Advance Selling, Competition and Brand Substitutability

Oksana Loginova∗

July 23, 2018

Abstract

This paper studies the impact of competition on the benefits of advance selling. I con-struct a two-period price-setting game with two firms that produce different brands servingheterogeneous consumers. Some consumers prefer one brand, others prefer the other brand.Consumers derive common value from their preferred brand, but they differ in how stronglythey dislike their less preferred brand. One of the firms can offer consumers the opportunityto pre-order its product in advance of the regular selling season. I calculate the benefits ofadvance selling when this firm faces competition from the other firm in the regular sellingseason and when it does not. Competition is shown to enhance the benefits of advanceselling when in the advance selling season consumers are uncertain about which brand theywill prefer. Comparative statics analysis with respect to brand substitutability reveal someinteresting results.

Keywords: advance selling, price competition, strategic consumers, valuation uncertainty,consumer heterogeneity, substitutability of brands.

JEL codes: C72, D42, D43, L12, L13, M31.

∗Department of Economics, University of Missouri, 118 Professional Bldg, Columbia, MO 65211, USA; E-mail:loginovao@missouri.edu; Phone: +1-573-882-0063.

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1 Introduction

Advance selling is a standard practice in the travel and leisure industries. Consumers oftenpurchase airline tickets, cruises, vacation packages, tickets for sporting events, concerts andBroadway shows in advance. With recent developments in the Internet and information technol-ogy, advance selling has also been adopted in retail, including books, movies and CDs, software,video game consoles and smartphones. In the automobile industry, car manufactures frequentlyoffer bonus cash deals to those who place an order prior to the launch of a new model.

The essence of advance selling is that sales of a product precede the date when the prod-uct becomes available for consumption. Because advance orders are pre-committed, advanceselling helps the firm partially resolve uncertainty about future demand for its product, thus al-lowing for better production and distribution capacity planning. By placing an advance order,consumers avoid the risk of facing a stock-out, as it guarantees delivery of the product in theregular selling season. At the same time, consumers have to make their purchasing decisionswhile still uncertain about their valuations for the yet-to-be-seen product, which may allow thefirm to sell to more consumers.

The theoretical literature on advance selling is small but growing. Most of the papers showhow a monopolist can take advantage of advance selling (see Xie and Shugan, 2001; Boyaciand Ozer, 2010; Nocke, Peitz, and Rosar, 2011; Prasad, Stecke, and Zhao, 2011; Li and Zhang,2013; Loginova, Wang, and Zeng, 2017, inter alia). Monopolistic models are well-suited forunique experiential products such us a Justin Bieber concert, a new Harry Potter book or the nextNeed for Speed game. These products have a well defined customer base of devoted followerswho would not consider Tolkien a replacement for Rowling or Mario Cart a replacement forNeed for Speed.

Many other products have close substitutes. Both American and Delta offer multiple flightsbetween New York and Chicago. On any given night, there are dozens of Broadway shows thattarget similar audiences. A consumer who is planning to purchase a new SUV can pre-ordernext year’s Chevrolet Equinox and enjoy a special introductory price, or he can wait until thelaunch of the model to compare it directly to a Honda CR-V or a Toyota RAV4.

In this paper I inject competition by considering an alternative brand to the firm’s productin the regular selling season. My main goal is to investigate the impact of competition on thebenefits of advance selling. Stated differently, does a firm have stronger or weaker incentivesto advance sell when it faces competition from another firm in the regular season compared towhen it does not? Our first instinct may be to say that competition increases the benefits ofadvance selling because of the business-stealing effect. However, the effect of advance sellingon price competition in the regular selling season must also be taken into account. The latterdepends on which consumer types purchase in advance.

The focus herein is on advance selling of tangible products, e.g., new car models, newversions of game consoles, tablets and smart phones. In such markets: (1) Consumer valuationuncertainty can only be resolved when the product becomes available for consumption. In theaforementioned Chevrolet Equinox example, a test drive is not available at the time of pre-order.(2) Advance selling is a marketing strategy (as opposed to standard practice in service industriessuch as airlines, car rentals, concerts and hotels). (3) There are no hard capacity constraintsunlike, say, in the entertainment industry where the number of seats in the venue is limited. (4)

2

Retailers often cannot commit to future prices in advance.With these observations in mind, I build a two-period price-setting game in which two firms,

A and B, produce and sell two different brands. Consumers are heterogeneous in two dimen-sions. First, half of the consumers prefer brand A and the other half prefer brand B (type Aand type B consumers). Second, while consumers receive the same level of utility from theirpreferred brand, they differ in how strongly they dislike their less preferred brand. I considerthe situation in which one firm (firm A) can offer consumers the opportunity to pre-order itsproduct in advance of the regular selling season. Under the assumption that firm A cannotcredibly commit to a particular regular selling season price in advance, I calculate firm A’s ben-efits of advance selling when it faces competition from firm B in the regular selling season (thecompetitive setting) and when it does not (the monopoly setting).1

Does competition enhance the benefits of advance selling? The answer depends on whatconsumers know about their valuations for the two brands in the advance selling season. InModel I (Section 3) a consumer is uncertain about which brand she will prefer. I find thatcompetition enhances the benefits of advance selling. This is because consumers with weakbrand preferences (low disutilities from consuming their less preferred brand) are more likely topre-order, which relaxes price competition in the regular selling season.

In Model II (Section 4) a consumer knows which brand she will prefer, but is uncertain abouther disutility from consuming the alternative. I show that in both competitive and monopolysettings firm A chooses not to implement advance selling. The intuition behind this result isas follows. Type B consumers have stronger incentives to wait until the regular selling seasonthan type A consumers, which reduces firm A’s price in the regular selling season. To encourageconsumers who prefer brand A to buy in the advance selling season, firm A must set its pre-orderprice below the (already low) regular selling season price. Generating positive demand in theadvance selling season turns out to be too costly.

In the course of the equilibrium analysis, I uncover the interplay between consumer valua-tion uncertainty (Models I and II), competition (whether firm A faces competition in the regularselling season) and the level of brand substitutability.2 The most interesting results are:

• In the competitive setting of Model I, firm A has stronger incentives to advance sell whenthe level of brand substitutability is high. The reverse holds in the monopoly setting.

• For some parameter values, in Model I firm A is better off in the competitive setting thanin the monopoly setting.

• Competition enhances the benefits of advance selling when consumers are initially uncer-tain about their preferred brand (Model I). The enhancement is greater when the level ofbrand substitutability is high.

1In the monopoly setting with only brand A available for consumption, type A consumers are fans of firm A’sproduct, while type B consumers care less about it.

2I assume that consumers draw their disutilities from the interval [0, d], so d is used as a measure of brandsubstitutability. The lower (higher) is d, the higher (lower) is the level of substitutability between the two brands. Inthe marketing literature, the most commonly used measures of brand substitutability are brand switching probabilitiesand cross elasticities. See, for example, Bucklin, Russell and Srinivasan (1998).

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• There are no benefits to selling in advance when consumers know their preferred brandbut are uncertain about their disutilities from consuming the alternative (Model II).

