer miles. This shows the carriers' interest on selling through its own websites. In resonse, Orbitz.com and Exedia.com adoted similar olicies. Based on all the multile ways in which fares can otentially differ for exactly the same ticet, we have to come u with a clean measure of a ticet's fare. The best candidate is each carrier website fare which is directly under the carrier's control and is free of any additional fees imosed by CRS, travel agencies or the same carrier if sold offline. For all the carriers in our samle, the fare found in Exedia.com is $5 more than each carrier s website fare, thus obtaining the carriers' website fare is straight forward. Moreover, it is interesting to now ASTA reorted that in 22 the biggest on-line travel agency was Exedia.com, with a maret share of 28.7 ercent, followed by Travelocity.com (28.5 ercent) and Orbitz.com (2.3 ercent). 9 Additional fees common to all include taxes, secial surcharges, segment fees and Setember security fees.

12 Regarding online sales, we now that they have been growing significantly during the last coule of years. The ASTA s reort in 25 citing PhoCusWright Inc. as the source, state that for leisure and unmanaged air sales, the overall online sales as a ercentage of total sales went u from 3.8 ercent in 2 to 56.2 ercent in 24. Of these sales, 38.3 ercent corresond to online travel agencies and 6.7 ercent to sales through the airlines web sites. III. The Emirical Model 3.. A Oligooly Model of Costly Caacity and Demand Uncertainty In this section we derive a simle oligooly model under caacity constraints and demand uncertainty. The redictions of this basic model were already obtained in a more formal environment in Dana (999b). The current derivation extends naturally to our formulation of demand uncertainty and testing rocedure in the emirical section. Let the total number of demand states be H +. The uncertainty in the demand comes from the fact that each carrier does not now ex ante which demand state may occur. Let h be the number of consumers who will arrive at the demand state h, where h =,, H and h h+. This ordering imlies that all the travelers who arrive at demand state h will also arrive at a higher-numbered demand state h+. ow, define a batch as the additional number of travelers that arrive at each demand state when comared to the immediate lower demand state, so batch h will be given by h - h- and the first batch is just. Consider the case where consumers reservation values for homogeneous airlane seats are uniformly distributed [, ], then the demand at state h is given by: Dh ( ) = h () Each demand state h occurs with robability ρ h. Given that all demand states have at least otential travelers, the robability of having otential travelers arriving is Pr = H = ρ = κ κ. In general, the robability that at least h otential travelers arrive is the summation of the robabilities of demand states that have at least h customers, Pr h = κ = ρ h κ. This imlies that the robability that h otential consumers arrive is always as high as the one that h- otential consumers arrive, Pr h Pr h+. Following Prescott (975), the only cost for the carriers is a strictly ositive cost λ incurred on all units, regardless whether these units are sold or H

13 not. This cost can be interreted as the unit cost of caacity (or shadow cost), or the cost of adding an additional seat in the aircraft. Unlie Dana (999b), we assume that the unit marginal cost of roduction incurred only on the units that are sold is zero. Define the effective cost of caacity (ECC) as ECC h = λ/pr h. This ECC adjusts the unit cost of caacity by the robability that this unit is sold. Since some of the seats will be sold only at higher-numbered demand states, if these units are sold, the effective cost of caacity reflects the costs that should be covered whether or not they are sold. If the unit cost of caacity is $, but this unit is sold only half of the times, if it gets sold, the cost that should be covered is $2. The number of identical carriers in the maret is M. When the demand state is h= with the corresonding firm s effective cost of caacity ECC, the standard symmetric ash equilibrium solution of a Cournot oligooly cometition is: + M ECC = M + (2) ( ECC ) M δ = D ( ) = ( M + ) where is the equilibrium rice, and δ is the total amount of seats sold. ote each firm would allocate δ /M number of seats at rice. From the second art of (2) we obtain that the otential number of assengers that arrive at demand state h= is: + M = ECC M ( ) δ [ ] When the demand state is h =, according to (), the total demand at rice is given by: D = ( ) ote that D ( ) D ( ) since, i.e., the total amount of seats demanded at rice when h = is at least as large as the re-allocated number of seats δ. Dana (999b) uses roortioning rationing to assign seats at. This means that everybody has a equal chance δ /D ( )= / to get a seat at. The residual demand, therefore, is: (3) (4) In our setting this basically means that the only relevant cost for the carriers is the one incurred when deciding whether or not to hold inventories for an additional seat. The cost that is assumed to be zero is eanuts (or retzels and soft drins lus any other marginal cost, i.e. baggage transortation). In the hotel examle these marginal costs may include cleaning the room, changing towels, sheets and in many cases the breafast. 2

