entry deterrence of capacitated competition using price ... · we also study the timing difference...

Entry Deterrence of Capacitated Competition UsingPrice and Non-Price Strategies

Huaqing WangSchool of Business, Department of Business Administration, St. Thomas University, Miami Gardens, Florida 33054, USA,

[email protected]

Haresh GurnaniSchool of Business, Center for Retail Innovation, Wake Forest University, Winston-Salem, North Carolina 27106, USA,

[email protected]

Murat ErkocCollege of Engineering, Department of Industrial Engineering, University of Miami, Coral Gables, Florida 33124, USA,

[email protected]

W e analyze the role of pricing and branding in an incumbent firm’s decision when facing competition from anentrant firm with limited capacity. We do so by studying two price competition models (Stackelberg and Nash),

where we consider the incumbent’s entry-deterrence pricing strategy based on a potential entrant’s capacity size. In anextension, we also study a branding model, where the incumbent firm, in addition to pricing, can also invest in influenc-ing market preference for its product. With these models, we study conditions under which the incumbent firm may blockthe entrant (i.e., prevent entry without any market actions), deter the entrant (i.e., stop entry with suitable market actions)or accommodate the entrant (i.e., allow entry and compete), and how the entrant will allocate its limited capacity acrossits own and the new market, if entry occurs. We also study the timing difference between the two different dynamics ofthe price competition models and find that the incumbent’s first-mover advantage benefits both the incumbent and theentrant. Interestingly, the entrant firm’s profits are not monotonically increasing in its capacity even when it is costless tobuild capacity. In the branding model, we show that in some cases, the incumbent may even increase its price and suc-cessfully deter entry by investing in consumer’s preference for its product. Finally, we incorporate demand uncertaintyinto our model and show that the incumbent benefits from demand uncertainty while the entrant may be worse offdepending on the magnitude of demand uncertainty and its capacity.

Key words: capacity; competition; pricing; branding; game theoryHistory: Received: September 2013; Accepted: July 2015 by Kalyan Singhal, after 2 revisions.

1. Introduction and Literature Review

1.1. IntroductionCapacity, a key asset of a firm, plays an important rolewhen it comes to making a decision to enter into anew market and also has been considered as a vitalcompetitive tool. Consider the case of Arcelik, a sub-sidiary of Koc� Holding, which was the dominant sup-plier of appliances in Turkey. Arcelik was the sixthlargest European manufacturer of household appli-ances with domestic market share in Turkey exceed-ing 60–70% in major white goods categories. Therewas considerable debate within the company regard-ing how much emphasis to place on internationalsales (Quelch and Root 1997). Due to its tight capacity,Koc� management was considering its decisions forfuture international expansion and, more impor-tantly, how to balance its domestic market share

against further foreign expansion to optimize its totalprofits. Another example is of Vina San Pedro (VSP)which was the third-largest vineyard in Chile (Ran-gan et al. 2000). VSP had recently expanded its capac-ity and although its sales had previously been almostexclusively domestic, exports accounted for much ofthe growth in recent years. VSP’s managers were con-sidering how fast to push into both foreign anddomestic markets and how to balance its capacityallocation between both markets.There are other examples of companies that balance

their capacity allocation to certain markets in order toremain competitive and profitable across all markets.For example, Maersk Line, the world’s largest ship-ping line, announced that it would reduce its capacityon the Asia-Europe trade lanes by 9% and expandelsewhere in an effort to restore profitability (2012). Inthe words of Maersk Line CEO, Søren Skou: “...with

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Vol. 25, No. 4, April 2016, pp. 719–735 DOI 10.1111/poms.12500ISSN 1059-1478|EISSN 1937-5956|16|2504|0719 © 2015 Production and Operations Management Society

this adjustment we are able to reduce our Asia-Europe capacity and improve vessel utilizationwithout giving up any market share we have gainedover the past two years.” (Maerskline.com 2012).The examples above suggest that firms are con-

stantly seeking to explore new markets in search ofhigher profits. This poses a challenge for incumbentfirms facing potential competition as they have todesign suitable strategies to deter entry. One of thestrategies is the use of pricing to deter or to competewith the entrant. In an example in the airline industry,Goolsbee and Syverson (2008) study the relationshipbetween incumbents’ prices and the presence of anentrant airline (Southwest) on a route and they showthat incumbents cut prices significantly once South-west threatened entry.While competitive pricing may be one such option,

in some cases, the incumbent firm may be better byemphasizing the difference of its brand from theentrant with non-price strategies. Empirical evidencesand real world examples abound. Bunch and Smiley(1992) find that incumbent firms often use advertisingas a way to build consumer loyalty in order to deterentry. Kadiyali (1996) studied Kodak’s pricing andadvertising strategies for entry, deterrence, andaccommodation and showed that it engaged in limitpricing and limit “advertising” before Fuji’s entry inthe film market. However, with an increase in Fuji’smarket share, Kodak was compelled to take anaccommodating stance toward Fuji.Ocean Park was the only amusement park in Hong

Kong until 2005, when Hong Kong Disney entered itsmarket. The park moved to highlight the differenceswith Disney rather than compete on price only(Young et al. 2006). As the CEO Zeman stated, “Dis-neyland is about fantasy. ... It’s make-believe, ...,Ocean Park is educational. It’s about the environment,sea mammals, conservation.” (Wiseman 2007), “Wehave no intention of trying to out Disney.” By follow-ing this strategy, the park was able to be listed asHong Kong’s Top 5 Tourist Attractions in 2007. Assuch, firms may adopt a non-price strategy in the faceof potential threat from an entrant.In this study, we consider a competition problem

between two firms that operate in their own indepen-dent markets and one firm - the entrant - which deci-des whether to enter into the other firm’s (incumbent)market. In our model, the incumbent firm has suffi-cient capacity to operate as monopoly in its market;the entrant, however, has limited capacity and uponentry, has to allocate it across the two markets to max-imize total profit.We consider two different models of price competi-

tion - Stackelberg and Nash. In the Stackelberg game,we assume that the incumbent, as the prevailing firmin its own market, is the price leader and sets its price