In Section 5, I relax the assumption that only one firm can advance sell. The two firmssimultaneously and non-cooperatively choose whether to implement advance selling, resultingin four separate scenarios. Under the assumptions of Model I the game formed by these fourscenarios has two asymmetric pure-strategy Nash equilibria: only firm A (firm B) implementsadvance selling. Thus, all the results obtained in Section 3 continue to hold. Next I find that ifconsumer valuation uncertainty is as in Model II, the game has two symmetric equilibria. In thefirst equilibrium both firms implement advance selling and all consumers pre-order, while in thesecond one neither firm implements advance selling.

1.1 Related Literature

Most of the theoretical studies that examine advance selling are single-firm models. Only fivepapers study advance selling in a competitive environment. Out of these five, Shugan and Xie(2005), Cachon and Feldman (2017), and Moller and Watanabe (2016) are most relevant, as theyexplore the impact of competition on advance selling and take into account strategic behavior ofconsumers.

In Shugan and Xie (2005), two firms simultaneously decide whether to advance sell or spotsell, and at which prices. If a firm advance sells, consumers can only purchase the product inthat period.3 Consumers observe the prices and decide from which firm to purchase accordingto a market share attraction model (see, for example, Bell, Keeney, and Little, 1975). Providedthat the firms’ marginal costs are sufficiently low, the authors show that advance selling by bothfirms is an equilibrium. Moreover, each firm obtains a higher profit when both firms advancesell than when they both spot sell.

In contrast, Cachon and Feldman (2017) show that even though advance selling by bothfirms may occur in equilibrium, the firms would in most cases benefit if they could commit tosell on the spot market. In their setup two firms simultaneously announce advance period pricesand then spot period prices. Under the first demand specification, consumers initially are unsureabout which firm they will prefer as well as how strongly they will value their preferred firm.Specifically, the value from their preferred firm can be high or low, while the value from theother firm is zero. Consequently, the firms optimally set their prices at the high value in the spotperiod. The authors find that the possibility of advance selling often hurts the firms. This isbecause consumers are more homogeneous in the advance period, which intensifies price com-petition and drives down the advance period prices. Under the second demand specification, afraction of consumers arrive in the advance period and are loyal to one of the two firms, whereasthe rest arrive in the spot period. Because consumers buy in advance only if they get a gooddiscount, competition in the spot period forces the firms to further lower their advance periodprices, even though they do not directly compete in the advance period. Again, competitionlowers the attractiveness of advance selling.

3In my model if a consumer did not purchase the product from firm A in the advance selling season, she can stillpurchase it in the regular selling season. Thus, firm A’s regular selling season price influences the pre-order price.

4

Unlike the current manuscript and the above two papers, Moller and Watanabe (2016) endowfirms with the ability to commit to future prices. There are two differentiated products. Inthe advance purchase period each consumer receives an imperfect signal about the identity ofhis/her preferred product. More “choosy” consumers derive a higher consumption value fromtheir preferred product and a lower consumption value from the other product. The authorscompare the case in which the products are sold by two competing firms with the case in whichthe products are sold by one firm. They show that advance purchase discounts are larger in theoligopolistic scenario. Competition results in welfare reduction, as it increases the fraction ofconsumers who buy in the advance purchase period, without full knowledge of their preferences.

It is worth noting that Cachon and Feldman (2017), Moller and Watanabe (2016), and ModelI of the present manuscript all assume that consumers are uncertain about the identity of theirpreferred brand/firm. In contrast, consumers in Model II of the present manuscript know theirpreferred brand, but are uncertain about their valuation from consuming the alternative. This isa reasonable assumption in settings where the products are derivatives of the existing products(e.g., new versions of Apple iPhone and Samsung Galaxy).

Tang, Rajaram, Alptekinoglu, and Ou (2004) and McCardle, Rajaram, and Tang (2004)consider situations in which firms and retailers have to purchase inventories before a short sellingseason with uncertain demand. Hence, advance selling enables retailers to lock in a portion ofthe customer demand and use the advance orders to develop more accurate forecasts and supplyplans. Tang, Rajaram, Alptekinoglu, and Ou (2004) assume two brands, A and B. A singlefirm (firm A) launches the advance booking discount (ABD) program by offering a discount tothe fixed regular selling season price.4 The number of consumers who pre-order brand A is anexogenously given increasing function of the discount. The authors evaluate the benefits of theABD program, characterize the optimal discount price, and analyze how it is affected by thedegree of demand uncertainty and the correlation between the demands for brand A and brandB. McCardle, Rajaram, and Tang (2004) extend Tang, Rajaram, Alptekinoglu, and Ou (2004)by assuming that each of the two retailers simultaneously and independently chooses whetherto launch an ABD program. The authors show that if it is optimal for one firm to adopt an ABDprogram, the unique equilibrium has both firms adopting an ABD program.

Several papers study advance selling in a monopoly setting and, like the present manuscript,involve uncertainty of consumer valuations. Xie and Shugan (2001) consider a variety of sit-uations, including limited capacity, refunds, and exogenous credibility. Their findings suggestthat given buyer uncertainty and sufficiently low marginal costs, the monopolist can potentiallyearn greater profit by either increasing sales to more buyers or by allowing a premium advanceprice. Fay and Xie (2010) explore two pricing mechanisms employed by a two-product mo-nopolist: advance selling and probabilistic selling. Under advance selling the seller encouragesconsumers to make decisions before their valuations for the two products are known (thus ho-mogenizing consumers), while under probabilistic selling, the seller motivates consumers withweak product preferences to choose the uncertain option (thus separating heterogeneous con-sumers). Nocke, Peitz, and Rosar (2011) assume that the seller can credibly commit to futureprices. By offering an advance-selling discount, the monopolist induces consumers with highexpected valuations to purchase the product in advance. Moller and Watanabe (2010) consider

4This setup is similar to the present manuscript in the sense that only one firm can advance sell.

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advance purchase discounts and clearance sales. They determine how the comparison of thesetwo pricing strategies depends on price commitment, the availability of temporal capacity lim-its, the rationing rule, and resale. Chu and Zhang (2011) allow the firm to control the releaseof information about the product at pre-order. They show that the seller may want to releasesome information or none, but never all. In Prasad, Stecke, and Zhao (2011) random (possiblycorrelated) numbers of consumers arrive in the advance and regular selling seasons. The authorsfind that implementing advance selling is not always optimal. Several papers examine advanceselling in an environment where consumers deviate from rational decision making (Zhao andStecke, 2010; Naisiry and Popescu, 2012; Lim and Tang, 2013). Yu, Ahn, and Kapuscinski(2015) analyze the role of advance selling as a signal of product quality. Yu, Kapuscinski,and Ahn (2015) study the impact of the interdependence of consumers’ valuations on advanceselling.

Contribution of this paper to the literature on advance selling is three-fold. First, I factorbrand substitutability into the model, which none of the other theoretical papers have done.Second, I consider two different types of consumer valuation uncertainty. Third, the existingliterature arrives at opposing conclusions about the welfare implications of advance selling, orthe effect of competition on the firms’ incentives to advance sell. Indeed, while some of thepapers reviewed above conclude that firms can extract greater profits when advance selling ispossible (Shugan and Xie, 2005), others find that advance selling decreases the firms’ profits(Cachon and Feldman, 2017) and results in welfare reduction (Moller and Watanabe, 2016).This paper discovers that brand substitutability and the nature of consumer valuation uncertaintyplay an important role in determining the equilibrium outcome, thus shedding some light on thecontroversy in the literature.

