Revenue management under joint pricing and capacity allocation competition

Accepted Manuscript

Revenue Management under Joint Pricing and Capacity Allocation Competition Xuan Zhao, Derek Atkins, Ming Hu, Wensi Zhang PII: DOI: Reference:

S0377-2217(16)30661-0 10.1016/j.ejor.2016.08.025 EOR 13911

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European Journal of Operational Research

Received date: Revised date: Accepted date:

31 March 2014 20 June 2016 9 August 2016

Please cite this article as: Xuan Zhao, Derek Atkins, Ming Hu, Wensi Zhang, Revenue Management under Joint Pricing and Capacity Allocation Competition, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.08.025

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Highlights • We model competition between firms in both discount and full-fare markets.

• We allow each firm to make cross-functional pricing and capacity allocation decisions. • We endogenize consumer switching behavior upon a sellout in the full-fare market.

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• We consider the strategic decision of whether to launch early sales.

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Revenue Management under Joint Pricing and Capacity Allocation Competition

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Xuan Zhao School of Business and Economics, Wilfrid Laurier University, Waterloo, ON N2L3C5, Canada, [email protected]

Derek Atkins

Sauder School of Business, University of British Columbia, Vancouver, BC V6T1Z2, Canada, [email protected]

Ming Hu

Rotman School of Management, University of Toronto, Toronto, ON M5S3E6, Canada, [email protected]

Wensi Zhang

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School of Business and Economics, Wilfrid Laurier University, Waterloo, ON N2L3C5, Canada, [email protected]

We investigate joint pricing and capacity allocation decisions of a duopoly, each of which competes in selling a fixed amount of substitutable perishable goods in both early discount and regular full-fare markets with demand uncertainty. Upon a firm’s stockout, unsatisfied demand spillovers to the competitor, depending on the substitutability of products offered by the two firms. We first show that there exists a pure-strategy Nash

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equilibrium. For identical firms, there exists a unique pure-strategy Nash equilibrium that is symmetric. Then we study how firms’ pricing and capacity allocation decisions depend on operational and market conditions. In particular, we show that more cut-throat competition in the full-fare market never leads to a rise in

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the price and a fall in the protection level simultaneously in the full-fare market. We demonstrate a similar result of comparative statics for the competition intensity in the discount market. In comparison with a

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monopolist’s optimal decision, a duopoly equilibrium never results in a situation that definitely benefits the full-fare consumers in both dimensions of price and availability. Finally, we examine how the adoption of early discount, as practiced in the airline industry, may benefit competing firms. When one of the two firms

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first introduces early discount sales, we find that there is a “free-ride” effect for the other firm. Interestingly, the “free-rider” can benefit more than the adopter, especially with stronger competition.

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Key words : game theory, capacity allocation, pricing, competition, early discount

1.

Introduction

The deregulation of the airline industry in 1978 intensified competition and stimulated innovation in airline operations, including the widespread adoption of revenue management (RM). Initially, RM was designed to advance-sell discount fares at a fixed price subject to restrictions such as no-refunds and a minimum stay to leisure customers who might otherwise have chosen other transportation modes or not travelled (Talluri and van Ryzin

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2004, §10.1). An early company to apply RM was British Airways, offering early-bird rebates for customers who booked tickets at least twenty-one days before takeoff. Several other airlines followed, the most notable adoption credited to American Airlines (Jerenz 2008, p8, §10.1.1, Talluri and van Ryzin 2005). Early discounts with limited quantities is also popular in other industries – retail, event ticket, automobile, etc. The key decision

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was what capacity to reserve for full-fare customers not committing to an early purchase. The trade-off at the heart of RM was captured in a classic paper by Littlewood. He proposed a two fare class model in which a monopoly firm balances advance sales to discount customers, assumed abundant, and protecting capacity for later uncertain demand from the full-fare segment (Littlewood 1972). Both full and discount prices were given and fixed (see also Netessine and Shumsky 2005). This simple model gave real clarity about

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the essential trade-off. Over the years the spread of RM to hospitality, car rental, and other industries has required many generalizations of the basic model. For example the hospitality industry must deal with multiple stay lengths instead of entirely perishable room capacity and car rentals have flexibility to transship capacity (cars) between outlets at short notice. Airlines now change reserved seating and prices frequently as booking

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proceeds and allows for multiple flight legs. In this mˆel´ee it seems of value to return to the basic Littlewood model but relax it in two fundamental ways to understand the role that

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key assumptions play. Two obvious relaxations are introducing competition and making pricing a decision. Our aim is to contribute to the fundamentals of RM by considering

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the impact of joint pricing and capacity decisions on firm revenues and consumer welfare under competition.

Although we use airlines to tell the story, we do not mean to imply that our results

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apply holus bolus to today’s airlines without major modifications. Our results are better viewed in terms of airlines frozen in Littlewood’s time or at best agnostic to any particular

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industry. We are simply trying to transform a powerful but simple model (Littlewood) in a minimalist way to include competitive price-setting to see what insights this gives. We employ a stylized two-period model of a duopoly selling a discount fare in the first period and a full-fare in the second. Both discount and full-fare passengers see the two firms as imperfect substitutes. As others have shown (e.g., Netessine and Shumsky 2005), even a parsimonious two-period model can offer rich insights without losing tractability. As in Littlewood, discount seats are pre-sold to a booking limit and the rest reserved for the full-fare market. If insufficient seats are reserved unsatisfied high-fare customers may

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spill to the competitor. This spill behavior results in capacity competition between firms. We remark at this point that the degree of spill must be inextricably linked to product substitutability, the intensiveness of the competition. We model this link, but more on this later. The two key decisions for the full-fare market are: 1) the full-fare price (the marketing decision) and 2) the number of full-fare seats reserved (the operational decision).

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Emphasizing our debt to Littlewood, we inherit the assumption that discount seats always sell out due to the abundant bargaining-hunters (see also Cachon and Swinney 2009). Thus discount sales equal total capacity less reserved full-fare seats. This allows discount fares to be derived as market clearing prices. To summarize our main research questions:

• Is there an equilibrium strategy for this RM game and if so how is it affected by

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market parameters? For example, does intensified competition increase or decrease pricing and capacity decisions?

• What is the impact of removing competition (airline merger?) How does the compet-

itive equilibrium differ from a monopoly solution?

• How are revenues affected if one airline implements RM (offers a discount fare), but

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the other stays with only a single full fare? Results and Insights

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We first establish existence and uniqueness of a pure-strategy Nash equilibrium for this RM game, then investigate how it changes with operational and economic market conditions.

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In equilibrium firms increase full-fare prices and protection levels as spill rates increase. However, the intensity of competition has a more complicated impact. On the one hand, intensified price competition in the full-fare market leads to lower prices and lower protec-

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tion levels; on the other hand, more intensified competition increases spill calling for higher prices and protection levels. Fierce competition never results in simultaneous increased

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prices and decreased protection levels. Similar results occur in the discount market. Compared to monopoly, the competitive equilibrium will never lead to an increased price

and a decreased protection level simultaneously for the full-fare market. Benchmarked against a monopoly, competition never hurts the full-fare consumers in both price and availability simultaneously. Given two airlines serving only the full-fare market, but one unilaterally offers early discount sales to a distinct lower-fare market, we find the second firm has a free-rider benefit, possibly larger than the adopter’s benefit. Somewhat surprisingly, early discount

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sales offered by both firms is Pareto-dominant over the scenario that neither firm offers discounts. Interesting to see through numerical work that intensified price competition hurts the airline that processes stronger market power, but not necessarily the weaker airline. Airlines behave more conservative when they face stronger price competition: they reserve less

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protection levels and charges less when demand is more variable. Less correlated demands are good news for airlines competing in capacity allocation. Entry leads to increased highfare seats and lower high fare prices, but less seats for low-fare markets.

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Literature Review

Based on the foundation laid by Littlewood (1972) the past several decades have witnessed

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tremendous development in RM research (Talluri and van Ryzin 2004). Three streams have been fast growing: competition, dynamic pricing, and consumer behavior. This work has a combination of the previous two features, taking into account of consumer spillover behavior upon stockout.

Research on horizontal competition has drawn considerable attention from opera-

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tions management researchers. With exogeneous prices, Netessine and Rudi (2003) study newsvendors selling a product in a single sales season and investigate how consumer switching (at an exogenous rate) upon a stock-out affects inventory decisions. McCardle et

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al. (2004) consider competition between retailers offering advanced discount programs, but without considering competition in the subsequent full-fare market. Grauberger and

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Kimms (2014) develop algorithm to compute equilibrium booking limits for multiple competing airlines. Zhao and Atkins (2008) extend Netessine and Rudi (2003) to competition

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on both price and inventory. Our paper further extends Zhao and Atkins (2008) to joint inventory and price competition over two periods. More recently, work on joint pricing and inventory control (for example, Cizaire and

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Belobaba (2013), Yu (2014)) considers combined pricing, resource allocation, or overbooking RM problem with demand uncertainty. Raz and Porteus (2006), and Salinger and Ampudia (2011) study the joint pricing and inventory control problem for the newsvendor. Those papers do not consider competition. Mookherjee and Friesz (2008) study pricing, allocation and overbooking in dynamic service network competition with demand uncertainty. Adida and Perakis (2010) study a make-to-stock manufacturing system where two firms compete through dynamic pricing and inventory control in the presence of demand uncertainty. None of the above consider the issue of spillover.

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The RM literature on strategic consumers often studies a two-period model with discounted price in the second period; see, e.g., Zhang and Cooper (2008), Gallego et al. (2008), and Liu and van Ryzin (2008). In contrast, our paper studies mark-up pricing strategy with the discount price offered earlier in the first period, which is more commonly used in the airline industry.