first and then the entrant, as follower, sets price in itsown market as well as in the incumbent’s market. Inthe Nash game, both firms set prices simultaneously.The demand of each firm’s product when entry occursin the incumbent’s market is determined by the firm’sown price and the substitution effect of the otherfirm’s price. In both price competition models, westudy the effects of the capacity size on entry deter-rence. Specifically, we study (a) how the entrant’scapacity size would influence the incumbent’s strategyto block the entrant (i.e., prevent entry without anymarket action), deter the entrant (i.e., stop entry bylowering its price) or accommodate the entrant (i.e.,allow entry and compete); (b) how the entrant wouldallocate its limited capacity into the two markets whenentry occurs; (c) the implications of the alternative tim-ings for market outcomes and firm profits.The study of these models present two interesting

observations. While it is intuitive that the incumbentfirm would be better off if it were to act as leader andpreempt entry (or better compete) by suitably choos-ing its price, our analysis indicates that the entrantfirm, as follower, is also better off by responding tothe incumbent’s price as compared to setting pricessimultaneously. We show that this is due to thehigher prices set by the two firms in the Stackelberggame as compared to the Nash game. Another coun-ter-intuitive result is that in contrast to expectations,the profit for the entrant firm is not monotonicallyincreasing in its capacity. Although the competitivethreat to the incumbent firm increases in the size ofentrant firm, the entrant firm does not necessarilybenefit from high capacity even when it is costless tobuild capacity. Essentially, with large capacity, theentrant cannot credibly commit to selling lower quan-tity in the incumbent’s market, and in response, theincumbent engages in more intense price competitionleading to lower profits for both firms.In an extension to the pricing models, we also con-

sider the incumbent firm’s branding strategy. Unlikethe pricing model where we assume the incumbent’smarket size is fixed, in the branding model, theincumbent can increase the size of its customer basethrough costly investment. As such, the incumbenthas another mechanism, besides price, with which todeter entry. Interestingly, we note that in contrast tothe price only models, when branding is used, theincumbent may even increase its price and success-fully deter entry by investing in branding. This coun-ter-intuitive result is shown to be consistent withempirical evidence (see Thomas 1999).Finally, we extend the model to consider the case

of demand uncertainty in the incumbent’s market.We show that the incumbent can benefit fromdemand uncertainty in its market and is more likelyto maintain its monopoly status and deter entry. In

Wang, Gurnani, and Erkoc: Entry Deterrence of Competition720 Production and Operations Management 25(4), pp. 719–735, © 2015 Production and Operations Management Society

contrast, our results also show that the entrant sellsless quantity and makes lower profit in the incum-bent’s market.

1.2. Literature ReviewOur paper studies the competition and strategicbehavior of a capacitated entrant and an uncapaci-tated incumbent. The entrant is a monopolist in itsown market and considers entry into an incumbent’smarket. As such, the entrant faces the problem of opti-mally allocating its capacity between the two markets.In response, the incumbent adopts strategies to com-pete with the entrant. There are three streams ofresearch directly related to our study.The first stream investigates capacity allocation in

different markets. In fact, capacity allocation is a verybroad research area and appears in mechanismdesign (e.g., Maskin and Riley 1989, Cachon andLariviere 1999) and in the operations research litera-ture (e.g., Wein 1989). Our research interests, how-ever, focus on an entrant’s capacity allocation todifferent markets. In this research area, for example,Ding et al. (2007) and Ahmed et al. (2012) study amultinational firm that allocates capacity in either oneof the two markets or in both markets under uncertainexchange rates. However, they assume that there isno incumbent competitor in the market. In our model,we allow the incumbent’s market to be a duopoly.Specifically, we study the actions both of an entrantand of an incumbent in a duopoly price competitionsetting.The second stream of relevant literature, accord-

ingly, are the papers that study problems related toprice duopoly with capacity constraints. Indeed, it isa well-studied problem in economics literature with aseminal study by Levitan and Shubik (1972). Singhand Vives (1984) analyze a Bertrand price competi-tion in a differentiated duopoly model with no capac-ity constraint. Osborne and Pitchik (1986) study theequilibrium in a model of price-setting duopoly inwhich each firm has limited capacity. Van Mieghemand Dada (1999) study an competitive price post-ponement model under capacity-constrained condi-tion. However, these studies only considercompetition in a single market. We contribute to thisline of literature by considering the interaction offirms across the two markets, one of which is a duo-poly and the other a monopoly. We study two differ-ent dynamics of price competition in the duopolymarket and show that the dynamics of the price gamehave a significant impact on the two players’ behav-ior in both markets.Gal-Or (1985) investigates the conditions under

which a player will prefer to be the leader or a fol-lower in a game of sequential moves (with comple-mentary or substitutable products) under price

competition and under quantity competition. In ourstudy, we also analyze the strategies and payoffs ofa player in a first-mover or simultaneous game.However, we differ in our analysis by consideringtwo different players: one player (the entrant) withconstrained capacity serving two markets and theother (the incumbent) operating in its home marketwith unconstrained capacity. By investigating theimpact of two different dynamics of pricing gameson both incumbent’s and entrant’s pricing strategiesand profits, this study contributes to the stream ofresearch on the first-mover advantage.The third stream of literature that is relevant to our

study is the one which studies the problems of strate-gic entry-deterrence in economics. A large stream ofthe economics literature has been concerned withstrategic entry-deterrence since Spence (1977) andDixit’s (1979, 1980) seminal papers that study the useof limit pricing. Entry-deterrence models in this lineemphasize the use of limit pricing in price competi-tion or limit quantity in quantity competition as theincumbent’s strategic tool for deterring entry.We have some common research goals in this

regard but our research differs from most previousanalyses of entry-deterrence in some importantaspects relating to the role of both the incumbentand the entrant firm. Research from the entrant’spoint of view in the entry-deterrence model hasbeen relatively neglected. Previous literatureassumes that the entrant has no resource constraint.However, as the examples discussed earlier in thesection suggest, entry decision also involves alloca-tion of capacity across different markets. Specifically,we focus on the implications for adopting differentpricing (or branding) strategies when the incumbentfirm faces competition from an entrant withconstrained-capacity. In addition, unlike previousentry-deterrence models that pay little attention tothe status in the entrant’s own market, our modelconsiders a scenario where the entrant has to makecapacity allocation decisions. This allows us tounderstand the inter play between the incumbent’sentry-deterrence policies and the pricing and capac-ity allocation strategies employed by the entrant inits own market. As discussed in the Koc� Holdingexample in the previous section, this setting is morerealistic since it is rare for an entrant to enter intothe incumbent’s market with no consideration for itsown market operations. Our results show that theentrant’s profit in its own monopoly market isindeed affected by the dynamics of competition inthe incumbent’s market.Previous literature considers one mechanism such

as price or output level (quantity) in the post-entrygame. Dixit (1980), Spulber (1981), and Bulowet al. (1985) examine the entry problem when the