2 Setup

There are two firms: firm A sells brand A and firm B sells brand B. The firms possess identicalconstant returns to scale technologies with marginal cost c = 0.

The market consists of continuum consumers of mass one. Consumers have unit demandsand are heterogeneous in two dimensions. First, half of the consumers prefer firm A’s productand the rest firm B’s product. I will refer to the former group as type A consumers and to thelatter as type B consumers. Second, consumers differ in how strongly they dislike their lesspreferred brand. Specifically, a consumer is willing to pay v for her preferred brand and v − xfor the alternative. I assume that x is an i.i.d. draw from a uniform distribution on the interval[0, d], where d ≤ v. In essence, parameter dmeasures brand substitutability. The higher (lower)is d, the lower (higher) is the level of substitutability between the two brands.

There are two periods. Period 1 is the advance selling season and period 2 is the regularselling season. I analyze two models that differ in the nature of uncertainty that consumers facein the advance selling season. In Model I a consumer knows her x, but is uncertain about whichbrand she will prefer. In Model II a consumer knows her preferred brand, but is uncertain abouther x. Consumers fully learn their valuations for the two brands in the regular selling seasonwhen the brands become available for consumption.

Within each model, I consider two settings: competitive and monopoly. In the competitive

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setting, one firm (firm A) decides whether to offer consumers the opportunity to pre-order itsproduct in period 1 and at which price, knowing that in period 2 it will compete with the otherfirm (firm B).

In the monopoly setting, firm A is the only firm serving the market. I maintain the assump-tion that half of the consumers are willing to pay v for firm A’s product, while the other halfv − x, where x is uniformly distributed on [0, d]. Basically, type A consumers are consumerswho like brand A, while type B consumers are those who do not like firm A’s product that much.This demand is exactly what firm A would face in the competitive setting if firm B set its priceso high that no consumers wished to purchase brand B.

In both settings I assume that in the advance selling season firm A cannot commit to aparticular regular selling season price. Let pA1 and pA2 denote the prices of firm A in periods 1and 2, and let pB2 denote the price of firm B in period 2.

2.1 Benchmark: No Advance Selling

Models I and II are identical when firm A cannot implement advance selling, because in bothmodels consumer uncertainty is fully resolved before any purchases are made. I will refer tothis model as the no advance selling benchmark. The timeline for the competitive setting is:

1. Consumers learn their valuations for the brands.

2. Firms A and B simultaneously and non-cooperatively choose their regular selling seasonprices pA2 and pB2 .

3. Each consumer decides which brand to purchase, if any.

Proposition 1 reports the Nash equilibrium prices. Not surprisingly, as the brands become lesssubstitutable (d increases), price competition subsides.

Proposition 1 (Benchmark, competitive setting). In the competitive setting of the no advanceselling benchmark, the Nash equilibrium prices are pA2 = pB2 = d. Each consumer purchasesher preferred brand. Equilibrium profits for firm A and firm B, respectively, are

ΠA = ΠB = d/2.

In the monopoly setting of the no advance selling benchmark, firm A chooses pA2 to maxi-mize its profit.

Proposition 2 (Benchmark, monopoly setting). In the monopoly setting of the no advance sell-ing benchmark:

(i) When d ≤ v/3, firm A charges pA2 = v − d, all consumers buy, and firm A’s equilibriumprofit is ΠA = v − d.

(ii) When d ∈ (v/3, v], firm A charges pA2 = (v + d)/2. All type A consumers buy, and type Bconsumers with x ≤ (v − d)/2 buy. Firm A’s equilibrium profit is

ΠA =v

4+v2

8d+d

8.

7

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

d

equi

libriu

m p

rofit

s

ΠA

monopoly setting

ΠA = ΠBcompetitive setting

Figure 1: Equilibrium profits in the no advance selling benchmark

All proofs are relegated to the Appendix. In contrast to the competitive setting, in themonopoly setting firm A’s profit decreases in d. This is because the differentiation effect ofd – that price competition subsides as the brands become less substitutable – is “turned off” inthe monopoly setting. Figure 1 depicts the relationships between the firms’ equilibrium profitsin the two settings of the benchmark and the brand substitutability parameter d.5

We will use the results of Propositions 1 and 2 in the next two sections, when calculatingthe relative benefits of advance selling

firm A’s profit in Model I (Model II)firm A’s profit in the benchmark

∣∣∣∣competition

andfirm A’s profit in Model I (Model II)

firm A’s profit in the benchmark

∣∣∣∣monopoly

in the competitive and monopoly settings, respectively. When the first ratio is larger (smaller)than the second, we say that competition enhances (diminishes) the benefits of advance selling.

5Recall that d ∈ [0, v]. In Figure 1 v is normalized to 1, so the range for d becomes [0, 1].

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3 Model I

In Model I consumers are initially uncertain about which brand they will prefer in the regularselling season. Thus, in period 1 they are heterogenous with regard to x, but not the brand.

3.1 Competitive Setting

The timeline for the competitive setting is:

1. At the beginning of period 1, each consumer learns her x.

2. Firm A sets a pre-order price for its product, pA1 .

3. Consumers decide whether to pre-order the product from firm A.

4. At the beginning of period 2, each consumer learns her preferred brand.

5. Firms A and B simultaneously and non-cooperatively choose their regular selling seasonprices pA2 and pB2 .

6. Those consumers who did not pre-order in period 1 decide which brand to purchase, ifany.

Obviously, consumers with high values of x (strong brand preferences) have greater incen-tives to wait until the regular selling season to learn their preferred brand than those with lowvalues of x (weak brand preferences). Hence, suppose that consumers with x ∈ (x, d] are lefton the market in period 2. The next lemma establishes the Nash equilibrium prices in period 2as functions of the threshold x.

Lemma 1 (Nash equilibrium prices in period 2 given x). Suppose that consumers with x ∈(x, d] are on the market in period 2. The Nash equilibrium prices are:

(i) When d ≤ v/2,

pA2 (x) = pB2 (x) =

d+ 2√dx− x2, if x ≤ d

2

2d, if x ∈(d2 , d).

(ii) When d ∈ (v/2, v],

pA2 (x) = pB2 (x) =

d+ 2√dx− x2, if x ≤ d

2 −√2vd−v2

2

v, if x ∈(d2 −

√2vd−v2

2 , d)

.

It is easy to verify that the prices in Lemma 1 are (weakly) increasing functions of x. This isintuitive. The higher is x, the stronger are the brand preferences of those consumers who remainon the market in period 2, which relaxes price competition.

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After the announcement of pA1 , consumers form their expectations about the firms’ regularselling season prices – that pA2 and pB2 will be the same price p2. If a consumer pre-orders theproduct from firm A, her payoff will be v−pA1 if brand A is her preferred brand, and v−x−pA1otherwise. Thus, the consumer’s expected payoff is

v − x

2− pA1 .

If the consumer waits until period 2 to purchase her preferred brand (provided that p2 ≤ v), shereceives payoff

v − p2.