14 3 ( ) ( ) ( ) ( ) D D R = = δ (5) Again, the symmetric ash equilibrium solutions if the demand function is R ( ) in (5) will be: ( ) ( ) ) ( + = + + = M ECC M M ECC M δ (6) Comare (2) and (6), we can see that given that Pr Pr. In this case, from the second art of (6) we obtain that the otential number of assengers that arrive at demand state h = is given by: [ ] ) ( ECC M M + + = δ (7) If the demand state is h = 2, we are interested in the residual demand after those travelers who have bought ticets at rice and, denoted as R 2 (, ). To find out R 2 (, ), we start with the residual demand after those who bought ticets at, denoted as R 2 ( ), which can be obtained from (6): ( ) ( ) 2 2 R = (8) Travelers who are still in the maret after the ticets at have been sold out will now have the chance to urchase ticets at. The number of otential consumers who will demand ticets at is R 2 ( ), given by (8), and the number of ticets available at rice is R ( ), given by (5), R 2 ( ) R ( ). We aly the roortional rationing again to get the residual demand R 2 (, ): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , R R R R = = = (9)

15 The symmetric ash equilibrium solution for the residual demand function R 2 (, ) in (9) is given by: 2 + M ECC = M + 2, ( ) ( ECC ) 2 δ 2 = M 2 () ( M + ) It is imortant to mention that here carriers are assumed to not observe the seat availability of their cometitors. Once carriers sell their ortion δ /M for the first batch of otential travelers they tae the next ste which is ricing the second batch of consumers. This assumtion guarantees that any given carrier does not try to allocate its entire caacity to the first batch at the exense of their cometitors. At the end of the derivation once we generalize the findings for a continuum of demand states, this assumtion will be no longer needed. This Cournot ricing strategy at each of the batches may allow the ossibility that cometitors behave strategically as in a reeated Cournot game where in each subsequent stage of the game firms face each time higher costs given by ECC. Since this is a finitely reeated game, we just obtain the subgame erfect ash equilibrium by bacward induction. Firms will not be able to collude since each subgame is layed as a static Cournot game. Proosition generalizes revious discussions to any number of demand states. Proosition : Let aggregate demand function be given in (). R (, L, ), is the residual demand when demand state is and travelers who have bought ticets at lower rices,, - have left the maret (as in Eden (99)). We have: R L () (,,, ) ( ) = Proof: When the demand state =, according to (5), the roosition holds. 2 We will rove: if the roosition holds at demand state, then it must hold at demand state +. Suose the roosition at demand state holds. When demand state is +, according to (9), the residual demand after travelers who have bought ticets at lower rices of,, - have left the maret is given by: R L. (2) (,,, ) ( ) + = + The continuum of demand states is lie an infinitely reeated game. If collusion is achieved in this scenario, we just require collusion ayoffs in each stage game to be a function only of the same stage ayoffs for the results in this section to hold. Again, for a stricter derivation of the same results see Dana (999b). 2 According to (9), the roosition also holds for = 2. 4