Wang, Gurnani, and Erkoc: Entry Deterrence of CompetitionProduction and Operations Management 25(4), pp. 719–735, © 2015 Production and Operations Management Society 721

post-entry game is Cournot competition. Basu andSingh (1990) study entry when the post-entry game isStackelberg quantity competition. Allen et al. (2000)consider a model of Bertrand price competition in thepost-entry game. However, in our study, by buildingon a case where post-entry game is Stackelberg pricecompetition, we allow the incumbent to have twolevers: price and branding, both of which affectdemand and thus affect the entrant’s entry and pric-ing decisions. Chayet and Hopp (2005) study firms’post-entry behavior in a simultaneous price gamewith delivery time determined endogenously. In theirmodel, the cost of expected leadtime is part of the fullprice that affects the customer’s market clearingdemand allocation decision. However, our study dif-fers from their paper in three important ways. First, intheir model, the cost of expected leadtime is decidedby a queuing model, where the only decision variableis the service rate. As a result, the expected leadtimein itself is not a decision variable. Moreover, the lead-time expression in their model is only an approxima-tion based on additional assumptions. In contrast, weconsider both price and non-price factors in ourmodel that affects the entry deterrence game. A sec-ond major difference in our study is that the entrantfirm has to decide on how to allocate its limited capac-ity across its own market and the incumbent’s market.This is not possible to model using a queuingapproach as the incumbent and the entrant competeonly in a single market. As such, Chayet and Hopp(2005) do not consider capacity allocation across twomarkets. Moreover, while higher capacity for theentrant firm does not always lead to higher profits inour model, this is not the case using standard queuingmodels. Finally, in addition to the simultaneous pricecompetition, we also study Stackelberg price competi-tion in the post-entry game.The rest of the study is organized as follows. In

Section 2, we present the model assumptions anddefine the notation. In Section 3, we introduce theStackelberg competition model, and in Section 4,the Nash competition model is discussed and wecompare and contrast the differences between thetwo pricing models. In Section 5, we extend theanalysis to consider the branding model, and inSection 6, demand uncertainty in the incumbent’smarket. Finally, in Section 7, we conclude and dis-cuss future research. All proofs are presented in theAppendix S1.

2. Model Formulation

The base model comprises of two firms operating intheir own markets: the entrant’s market and theincumbent’s market. Both markets are independent,but one firm - a potential entrant - plans to enter into

the other firm’s market. The entrant firm has limitedcapacity (k), and upon entry, would need to allocateits capacity across the two markets. Moreover, thefirms would engage in price competition if entry doesoccur.We study two models of price competition in this

paper. First, we model competition as a Stackelberggame, where the entrant firm sets its price afterobserving the incumbent’s prevailing price in its ownmarket. In the second case, we model simultaneousprice competition as a Nash game where the twofirms set prices simultaneously. In the literature onprice (or quantity) competition between firms withdifferentiated goods, it is common to find lineardemand models (see Singh and Vives 1984, Vives1984, H€ackner 2000, Symeonidis 2003, Choi andCoughlan 2006, etc.) that we use in our study . Werefer to these studies for discussion of the lineardemand model which is derived from a more struc-tured approach of a utility maximizing consumer (seeIngene and Parry 2004; Cai et al. 2012). A detailed listof notation is given in Table 1.

Table 1 Notation in Competition Models

Notation Explanation

i Firm index i = 1 for entrant in its ownmarket, i = 2 for incumbent i = t for entrant inincumbent’s market

j Index j = n for Nash competition, j = s forStackelberg competition j = m for BlockadedEntry, j ¼ m0 for Ineffectively Impeded

k Entrant’s capacitypn1 , p

s1 Entrant’s price in its own market in Nash,

Stackelbergpji pnt , p

st Entrant’s price in incumbent’s market in Nash ,

Stackelbergpn2 , p

s2 Incumbent’s price in Nash, Stackelberg

pm2 , pm02 Incumbent’s price in Blockaded Entry region,

Effectively Impeded Entry regionqn1 , q

s1 Entrant’s quantities sold in its own market in Nash,

Stackelbergqji qnt , q

st Entrant’s quantities sold in incumbent’s market in

Nash, Stackelbergqn2 , q

s2 Incumbent’s quantities sold in its own market in

Nash, Stackelbergqm2 , q

m02 Incumbent’s quantities sold in its own market in

Blockaded Entry, Effectively Impeded Entrypn1 , p

s1 Entrant’s Total profit in Nash, Stackelberg

pnt , pst Entrant’s profit in the incumbent’s market in Nash,

Stackelbergpji pn2 , p

s2 Incumbent’s profit in Nash, Stackelberg

pm2 , pm02 Incumbent’s profit in Blockaded Entry, Effectively

Impeded Entryai Market i’s sizeci Firm i’s unit production cost, ci � 0k jx k nx , k

sx x 2 {1, 2, 3}. Threshold value of capacity in

Nash and Stackelberg competitionl Market preference, 0 ≤ l ≤ 1h Product’s substitution effect on demand 0 < h < 1t Entrant’s transportation cost, t > 0