Therefore, the consumer will optimally pre-order the product from firm A if and only if

v − x

2− pA1 ≥ v − p2,

orx ≤ 2(p2 − pA1 ).

Summarizing the above,

x =

0, if pA1 > p2

min{d, 2(p2 − pA1 )}, if pA1 ≤ p2.(1)

For a given value of pA1 , a rational expectations equilibrium consists of prices pA2 and pB2 ,consumer expectations p2, and a threshold x such that: (i) pA2 and pB2 are the Nash equilibriumprices given x (Lemma 1); (ii) x describes the optimal consumer behavior given p2 (equation1); and (iii) consumer expectations are fulfilled (pA2 = pB2 = p2).

The statement of the next lemma, Lemma 3, is relegated to the Appendix because of itslength. It gives the equilibrium values of x and p2 as functions of pA1 . As it turns out, bothare (weakly) decreasing functions. The intuition is as follows. As pA1 increases, the number ofconsumers who pre-order the product from firm A decreases. Because lower values of x leadto more intense price competition in period 2, p2 decreases. That a higher pre-order price leadsto lower regular selling season prices is an interesting result driven by the information structureassumed in Model I.

It remains to calculate the optimal pre-order price pA1 that maximizes firm A’s expectedprofit:

ΠA(pA1 ) = pA1x(pA1 )

d+ p2(p

A1 )d− x(pA1 )

2d, (2)

where the functions x(pA1 ) and p2(pA1 ) are from Lemma 3.

Proposition 3 (Model I, competitive setting). In the competitive setting of Model I, the equilib-rium prices, profits, and consumer behavior are:

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(i) When d ≤ v/2, firm A charges pA1 = 3d/2 thereby inducing all consumers to pre-order.6 Inperiod 2 the firms charge p2 = 2d. The firms’ equilibrium profits are

ΠA =3d

2and ΠB = 0.

(ii) When d ∈ (v/2, v], firm A charges pA1 = 3v/4 thereby inducing consumers with x ≤ v/2 topre-order. In period 2 the firms charge p2 = v. The remaining consumers purchase theirpreferred brand. The firms’ equilibrium profits are

ΠA =v

2+v2

8dand ΠB =

v

2− v2

4d.

When the level of brand substitutability is high (d ≤ v/2), price competition in period 2 isintense (part (i) of Lemma 1). Low second-period prices imply firm A’s pre-order price must below to generate positive demand. When both pA1 and p2 are low, firm A’s profit is maximizedwhen the total number of consumers who purchase its brand,

x

d+d− x

2d,

is maximized. In equilibrium firm A charges pA1 = 3d/2, which induces all consumers topre-order: x = d.

When the level of brand substitutability is low (d > v/2), price competition in period 2 isless intense (part (ii) of Lemma 1). Maximizing (2) yields an interior solution in the sense thatfirm A’s equilibrium pre-order price attracts only a fraction of consumers: x < d.

Recall that in the competitive setting of the no advance selling benchmark (Proposition 1)the firms’ profits are increasing functions of d. In contrast, with advance selling in Model I firmA’s profit first increases, then decreases in d. Firm B’s profit increases in d. Figure 2 depicts thefirms’ equilibrium profits as functions of d.

With Propositions 1 and 3 in hand, we can calculate the relative benefits of advance sellingwhen firm A competes with firm B in the regular selling season:

firm A’s profit in Model Ifirm A’s profit in the benchmark

∣∣∣∣competition

=

3, if d ≤ v2

vd + v2

4d2, if d ∈

(v2 , v].

(3)

The ratio is strictly above one for all values of d: firm A always benefits from advance sellingin the competitive setting of Model I. Also note that the ratio (weakly) decreases in d. That is,the relative benefits of advance selling decrease as the two brands become less substitutable.

3.2 Monopoly Setting

Now we consider the monopoly setting. The results of the equilibrium analysis are summarizedin the proposition below.

6When no consumers are left on the market in period 2, we assume that the firms’ prices are pA2 (d) = pB2 (d) =min{2d, v}, obtained by substituting x = d into the expressions for pA2 (x) and pB2 (x) in Lemma 1.

11

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

d

equi

libriu

m p

rofit

s

ΠAmonopoly setting

ΠA

ΠB

competitive setting

Figure 2: Equilibrium profits in Model I

Proposition 4 (Model I, monopoly setting). In the monopoly setting of Model I, the equilibriumprices, profits and consumer behavior are:

(i) When d ≤ v/3, firm A sets the pre-order price sufficiently high to foreclose advance sales.In period 2 firm A charges pA2 = v − d. All consumers buy. Firm A’s equilibrium profit is

ΠA = v − d.

(ii) When d ∈ (v/3, v/2], firm A sets the pre-order price sufficiently high to foreclose advancesales. In period 2 firm A charges pA2 = (v + d)/2. All type A consumers buy, and type Bconsumers with x ≤ (v − d)/2 buy. Firm A’s equilibrium profit is

ΠA =v

4+v2

8d+d

8.

(iii) When d ∈ (v/2, v], firm A charges pA1 = 3v/4 inducing consumers with x ≤ v/2 to pre-order. In period 2 firm A charges pA2 = v. The remaining type A consumers buy. Firm A’sequilibrium profit is

ΠA =v

2+v2

8d.

12

When d ≤ v/2, the monopolist sets a high pre-order price so that all consumers wait until theregular selling season. This pricing strategy is payoff equivalent to not implementing advanceselling. Inability to set the regular selling season price in advance hurts the monopolist. If onlyfirm A could commit to pA2 = v! It then could charge pA1 = v − d/2, inducing all consumers(even the one with x = d) to pre-order. Firm A would earn

ΠA = v − d

2,

a higher profit than that in parts (i) and (ii) of Proposition 4.When d > v/2, the monopolist sells in both periods. Observe that firm A sets the same

prices and earns the same profit as in the competitive setting of Model I. The only differencebetween part (ii) of Proposition 3 and part (iii) of Proposition 4 is that in the competitive settingtype B consumers with x > v/2 are served by firm B.

Figure 2 depicts firm A’s equilibrium profit as a function of d. It is straightforward to showthat when

d ∈

((1 +

√12)v

11,v

2

]firm A’s profit is actually lower in the monopoly setting than in the competitive setting. Counter-intuitive at first, this result is driven by firm A’s inability to commit to a particular regular sellingseason price in advance. Recall that in the competitive setting firm A finds it optimal to pursuethe strategy of selling to all consumers in period 1 when d ≤ v/2. What if firm A adopts thesame strategy in the monopoly setting? Firm A will then have to change its pre-order price frompA1 = 3d/2 to pA1 = v− d, because consumers expect firm A’s regular selling season price to bepA2 = v − d. (In the absence of firm B, firm A sets its regular season price to reach everyone,including the type B consumer with x = d.) As d increases from 0 to v/2, 3d/2 surpasses v−d.

With Propositions 2 and 4 in hand, we can calculate the relative benefits of advance sellingwhen firm A is a monopolist:

firm A’s profit in Model Ifirm A’s profit in the benchmark

∣∣∣∣monopoly

=

1, if d ≤ v2

v/2+v2/(8d)v/4+v2/(8d)+d/8

, if d ∈(v2 , v].