16 5 Therefore, the residual demand after travelers who have bought ticets at lower rices of,, -, have left the maret is given by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) R R R R = = = ,,,,,,,,,, L L L L (3) ote ( ),, R L in (3) is from () and ( ),, R L + is from (3). Equation (3) roves Proosition. From the residual demand equation of (2), it is easy to get that: + + = M ECC M, ( ) ( ) ) ( + = M ECC M δ (4) For the general case, using the second art of (4) we obtain that the otential number of assengers that arrive at demand state h= is given by: [ ] ) ( + + = ECC M M δ (5) By recursive substitution, considering the construction of the ECC for each batch of travelers, and for a continuum and infinite number demand states we can obtain that the number of otential travelers that arrive at demand state h is given by: ω κ ρ λ δ ω κ ω d d M M h h ) ( + = (6) From these h consumers that arrive at demand state h, only h d κ δ κ are able to buy a seat. Moreover, notice that the rice aid by each grou ω is different and given by: + + = ω κ ω κ ρ λ d M M P [ ],h ω (7) This is just the continuum version of the first art of equation (4). 3 3 Equation (7) is analogous to the first equation in age 233 in Prescott (975), equation () in Eden (99), equation () in Dana (998) and more closely related to equation (5) in Dana (999b) for an oligooly case. The benefit from our equation (7) over Dana s (999b) is that by assuming a secific

17 We now just use this last equation to derive two testable imlications: ω ECC ω M = M + >, and ω ECC M ω = ( M + ) The first art of equation (8) tells us that when the ECC increases, rice also increases. The second art imlies that as the maret becomes more cometitive (larger M), the marginal effect of ECC on fares is greater. Therefore, for a given distribution of demand uncertainty more cometitive marets will show greater rice disersion. The exressions in equations (8) reduce to a monooly when M = and to a erfectly cometitive maret when M. ote that in a erfectly cometitive maret, (8) redicts that every dollar increase in the ECC is transferred to rices as no marus exist to absorb art this increase. 2 >. (8) 3.2 Modeling Demand Uncertainty Let s initially assume that carriers commit to an otimal distribution of rices for each flight before demand is nown. 4 By rice commitment we mean that when demand is low, a traveler who arrives early or arrives late will face the same rice as long as the carrier has not sold ticets in the meantime. Prices increase only if carriers have been selling ticets. Therefore, the information in the rice schedule can be imlicitly included in the functional form secified for the selling robability. This basically means that the robabilities are redetermined for each rice schedule and the secification of demand uncertainty. The rice schedule will be otimal and firms will not want to deart from it as long as they do not start learning about the state of the demand. As mentioned by Dana, useful information about the demand may only be available close to dearture or once it is too late for carriers to change fares. Furthermore, as long as carriers do not learn any useful information about the state of the demand during the trading rocess, we can relax the rice rigidity assumtion (Eden (99)). Starting with the simlest scenario where each demand state is equally liely with robability given by ρ h = α/m. This just means that demand states are uniformly distributed [, m/α] with m being the total number of seats in the aircraft and α. The last inequality assures that there is a ositive robability that the last seat gets sold. Following the intuition from functional form in the demand, rice can be isolated on the left hand side of the equation. Dana (999b) rovides a more general derivation of this result. 4 Later in the emirical section we will allow for some deviations from rice commitment. In articular, we allow the ossibility of current shocs affecting future rices by estimating a dynamic model of Arellano and Bond (99). 6