If entry does occur, we assume that the entrant’s(firm 1) demand functions (Figure 1) in the two mar-kets are given by:

qj1 ¼ a1 � p

j1; ð1Þ

qjt ¼ ð1� lÞa2 � p

jt þ hpj2; ð2Þ

where qj1 þ q

jt � k, j = s or n for Stackelberg and

Nash competition, respectively. We also assume thatdemand function for the incumbent (firm 2) is givenby:

qj2 ¼ la2 � p

j2 þ hpjt: ð3Þ

The total size of the incumbent’s market is equal toa2, and l represents the consumer’s preference for itsproduct. Similarly, h represents the substitution effecton demand. Each firm’s demand in market 2 isdecreasing in its own price level and increasing in thecompetitor’s price. The entrant firm also incurs trans-portation cost t per unit if it enters into the incum-bent’s market.If entry does not occur, the incumbent firm acts as a

monopoly in its market and its demand function issuitably modified. Essentially, the transformation inthe demand function is based on setting qt ¼ 0, andsubstituting the expression for p

jt back into q

j2 defined

above (see Ingene and Parry 2004):

qm2 ¼ ½lþ hð1� lÞ�a2 � ð1� h2Þpm2 : ð4ÞWith this demand function, the monopoly profitfunction for the incumbent firm is given by:

pm2 ¼ ðpm2 � c2Þ½ðlþ hð1� lÞÞa2 � ð1� h2Þpm2 �; ð5Þ

and the optimal monopoly price is:

pm2 ¼ ½lþ hð1� lÞ�a2 þ c2ð1� h2Þ2ð1� h2Þ : ð6Þ

3. Stackelberg Competition

In this section, we consider the case where theincumbent (firm 2) is the price leader and theentrant is the follower. The sequence of events areas follows:

(1) Facing the threat of potential entry, theincumbent firm sets the price in its own mar-ket;

(2) Observing the incumbent’s price, the entrantdecides whether to enter the market or not. Ifit decides to enter, it sets price pst in theincumbent’s market and ps1 in its own marketto maximize total profits. As is standard, wesolve the problem by backward induction. Allproofs are in the Appendix S1.

If entry occurs, each player’s optimization problemis given by:

maxps1;pst

: ps1 ¼ ðps1 � c1Þða1 � ps1Þ þ ðpst � t� c1Þ

� ½ð1� lÞa2 � pst þ hps2�;ð7Þ

s:t: a1 � ps1� �þ ð1� lÞa2 � pst þ hps2

� �� k; ð8Þ

and

maxps2

: ps2 ¼ ðps2 � c2Þðla2 � ps2 þ hpstÞ: ð9Þ

Starting with the entrant’s (firm 1) problem, whenthe capacity constraint is activated, we get:

ps1ðps2Þ ¼3a1 þ a2 þ ps2h� a2l� t� 2k

4ð10Þ

and

pstðps2Þ ¼a1 þ 3a2 þ 3ps2h� 3a2lþ t� 2k

4: ð11Þ

On substituting pstðps2Þ into qst ¼ ð1 � lÞa2 �pst þ hps2 we get:

qstðps2Þ ¼2k� t� a1 þ a2 þ ps2h� a2l

4: ð12Þ

Note that in Equation (12), qst is a function of ps2.There may exist some values of k such that the incum-bent can choose price ps2 to make qst � 0 (as long as itis more profitable for the incumbent to do socompared to other strategies), which means that the

Figure 1 The Competition Model


entrant would not enter the incumbent’s market. Asshown next, the capacity size k plays a critical role inshaping the incumbent’s selection of price andaccordingly its competition strategy.

PROPOSITION 1. In equilibrium, there exist three thresh-olds (ks1 \ ks2 \ ks3) in capacity size with four regionssuch that:

(a) Blockaded Entry region (k � ks1): The entrant doesnot enter the incumbent’s market and the incum-bent sets the monopoly price.

(b) Effectively Impeded Entry region (ks1 \ k � ks2):The incumbent is able to lower price and theentrant does not enter the incumbent market.

(c) Ineffectively Impeded Entry region (ks2 \ k � ks3):Both players compete in the incumbent’s market.

(d) Unlimited Capacity region (ks3 \ k): The capaci-tated problem switches to the uncapacitated case.

The four regions in capacity size for the entrant firmare as follows.

3.1. Blockaded Entry Region (k � ks1)In this region, the potential entrant’s capacity issmall and therefore if the incumbent uses its mono-poly price (pm2 ), the expression for qstðpm2 Þ becomesnegative. Consequently the entrant would notenter into the incumbent’s market. The incumbentcan make its monopoly profit by setting price

at pm2 ¼ c2 � c2h2þa2ðhþl�hlÞ2�2h2

and get profits pm2 ¼½a2ðhþl�hlÞ� c2ð1�h2Þ�2

4ð1�h2Þ . The entrant’s capacity is so low

that devoting any output to the incumbent market(where he faces competition) is not worthwhile.

3.2. Effectively Impeded Entry Region (ks1 \ k � ks2)In this region, if the incumbent uses its monopolyprice pm2 , the expression for qstðpm2 Þ is positive, whichmeans the entrant would be interested in enteringthe incumbent’s market. On substituting pm2 into

pstðp2Þ, we get pdt ¼ �2kþtþa1þ3a2þ3pm2h�3a2l

4 , which is the

entrant’s price in the incumbent market when theincumbent allows entry by using its monopoly price.

(Here we abuse the notation by creating pdt and pd2).Then, substituting pm2 and pdt into pd2 ¼ ðpm2 � c2Þðla2 � pm2 þ hpdt Þ, we get the incumbent’s profit. How-ever, if the incumbent lowers its price to stop entry,then substituting pst into the equation qst ¼ ð1� lÞa2 � pst þ hps2 and solving for ps2 that makes qst ¼ 0,we get:

pm0

2 ðkÞ ¼ a1 � a2 þ a2lþ t� 2k

h: ð13Þ

Using the above price in the incumbent’s monopolyprofit function pm2 , we get a modified monopoly profit

function pm0

2 ðkÞ. Further, if ks1 \ k � ks2, we havepd2 \ pm

02 ðkÞ, that is, the incumbent has the incentive to

lower its price to stop entry.

3.3. Ineffectively Impeded Entry Region(ks2 \ k � ks3)If the incumbent continues to lower price to stop entryin this region, its profit would be lower than the profitfrom allowing entry by setting the optimal price withcompetition. In this case, we need to calculate theincumbent’s profit with competition. Substituting pstinto ps2 ¼ ðps2 � c2Þðla2 � ps2 þ hpstÞ, we get:

maxps2

: ps2 ¼ ðc2 þ p2Þ½a2l� p2 þ 1

4hð�2kþ tþ a1

þ 3a2 þ 3p2h� 3a2lÞ�:ð14Þ

Solving for ps2, we get:

ps2 ¼4c2� 2khþ thþ a1hþ 3a2h� 3c2h

2þ 4a2l� 3a2hl

2ð4� 3h2Þ :

ð15ÞSubstituting ps2 back into Equations (7, 9, 10, and

11), we get ps1ðkÞ, ps2ðkÞ, ps1ðkÞ, pstðkÞ as given inTable 2. On comparing the profits ps2ðkÞ and pm

02 ðkÞ,

we find the threshold value ks2 such that if k [ ks2then pm

02 \ ps2ðkÞ. Therefore, instead of lowering its

price any further to stop entry, the incumbentwould allow entry and set the optimal price to com-pete with the entrant.