(4)

The ratio equals one when d ≤ v/2: firm A finds it optimal not to implement advance selling(parts (i) and (ii) of Proposition 4).

We can now answer the main question of the paper, “Does competition enhance the benefitsof advance selling?” Figure 3 plots the benefit ratios (3) and (4). We see that in Model I com-petition always enhances the benefits of advance selling. This is because in Model I advanceselling attracts consumers with weak brand preferences (low x), relaxing price competition inthe regular selling season.

4 Model II

In Model II, a consumer knows her preferred brand but is initially uncertain about her disutilityfrom consuming the alternative. We first analyze the competitive setting.

13

0.0 0.2 0.4 0.6 0.8 1.0

01

23

4

d

bene

fits

of a

dvan

ce s

ellin

g

monopoly

competition

Figure 3: Benefits of advance selling in Model I

Type B consumers have stronger incentives to wait until period 2 than type A consumers.Lemma 4 (in the Appendix) reports the Nash equilibrium prices when the market in period 2consists of the fraction 1 − β of type B consumers (i.e., all type A consumers and the fractionβ of type B consumers have pre-ordered the product from firm A) and when it consists of alltype B consumers and the fraction 1 − α of type A consumers (i.e., the fraction α of type Aconsumers have pre-ordered the product from firm A).

For a given price pA1 that firm A might be charging, Lemma 2 reports the firms’ equilibriumprices in period 2 and the optimal consumer purchasing behavior.

Lemma 2 (Rational expectations equilibrium given pA1 ). Suppose that firm A charges pA1 inperiod 1. In a rational expectations equilibrium:

(a) If pA1 ≤ d/9, all consumers pre-order. In period 2 the firms charge pA2 = d/3 and pB2 =2d/3.7

(b) If pA1 ∈ (d/9, d/3], only type A consumers pre-order. In period 2 the firms charge pA2 = d/3and pB2 = 2d/3. Type B consumers with x ≤ d/3 purchase brand A, while those withx ∈ (d/3, d] purchase brand B.

7When no consumers are left on the market in period 2, we assume that the firms’ prices are pA2 = d/3 andpB2 = 2d/3 (as in the first part of Lemma 4).

14

(c) If pA1 ∈ (d/3, d], the fraction

α =3(d− pA1 )

2d

of type A consumers pre-order. In period 2 the firms charge

pA2 =d(3− 2α)

3= pA1 and pB2 =

d(3− α)

3=d+ pA1

2.

Type B consumers with x ≤ αd/3 and the remaining type A consumers purchase brandA, while type B consumers with x ∈ (αd/3, d] purchase brand B.

(d) If pA1 > d, no consumer pre-orders. In period 2 the firms charge pA2 = pB2 = d and eachconsumer purchases her preferred brand.

It turns out that firm A maximizes its profit when it sets a high pre-order price so that allconsumers wait until period 2 (part (d) of the above lemma).

Proposition 5 (Model II, competitive setting). In the competitive setting of Model II, firm A setsthe pre-order price sufficiently high to foreclose advance sales. In period 2 the firms chargepA2 = pB2 = d. Each consumer purchases her preferred brand. The firms’ equilibrium profitsare

ΠA = ΠB =d

2.

Why is it the case that any pA1 that generates positive demand (parts (a), (b) and (c) of Lemma2) is inferior to not implementing advance selling (charging a high pre-order price so that noconsumer pre-orders)? In Model II, type A consumers have stronger incentives to pre-order theproduct than type B consumers, so there will be more type B consumers than type A consumersin period 2. This intensifies price competition. Consider, for example, pA1 ∈ (d/9, d/3] (part(b) of Lemma 2). Such a price induces type A consumers to pre-order. In period 2 the firmscompete for type B consumers. To attract customers, firm A has to lower its price. Firm Bresponds by lowering its price, too. As a result, pA2 = d/3 and pB2 = 2d/3. (If all type Aconsumers and all type B consumers were present in period 2, the equilibrium prices would behigher, pA2 = pB2 = d.)

Intense price competition in period 2, in turn, puts pressure on firm A’s pre-order price.Indeed, if firm A pursues a strategy of selling to some consumers in period 1, pA1 cannot behigher than pA2 . Thus, it becomes very costly for firm A to generate positive demand in period1. Firm A is better off charging a high pre-order price so that no consumer pre-orders.

The same result – that firm A finds it optimal not to implement advance selling – also arisesin the monopoly setting.

Proposition 6 (Model II, monopoly setting). In the monopoly setting of Model II, firm A setsthe pre-order price sufficiently high to foreclose advance sales. In period 2, firm A’s equilibriumprice and profit are the same as in the monopoly setting of the no advance selling benchmark(Proposition 2).

15

Firm B

Firm A

N Y

N d2 , d

2 0, 3d2

Y 3d2 , 0 0, 0

(a) d ≤ v/2

Firm B

Firm A

N Y

N d2 , d

2v2 −

v2

4d , v2 + v2

8d

Y v2 + v2

8d , v2 −

v2

4d 0, 0

(b) d ∈ (v/2, v]

Figure 4: The both-firms-can-advance-sell version of Model I

The proof of Proposition 6 (in the Appendix) is rather involved despite its simple conclusion.Again, firm A’s inability to commit not to lower its price in period 2 for type B consumers hurtsthe monopolist. If firm A could credibly commit to pA2 = v, then it would charge pA1 = v−d/2,inducing both types of consumers to pre-order. Firm A’s profit of

ΠA = v − d

2

would be higher than that in Proposition 2.To sum up, in both competitive and monopoly settings of Model II firm A finds it optimal

to not implement advance selling. This happens because in Model II type B consumers havestronger incentives to wait until the regular selling season than type A consumers, which impliesfirm A’s price in period 2 cannot be high. To encourage consumers who prefer brand A to pre-order, the firm must set its pre-order price below this low regular selling season price. Generatingpositive demand in period 1 is too costly.

5 Both Firms Can Advance Sell

In this section I consider the situation in which both firms have an opportunity to advance sell.Specifically, at the beginning of the game firms A and B simultaneously and non-cooperativelychoose whether to implement advance selling. This leads to four different subgames: (N, N), (Y,N), (N, Y) and (Y, Y), where N stands for no advance selling and Y stands for advance selling.I will refer to this new game as the both-firms-can-advance-sell version of Model I (Model II).

Let us start with the both-firms-can-advance-sell version of Model I. Proposition 1 tells usthe equilibrium in subgame (N, N) and Proposition 3 covers subgames (Y, N) and (N, Y). Itis easy to show that the firms find themselves in a pure Bertrand price war in subgame (Y,Y): pA1 = pB2 = c = 0. Figures 4(a) and 4(b) depict the payoff matrices for d ≤ v/2 andd ∈ (v/2, v], respectively, corresponding to parts (i) and (ii) of Proposition 3. These matricesmake it clear that the game has two asymmetric Nash equilibria: (Y, N) and (N, Y).

Allowing both firms to advance sell turns out to be unnecessary, as (Y, Y) cannot be anequilibrium under any set of parameter values. Therefore, all the results obtained in Section 3continue to hold.