18 Section 3., having m/α demand states is the same as having m/α = H + batches ( - )of travelers with the first batch showing u with the highest robability and the subsequent ones showing u each time with a lower robability than the revious one. Assume that the lowest demand state has one consumer buying a ticet (δ = ) and for subsequent demand states we have one additional buyer each time we move to the next higher demand state (δ = for all ). Because in every demand state there is at least one consumer buying a ticet, the robability of selling the first seat is equal to one. In all but the lowest demand state there are at least two travelers, so the robability of selling the second ticet is given by one minus the robability of the having the lowest demand state, that is α/m. In general, the robability that seat h gets sold is given by: Pr h α = h q( ), h {,2,..., m}, (9) m which is just one minus the robability of having any demand state with lower demand than state h given the carrier's rice distribution q(). In this equally liely demand states case, α is a constant that determines the rate at which the robability that the next seat gets sold diminishes. Assuming that each demand state is equally liely seems too restrictive. Given our construction of demand uncertainty, this would imly that having only one assenger flying is as liely as having the lane at half caacity and that the robability of selling one additional seat decreases linearly. To allow for more flexibility in the characterization of demand uncertainty we consider the case where ρ h = φ h, with φ being the df of a normal density that has mean μ and standard deviation σ. From the discussion so far we now that the robability of selling seat h is the summation of the robabilities of all demand states that have at least h travelers. For a continuum of demand states, this is given by seat h for the normal density will be: Pr h = ρ κ h dκ. Therefore, the robability of selling Pr h = φ κ dκ q( ) = Φ q( ), (2) with Φ being the cdf of a normal distribution. h h 3.3 Calibrating the Probability Density of Demand Uncertainty To obtain Pr h used in calculating the ECC, it is necessary to get the values for the arameters α in the uniform distribution and the mean, μ, and standard deviation, σ, in the normal 7

19 distribution. In this subsection we calibrate the values of these arameters to mimic the demand uncertainty conditions in each of the routes. A ey source of information for the calibration comes from the T- data from the Bureau of Transort Statistics. We use this dataset to obtain yearly occuancy rates, or load factors at time of dearture. This is done in three stes. First, for each of the routes in the samle, we calculate its load factor for the 8 routes in the samle for the eriod 99 to 25, based on the T- data. Second, each of these 8 series is used to estimate an ARMA model. Finally, the estimated ARMA model is alied to obtain the 26 value using a one-ste ahead forecast. 5 For routes where the ARMA model redicts a high load factor, meaning that most of the seats are exected to be sold, the calibration rocedure will assign higher robabilities to higher demand states. In this case the ECC is going to be relatively low for a large majority of the ticets. When the forecasted load factor is low, the robability of selling the last coule of seats is going to fall fast, meaning that the cost of stocing inventories is higher. The roblem with the information obtained from the T-, however, is that we have a measure of the forecasted value of the average number of ticets sold rather than of the forecasted value of the average number of ticets demanded. This arises because the demand state is censored when transformed to the number of ticets sold. Once the aircraft is sold out the T- no longer records higher demand states. To overcome this limitation let the underlying demand state h *, be distributed (μ, σ 2 ) with the observed number of seats sold h = h * if h < m or else h = m. Recall here that m is the maximum number of seats available in the airlane. Then the exected number of ticets sold is given by the first moment of the censored normal: E = ( h) = Pr( h = m) E( h h = m) + Pr( h < m) E( h h < m) m μ m μ φ( ( m μ) / σ ) ( ) Φ m + Φ σ σ σ Φ ( m μ) / σ The exression for E(h h<m) is obtained from the mean of a truncated normal density. The df and the cdf of the normal density are evaluated at the moment the flight sells out. Hence, the value Φ((m-μ)/σ) is interreted as the sold out robability. Using information on the robability that a flight sells out, based on the second dataset obtained from Exedia.com, and the exected number of ticets sold, obtained from the ARMA models, we can use (2) to obtain values for μ and σ. Calibrating the value of α in the uniform distribution is simler. We obtain the analog of equation (2), E(h)=- α/2, by using the truncated uniform distribution. This equation can be (2) 5 The details of the estimation are available uon request. 8