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Xu and Hopp (2006), Levin et al. (2009), and Liu and Zhang (2010) study multiperiod dynamic pricing and oligopolistic competition. Gallego and Hu (2014) consider a continuous-time version of the problem. Netessine and Shumsky (2005) consider capacity competition in RM, but not price competition. In contrast, our paper studies simultaneous inventory and price competition. As a unique feature, our results speak to the impact

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of competition on the consumer individual surplus or the service level (i.e., availability experienced by consumers).

Literature on understanding the impact of the early-discount-adopting on rivals is scant. Belobaba and Wilson (1997), for the airline industry, used a simulation approach to analyze a single origin/destination market with two airlines competing. They find that there is

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always a “first mover” advantage for initiating RM, and that the airline without RM can be hurt. Both airlines, however, see revenue increases once both adopt RM. In contrast

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to the above literature, we consider horizontal competition in both discount and regular market, as well as demand spill. We show that when one of the two firms first introduces

Contribution

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early discount sales, there is a “free-ride” effect for the other firm.

We contribute to the RM competition literature by explicitly modeling competition

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between firms in both discount and full-fare markets, allowing each to make pricing and capacity decisions. We take care to endogenize consumer spill as a result of a sell-out in a

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full-fare market and explore the impact of spill on firms decisions. This is used to explore the strategic decision of whether to launch early discount sales. This shows how the first mover affects the non-adopter and how the subsequent adoption of the second firm in turn affects the first mover. The paper is organized as follows: after a brief literature review, Section 3 describes the model; Section 4 conducts the equilibrium analysis and comparative statics; Section 5 explores the adoption of early discount sales; and Section 6 provides extensions and numerical results.

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3.

The Model

A duopoly offers differentiated substitutable products over two periods: an early discount period and a regular full-fare season. For airlines, leisure-class consumers come early hunting for bargains and business consumers come later, willing to pay the full price. We assume the full fare to be greater, typically observed in empirical studies (e.g., McAfee and te

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Velde 2007), and usually explained that later-arriving customers are willing to pay more. The early and regular markets are often treated distinctly in the RM literature because 1) the business travelers’ typically commit to purchase later; 2) the discount fare appeals to low-price hunters with time to shop around and compare prices and are prepared to accept restrictions such as no refunds and minimum stays specifically designed to fence

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off business travelers. As in the Littlewood model, we assume such bargain hunters to be abundant implying these tickets always sell out at the price offered (see also Cachon and Swinney 2009 for the same assumption of abundant bargain hunters). The second period is the regular sales period with uncertain demand. Firms need to decide the price and the capacity to protect for this period. Given the substitutable nature of the two products,

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once the first choice of regular season consumers is sold out, they may spill to their second choice.

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Initially, each firm i, i “ 1, 2, decides a price (pi ) to charge and the number of seats (Zi ) to reserve for the regular period. If the total capacity is Ci , the seats available for the

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discount period is Li “ Ci ´ Zi .

Events occur as follows: 1) firms simultaneously announce prices and seats reserved

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for the regular period (the discount period price will be derived from the assumption of abundant bargain hunters; see below); 2) the discount period market clears; 3) regular period demand uncertainties are resolved; if a firm sells out of seats, customers may spill

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to the other provider if product is available. 3.1.

Modeling the Early Discount Market

Denoting the early discount price as pLi , we assume a linear demand of firm i in the discount period of the form di ppLi , pLj q “ aL ´ bL pLi ` θL ppLj ´ pLi q, where aL , bL , θL ą 0. So

the revenue is pLi mintLi , di ppLi , pLj qu. The intercept aL can be interpreted as the potential market size for each firm when both products are offered free. With abundant bargain hunters, aL can be understood as being sufficiently large. Under this assumption, each firm

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will discount to the market clearing price such that the reserved capacity Li is cleared, i.e., aL ´ bL pLi ` θL ppLj ´ pLi q “ Li , j ‰ i, i “ 1, 2. Then we have pLi “ p0 ´ β 1 Li ´ β 2 Lj ,

j ‰ i, i “ 1, 2,

where p0 “ aL {bL , β 1 “ pbL ` θL q{rbL pbL ` 2θL qs ą β 2 “ θL {rbL pbL ` 2θL qs. The first two

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terms, p0 ´ β 1 Li , reflect the potential market price associated with the amount of products

made available by firm i. The third term reflects the adjustment made because competition

from firm j reduces the price of firm i. When more substitutable products are available, firm i has to decrease the price in order to attract consumers. As a result, the revenue that firm i gains from early discount sales is Li pLi . Modeling of Regular Market

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3.2.

We now turn to a description of the regular period. We assume the direct regular demand for firm i to be Di “ qi ppi , pj q ` i , where qi ppi , pj q “ ai ´ bi pi ` θppj ´ pi q, ai , bi , θ ą 0 and

random variables (i , j ) are independent and each has a continuous distribution with

increasing failure rate1 (analogous to the newsvendor problem; see Petruzzi and Dada 1999). Given this additive demand shock, the capacity reserved for regular demand can be

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written as Zi “ qi ppi , pj q ` zi , where zi is the “protection level” against the uncertainty i .

After the random demand is realized, unsatisfied consumers of firm j might, if products

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are available, spill to firm i. Therefore the indirect demand for firm i is pDj ´ Zj q` “ pj ´

zj q` . However, not all unsatisfied customers of firm j will want to switch; the proportion

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of those who do is termed as the “spill rate”, which depends on the substitutability of products. By applying representative consumer theory, we develop a mathematically simple and economically meaningful model that connects the spill rate to the linear demand

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function as γ ji rθ{pbj ` θqs. The first part, γ ji ď 1, captures the “operational spill rate”: a

reduction in spill due to actual costs related to switching after observing a stock-out, such

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as travel and search costs. The second part, θ{pbj ` θq, is the substitution rate as a result of product differentiation only. For example, when θ “ 0, there is no price competition,

products are not substitutes and hence spill is zero. When θ becomes very large, then the spill rate is bounded by the “operational spill rate” and so spill is just γ ji . This simple formula effectively considers not only the classical operational spill-over factor, but also the economic spill-over factor, the “degree of competition or product substitutability ” θ.

We provide a proof in the Appendix. 1

This is a stronger assumption than needed. In the numerical session, we also investigate the impact of correlation.

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Thus, the indirect demand for firm i is rγ ji θ{pbj ` θqspDj ´ Zj q` “ rγ ji θ{pbj ` θqspj ´ zj q` ,

and the total demand for firm i is Di ` rγ ji θ{pbj ` θqspj ´ zj q` “ qi ppi , pj q ` Dis , where

Dis :“ i ` rγ ji θ{pbj ` θqspj ´ zj q` denotes the effective stochastic demand of firm i in the regular period. Firm i’s revenue from regular period is pi pqi ppi , pj q ` ErmintDis , zi usq .

Combining sections 3.1 and 3.2, firm i’s problem is to choose pi and zi to maximize

where Li “ Ci ´ qi ppi , pj q ´ zi and pLi “ p0 ´ β 1 Li ´ β 2 Lj .

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π i ppi , zi q “ Li pLi ` pi pqi ppi , pj q ` ErmintDis , zi usq,

(1)

The games with strategic decision variables ppi , zi q and ppi , Zi q can be shown to be

equivalent (Zhao and Atkins 2008). We assume that the airlines’ strategy sets are compact

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, zimin ď zi ď ci u. To ensure the regular price exceeds the discount as tppi , zi q : p0 ď pi ď pmax i

price, we make a technical assumption that regular prices are bounded below by p0 , which

is the largest potential price in the early period. All other bounds on decision variables never restrict the players (Cachon and Netessine 2004).

We summarize the two periods in the following table.

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Market Arrival Price Size Casual Early Low Abundant Business Late High Uncertain Summary of Market Segments

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Table 1

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For convenience we summarize notations. Exogenous parameters:

Ci : total capacity level;

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p0 : market price potential for the discount market (base discount price);

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1.

β 1 : sensitivity of discount price per capacity level for early discount market;

β 2 : sensitivity of discount price per rival’s capacity level for early discount market;

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γ ji : rate of spill due to physical costs related to switching;

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θ: degree of substitutability or degree of competition in high-fare market between

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airlines.

Decision variables: 1.

pi : the price for regular market;

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zi : the “protection level” for regular market. Note this is a construct, not the real number of seats reserved. The difference is the demand for seats, qi ppi , pj q, that will

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be known once prices are known. This generalizes the protection level used in Littlewood to account for endogenized pricing. Being only a construct, note that it is entirely possible for zi to be negative. This simply means that you reserve less than the expected demand. As in the retail trade, stocking less than the expected demand is common when costs of unsold merchandize are very large.

2. 3. 4. 5.

qi ppi , pj q “ ai ´ bi pi ` θppj ´ pi q: deterministic portion of demand in regular market; Zi “ zi ` qi ppi , pj q: total capacity reserved for regular market;

pLi “ p0 ´ β 1 Li ´ β 2 Lj : the price for early discount market; Li “ Ci ´ qi ppi , pj q ´ zi : sales for early discount market;

Di “ qi ppi , pj q ` i : random demand for regular market.

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1.