16

Finally, consider the both-firms-can-advance-sell version of Model II. We already know thatin the setting where only one firm can implement advance selling, that firm will optimally choosenot to implement advance selling (Proposition 5). What happens in subgame (Y, Y)? Lemma 5(relegated to the Appendix) shows that the firms set their advance selling prices equal to d andeach consumer pre-orders her preferred brand. The resulting profits are

ΠA = ΠB =d

2.

It follows that the both-firms-can-advance-sell version of Model II has two symmetric equi-libria: one in which none of the firms implement advance selling and the other in which bothfirms implement advance selling. In other words, allowing both firms to advance sell adds an-other equilibrium to the one found earlier in Section 4.

6 Concluding Remarks

This paper examines the effect of competition on the advantages of advance selling driven byconsumer valuation uncertainty. I assumed two firms that produce different brands. Some con-sumers prefer one brand, and others prefer the other brand. Consumers derive the same valuefrom their preferred brand, but they differ in how strongly they dislike their less preferred brand.One of the firms can offer consumers the opportunity to pre-order its product in advance of theregular selling season. I calculated the benefits of advance selling when this firm faces compe-tition from the other firm in the regular selling season and when it does not, under two demandspecifications. In Model I consumers are initially uncertain about the identity of their preferredbrand; in Model II consumers are uncertain about their valuation of consuming the alternative.

The first message of the paper is that consumer valuation uncertainty and competition in-teract to shape the firm’s incentives to advance sell. I showed that the firm does not want toimplement advance selling when consumers know their preferred brand from the outset (ModelII), and that competition enhances the benefits of advance selling in Model I.

The second important message is that the level of brand substitutability plays an importantrole in determining the equilibrium outcome. Its effect on the firm’s incentives to advance sellcan be positive or negative, depending on the model and the setting. For example, I found thatin the competitive setting of Model I the firm has stronger incentives to advance sell when thebrands are more substitutable, while the reverse holds in the monopoly setting.

Finally, I performed a robustness check to demonstrate that in Model I even when bothfirms can advance sell, they never pursue the option simultaneously. In contrast, although theequilibrium obtained in Model II continues to be an equilibrium in the both-firms-can-advance-sell version of Model II, another equilibrium obtains in which both firms choose to implementadvance selling.

Appendix

Proof of Proposition 1: Suppose we found a symmetric Nash equilibrium in which pA2 = pB2 =p; each consumer buys her preferred brand and the firms equilibrium profits are ΠA = ΠB =

17

p/2. If firm A deviates to p− ε, it will attract type B consumers with x ≤ ε, so the firm’s profitwill go up by

(p− ε) ε2d,

which equalspε

2d,

if we ignore all terms of the order ε2 or higher. At the same time, firm A will lose

ε1

2

on its existing customers due to the price decrease. Equilibrium requires

pε

2d= ε

1

2,

p = d.

Proof of Proposition 2: Any pA2 ∈ [v − d, v] will induce type A consumers and type B con-sumers with x ≤ v − pA2 buy the product. Firm A faces the following optimization problem:

maxpA2 ∈[v−d,v]

pA2

(1

2+v − pA2

2d

).

Differentiating w.r.t. pA2 and setting the derivative to zero yields

1

2+v − 2pA2

2d= 0,

implying that when d ≤ v/3 the firm should charge pA2 = v − d and sell to all consumers.Otherwise, the firm should charge

pA2 =v + d

2,

earning

ΠA =v

4+v2

8d+d

8.

Proof of Lemma 1: Suppose we found a symmetric Nash equilibrium in which pA2 = pB2 = p ≥d (p ≥ d will be confirmed later), implying that type A consumers buy from firm A and type Bconsumers from firm B. Since only consumers with x ∈ (x, d] are on the market in period 2, afirm (say, firm A) should decrease its price by at least least x to attract any of type B consumers.Then p is the highest price such that a deviation to p − x − ε, ε ∈ (0, d − x] is unprofitable.Suppose firm A deviates to p − x − ε. It will attract type B consumers with x ∈ (x, x + ε], sothe firm’s profit will go up by

(p− x− ε) ε2d.

18

At the same time, firm A will lose

(x+ ε)d− x

2d

on its existing customers. This deviation is unprofitable if for ε ∈ (0, d− x]

(p− x− ε) ε2d≤ (x+ ε)

d− x2d

,

ε2 − ε(p− d) + dx− x2 ≥ 0.

The left-hand side attains its minimum at ε = min{d − x, (p − d)/2}. When ε = d − x, theabove inequality becomes

p ≤ 2d.

When ε = (p− d)/2, it becomes

(p− d)2 ≤ 4(dx− x2),

p ≤ d+ 2√dx− x2.

As long as p ≤ v, we have

p =

d+ 2√dx− x2, if x ≤ d

2

2d, if x ∈(d2 , d].

When d > v/2 and x > d/2−√

2vd− v2/2, p = v.

Lemma 3 (Rational expectations equilibrium given pA1 ). Suppose that firm A charges pA1 inperiod 1 and let

g(pA1 ) ≡ 1

17

(10d− 2pA1 + 4

√2d2 + 6dpA1 − 4(pA1 )2

).

Under the rational expectations equilibrium we have:

(i) When d ≤ v/2,

(a) pA1 ≤ 3d/2 induces all consumers to pre-order; in period 2 the firms charge p2 = 2d;

(b) pA1 ∈ (3d/2, 7d/4] induces consumers with x ≤ 2(2d − pA1 ) to pre-order; in period2 the firms charge p2 = 2d and the remaining consumers purchase their preferredbrand;

(c) pA1 ∈ (7d/4, (3 +√

17)d/4] induces consumers with x ≤ g(pA1 ) to pre-order; inperiod 2 the firms charge p2 = pA1 +g(pA1 )/2 and the remaining consumers purchasetheir preferred brand;

(d) pA1 > (3 +√

17)d/4 induces all consumers to wait until period 2; in period 2 thefirms charge p2 = d and each consumer purchases her preferred brand.

19

(ii) When d ∈ (v/2, v],

(a) pA1 ≤ v − d/2 induces all consumers to pre-order; in period 2 the firms chargep2 = v;

(b) pA1 ∈ (v − d/2, v − (d −√

2vd− v2)/4] induces consumers with x ≤ 2(v − pA1 )to pre-order; in period 2 the firms charge p2 = v and the remaining consumerspurchase their preferred brand;

(c) pA1 ∈ (v − (d −√

2vd− v2)/4, (3 +√

17)d/4] induces consumers with x ≤ g(pA1 )to pre-order; in period 2 the firms charge p2 = pA1 + g(pA1 )/2 and the remainingconsumers purchase their preferred brand;

(d) pA1 > (3 +√

17)d/4 induces all consumers to wait until period 2; in period 2 thefirms charge p2 = d and each consumer purchases her preferred brand.

Proof of Lemma 3: We first consider the case d ≤ v/2 that corresponds to part (i) of the lemma.

(a) Suppose pA1 ≤ 3d/2. It follows from Lemma 1 that when x = d, p2 = 2d. It is left to checkthat under these prices all consumers will pre-order the product. Indeed, by equation (1)

x = min{d, 2(p2 − pA1 )} = d,

as 2(2d− pA1 ) ≥ d holds for pA1 ≤ 3d/2.