3.4. Unlimited Capacity Region (ks3 \ k)To facilitate our capacity-constrained analysis, westudy the case where the entrant’s capacity isunconstrained (i.e., relax the capacity constraintcondition by eliminating Equation (8)). We employthis assumption to find the maximum capacity incapacitated scenario that the entrant could utilize tosatisfy demand in both markets. Solving Equations(7) and (9) by backward induction, we obtain theincumbent’s profit in the uncapacitated case pus2 .(Here we use pus2 to denote the incumbent’s profitin the uncapacitated Stackelberg competition case).If the incumbent continues to lower price, its profitwould be lower than pus2 . As such, the incumbentwill set price equal to pus2 and the game switches tothe uncapacitated case. On setting pus2 ¼ ps2ðkÞ, weget ks3, the threshold capacity from which the gameswitches to competition as if the entrant has unlim-ited capacity. In this region, allocating more capac-ity into the incumbent’s market would hurt theprofitability for the entrant, as the example ofMaersk Line shows in the introduction. We definethe following and list all the expressions in Table 2:


A ¼ thþ a1hþ 3a2h� c2ð4� 3h2Þ þ 4a2l� 3a2hl;

B ¼ c1hþ thþ a2h� c2ð2� h2Þ þ 2a2l� a2hl;

C ¼ tþ a1 � a2 þ a2l:

Figure 2 is based on a numerical analysis withparameters a1 ¼ 100, a2 ¼ 100; h ¼ 0:5; c1 ¼ 1; c2 ¼1:2; l ¼ 0:5 and t = 0.5. As shown in the figure, whenthe entrant’s capacity level goes beyond the BlockadedEntry region, it would be profitable if the entrantenters the incumbent’s market. However, the incum-bent can lower its price to deter entry for k\ ks2. Asthe entrant’s capacity increases, the incumbent wouldbe worse off if it lowers the price to stop entry. Conse-quently, the firms would compete on price, and theentrant would optimally allocate its limited capacityinto the two markets. It is interesting to note that theentrant firm’s profit is not monotonically increasingin its capacity size, as shown in the figure above. Thisis a consequence of the fact that as capacity increases,the entrant aims to sell more quantity in the incum-bent’s market by lower the selling price, and there-fore, the profit is not monotonically increasing. Thisleads to the next corollary.

COROLLARY 1. The profit for the entrant firm is notmonotonically increasing in its capacity, even whenexpanding capacity is cost free.

Corollary 1 notes the counter-intuitive result thathaving higher capacity may in fact lead to lower prof-its for both firms even when there is no cost of capa-city. As its capacity increases, the entrant sells morequantity by lowering its price and its profits peak atk ¼ ks

�. It would then prefer not to use more capac-

ity. However, it cannot credibly convey to the incum-bent firm that it would not release more quantity intoits market. Consequently, the incumbent wouldbelieve that the entrant, in order to get higher profit,would deviate at k ¼ ks

�resulting in more intense

price competition and lower profits for both firms fork [ ks

�.

When the entrant’s capacity is larger than ks3, if theincumbent continues to lower the price to competewith the entrant, its profit will be lower than the profitit can get by setting its price at the level in the un-capacitated scenario. Thus the incumbent will set theprice at pus2 when the entrant has unlimited capacityand the game switches to the uncapacitated case. Atk ¼ ks3, the incumbent’s profit in both scenarios areequal. However, our analysis shows that the entrant’sprofit at k ¼ ks3 in the two scenarios are not equal.This leads to the next proposition.

PROPOSITION 2. At the critical threshold ks3 dividing theuncapacitated and capacitated regions, (a) the entrant incapacitated scenario sells less in its own market than it doesin uncapacitated scenario, that is, qs1ðks3Þ\ qus1 ¼ a1�c1

2 ;However, the entrant in the capacitated scenario sells morein the two markets together than it does in uncapacitatedcase, that is: ks3 [ qus1 þ qust ; (b) the entrant’s profit in thecapacitated scenario is higher than the profit in theunconstrained case, that is, ps1ðks3Þ [ pus1 .

The intuition behind Proposition 2(a) is thatbecause of limited capacity, the entrant faces a trade-off between obtaining higher profit in the incum-bent’s market and sacrificing some of the monopolyprofits in its own market. As long as the entrant canobtain higher total profits, the entrant is willing to

Table 2. Equilibrium Expressions in Stackelberg Game

Blockaded Entry Effectively Impeded EntryIneffectively

Impeded Entry Unlimited

pj1 a1 � k a1 � k see Table 4* a1 þ c12

pjt – – see Table 4c1 þ t þ a2 � a2l

2þ h2

�B

2ð2� h2Þ þ c2

�

pj2c2ð1� h2Þ þ a2ðhþ l� hlÞ

2� 2h2�2k þ C

hA� 2kh

2ð4� 3h2Þ þ c2B

2ð2� h2Þ þ c2

qj1 k k see Table 4a1 � c1

2

qjt – – see Table 4Bh

4ð2� h2Þ þða2 þ c2h� a2lÞ � c1 � t

2

qj2a2ðhþ l� hlÞ � c2ð1� h2Þ

2

a2hðhþ l� hlÞ � ð1� h2Þð�2k þ CÞh

A� 2kh8

B

4

pj1 kða1 � k � c1Þ kða1 � k � c1Þ see Table 4 see Equation (40)*

pj2ða2ðhþ l� hlÞ � c2ð1� h2ÞÞ2

4ð1� h2ÞðC � 2k � c2hÞða2hðhþ l� hlÞ � ðC � 2kÞð1� h2ÞÞ

h2ðA� 2khÞ264� 48h2

B2

8ð2� h2ÞNote: *See Appendix S1.


sell more in the incumbent’s market and less in itsown market. Proposition 2(b) notes the interestingresult that the entrant with limited capacity (slightlyless than ks3) is able to make higher profit than theentrant with unlimited capacity (k ¼ ks3). This is aconsequence of the price competition between thetwo firms. In the case of unconstrained capacity forthe entrant, the incumbent firm faces risk of largersupply quantity of the entrant firms’ product, andconsequently, price competition becomes moreintense leading to lower profits. However, in the lim-ited capacity region, the supply is reduced resultingin higher prices and higher profits for both firms.