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Derived variables:

It is convenient to define some summary notation used in the rest of the paper. The expected sales quantity beyond mean is: Xi :“ ErmintDis , zi us. The first derivapiq

tives are Xi :“

BXi Bzi

pjq

“ PrpDis ě zi q and Xi piiq

zj q. The second derivatives are Xi

:“

piq

dXi dzi

:“

BXi Bzj

“ ´rγ ji θ{pbj ` θqs PrpDis ď zi , j ě pijq

“ ´fDis pzi q and Xi

:“

piq

dXi dzj

“ ´rγ ji θ{pbj `

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θqsfDis |j ězj pzi q Prpj ě zj q, where fx denotes the probability density function associated

with a random variable x. These differentiation results are derived similarly as in Rudi et al. (2001). We further assume that Dis has increasing failure rate (IFR), denoted by

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rDis :“ fDis pzi q{ PrpDis ě zi q. This is guaranteed by the assumption that pi , j q are indepen-

dent and have IFR distributions.

Equilibrium Analysis

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4.

We first aim to understand three questions: Is there a pure-strategy Nash equilibrium for

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the game? How does the equilibrium behave with changing economic conditions? How does a competitive equilibrium compare with a monopolist’s optimum? We find that there exists

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a pure-strategy Nash equilibrium for the game; for identical firms, the symmetric equilibrium is unique. Recall that a game is symmetric in the sense that the players’ strategies and payoffs are identical (see Cachon and Netessine 2004 for the definition of symmetric game); and Mahajan and van Ryzin (2001) argue that a symmetric equilibrium is more focal than an asymmetric equilibrium. In exploring the properties of the symmetric equilibrium, we find that both prices and protection levels increase with the cost related spill rate (γ). For other economic parameters, no matter whether they are inter-firm competition factors (θ and β 2 q or self-price and self-capacity sensitivity factors (b and β 1 q, an increase in any

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of these parameters does not simultaneously lead to higher prices and lower protection levels. Compared with a monopolist’s optimum, a competitive equilibrium never leads to an increased price and a decreased protection level simultaneously. 4.1.

Existence and Uniqueness

We establish the equilibrium existence and uniqueness as follows. We employ a technique which is needed for the equilibrium existence result.

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to check the quasi-concavity of a firm’s objective function with two decision variables,

Theorem 1 (Equilibrium Existence). π i ppi , zi q, i “ 1, 2, is jointly quasi-concave in

ppi , zi q and a pure-strategy Nash equilibrium in protection level and price exists. The interior

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Nash equilibrium solves: for i “ 1, 2, j ‰ i,

piq

´pLi ` β 1 Li ` pi Xi “ 0,

(2)

pbi ` θqpLi ` Li r´β 1 pbi ` θq ` β 2 θs ` rqi ppi , pj q ` Xi s ´ pbi ` θqpi “ 0.

(3)

Equation (2) is derived from the first order condition Bπ i {Bzi “ 0, with each term reflect-

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ing how firm i’s total revenue changes with the protection level zi . An increase in zi can lead to changes in π i in two ways: 1) the first two terms reflect that a higher zi not only

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negatively affects the first-period revenue by reducing the capacity level for early discount (Li q but also positively affects the first-period revenue by raising the realized discount

price (pLi q; 2) the third term reflects that a higher zi directly increases the second period

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revenue.

Equation (3) is obtained from the first order condition Bπ i {Bpi “ 0, showing how the

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total revenue changes in price pi : 1) the first two terms reflect that a higher pi positively affects the first-period revenue by increasing the capacity level for early discount and at

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the same time negatively affects the first-period revenue by decreasing the discount price; 2) the third and fourth terms reflect that a higher pi increases the per-unit second-period

revenue but also decreases the deterministic portion of the second-period demand. Writing out pLi explicitly and rearranging terms, Equation (2) becomes p0 ´ 2β 1 Li ´ piq

piq

β 2 Lj “ pi Xi . If β 1 “ β 2 “ 0, then Equation (2) becomes p0 “ pi Xi , a similar result to

Littlewood (1972). So the consideration of the low-fare competition (β 1 ě β 2 ě 0) leads

firm i to increase the protection level, when everything else remains unchanged. Similarly, writing out pLi explicitly and rearranging Equation (3), we get t´pbi ` θqβ 2 Lj ´ Li rβ 1 p2bi `

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θq ` pβ 1 ´ β 2 qθsu ` pbi ` θqp0 ` Xi “ ´qi ppi , pj q ` pbi ` θqpi . The first part on the left-hand-

side collects all the terms related to the discount-market competition and is negative (as β 1 ě β 2 q, so the competition decreases the full-fare price, when keeping other factors the

same. The equilibrium is solved by four simultaneous equations:

p1q

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´pL1 ` β 1 L1 ` p1 X1 “ 0,

pb1 ` θqpL1 ` L1 r´β 1 pb1 ` θq ` β 2 θs ` rq1 pp1 , p2 q ` X1 s ´ pb1 ` θqp1 “ 0; p2q

´pL2 ` β 1 L2 ` p2 X2 “ 0, pb2 ` θqpL2 ` L2 r´β 1 pb2 ` θq ` β 2 θs ` rq2 pp2 , p1 q ` X2 s ´ pb2 ` θqp2 “ 0.

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When two airlines makes identical decisions, i.e., p1 “ p2 and z1 “ z2 , then the solution is unique.

Corollary 1 (Equilibrium Uniqueness). For identical firms, there exists a unique

4.2.

Comparative Statics

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symmetric equilibrium.

One of the features of the paper is that we explicitly derive the spill rate of the regular sales

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market as a function of substitutability between products offered by the two firms. With an interior symmetric equilibrium (omitting subscripts hereafter for notation simplicity because we only consider symmetric equilibrium), we investigate the impact of operational

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and economic parameters on equilibrium, which include the switching-cost-related spill rate pγq, the substitution factor between products pθq, and the relevant competition parameters

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in the early discount market pβ 1 and β 2 q.

Result 1 (Effect of Switching-Cost Spillover Rate). For identical firms, the

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symmetric equilibrium increases in γ, i.e., dp{dγ ą 0 and dz{dγ ą 0.

When full-fare customers are less sensitive to the switching cost (that is, γ is higher),

firms charge higher prices and set higher protection levels for the full-fare market. Result 2 (Effect of Competition Intensity in Regular Market). For iden-

tical firms and under the symmetric equilibrium, 1. When airlines are closer substitute such that the switchover effect dominates, then dp{dθ ą 0, and dz{dθ ą 0. If dz{dθ ă 0, then dp{dθ ă 0. Furthermore, dp{dθ ą 0,

dz{dθ ă 0 never occurs.

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2. If dz{db ă 0, then dp{db ă 0. If dp{db ą 0, then dz{db ą 0. Alternatively, dp{db ą 0, dz{db ă 0 never occurs.

The substitution factor θ, linked to consumer switching behavior, imposes two opposing externalities on the full-fare market. On one hand, an increased θ intensifies price competition, resulting in lower prices (p) and fewer protection levels (z) for full-fare cus-

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tomers. On the other hand, a larger θ increases the spill rate, encouraging firms to set higher protection levels and higher prices. When the latter effect dominates, intensified competition leads to higher full-fare prices as well as enhanced protection levels. Fiercer competition between products (larger θq will never lead to simultaneously increased prices and decreased protection levels for the full-fare period, a situation that definitely hurts the regular consumers.

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The impact of b on a symmetric equilibrium is also indeterminant, but for different reasons. On one hand, a larger b means that regular customers are more price-sensitive and less likely to spill, resulting in a lower regular price and fewer protection levels. On the other hand, a larger b can make it more attractive to sell in the discount market by charging a higher regular price ppq and allocating more capacities to the discount market

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because L “ c ´ a ` bp ´ z is increasing in bp. When the former effect dominates and revenue

from the discount market is insensitive to decisions from the regular market (either β 1 or

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β 2 is relatively small), then, as b increases, firms should provide fewer protection levels and set lower regular prices for the regular market. Otherwise, when the discount price is more

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influenced by decisions in the regular market (either β 1 or β 2 is relatively large), then the the equilibrium regular prices increase and the equilibrium protection levels increase as well. More price-sensitive consumers (larger bq will not lead to simultaneously increased

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prices and decreased protection levels. Result 3 (Effect of Competition Intensity in Discount Market).

For

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identical firms and under the symmetric equilibrium, if dz{dβ 2 ă 0, then dp{dβ 2 ă 0. If

dp{dβ 2 ą 0, then dz{dβ 2 ą 0. Alternatively, dp{dβ 2 ą 0, dz{dβ 2 ă 0 never occurs. Similar

results hold for β 1 .

Intensified competition in the early sales period leads to a less profitable early discount

market (pL “ p0 ´ pβ 1 ` β 2 qL), which provides incentive for firms to provide larger capac-

ities (Z “ a ´ bp ` z) for the regular sales market. Interestingly, a larger β 2 leads to a

higher full-fare price which reduces Z and a higher protection level which increases Z; or

a lower protection level associated with a lower price. The other competition factor in the

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early discount market, β 1 , also affects decisions for the regular market in the same fashion. Fiercer competition within the discount market (larger βsq will never lead to simultaneously increased full-fare prices and decreased protection levels for the full-fare period, a situation that definitely hurts the regular consumers.

Competition Dynamics

5.1.

Effect of Competition

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5.

An important consideration of the impact of competition is the question of how pricing and seat reservation decisions in a duopoly differ from those made by a single organization, a monopolist who manages both products. Let pM and z M represent the strategies of such a monopolist who maximizes the joint profits of the two products.

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Proposition 1. Assuming a highly competitive low-fare market with an exogenous competitive price p0 ,

1. with only price competition (γ “ 0), a duopoly sets a lower price and protection level for full fare customers compared to a monopoly;

2. with both types of competition, and identical carriers, then pM ă p and z M ą z cannot

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occur simultaneously.