(b) Suppose pA1 ∈ (3d/2, 7d/4]. It follows from Lemma 1 that when x = 2(2d − pA1 ), whichdecreases from d to d/2 as pA1 increases from 3d/2 to 7d/4, p2 = 2d. We need to checkthat it is optimal for consumers with x ≤ 2(2d−pA1 ) to pre-order the product. By equation(1)

x = min{d, 2(p2 − pA1 )} = 2(2d− pA1 ),

as 2(2d− pA1 ) < d holds for pA1 ∈ (3d/2, 7d/4].

(c) Suppose pA1 ∈ (7d/4, (3 +√

17)d/4]. In this case x and p2 are found from the system ofequations p2 = d+ 2

√dx− x2

x = 2(p2 − pA1 ).

Substituting p2 = pA1 + x/2 into the first equation allows us to solve for x:

x =1

17

(10d− 2pA1 + 4

√2d2 + 6dpA1 − 4(pA1 )2

).

As pA1 increases from 7d/4 to (3+√

17)d/4, x decreases from d/2 to (√

17−1)d/(2√

17)and p2 = pA1 + x/2 decreases from 2d to (4 +

√17)d/

√17.

20

(d) Suppose pA1 > (3 +√

17)d/4. The above system of equations does not have a solution. Inthis case x = 0, implying p2 = d (Lemma 1). We need to check that under these pricesno consumer will pre-order the product. Indeed, by equation (1)

x = 0,

as pA1 > p2 = d holds for pA1 > (3 +√

17)d/4.

Proof of part (ii) of the lemma is very similar to that of part (i).

Proof of Proposition 3: We first consider the case d ≤ v/2 that corresponds to part (i) of theproposition.

(a) Suppose pA1 ≤ 3d/2. Then (2) is simply

ΠA(pA1 ) = pA1 .

The profit is maximized at pA1 = 3d/2, ΠA(3d/2) = 3d/2.

(b) Suppose pA1 ∈ (3d/2, 7d/4]. Then (2) becomes

ΠA(pA1 ) = pA12(2d− pA1 )

d+ 2d

d− 2(2d− pA1 )

2d= −2(pA1 )2

d+ 6pA1 − 3d.

The function attains its maximum at pA1 = 3d/2, ΠA(3d/2) = 3d/2.

(c) Suppose pA1 ∈ (7d/4, (3 +√

17)d/4]. Then (2) becomes

ΠA(pA1 ) = pA1g(pA1 )

d+

(pA1 +

g(pA1 )

2

)d− g(pA1 )

2d.

Numerical calculations show that for any value of pA1 ∈ (7d/4, (3+√

17)d/4], ΠA(pA1 ) <3d/2.

(d) Suppose pA1 > (3 +√

17)d/4. Then (2) is simply

ΠA(pA1 ) =d

2.

This is less than 3d/2.

It follows that when d ≤ v/2, firm A should set its pre-order price to 3d/2, which induces allconsumers to pre-order. Firm A’s equilibrium profit is 3d/2 and firm B’s profit is zero. Next,we consider the case d ∈ (v/2, v] that corresponds to part (ii) of the proposition.

(a) Suppose pA1 ≤ v − d/2. Then (2) is simply

ΠA(pA1 ) = pA1 .

The profit is maximized at pA1 = v − d/2, ΠA(v − d/2) = v − d/2.

21

(b) Suppose pA1 ∈ (v − d/2, v − (d−√

2vd− v2)/4]. Then (2) becomes

ΠA(pA1 ) = pA12(v − pA1 )

d+ v

d− 2(v − pA1 )

2d= −2(pA1 )2

d+

3v

dpA1 +

v(d− 2v)

2d.

The function attains its maximum at pA1 = 3v/4, ΠA(3v/4) = v/2 + v2/(8d). This isgreater than v − d/2.

(c) Suppose pA1 ∈ (v − (d−√

2vd− v2)/4, (3 +√

17)d/4]. Then (2) becomes

ΠA(pA1 ) = pA1g(pA1 )

d+

(pA1 +

g(pA1 )

2

)d− g(pA1 )

2d.

Numerical calculations show that for any value of pA1 ∈ (v − (d −√

2vd− v2)/4, (3 +√17)d/4], ΠA(pA1 ) < v/2 + v2/(8d).

(d) Suppose pA1 > (3 +√

17)d/4. Then (2) is simply

ΠA(pA1 ) =d

2.

This is less than v/2 + v2/(8d).

It follows that when d ∈ (v/2, v], firm A should set its pre-order price to 3v/4. This priceinduces consumers with x ≤ v/2 to pre-order. In period 2 firm A and firm B charge p2 = v.The remaining consumers purchase their preferred brand. The firms’ equilibrium profits areΠA = v/2 + v2/(8d) and ΠB = v/2− v2/(4d).

Proof of Proposition 4: The proof is completed in four steps.

1. We start with period 2 and assume that consumers with x ∈ (x, d] are left on the market.Firm A can charge pA2 = v and sell only to type A consumers. In this case it will earn

vd− x

2d.

If the firm charges pA2 = v − d, it will sell to all consumers and earn

(v − d)d− xd

.

Finally, the firm can charge pA2 ∈ [v− d, v− x] and sell to all type A consumers and typeB consumers with x ∈ (x, v − pA2 ], earning

pA2

(d− x

2d+v − pA2 − x

2d

).

The above function is maximized at

pA2 =

v+d−2x2 , if x ≤ 3d−v

2

v − d, if x ∈(3d−v2 , d

].

22

It can be shown that pA2 = v maximizes firm A’s profit in period 2 when d ∈ (v/2, v] andx > (d −

√2vd− v2)/2; pA2 = v − d is optimal when d ≤ v/3 or when d ∈ (v/3, v/2]

and x ∈ ((3d− v)/2, d]. Otherwise, the firm is better off charging pA2 = v+d−2x2 .

2. In period 1, a consumer pre-orders the product if and only if her expected payoff fromdoing so exceeds her expected payoff from postponing the purchase,

v − x

2− pA1 ≥

1

2(v − pA2 ) +

1

2max{0, v − x− pA2 }.

3. Let us check that the prices and consumer purchasing behavior postulated in Proposition4 satisfy the requirements of the rational expectations equilibrium.

(i) Let d ≤ v/3. Suppose firm A charges a high pre-order price so that no consumerpurchases in period 1 (x = 0). Then pA2 = v − d will maximize the firm’s profit inperiod 2 (see Step 1). Firm A’s profit in this case is as in Proposition 2(i)

ΠA = v − d.

(ii) Let d ∈ (v/3, v/2]. Suppose firm A charges a high pre-order price so that no con-sumer purchases in period 1 (x = 0). Then pA2 = (v+d)/2 will maximize the firm’sprofit in period 2 (see Step 1). Firm A’s profit in this case is as in Proposition 2(ii)

ΠA =v

4+v2

8d+d

8.

(iii) Let d ∈ (v/2, v]. Suppose firm A charges pA1 = 3v/4 in period 1 and consumerswith x ≤ v/2 pre-order the product. Then pA2 = v will maximize the firm’s profit inperiod 2 (see Step 1). Pre-ordering the product is optimal if and only if (see Step 2)

v − x

2− 3v

4≥ 0,

orx ≤ v

2.