4. Nash Competition

In this section, we consider the simultaneous pricesetting competition game. While it is less likely thattwo firms will post their prices simultaneously, it ismay be pertinent in some pricing settings, such ashealth insurance markets where firms offer contract

throughout a given enrollment period (Sweeting andSweeney 2015) or markets where suppliers quoteprices to an upstream buyer without knowing eachother’s prices. For example, Monsanto is the manu-facturer of the patent-protected NutraSweet, a sweet-ener used in soft drinks. Anticipating the expirationof the Monsanto’s patent, the Holland Sweetenerplanned to enter into the Monsanto’s monopolisticmarket in the United States. Just prior to HollandSweetener’s entry, Coke and Pepsi signed long-termcontract with Monsanto for the continued supply ofNutraSweet, leading to a combined savings of$200 million for Coke and Pepsi (Brandenburger andNalebuff 1995). Allenby et al. (2014) also argue that aNash equilibrium pricing game provides a reason-able basis for equilibrium calculations in many patentsituations with a differentiated products demandsystem.The demand and profit functions are exactly the

same as in the Stackelberg game but market dynamicsare different. The sequence of events in the Nash

Figure 2 Stackelberg Competition


game are as follows: the incumbent (firm 2) and theentrant (firm 1) simultaneously choose price pn2 and pntand the entrant also chooses price pn1 in its own mar-ket to maximize their profits.Since the structure in Nash competition is similar to

that in Stackelberg competition, we only give a briefexplanation in this section. All details are in theAppendix S1. With the market dynamics in Nashcompetition, we solve the following problems simul-taneously:

maxpn1;pnt

: pn1 ¼ ðpn1 � c1Þða1 � pn1Þ þ ðpnt � t� c1Þ

� ½ð1� lÞa2 � pnt þ hpn2 �;ð16Þ

s:t:½a1 � pn1 � þ ½ð1� lÞa2 � pnt þ hpn2 � � k; ð17Þand

maxpn2

: pn2 ¼ ðpn2 � c2Þðla2 � pn2 þ hpnt Þ: ð18Þ

We define the following and list all the expressionsin Table 3:

D ¼ 2a1 � 2a2 � c2h� a1h2 þ tð2� h2Þ þ 2a2l� a2hl;

E ¼ 2a2 þ c2h� c1ð2� h2Þ � tð2� h2Þ � 2a2lþ a2hl:

The results are similar to those of the Stackelbergcompetition and are given in Proposition 3.

PROPOSITION 3. In equilibrium, there exist three thresh-olds (kn1 \ kn2 \ kn3) in capacity size with four regionssuch that:

(a) Blockaded Entry region (k � kn1): the entrant wouldvoluntarily not enter the incumbent’s market.

(b) Effectively Impeded Entry region (kn1 \ k � kn2):the entrant is interested in entering the incum-

bent’s market but the incumbent can lower itsprice to stop entry.

(c) Ineffectively Impeded Entry region (kn2 \ k � kn3):Both players compete in the incumbent’s marketand the entrant will optimally allocate its limitedcapacity into the two markets to maximize its totalprofits.

(d) Unlimited Capacity region (kn3 \ k): The capaci-tated problem switches to the uncapacitated case.

See Figure 3 for an illustration of the results (thenumerical parameters are the same as in Figure 2).Similar to the case of Stackelberg competition, theentrant firm’s profits are again not monotonicallyincreasing in its capacity. Although the structure ofthresholds of the Nash game are similar to those ofStackelberg competition, the next proposition showssome important differences.

PROPOSITION 4. At the threshold capacity (kn3) in Nashcompetition, (a) the entrant in capacitated scenario sells thesame quantities in each market as it does in the uncapacitatedscenario, that is: qn1ðkn3Þ ¼ qun1 and qnt ðkn3Þ ¼ qunt ; (b) Theentrant gets the same profit in each market in both scenarios,that is, pn1ðkn3Þ ¼ pun1 and pnt ðkn3Þ ¼ punt .

The results above are in contrast to the findings inProposition 2 for the Stackelberg game. Essentiallywith simultaneous pricing in the Nash game, theequilibrium prices are continuous in the entrant’scapacity size and therefore, the profits are also contin-uous at the critical threshold level.

4.1. Comparison Between Competition ModelsIn this section, we compare the results from the twocompetition models. We focus our analysis on theimpact of different market dynamics on both firms’

Table 3 Equilibrium Expressions in Nash Game

Blockaded and Effectively Impeded Entry Ineffectively Impeded Entry Unlimited

pj1 Same as in Stackelberg (See Table 2)a1 � D þ kð4� h2Þ

8� 3h2a1þc1

2

pjt4k � 2t � 2a1 � 6a2 � 3c2hþ 6a2l� 3a2hl

�8þ 3h22c1 þ 2t þ 2a2 þ c2h� 2a2lþ a2hl

4� h2

pj2�2khþ A

8� 3h2þ c2

2c2 þ c1hþ thþ a2hþ 2a2l� a2hl

4� h2

qj1D þ kð4� h2Þ

8� 3h2a1 � c1

2

qjt�D þ 2kð2� h2Þ

8� 3h2E

4� h2

qj2A� 2kh

8� 3h2B

4� h2

pj1 see Table 5*E 2

ð4� h2Þ2þ ða1 � c1Þ2

4

pj2ðA� 2khÞ2ð8� 3h2Þ2

B2

ð4� h2Þ2

Note: *See Appendix S1.


profits, incumbent’s strategic regions, and theentrant’s capacity allocation behavior.