Proposition 1 is based on a special case of exogenous low-fare price. With endogenous

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low-fare price, we have the following.

Proposition 2 (Centralized Decision Making). For policy makers, allowing merg-

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ers between duopoly firms who compete sequentially in both discount and regular markets will not lead to lower prices and more protection levels simultaneously for the regular mar-

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ket, i.e., pM ă p and z M ą z cannot occur simultaneously. This is a generalized result and echoes Proposition 1.2, which shows that the impact of

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merger on the high-fare decisions is robust and consistent regardless of introducing low-fare competition or not. Analytically, we can exclude one possibility after the merger, which is the most desirable scenario in which the monopoly benefits the full-fare consumers by simultaneously decreasing price and increasing protection level. The message from this result further supports the idea that scrutiny needs to be made by policy makers for a merger between firms or formation of alliances in the format of joint revenue maximization. How does the number of firms n in the market affect the equilibrium RM strategy? We n ř revise the model by letting Li “ a ´ bpi ` θj ppj ´ pi q. Note that the demand function j“1

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allows the market grow with the number of firms, we do not restrict the market to be constant as the number of firm increases. To see the effect of n on RM strategy we assume symmetry, i.e., θj “ θ and Li “ a ´ bpi ` pn ´ 1qθppj ´ pi q. Proposition 3. Assuming a highly competitive low-fare market with an exogenous competitive price p0 , with only price competition (γ “ 0), the equilibrium has dpi {dn ă 0 and

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dzi {dn ă 0. Furthermore, in symmetric equilibrium Bπ i {Bn ă 0.

With only price competition, increasing number of competing airlines reduces the high fare price, the protection level, as well as the airlines’ revenues. We will further examine the impact of n for the general model in subsection 6.4. Adoption of Early Discount Sales

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5.2.

RM has its origins in the rise of capacity-controlled discount fares after the deregulation of the U.S. airline industry. Before deregulation, the service options offered by commercial airlines were unsegmented without any restrictions. The period after deregulation in the United States was characterized by successive innovations in creating discounted products,

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e.g., “Super Saver” fares introduced by American Airlines in 1975 and “Max Saver” fares introduced by Texas Air Corporation in 1987 (Talluri and van Ryzin 2004). These fares

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had a seven-day advance-purchase requirement and minimum-stay conditions and required round-trip travel. In addition to understanding the status quo of industry wide practice of adopting early discount sales, it is of interest to academicians, industry practitioners, and

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policy makers to understand the strategic implications of introducing early sales discount fares in a competitive market.

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The main question is whether sellers should opt for the conservative but lower revenue path and deploy their capacity in a discount-like market or retain capacity for a higher

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returns in an uncertain regular market. For instance, if early sales are introduced, event organizers (concerts, etc.) need to choose the percentage of seats to sell in advance to loyal patrons. At the strategic level, whether introduction of such early sales is beneficial to firms under competition remains a less-understood question. So far we assumed both firms adopt early discount sales, and that the two markets of early sales and regular sales are distinct. This assumption will not affect the key insights as will be discussed in subsection 6.5. For a monopoly, the early discount sales enable a firm to attract customers who otherwise might not consider the product at all. Thus, when

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inventories are optimally rationed and prices optimally set, offering early discount sales benefits the firm. However, can the practice of offering early discount sales by both firms be sustained as a market equilibrium in a strategic decision game? Does offering early discount sales benefit firms competing with each other? To answer these questions for a symmetric duopoly, we

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need to compare our base model with two other cases: neither firm offers early discount (NN case) and only one firm offers early discount (YN case).

When both firms sell only to the regular market (NN case), each firm sets a regular retail price and sell the entire capacity (Ci q to the single full-fare market. As the early discount market is distinct from the regular market, the total regular demand of a firm is

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N , where still qi ppi , pj q ` Dis . Each firm i “ 1, 2 chooses a regular price pi to maximize π N i N πN “ pi pqi ppi , pj q ` ErmintDis , zi usq, i

and zi “ Ci ´ qi ppi , pj q.

(4)

If only one firm offers the early discount, that firm chooses p1 and z1 to maximize π Y1 N ,

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where

π Y1 N “ L1 pL1 ` p1 pq1 pp1 , p2 q ` ErmintD1s , z1 usq

(5)

maximize π Y2 N , where

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and z1 “ C1 ´ q1 pp2 , p1 q ´ L1 . The other firm that only offers regular sales chooses p2 to

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π Y2 N “ p2 pq2 pp2 , p1 q ` ErmintD2s , z2 usq

(6)

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and z2 “ C2 ´ q2 pp2 , p1 q.

Since profit functions in (4) and (6) are continuous in the price decision and the profit

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function in (5) is continuous in both price and protection level decisions, a Nash equilibrium (not necessarily in pure strategies) exists for both NN and YN games (Theorem 1.3 in Fudenberg and Tirole 1991). In some situations, we find that a pure-strategy Nash equilibrium exists and we are able to analytically investigate the effect of offering early discount sales. Theorem 2 (Free Riding Effect). Suppose there is no spill pγ “ 0q. There exists a

pure-strategy Nash equilibrium in the N N and Y N scenarios. As long as the firm who first offers early discount sales benefits, the other firm also benefits.

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This result considers the situations in which firms aggressively use reward or other loyalty programs such that the spill between firms is rare; for example, instead of switching to another firm with a distinct reward program, unsatisfied consumers would rather shop with the same firm but go for another time and date. In this situation, if the market condition favors one firm’s choice of offering early discount sales, then the competitor who

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does not offer early discount also benefits as a free rider. This is because after the firm profitably allocates some capacity to the early sales period, it raises its regular price, which alleviates price competition in the regular sales market and hence benefits the competitor in the absence of capacity competition. Similar insights should hold for the general model with capacity competition to some extent: if the firm who profitably offers the early sales

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also reduces its protection levels, capacity competition is alleviated as well and hence the competitor benefits even in the existence of capacity competition. In the absence of spill, if the competitor follows and also offers an early discount (moving from Y N to Y Y scenario), it usually increases its regular price but reduces its protection levels. While the first change could enhance the revenue of the early adopter of early

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and regular sales, the second change could intensify price competition in the discount market and hence hurt the early adopter’s overall profitability. With spill, the sequential adoption of early sales by the competitor leads to several more effects on the early adopter’s

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profitability. For example, after the late adopter allocates capacity to the discount market, price competition in the discount market is intensified (reducing the early adopter’s revenue

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from the discount market) but there is more of the late adopter’s unsatisfied regular demand switching to the early adopter (enhancing the early adopter’s revenue from the regular

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market). To further understand these trade-offs, we supplement the above analytical results with numerical experiments in the next section.

Numerical Results

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6.

In this section, we explore results that are not analytically available. We assume p1 , 2 q to be distributed as a joint normal N pµ1 , µ2 , σ 1 , σ 2 , ρq. Note that we have asummed ρ “ 0 so

far in this paper, but we shall investigate correlated demands in this numerical section. To ensure we used parameters to calibrate our model that made reasonable sense in practice we used data from Bilotkach et al. (2009). Focusing on two key carriers in the New York (John F. Kennedy) – London (Heathrwow) city pair route: American Airlines (AA) as carrier 1 and Virgin Atlantic (VS) as carrier 2, Bilotkach et al. (2009) collected data through

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Expedia.com, from September 22 until November 21, 2005 for 60 days. They claimed that the data accounted for 29% of tickets booked via on-line travel agents, which in turn accounted for 25% of tickets actually booked by the traveling public. Given the number of fare quotes and number of flights for the two airlines during 60 days, a reasonable estimate of the potential demand for each flight is: a1 ` µ1 “ 433, and a2 ` µ2 “ 468. Assuming the

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random demand accounts for around 1/3 of the total demand potential (this assumption will not affect the key insights), we have µ1 “ 150 and µ2 “ 180. Bilotkach quotes an average price across market as $620. From a few checks of our own, we take the low fare price to

be about half of the average price (again this will not affect the key insights), so p0 “ 300. This gives upper bounds on the price sensitivity parameters: b1 ă 433{300 “ 1.4 and b2 ă

468{300 “ 1.6. To capture the asymmetric market power, we chose VS, as it has a larger

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potential demand and then endow it with a less sensitive consumer base. So we choose b1 “ 1 and b2 “ 0.8. The price competition factor θ can then be chosen as values lower than,

about equal to or greater than these price sensitivity parameters, i.e., θ “ r0.5, 1, 1.5, 2s. We

also consider demand spill γ “ r0, 0.25, 0.75, 1s and present cases with γ “ 0 representing no spill and γ “ 1 representing complete spill. To incorporate the effect of demand uncertainty,

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we take a coefficient of variation cv “ 0.1 (thus σ 1 “ 15 and σ 2 “ 18) to represent the less

variable scenario and cv “ 0.3 (thus σ 1 “ 45 and σ 2 “ 48) or cv “ 0.5 (thus σ 1 “ 75 and

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σ 2 “ 90) to represent the more variable scenario. Although our theory assumed independent demand here we can also consider two cases of demand correlation, ρ “ 0 as uncorrelated

demand and ρ “ 0.6 correlated demand. The equipment used by AA are Boeing 777-300

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with capacity C1 “ 307, and by VS, Airbus 330-300, with C2 “ 334. With p0 “ 300, then

to guarantee a positive low fare price if the airline allocates all capacity to the low fare,

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we have the lower bound of β’s: β 1 ą 300{307 “ 0.98; and β 2 ą 300{334 “ 0.90. We stress

again that this model calibration is purely so that our numerical examples are reasonable. Capacity Allocation and Effect of Market Power

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6.1.