Firm A’s profit in this case is

ΠA =v

2+v2

8d.

4. Tedious algebra and numerical calculations reveal that any other combination of pricesand consumer purchasing behavior – pA1 , pA2 and x – that satisfies the requirements of therational expectations equilibrium lead to a lower profit for firm A.

Lemma 4 (Nash equilibrium prices in period 2). Suppose the market in period 2 consists of thefraction 1− β of type B consumers, then the firms’ Nash equilibrium prices are:

pA2 =d

3and pB2 =

2d

3.

23

Consumers with x ≤ d/3 will purchase brand A and those with x ∈ (d/3, d] will purchase brandB. If the market consists of all type B consumers and the fraction 1 − α of type A consumers,then

pA2 =d(3− 2α)

3and pB2 =

d(3− α)

3.

Type B consumers with x ≤ αd/3 and type A consumers will purchase brand A, while type Bconsumers with x ∈ (αd/3, d] will purchase brand B.

Proof of Lemma 4: Suppose the market in period 2 consists of the fraction 1 − β of type Bconsumers. Then consumers with x ≤ x will purchase brand A and the rest brand B, where xsatisfies

v − x− pA2 = v − pB2 ,

x = pB2 − pA2 .

Firm A’s and firm B’s profits in period 2 will be

pA2(1− β)(pB2 − pA2 )

2d

and

pB2(1− β)(d− pB2 + pA2 )

2d,

respectively. Differentiating the profit of firm A w.r.t. pA2 and the profit of firm B w.r.t. pB2 , thensetting the derivatives to zero yields the system of equations pB2 − 2pA2 = 0

d− 2pB2 + pA2 = 0

from which the firms’s NE prices can be found, pA2 = d/3 and pB2 = 2d/3.If the market consists of all type B consumers and the fraction 1− α of type A consumers,

then type B consumers with x ≤ x and type A consumers will purchase brand A and the restbrand B, yielding

pA2

(pB2 − pA2

2d+

1− α2

)to firm A and

pB2d− pB2 + pA2

2d

to firm B. Differentiating the profit of firm A w.r.t pA2 and the profit of firm B w.r.t to pB2 , thensetting the derivatives to zero yields the system of equations pB2 − 2pA2 + (1− α)d = 0

d− 2pB2 + pA2 = 0

24

from which the firms’s NE prices can be found, pA2 = d(3− 2α)/3 and pB2 = d(3− α)/3.

Proof of Lemma 2: In each case we will verify that the firms’ prices and consumer purchasingbehavior constitute a rational expectations equilibrium.

(a) Suppose pA1 ≤ d/9 and all consumers pre-order the product from firm A. Lemma 4 impliespA2 = d/3 and pB2 = 2d/3. Obviously, pre-ordering is optimal for type A consumers. Itis left to check that type B consumers are also better off purchasing in period 1:

v − d

2− pA1 > v − 1

3

(pA2 +

d

6

)− 2

3pB2 = v − 1

3

(d

3+d

6

)− 4d

9,

pA1 <d

9.

(b) Suppose pA1 ∈ (d/9, d/3] and only type A consumers pre-order. Lemma 4 implies pA2 =d/3 and pB2 = 2d/3. In period 2, type B consumers with x ≤ d/3 optimally purchasebrand A, while those with x ∈ (d/3, d] brand B. In period 1, pre-ordering the product isoptimal for type A consumers (obviously). Thus, it is left to check that type B consumersare better off purchasing in period 2:

v − d

2− pA1 < v − 1

3

(pA2 +

d

6

)− 2

3pB2 = v − 1

3

(d

3+d

6

)− 4d

9,

pA1 >d

9.

(c) Suppose pA1 ∈ (d/3, d] and fraction α = 3(d − pA1 )/(2d) of type A consumers pre-order.Lemma 4 implies

pA2 =d(3− 2α)

3= pA1

and

pB2 =d(3− α)

3=d+ pA1

2.

In period 2, type B consumers with x ≤ αd/3 = (d − pA1 )/2 and the remaining typeA consumers optimally purchase brand A, while type B consumers with x ∈ (αd/3, d]brand B. In period 1, type A consumers are indifferent between pre-ordering the productor purchasing brand A in period 2 (obviously), so mixing is optimal. To show that typeB consumers are better off purchasing in period 2, it will be sufficient to show that theirpayoff from pre-ordering the product is below what they can get if they (sub-optimally)purchase from brand B in period 2:

v − d

2− pA1 < v − pB2 = v − d+ pA1

2,

pA1 >pA12.

25

(d) Suppose pA1 > d and no consumer pre-orders. Lemma 4 implies pA2 = pB2 = d. In period1, all consumers optimally choose not to pre-order.

Proof of Proposition 5: We calculate firm A’s profit under each item of Lemma 2. It is easy toshow it is maximized under item (d).

Proof of Proposition 6: Suppose firm A sets a high pre-order price so that no consumer pur-chases in period 1. Then firm A’s optimal price in period 2 and the resulting profit will be as inProposition 2.

Alternatively, firm A can sell to type A consumers in period 1 and to type B consumers inperiod 2. In this case pA2 solves the following maximization problem:

maxpA2 ∈[v−d,v]

pA2v − pA2

2d.

If d < v/2, the maximum is achieved at pA2 = v − d. To induce type A to pre-order, pA1 cannotexceed v − d. The resulting profit is ΠA = v − d. If d ∈ (v/2, v], the maximum is achievedat pA2 = v/2 (type B consumers with x ≤ v/2 will buy the product). To induce type A topre-order, pA1 cannot exceed v/2. The resulting profit is

ΠA =v

4+v2

8d,

which is below the one in Proposition 2.Finally, suppose firm A sells to all consumers in period 1. In this case pA2 = v−d if d < v/2

and pA2 = v/2 if d ∈ (v/2, v] (obtained as a limit as β → 1). Consider d < v/2 first. To induceall consumers to pre-order, pA1 cannot exceed v− d. The resulting profit is v− d. Now considerd ∈ (v/2, v]. Let us find the value of pA1 that makes type B indifferent between pre-ordering theproduct and waiting until period 2:

v − d

2− pA1 =

v

2d

(v − v

4− v

2

),

pA1 = v − d

2− v2

8d.

The resulting profit is

ΠA = pA1 = v − d

2− v2

8d,

which is below the one in Proposition 2.

Lemma 5 (Model II with both firms advance selling). Consider subgame (Y, Y) of the both-firms-can-advance-sell version of Model II. In period 1 firm A and firm B charge pA1 = pB1 = d;each consumer purchases her preferred brand. In period 2 the firms charge pA2 = pB2 = d. Theresulting profits are

ΠA = ΠB =d

2.

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Proof of Lemma 5: Suppose we found a symmetric Nash equilibrium in which pA1 = pB1 = p;each consumer buys her preferred brand and the firms’ equilibrium profits are ΠA = ΠB = p/2.If firm A deviates and decreases its price by d/2, it will lose

d

4

on its existing customers. At the same time, it will attract all type B consumers, thus increasingits profit by

1

2

(p− d

2

).

Equilibrium requires1

2

(p− d

2

)=d

4,

p = d.

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