20 used directly to get α. In this case since we only have to calculate one arameter, the sold-out robabilities are no longer needed. The cost of requiring less information is to have less flexible characterization in which one single arameter α affects both the mean and the variance of the distribution of demand states. 3.4 Estimated Equation and Interretation Following a similar aroach as Stavins (2), we estimate a reduced-form model of log airfare on ECC, maret concentration, carrier's maret share and route-secific factors. The ey new variable in our analysis is the ECC that measures the effect of costly caacity and demand uncertainty by adjusting the unit cost of caacity by the robability that the ticet gets sold. The construction of the dataset also allows us to control for all other relevant ticet-secific characteristics as exlained in Section II. The equation to be estimated is given by: ln FARE ijt = β + (δ + δ HHI j ) ECC ijt + β DAYADV ijt + β 2 DIST j + β 3 DISTSQ j + β 4 ROUSHARE ij + β 5 HHI j + β 6 DIFTEMP j +β 7 DIFRAI j + β 8 DIFSU j + (22) β 9 AVEHHIC j + β AMEAPOP j + γ HUB ij + γ 2 SLOT j + u i + ν ijt where the subscrit i refers to the flight, j to the route, and t is time. Dummy variables have estimated coefficients denoted by γ, otherwise β. u i denotes the unobservable flight secific effect and ν ijt denotes the remainder disturbance. Different error structures will be assumed along the emirical section. Each observation in the samle reresents a unique ticet for a carrier on a route. By route we mean a combination of dearture and arrival airorts on a one-directional tri. FARE ijt is rice aid in US dollars. From Table, the samle mean fare is $29, with a minimum of $54 for an American Airlines flight from Dallas Fort Worth, TX to Houston International, TX when at least 8 ercent of the lane was emty. The maximum is $,224 in a United Airlines flight from Philadelhia International, PA to San Francisco International, CA when there are less than 9 ercent of the seats available. The ey variable in the analysis is ECC which is obtained from ECC = λ/pr h. In articular, when the distribution is uniform as defined in (9), we should have: ECC ijt λ λ = =, (23) Pr α h j ijt hijt m ij 9

21 where m ij is the total number of seats in the aircraft and h ijt is the number of seats that have already been sold at time t. α j is the mean of the uniform distribution. ECC is measured in the same units as FARE, nevertheless to be able to interret the magnitude of the coefficient; we initially normalize λ to be equal to one. For the normal density case as resented in (2), ECC is given by: ECC ijt λ = = λ Pr h ijt hijt mij 2πσ ex 2 j 2 2 ( ( κ μ ) / 2σ ) j dκ The values for μ j and σ j are allowed to change across routes, so they are indexed by route j. h ijt and m ij are directly observable from our dataset. ow we tae a loo at three different cases where the ECC should lay no role in the ricing decisions and analyze how our construction of this measure resond in each of these cases. In other words, these are the cases where the model of section 3. should redict no rice disersion due to costly caacity and demand uncertainty. (i) For routes where we exect higher load factors, costly caacity will lay a less imortant role. On the limit, when we exect to sell all the seats in the aircraft in every occasion E(h) =. In the case for uniform density α j =, and from (9) we get that the robability of selling the next seat does not decrease with the cumulative number of seats sold, Pr h =. For the normal density case μ j. In both situations, there will be no rising ECC as more seats are sold. Holding inventories of additional seats will have no cost since we now for sure that they will be sold. In summary, lim E ( h) ECC = λ. (ii) A similar henomenon would haen if aircrafts had infinite caacity, i.e. no caacity constraints. This can be interreted as carriers being able to adjust the size of the aircraft anytime before dearture at no additional cost. An alternative interretation could be that the good is not erishable; if the good is not sold today, it can be sold anytime in the future. Characteristic that does not hold for airline travel since once the lane dearts; carriers can no longer sell ticets. Again, we have lim = λ for both the uniform and the normal. m ECC (iii) Finally, in the case of no demand uncertainty, carriers would just set their caacity levels to match to the certain number of travelers, hence the ECC would lay no role, i.e., lim σ ECC = λ for the normal, but no demand uncertainty holds also for the uniform. In all three scenarios the rice that an airline charges would be same for every seat, and there will be no rice disersion. That is why models omitting demand uncertainty in their interretations lie Borenstein and Rose (994) or Stavins (2) would lead to interret this (24) 2