We use the less variable scenario and uncorrelated demands to illustrate the first set of results; the conclusions are similar in other cases. Figure 1 shows that as price competition θ increases both the protection level zi (the upmost in the bar) and the discount seats

available Li (the lowest part in the bar) decrease. This might seem a bit contradictory, so we explain what is happening. Think of the aircraft Ci as holding three groups of people: discounters Ci ´ pzi ` qi q, the predictable full fare passengers qi (the middle part in the bar)

and finally seats protected for the unknown random full fare passengers zi , which might

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be partially unused. Now as θ increases, regular prices will decrease and regular sales qi increased. So these extra regular seats must have come from either the discounters or the protected seats. But we see above they have come partially from both. The split between how much one is reduced compared with the other remains determined by the fundamental balancing at the margin (last seat) of Littlewood: the “sure thing” of a discount fare

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versus the “expected” loss from a random demand. We also observe that the protection level does not decline so fast when spill is present. As θ increases, the spill rate θ{pbi ` θq increases, which essentially grows the random demand by spills from the competitor. So

it is advantageous to increase the protection level, not enough to make it increase, but enough to mitigate its decline.

Finally we comment on the impact shown in these graphs from asymmetric market

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power. The difference in protection levels (Figure 1(a)) comes straight from the market potential differences. The price difference (Figure 1(b)) closes as the competition increases as we would expect. The reduced power of airline 1 makes more seats available for discount,

Seat Allocation Over Classes, Prices and Revenues Across Airlines

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Figure 1

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ED

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but both airlines have an almost identical discount fare.

Figure 1(b) also shows that a higher spill rate in the high-fare market always benefits

airlines, but not dramatically. Market power naturally gives airline 2 more revenue, but stronger price competition erodes this advantage; airline 2 revenue always declines. In fact airline 1 can actually increase revenue for modest increases in competition. 6.2.

Effect of Demand Uncertainty

We show results only for airline 1 and independent demand case but the observations hold also for correlated demands and airline 2. When price competition is less intense,

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higher variability leads airlines to reserve more protection seats (Figure 2(a)) and charge higher regular prices (Figure 2(b)). The opposite is true when the price competition is more intense. Facing stronger price competition, it appears that airlines should be “more conservative” in a sense that more variability in demand should be matched with reduced protection levels reserved for high fare and decreased prices as well; however facing weaker

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price competition, airlines should be “less conservative” in a sense that more variability

Impact of Demand Variation on Seats Allocation, Prices and Revenues

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Figure 2

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should lead to enhanced protection levels reserved for high fare and increased prices.

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Not surprisingly, less variable demands enhance the revenue of both airlines (Figure 2(b)). We see that not only do full fares fall with increased competition, but that this is

6.3.

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accentuated with increased variability. Effect of Correlation

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We focus on the less variable case and airline 1, but again, the observations hold for airline 2 and more variable cases as well. When the two airlines’ high-fare demands are positively correlated, ρ “ 0.6, they tend to reserve lower protection levels, and charge lower high-fare prices (Figure 3). As a result, the number of seats left for low-fare consumers is higher when the demand is more correlated, with resulting lower prices for low fare. For both airlines, less dependent demands result in higher total revenue (Figure 3(b)). The positive correlation reduces the value of capturing spill as both tend to have “feast or famine” at the same time.

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6.4.

Impact of Correlation on Seat Allocation, Prices and Revenues

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Figure 3

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Effect of The Number of Competitors

Here we consider a situation with multiple symmetric airlines and investigate the impact of the number of competitors on each airlines decisions and performances. We find that as the number of competitors increase, firm increases the protection levels for stochastic high-fare

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demand, as well as the seats for deterministic high-fare demand, and as a result, less seats are left for low-fare consumers (Figure 4(a)). Not surprisingly, airlines prices for high-fare

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market decrease and their revenues drop as well (Figure 4(b)). The same observations hold

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CE

PT

for different levels of coefficient of variations.

Figure 4

Impact of Number of Competitors in Seat Allocation, Prices and Revenues

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6.5.

Adopting Discount Sales With General Spill Rate

We report on more variable and correlated demand cases and for airline 2, with γ “ 1. The

observations hold for other cases as well. According to Figure 5, when airline 2 offers an early discount (moving from N N to N Y ), it decreases its protection level and increases its price for the high-fare market. When the competitor, airline 1, offers early discount (from

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N N to Y N ), airline 2 also decreases its protection level and increases its price for high fare, but at a relatively smaller magnitude. Both airlines offering early discount (Y Y ) results in the highest high-fare prices. By allocating some capacities to the discount market, airlines

Airline 2 First Adopting Early Discount: Impact on Its Seat Allocation, Price and Revenue

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Figure 5

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reduce their protection levels but increase their prices for the high-fare market.

Figure 6

Impact of Adopting Early Discount on Airline 1

When both airlines offer low-fare tickets (Y Y ), airline 2 reserves less seats for low fare than the case with only airline 2 offers low-fare tickets (N Y ). Low-fare customers will

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expect lower prices under Y Y than under N Y . In terms of revenues, both airlines achieve the best earnings when they both provide an early discount, Y Y is a win-win game (Figures 5 and 6). Both earn the least if they only target the high-fare market, so N N is a lose-lose game. For both airlines, it is beneficial to initiate an early discount when the price competition in the high-fare market is less

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intense. When the competition is more intense, it is smart to benefit from the competitor’s adoption of an early discount.

We also consider the case that after the introduction of early discount sales, the regular sales market may be cannibalized. In particular, we modify the potential market size of the discount market by simply adding a certain percentage of regular sales customers who

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switch over to the early discount market. We find that the Y Y scenario with a certain percentage of cannibalization rate is always better than the N N scenario, although it is obviously worse than the Y Y scenario without any cannibalization. This implies that if the regular and discount markets are sufficiently separate, then one of the main insights from distinct market cases, that offering early discounts is win-win for both firms, still

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holds. Note that in this experiment, we assume those who switch but do not get fulfilled at the discount price are lost and can not return to the regular sales market. In reality,

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with a limited amount of early discounted inventories, the detrimental effect of the early discounted market cannibalizing the regular sales market is capped, especially if most

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unsatisfied regular customers will return to purchase at the regular price. We further find that in contrast to the free-riding effect that is established without

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cannibalization (Theorem 2), when the first mover profitably offers the discount product (from N N to Y N ) with cannibalization, the second firm, whose regular customers can

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switch, may get hurt when the degree of competition is not intense: since the competition is not intense in the regular market, the first mover can allocate more capacity to the early sales market and lure some of the competitor’s regular customers. The first mover’s advantage still holds even in the existence of cannibalization. Given the fact that the cannibalization effect is capped by the limited capacity allocated to the early discount market, the insights we draw under the assumption of no cannibalization remain to a large extent in the context of cannibalization if the total capacity is limited, which is the most relevant scenario in RM.

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7.

Conclusions

This paper has investigated capacity allocation and pricing decisions of competing firms who sell a fixed capacity of perishable products over two sequential markets – an early discount market and a regular uncertain sales market. It seeks to understand how the market equilibrium behaves with changing economic conditions and how the option of

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offering early discount affects firms’ decisions and revenues. We first had to get the technical material out of the way; establishing existence of a purestrategy Nash equilibrium and showing that for identical firms, there is a unique symmetric equilibrium. We then considered some comparative statics of the equilibrium. We find that when regular customers are less sensitive to the cost of spilling to the competitor after a sellout of seats (a larger γ), firms reserve more seats (protection levels) and charge a

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higher price for regular customers. The substitution factor, θ, measures the substitutability between the two firms and equivalently, the likelihood of a spill to the competitor upon stockout. We found that intensified competition never leads to an increased price and a decreased protection level. Therefore, intense competition, due to easier comparison shopping facilitated by search engines and travel websites, never leads to a situation that

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definitely hurts the regular consumers in both dimensions of price and availability. Similar observations are found for the competition factor in the discount market.

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To further explore the impact of competition, we compare the competitive equilibrium with the optimal decision of a monopolist selling both products. It is obvious that the

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revenue will improve, but what is the implication for consumers? We find that the monopolist solution will never result in a reduced price and an enhanced protection level for the regular market. Therefore, mergers between two firms never lead to a situation that defi-

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nitely benefits regular customers in both dimensions of price and availability. This would be particular of interest to policy makers, as the airline industry is currently undertaking

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a new wave of merger and acquisition. We then further explore the impact of offering early discount sales by a firm and its

competitor. This strategic level decision making is of historical and contemporary interest. We find that there is a “free-rider” effect when an airline offers early discount sales. The other firm could benefit more than the adopter especially with stronger competition. The combined effect of both firms offering early discounts is a Pareto-dominant situation over both firms not offering an early discount. We also consider the case when the regular sales market is cannibalized after the introduction of an early discount sale, the insights we

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draw under the assumption of no cannibalization remain to a large extent in the context of cannibalization. Through numerical experiments, we further find that intense competition hurts the firm with stronger market power, but might benefit the one with weaker market power. Stronger price competition leads firms to be more conservative by reducing the protection levels

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and prices when demand is more variable; weaker price competition results in less conservative firms who enhance protection levels and prices when demand is more variable. More correlated demand between firms leads to reduced protection levels and prices for high-fare market. Independent demands gain more benefit from demand spill and improved revenues.