PROPOSITION 5. (a) The region for Blockaded Entry inboth Stackelberg and Nash competition is the same, that is,ks1 ¼ kn1 ; However, the region for Effectively ImpededEntry is larger in the Nash game, that is, ks2 \ kn2 . Finally,the region with equilibrium decisions under uncapacitatedcase occurs later in Stackelberg competition, that iskn3 \ ks3; (b) When competition occurs, both the entrant’sand the incumbent’s profit are higher in Stackelbergcompetition than in Nash competition, that is: whenk [ ks2, p

s1ðkÞ [ pn1ðkÞ and ps2ðkÞ [ pn2ðkÞ.

Proposition 5(a) is an interesting result as theincumbent firm, as Stackelberg leader, is willing tocompete with the entrant for lower values of theentrant’s capacity size (as compared to the Nashgame). This is again attributed to the higher prices forboth firms in the Stackelberg game.In Stackelberg competition, as expected, the incum-

bent is better off since it has the first-mover advantage.

However, it is counter-intuitive that the entrant’s profitis also higher in Stackelberg competition as comparedto Nash competition. In the Stackelberg game, theincumbent, as leader, is able to affect lower price com-petition by setting a higher price which induces a highprice response from the entrant as well. In contrast, inthe simultaneous game, it is not possible to make acredible high price commitment leading to moreintense price competition and lower profits for bothfirms as compared to the Stackelberg game. As such,the first-mover advantage for the incumbent firm ben-efits both players. In a different problem of supplierencroachment, Arya et al. (2007) show a similar resultin that both the supplier and the retailer (andconsumers as well) are better off under sequential (lea-der-follower) encroachment than under simultaneousmoves due to lower wholesale price charged by thesupplier. In contrast, in our study, prices are higher inthe Stackelberg game which would make consumersworse off. We also note that in the Stackelberg game,the quantity sold by the two firms in the incumbent’smarket are higher even though prices are higher.

Figure 3 Nash Competition


COROLLARY 2. In the Stackelberg game, the leader(incumbent firm) and the follower (entrant firm) sell lessin their own markets, whereas the follower (entrant firm)sells more in the incumbent’s market compared to theNash game, that is, qs1ðkÞ � qn1ðkÞ, qs2ðkÞ � qn2ðkÞ andqstðkÞ � qnt ðkÞ; 8 k.

These results are quite surprising as in a standardCournot competition model, the leader sells higherquantity due to its first-mover advantage. However,as we note above, this is not the case in the Bertrandgame as prices (quantities) are higher (lower) in theStackelberg game compared to the simultaneousNash game.

COROLLARY 3. At the threshold capacity where the capa-citated case switches to uncapacitated case, the entrantsells less in its own market but more in the incumbent’smarket in Stackelberg competition than in Nash competi-tion, that is, qs1ðks3Þ\ qn1ðkn3Þ and qstðks3Þ [ qnt ðkn3Þ.

Corollary 3 notes that the entrant strategicallyallocates more of its limited capacity into theincumbent’s market in Stackelberg competition thanin Nash competition because of the higher profit itcan get in the incumbent’s market. Consequently, atthe threshold capacity (ks3) in Stackelberg competi-tion, the limited capacity has some impact on theentrant’s own market, that is, the entrant will sellless than its monopoly quantity in Stackelberg com-petition. However, at the threshold capacity (kn3) inNash competition, the entrant will sell the mono-poly quantity in its own market, and therefore, lim-ited capacity has no impact on the entrant’s ownmarket.

5. Branding

In this section, we consider the case when theincumbent firm can use an additional mechanism(besides price) to stop the entrant from potentialentry or to better compete with it. We make the con-sumers’ market preference (l), a given exogenousparameter in previous sections, as a decision vari-able. That is, in addition to its pricing strategy, theincumbent can invest in product branding to changeconsumer’s preference for its product. While thetotal market size a2 is fixed, using branding, theincumbent firm can build stronger preferencetowards its product. We use an additional super-script b to denote decisions in the branding model.Let s be the coefficient in the incumbent’s investmentfunction, that is, the cost of branding, cðlÞ ¼ 1

2 sl2.

Without loss of generosity, we will analyze thebranding scenario based on the Stackelberg competi-tion model in Section 3.

We start with the case of monopoly in brandingmodel. From Equation (5), we obtain the incumbent’sprofit function as monopoly with branding as fol-lows:

maxpm2;lm

: pm2 ¼ ½ðlm þ hð1� lmÞÞa2 � ð1� h2Þpm2 �

� ðpm2 � c2Þ � sðlmÞ22

;

ð19Þ

where lm and pm2 are the incumbent’s market prefer-ence and price in the monopoly case with branding.It can be shown that if s[ a22ð1� hÞ=2ð1þ hÞ, theprofit function is jointly concave and therefore thefirst-order conditions are sufficient to characterizethe optimal decisions:

lm ¼ a2½a2h� c2ð1� h2Þ�2sð1þ hÞ � a22ð1� hÞ;

pm2 ¼ sa2hþ c2ð1� hÞ½sð1þ hÞ � a22ð1� hÞ�ð1� hÞ½2sð1þ hÞ � a22ð1� hÞ�

ð20Þ

and the incumbent’s monopoly profit is

pm2 ¼ s½a2h� c2ð1� h2Þ�22ð1� hÞ½2sð1þ hÞ � a22ð1� hÞ� : ð21Þ

5.1. Stackelberg Competition Model withBrandingIn the case of entry, the sequence of events are as fol-lows: (1) The incumbent invests in branding lb andsets price pb2, (2) The capacitated entrant decideswhether to enter the incumbent’s market and accord-ingly, sets its price and allocates capacity.We proceed by backward induction to solve this

problem. As in the previous sections, we show thatthe optimal strategy is divided into four regions basedon the entrant firm’s capacity. But now, the incum-bent uses both branding investment as well as pricingstrategy to thwart entry or to better compete with theentrant.

PROPOSITION 6. In equilibrium, there exist three thresh-olds (kb1 \ kb2 \ kb3) in capacity size with four regionssuch that:

(a) Blockaded Entry region (k � kb1): the incumbentwill choose monopoly decisions lm and pm2 and getmonopoly profits pm2 .