Future research in this area is potentially rich. One possibility is to model price com-

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petition across periods or recapture across periods. Another is to adopt a choice based framework to allow splitting of demand across firms and product classes. All these extensions are significant and we might need to simplify other parts of the model to obtain any insights. We are eager to explore these significant extensions.

A.

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APPENDIX.

Derivation of the Spillover Rate.

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We need the following assumptions. 1) Without loss of generality, we assume Eri s “ 0. 2)

If the second firm is sold out as well, consumers leave the market and remain unsatisfied.

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3) Firms do not have the opportunity to adjust prices for the spillover demand. We further use the representative consumer approach to derive the substitution rate. A representative

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consumer is a “fictitious agent whose utility embodies aggregate preference for diversity,” and the total amount of the product that the representative consumer will buy by maximizing the aggregate surplus will be the total demand for that product (Anderson et al.,

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1992).

The actual direct random demand is Di “ qi ppi , pj q ` i . Let the utility function of a

representative consumer for the economy with two substitutable products be U pq1 , q2 q “

α1 q1 ` α2 q2 ´ 12 pν 1 q12 ` 2λq1 q2 ` ν 2 q22 q with λ ą 0 representing substitutable goods (Shubik and Levitan 1980, Vives 1999). The first-order conditions for the maximization of the con-

sumer surplus maxq1 ,q2 U pq1 , q2 q ´ p1 q1 ´ p2 q2 lead directly to the linear demand function q1 pp1 , p2 q “

pα1 ´p1 qν 2 ´λpα2 ´p2 q ν 1 ν 2 ´λ2

and q2 pp2 , p1 q “

pα2 ´p2 qν 1 ´λpα1 ´p1 q . ν 1 ν 2 ´λ2

For simplicity of presenta-

tion, we use qi ppi , pj q “ ai ´ pbi ` θqpi ` θpj , and the one-to-one relationship between the

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26 2λ 1λ two sets of notations is given by a1 “ αν11νν22´α , a2 “ αν21νν12´α , b1 “ ν 1νν22´λ , b2 “ ν 1νν12´λ and ´λ2 ´λ2 ´λ2 ´λ2

θ “ ν 1 ν 2λ´λ2 .

We assume that if product 2 is unavailable, the random demand of firm 1 is additive

as well: D1 “ q 1 pp1 q ` 1 . Similar to Assumption 1, we have Eri s “ 0. To derive q 1 pp1 q,

we use the representative consumer theory as well. When the representative consumer’s

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only choice is product 1, the total amount that the representative consumer buys is the solution of maxq1 U pq1 , q2 “ 0q ´ p1 q1 . Notice that, here, we have used the same p1 as the

case that firm 2 exists. This is based on Assumption 3 above: In the same selling season, firms do not have the chance to change the price for the spillover demand. Then the optimal 1q q 1 pp1 q “ pα1ν´p . 1

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In order to derive the spillover rate from firm 2 to firm 1, we need to consider amount of

consumers whose first choice is firm 2 and second choice is firm 1 and are willing to purchase product 1 if product 2 is unavailable. The demand of firm 2 that will switch to firm 1 if firm 2 stocks out is D1 ´ D1 . Then, the proportion of the expected switching demand

over the expected total demand of firm 2 gives the substitution rate as ErD1 ´ D1 s{ErD2 s,

which, by Assumption 1, becomes rq 1 pp1 q ´ q1 pp1 , p2 qs{q2 pp2 , p1 q. “ νλ1 “ b2θ`θ . Similarly,

Proofs.

Proof of Theorem 1.

q2 pp2 q´q2 pp2 ,p1 q q1 pp1 ,p2 q

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B.

q 1 pp1 q´q1 pp1 ,p2 q q2 pp2 ,p1 q

“ νλ2 “ b1θ`θ .

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Thus

A bivariate function gpx1 , x2 q is jointly quasiconcave in two vari-

ables iff every “vertical slice” of the function is quasiconcave, or more formally, iff function

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gpx1 , x2 q is quasiconcave given mx1 ` x2 “ k for any real values m, k (Zhao and Atkins, 2008).

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Given this, we need to show that π i ppi q “ pLi Li ` pi rqi ppi , pj q ` Xi s is quasiconcave in pi

where zi “ k ´ mpi . Define ∆ppi q :“ dπ i ppi q{dpi , then

piq

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∆ppi q “ pbi ` θ ` mqpLi ` Li r´β 1 pbi ` θ ` mq ` β 2 θs ` qi ppi , pj q ` Xi ´ pbi ` θqpi ´ mpi Xi d∆ppi q dpi

piq

piiq

piq

“ 2pbi ` θ ` mqr´β 1 pbi ` θ ` mq ` β 2 θs ´ 2pbi ` θq ` Xi r´2m ` m2 pi Xi {Xi s

Notice that Υpθq “ 2pbi ` θ ` mqr´β 1 pbi ` θ ` mq ` β 2 θs ´ 2pbi ` θq is concave and that piiq

piq

both Υp0q and Υ1 p0q are negative, so Υpθq ď 0. Also note that rDis “ ´Xi {Xi ě 0 and is

increasing. If m ě 0, then π i ppi q is strictly concave in pi , done. piiq

piq

If m ă 0, then let n “ ´m ą 0. If nr2 ´ npi p´Xi {Xi qs ă 0, then π i ppi q is strictly conpiiq

piq

cave in pi , done. Otherwise, it can be shown (using IFR of Dis ) that nr2 ´ npi p´Xi {Xi qs piq

piiq

piq

is strictly decreasing and Xi nr2 ´ npi p´Xi {Xi qs strictly decreases in pi . So d∆ppi q{dpi

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changes sign at most once: it can either go from positive to negative or always be negative. Thus once ∆ppi q turns negative, it remains negative, and π i ppi q is quasiconcave in pi and

a pure strategy Nash equilibrium exists.

The interior solutions are solved by first-order conditions (2)–(3). Now we discuss possible boundary solutions. As Bπ i {Bzi |zi “0 ą 0, firms will always allocate positive levels of

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protection levels to the regular market. Bπ i {Bzi |zi “Ci “ ´pLi ` pi PrpDis ą Ci q ă 0 if we

assume PrpDis ą Ci q is sufficiently small. Then the solution will be interior. Otherwise,

the protection level solution will be at boundary and the unique pi can be solved from

(3). Bπ i {Bpi |pi “p0 “ pbi ` θqp´2β 1 Li ´ β 2 Lj q ` θLi β 2 ` qi ppi , pj q ` Xi . If Bπ i {Bpi |pi “p0 ă 0,

then the best solution is the boundary solution pi “ p0 and zi can be solved from (3). As Proof of Corollary 1.

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Bπ i {Bpi |pi Ñ8 ă 0, the solution to price will not be unbounded.

Quasiconcavity also guarantees the existence of a symmetric equi-

librium; that is when zi “ zj “ z and pi “ pj “ p. We prove that this equilibrium is unique. For a symmetric equilibrium, p2q becomes

piq

´ p0 ` pc ´ a ` bpqp2β 1 ` β 2 q “ p2β 1 ` β 2 qz ´ pXi .

(A1) piq

piiq

A1 pzq “ Xi

pijq

` Xi

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For simplicity, rewrite pA1q as zppq “ pp Apzq ` bq ` K, where we define Apzq :“ Xi ą 0, d ă 0, d :“ 2β 1 ` β 2 ą 0, and K :“ pc ´ aq ´ p0 {d.

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It is easy to check that given any p, there exists a unique zppq that solves equation pA1q.

Now we need to prove that a unique p solves equation p3q where z “ zppq, i.e.,

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pb ` θqpL ` Lr´β 1 pb ` θq ` β 2 θs ` qppq ` Xi pzppqq ´ pb ` θqp “ 0,

(A2)

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where pL “ p0 ´ pβ 1 ` β 2 qL, qppq “ a ´ bp and L “ c ´ a ` bp ´ zppq. Define Jppq to be the

LHS of pA2q.

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If p “ p0 , then z will be uniquely solved from pA1q; As Jppq|pÑpmax ă 0, there is no

unbounded solution to the problem. Next we prove that if the solution is interior, i.e., Jpp0 q ą 0, there is a unique p that solves Jppq “ 0, at dJppq{dp ă 0. piq

pjq

Note that dJppq{dp “ ´p2b ` θq ´ br2pb ` θqβ 1 ` bβ 2 s ` rXi ` Xi ` 2pb ` θqβ 1 ` bβ 2 sz 1 ppq, piq

pjq

where z 1 ppq “ dzppq{dp ą 0 and it can be show that in the symmetric case, Xi ` Xi decreases2 with z. piq

pjq

i q{η Cpzq “def Xi ` Xi “ tPrpi ě zq ` ∫zi “0 ∫8j “z`pz´i q{η f pi , j qdi dj u´ηt∫zi “0 ∫z`pz´ f pi , j qdi dj uC 1 pzq “ j “z ´ ∫8j “0 f pz, j qdj ` ∫8j “z f pz, j qdi dj ` η ∫zi “0 fi pi , zqdj ´p1 ` ηqp1 ` 1{ηq ∫zi “0 f pi , z ` pz ´ i q{ηqp1 ` 1{ηqdi ă 0,where η “ γ ji θ{pbj ` θq.

2

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28

Now take derivative of zppq from both sides to get z 1 ppq “ pA1 pzqz 1 ppq{d ` Apzq{d ` b, or z 1 ppq “ r

Apzq ´A1 pzq ` bs{p1 ` p q. d d

(A3)

Using a common property of fractions we have maxtb, Apzq{p´A1 pzqpqu ě z 1 ppq ě

mintb, Apzq{p´A1 pzqpqu ě 0. For the symmetric case, IFR of Dis means that ´A1 pzq{Apzq is

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increasing; therefore Apzq{p´A1 pzqpq is decreasing, and there are two possible regions for

z 1 ppq. Zone One is Apzq{p´A1 pzqpq ě z 1 ppq ě b and Zone Two is Apzq{p´A1 pzqpq ď z 1 ppq ď b. piq

pjq

In Zone One: As z 1 ppq ď b , and Xi ` Xi ă 2, dJppq{dp ď ´p2b ` θq ´ br2pb ` θqβ 1 `

bβ 2 s ` r2 ` 2pb ` θqβ 1 ` bβ 2 sb “ ´θ ď 0. Jppq starts positive and straightly decreases, and a unique p solves Jppq “ 0.

In Zone Two: We shall prove that z 1 ppq is decreasing in p, i.e., z 2 ppq ă 0. As in Zone Two

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Apzq ` A1 pzqpb ě 0 which implies

A1 pzqApzq ` ppA1 pzqq2 b ď 0.

(A4)

IFR of Dis means that an increasing ´A1 pzq{Apzq is equivalent to A1 pzq2 ě A2 pzqApzq.

Hence (A4) becomes A1 pzqApzq ` pA2 pzqApzqb ď 0 or A1 pzq ` pA2 pzqb ď 0, which indicates

M

A1 pzq ` pA2 pzqz 1 ppq ď 0 as z 1 ppq ă b in Zone Two.

1

1

1

ppq ` Taking derivative of (A3) from both sides to give z 2 ppqp1 ` p p´Adpzqq q “ 2 A pzqz d 2 pzqpz 1 ppqq2

d

ă pA1 pzq ` pA2 pzqz 1 ppqqz 1 ppqq{d. A sufficient condition for z 2 ppq ď 0 is that

ED

pA

A1 pzq ` pA2 pzqz 1 ppq ď 0, which is the case in Zone Two as shown above. As both z 1 ppq and pjq

piq

Xi ` Xi

is decreasing in p, dJppq{dp can initially be positive but decreases to negative

0. ˝

PT

and stays negative. In either case there is a unique p that solves Jppq “ 0 at dJppq{dp ă

CE

Proof of Result 1.

A symmetric equilibrium solves p2q ´ p3q where pi “ pj “ p, zi “ zj “

z and makes dJppq{dp ă 0 (see proof 2 of Theorem 1). Taking the derivative of p3q and p2q

AC

w.r.t. γ, we have

piq

pjq

´dp{dγr2bpb ` θqβ 1 ` β 2 b2 ` p2b ` θqs ` dz{dγr2β 1 pb ` θq ` β 2 b ` Xi ` Xi s “ ´dXi {dγ piq

piiq

and dp{dγrbp2β 1 ` β 2 q ` Xi s ´ dz{dγrp2β 1 ` β 2 q ´ ppXi Then

pijq

piq

` Xi qs “ ´pdXi {dγ.

piq pjq ´dXi {dγ 2β 1 pb ` θq ` β 2 b ` Xi ` Xi ´pdXipiq {dγ ´p2β 1 ` β 2 q ` ppXipiiq ` Xipijq q

ą0 dp{dγ “ piq pjq 2 ´r2bpb ` θqβ 1 ` β 2 b ` p2b ` θqs 2β 1 pb ` θq ` β 2 b ` Xi ` Xi piq piiq pijq rbp2β 1 ` β 2 q ` Xi s ´rp2β 1 ` β 2 q ´ ppXi ` Xi qs

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29

and ´r2bpb ` θqβ 1 ` β 2 b2 ` p2b ` θqs ´dXi {dγ piq piq bp2β 1 ` β 2 q ` Xi ´pdXi {dγ

ą 0. dz{dγ “ piq pjq 2 ´r2bpb ` θqβ 1 ` β 2 b ` p2b ` θqs 2β 1 pb ` θq ` β 2 b ` Xi ` Xi piq piiq pijq bp2β 1 ` β 2 q ` Xi ´rp2β 1 ` β 2 q ´ ppXi ` Xi qs piq

piiq

pijq Xi qsdJppq{dp ą 0.

˝

Proof of Result 2.

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Note that dXi {dγ ą 0, dXi {dγ ą 0, and the denominator is ´rp2β 1 ` β 2 q ´ ppXi

`

1 and 2. Taking derivatives of p3q and p2q w.r.t. θ, we have piq

pjq

´dp{dθr2bpb`θqβ 1 `β 2 b2 `p2b`θqs`dz{dθr2β 1 pb`θq`β 2 b`Xi `Xi s “ pp´pL q`Lpβ 1 ´β 2 q´dXi {dθ, piiq

piq

Then

piq

(A5)

piq pjq pp ´ pL q ` Lpβ 1 ´ β 2 q ´ dXi {dθ 2β 1 pb ` θq ` β 2 b ` Xi ` Xi piq piiq pijq ´pdXi {dθ ´p2β 1 ` β 2 q ` ppXi ` Xi q

dp{dθ “ pjq piq 2 ´r2bpb ` θqβ ` β b ` p2b ` θqs 2β pb ` θq ` β b ` X ` X 1 2 1 2 i i piq piiq pijq rbp2β 1 ` β 2 q ` Xi s ´rp2β 1 ` β 2 q ´ ppXi ` Xi qs

M

and

pijq

` Xi qs “ ´pdXi {dθ.

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dp{dθrbp2β 1 ` β 2 q ` Xi s ´ dz{dθrp2β 1 ` β 2 q ´ ppXi

(A4)

PT

ED

´r2bpb ` θqβ 1 ` β 2 b2 ` p2b ` θqs pp ´ pL q ` Lpβ 1 ´ β 2 q ´ dXi {dθ piq piq ´pdXi {dθ bp2β 1 ` β 2 q ` Xi . dz{dθ “ piq pjq ´r2bpb ` θqβ 1 ` β 2 b2 ` p2b ` θqs 2β 1 pb ` θq ` β 2 b ` Xi ` Xi piq piiq pijq bp2β 1 ` β 2 q ` Xi ´rp2β 1 ` β 2 q ´ ppXi ` Xi qs piq

piiq

pijq

CE

Note that dXi {dθ ą 0, dXi {dθ ą 0, and the denominator is ´rp2β 1 ` β 2 q ´ ppXi

`

Xi qsdJppq{dp ą 0. It turns out that the sign of p ´ pL ` Lpβ 2 ´ β 1 q ´ dXi {dθ determines

AC

the result of the analysis. It includes three terms. The first term, pp ´ pL q, represents the

decreased benefit of selling to the regular market when price competition intensifies. The

second term, Lpβ 1 ´ β 2 q, represents the decreased benefit from the discount market when

price competition in the regular market grows. The third term, dXi {dθ, represents the

rate at which regular sales increase with θ. If p ´ pL ` Lpβ 1 ´ β 2 q ´ dXi {dθ ď 0, i.e., the

switchover effect of θ dominates, then we have dp{dθ ą 0 and dz{dθ ą 0. Otherwise, from

pA5q, knowing its RHS is negative, if dz{dθ ă 0, then dp{dθ ă 0. However, dz{dθ ă 0 and piq

dp{dθ ă 0 invalid pA5q. The result on b is similarly derived, using dXi {db ă 0 and dXi {db ă

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30 piq

pjq

0. Note that ´dp{dbr2bpb ` θqβ 1 ` β 2 b2 ` p2b ` θqs ` dz{dbr2β 1 pb ` θq ` β 2 b ` Xi ` Xi s “ piq

pr2 ` 2β 1 pb ` θq ` β 2 bs ´ pL ` Lβ 1 ´ dXi {db. and dp{dbrbp2β 1 ` β 2 q ` Xi s ´ dz{dbrp2β 1 ` piiq

β 2 q ´ ppXi

pijq

piq

` Xi qs “ ´p2β 1 ` β 2 qp ´ pdXi {db. ˝

Proof of Result 3.

The result for β 1 and β 2 can be similarly derived, since piq

pjq

´dp{dβ 2 r2bpb ` θqβ 1 ` β 2 b2 ` p2b ` θqs ` dz{dβ 2 r2β 1 pb ` θq ` β 2 b ` Xi ` Xi s “ bL and piiq

pijq

` Xi qs “ ´L.

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piq

dp{dβ 2 rbp2β 1 ` β 2 q ` Xi s ´ dz{dβ 2 rp2β 1 ` β 2 q ´ ppXi

piq

pjq

´dp{dβ 1 r2bpb ` θqβ 1 ` β 2 b2 ` p2b ` θqs ` dz{dβ 1 r2β 1 pb ` θq ` β 2 b ` Xi ` Xi s “ 2pb ` θqL piq

piiq

and dp{dβ 2 rbp2β 1 ` β 2 q ` Xi s ´ dz{dβ 2 rp2β 1 ` β 2 q ´ ppXi Proof of Proposition 1.

pijq

` Xi qs “ ´2L. ˝

1. The price-competition-only game (γ “ 0q is,

maxπ i , where π i “ pci ´ Li ´ zi qp0 ` pi pLi ` E minti , zi uq and Li “ a ´ pb ` θqpi ` θpj . pi ,zi

For any given pj , a duopoly will set the optimal price by solving the following equation a`pb`θqp0 `θpj `E minti ,zi ppi qu 2pb`θq

where zi ppi q is solved from p0 “ pi Prpi ą zi q.

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pi “

Li ´ zi qp0 ` pi pLi ` E minti , zi uqs, which solves pi “ p0 “ pi Prpi ą zi q.

ř

rpci ´ pi ,zi pi ,zi i“1,2 a`pb`θqp0 `θpj `E minti ,zi ppi qu`θppj ´p0 q where 2pb`θq

A monopoly will determine the optimal price for i to maximize maxEπ “ max

Note that the LHS of both equations increases with pi . The monopoly’s equation has a

M

D larger RHS than the duopoly case. We then conclude that given any pj , pM i ppj q ą pi ppj q.

M D Similarly pM j ppi q ą pj ppi q. All of them are up-ward sloping (e.g., pi ppj q{dpj ą 0), so the

ED

D D M intersection of pM i ppj q and pj ppi q is larger than that of pi ppj q and pj ppi q. As

0 for both cases, we have the required.

dzi dpi

“ pi r1pzi q ě i

2. With both price and capacity competition game, pi ,zi

PT

maxπ i where π i “ pci ´ Li ´ zi qp0 ` pi pLi ` Xi q. The interior optimal solution solves piq

´p0 ` pi Xi “ 0 and pb ` θqp0 ` a ´ 2pb ` θqpi ` θpj ` Xi “ 0. Given any pj , zj , we define the

CE

explicit best response protection level as ziD “ hD ppi q, and the best response price function

as g D pzi q “

pb`θqp0 `a`θpj `Xi . 2pb`θq

AC

If the two flights are operated by a monopoly or alliance, the problem becomes, ř max rpci ´ Li ´ zi qp0 ` pi pLi ` Xi uqs where Li “ a ´ pb ` θqpi ` θpj . An interior optimal zi ,pi i“1,2

piq

piq

solution solves ´p0 `pi Xi `pj Xj “ 0 and pb`θqp0 `a´2pb`θqpi `θpj `Xi `θppj ´p0 q “ 0.

Given any pj , zj , define the explicit best response protection level as ziM “ hM ppi q, and

the best response price function as g M pzi q “ Note that

g D pzi q.

dg D pzi q dzi

“

dg M pzi q , dzi

pb`θqp0 `a`θpj `Xi `θppj ´p0 q . 2pb`θq

so the two lines in the (p, z) space are parallel and g M pzi q ą

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31

Both hM ppi q and hD ppi q start at the same point (p0 , 0) and hM ppi q ă hD ppi q. Let the

intersection for the monopoly case be ziM ppj , zj q, pM i ppj , zj q, and the unique intersection

for the duopoly be ziD ppj , zj q, pD i ppj , zj q. Only three possibilities occur as shown in Figures

Illustration 1

Figure B.2

Illustration 2

PT

CE AC Figure B.3

ED

M

Figure B.1

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1-3, we get the required. ˝

Illustration 3

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32

Proof of Proposition 2. The objective function of a monopolist is ř maxp1 ,p2 ,z1 ,z2 rLi pLi ` pi pqi ppi , pj q ` ErmintDis , zi usqs. It can be shown that the optimal i“1,2

solution is interior to the feasible range. There will be four first-order conditions, but we require only the first-order conditions with respect to zi . It is easy to derive them as p1q

p1q

p1q

p1q

M M pM 1 X1 |z1M ,z2M ` p2 X2 |z1M ,z2M “ p0 . As X2 ď 0, then we can write p1 X1 |z1M ,z2M ě p0 .

p1q

p1q

p1q

p1 X1 |z1 ,z2 “ p0 . Thus pM 1 X1 |z1M ,z2M ě p1 X1 |z1 ,z2 . p1q

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as

If β 1 “ β 2 “ 0, then the first order condition for the duopoly, equation (2), can be written

Now if ziM ą zi and we know that X1 is decreasing in z1 and z2 , then we must have that

pM i ą pi . From Result 3, we know that increasing β 1 and β 2 cannot lead simultaneously to

a higher p and a lower z. So with β 1 ą 0 and β 2 ą 0, the result still holds. ˝

Proof. As B 2 π i {Bpi Bzi ě 0, the game is supermodular (Topkis

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Proof of Proposition 3.

1998), and has decreasing differences in ppi , nq and pzi , nq for i “ 1, 2. So the supermodular

game has the equilibrium point decreases with n. Firms are also worse off since θppj ´ pi qppi ´ p0 q `

Bπ i Bpj Bpj Bn

Proof of Theorem 2.

˝

“

N 1. We need to show that π N “ pi rqi ppi , pj q ` E minti , zi us is quai

M

siconcave in pi .

ă 0 in the symmetric equilibrium.

Bπ i Bn

ED

N dπ N i “ qi ppi , pj q ` E minti , zi u ` pi r´pbi ` θq ` pbi ` θq Prpi ą zi qs. dpi N d2 π N i “ 2r´pbi ` θq ` pbi ` θq Prpi ą zi qs ` pi pbi ` θqr´pbi ` θqfi pzi qs dp2i

“ ´2pbi ` θq ` pbi ` θq Prpi ą zi qr2 ´ pi pbi ` θqfi pzi q{ Prpi ą zi qs.

Using IFR of i , similar arguments as Theorem 1 complete the proof.

PT

2. We only need to prove that π Y1 N “ L1 pL1 ` p1 rq1 pp1 , p2 q ` E mint1 , z1 us is quasiconcave

in p1 , given z1 “ k ´ mp1 .

“ pb1 ` θ ` mqpL1 ´ β 1 L1 pb1 ` θ ` mq ` rq1 pp1 , p2 q ` E mint1 , z1 us `p1 r´pb1 ` θq ´

CE

N dπ Y 1 dp1

m Prp1 ą z1 qs. N d2 π Y 1 2 dp1

“ ´2β 1 pb1 ` θ ` mq2 ` 2r´pb1 ` θq ´ m Prp1 ą z1 qs ´ p1 m2 f1 pz1 q

AC

“ ´2β 1 pb1 ` θ ` mq2 ´ 2pb1 ` θq ` Prp1 ą z1 qr´2m ´ m2 p1 f1 pz1 q{ Prp1 ą z1 qs.

Using IFR of i , similar arguments as Theorem 1 complete the proof.

3. First, firm one benefits from offering early discount sales if Bπ Y1 N {Bz1 |z1 “C1 ´q1 ă 0.

Using (2) and the fact that L1 “ L2 “ 0, we have Bπ Y1 N {Bz1 |z1 “C1 ´q1 “ ´p0 ` p1 PrpD1 ą C1 q.

Substituting p1 “ p0 ` ErmintD1 , C1 us{pb1 ` θq (solved from (3)), we get the required.

N Now, we prove that it also benefits the competitor. Note that with γ 12 “ 0, π N and π Y2 N 2

N NN YN YN are essentially the same, denote them by π 2 . To compare π 2 ppN 1 , p2 q with π 2 pp1 , p2 q,

we need to see how π 2 changes with p1 and p2 .

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33 N Bπ 2 {Bp2 “ 0 for any best response p2 and Bπ 2 {Bp1 ą 0, so firm two benefits if pY1 N ě pN 1 .

The rest of the proof includes three steps. First we show that pY1 N increases with p0 ,

N YN N then that its smallest value is pN ě πN 1 , and finally we show π 2 2 .

Step 1. The first-order condition of (5) is

pb1 ` θqp0 ´ 2β 1 pb1 ` θqL1 ` q1 ` X1 ´ pb1 ` θqp1 “ 0,

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(A6)

´p0 ` 2β 1 L1 ` p1 Prp1 ą z1 q “ 0.

(A7)

Given p2 , let pr1 and z1r be the best response functions solved from (A6)-(A7). Take derivatives of (A6)-(A7) w.r.t. p0 , we have

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dpr1 2β 1 pb1 ` θq ` Prp1 ą z1 q dz1r 1 “ ` , dp0 2pb1 ` θqrβ 1 pb1 ` θq ` 1s dp0 2rβ 1 pb1 ` θq ` 1s 2β 1 ` p1 f1 pz1 q 1 dpr1 dz1r “ ` . dp0 2β 1 pb1 ` θq ` Prp1 ą z1 q dp0 2β 1 pb1 ` θq ` Prp1 ą z1 q

(A8) (A9)

It can be shown that at the optimality (pr1 , z1r ), in the (dpr1 {dp0 , dz1r {dp0 ) quadrant, the

first line (A8) has a smaller (positive) slope, a smaller (positive) vertical intercept and a larger (absolute) horizontal intercept (which is negative) than the second line (A9), so (A8)

M

and (A9) joint at the third quadrant, i.e., dpr1 {dp0 ą 0 and dz1r {dp0 ă 0. Since dpr2 {dp0 “ 0,

we have dpY1 N {dp0 ą 0 and dz1Y N {dp0 ă 0.

ED

Step 2. For YN scenario, when p0 is very small, z1 will be set to the largest possible

value. Define p0 to be the value of p0 such that the optimal solution of firm one is the

PT

boundary solution z1 “ C1 ´ L1 . Putting this into (A6)-(A7), (A10)

´p0 ` 2β 1 L1 ` p1 Prp1 ą z1 q “ 0;

(A11)

CE

pb1 ` θqp0 ´ 2β 1 pb1 ` θqL1 ` q1 ` X1 ´ pb1 ` θqp1 “ 0,

AC

this can be simplified to pb1 ` θqp1 Prp1 ą z1 q ` q1 ` X1 ´ pb1 ` θqp1 “ 0, where z1 “ C1 ´ L1 , equivalent to the reaction function of firm one in the NN scenario. Solving the game we

N N will have pYi N “ pN and z1Y N “ C1 ´ LN under p0 “ p0 . i 1

N Step 3. For any p0 ě p0 , we have pY1 N ě pN 1 . Firm two is also better off. ˝

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