(b) Effectively Impeded Entry region (kb1 \ k � kb2):the incumbent will choose lm

0and pm

02 to prevent

entry and earn modified monopoly profits pm0

2 .(c) Ineffectively Impeded Entry region (kb2 \ k � kb3):

the incumbent will allow entry and choose optimalbranding investment lbðkÞ and get profits pb2ðkÞ.

(d) Unlimited Capacity region (kb3 \ k): The con-strained-capacity problem switches to the uncon-strained capacity case.


There are interesting differences between thebranding model and the price-only model as shownin Figure 4. In the Effectively Impeded Entry region,the incumbent invests in increasing l and may alsoincrease its price to successfully stop entry. Thiscounter-intuitive result is consistent with the empir-ical evidence in Thomas (1999). He finds that simi-lar to the Effectively Impeded Entry region in ourstudy, the ready-to-eat cereal industry is character-ized by rapidly increasing real prices with highadvertising. By investing in market preference andconsequently increasing market share, the incum-bent is closer to be a monopoly and therefore cantake the advantage of being a monopoly to increaseits price to cover its investment cost. However, ifthe coefficient s in the cost function of branding issufficient high (s [ a22ð1�hÞ

h ), then the incumbent willin fact lower its price to stop entry while still investin increasing l. But when s is sufficiently large, theincumbent’s ability to keep its monopoly status andto deter entry becomes weaker. For example, based

on the same numerical parameters (h = 0.5 anda2 ¼ 100), s should be at least greater than 10,000.As a result, if s = 10,050, then the Blockaded Entryregion does not exist and the Effectively ImpededEntry region is small. To summarize, using brand-ing as an additional lever, as long as the cost iswithin a reasonable range, the incumbent firm isable to deter entry by increasing price and increaseits investment in consumer’s preference for itsproduct.But the investment in l becomes increasingly costly

when the entrant’s capacity is sufficiently large as itcan also lower its price to gain entry. Consequently, atsome threshold, it becomes optimal for the incumbentto allow entry and to compete. In this region of Ineffec-tively Impeded Entry, investment in l and the pricepb2ðkÞ are both decreasing in the entrant’s capacity size.In the branding model, by making the market pref-

erence (l) endogenous as an additional lever for theincumbent to deter entry, we find that the thresholdvalues kb1 and kb2 are higher than those in pricing

Figure 4 Branding Model


model. That is, the incumbent is more likely to block-ade and effectively impede entry by using both price andnon-price strategy than by using price-only strategy.This can be analytically shown using the implicit solu-tions in the Proof of Proposition 6 and comparingthem with the threshold values in the pricing model,that is, kb1 [ ks1 and kb2 [ ks2.The analysis above assumed that the objective func-

tion is jointly concave. If s does not satisfy the jointlyconcave condition, that is, s\a22ð1� hÞ=2ð1þ hÞ, thenwe cannot have the interior solution as given inEquations (20) and can only solve the problemnumerically. However, intuitively, when s issufficiently low, the incumbent would invest more inmarket preference (l). As a result, the ability tokeep its monopoly status and to deter entry becomesstronger.

6. Demand Uncertainty

Since demand uncertainty plays an important role inentry deterrence games, we incorporate demanduncertainty into the branding model and study howdemand uncertainty in the incumbent’s market affectsdecision making for both firms. In order to under-stand the effect of uncertainty in a new market (whichis the incumbent’s market), we assume that demand inthe entrant’s own market is deterministic. We use anadditional superscript^ to denote the case of demanduncertainty. The base size in the incumbent’s market isa2 þ n, where ξ captures the demand uncertainty withmean 0 and standard deviation r. Note that since l isendogenous, the investment decision in l is alsoaffected by demand uncertainty. Thus the demandfunction in the incumbent’s market is given by

q̂bt ¼ ð1� lÞða2 þ nÞ � p̂bt þ hp̂b2 and

q̂b2 ¼ lða2 þ nÞ � p̂b2 þ hp̂bt :ð22Þ

Due to demand uncertainty in the incumbent’s mar-ket, the entrant faces the risk of not being able tosell all its capacity. Therefore, another importantfactor we incorporate into our model is to allow theentrant to pre-commit to a production quantity thatit will target to sell (Qb

1) in both markets. Dependingon the resolution of demand uncertainty, it is con-ceivable that part of the pre-committed quantityremains unsold. Alternately, if demand in theincumbent’s market is high, the entrant would beable to sell everything. We consider both these casesin the analysis and first determine the optimal pre-committment quantity decision.

6.1. Quantity Pre-Commitment DecisionWith the above new factors being considered in themodel of demand uncertainty, the sequence of

events are as follows: (1) Facing the threat of poten-tial entry, the incumbent invests in market prefer-ence (l), (2) After the incumbent makes itinvestment decision, the entrant pre-commits aquantity (Qb

1) to sell in both markets and incurs cost(c1Qb

1), (3) Uncertainty in demand is resolved (ξ), (4)The incumbent sets its price (p̂b2), and finally, (5) Theentrant sets price (p̂bt ) and (p̂b1) upon entry andallocates its pre-committed quantity across the twomarkets.As is standard, we solve the problem by backward

induction. After the realization of ξ, the entrant’sobjective function is given by

maxp̂b1;p̂

bt

: p̂b1 ¼ p̂b1ða1 � p̂b1Þ þ ðp̂bt � tÞ½ð1� lÞða2 þ nÞ

� p̂bt þ hp̂b2�;ð23Þ

s:t: a1 � p̂b1

h iþ ð1� lÞða2 þ nÞ � p̂bt þ hp̂b2

h i�Qb

1: ð24Þ

The incumbent’s objective function is given by

maxp̂b2

: p̂b2 ¼ ðp̂b2 � c2Þ½lða2 þ nÞ � p̂b2 þ hp̂bt ðp̂b2Þ�: ð25Þ

Note that the entrant’s production cost c1 wouldonly occur at stage 2, where the entrant pre-com-mits its quantity to be sold in both markets. Theoptimal solution to both firms’ problem depends onwhether the pre-commitment constraint equation(24) is activated and we consider the two casesseparately.

6.1.1. Case 1: Pre-Commitment Constraintequation (24) is Activated. Since the constraint isassumed to be activated, we optimize the entrant’sobjective function given in equation (23) and getp̂b1ðQb

1Þ and p̂bt ðQb1Þ as a function of Qb

1. On substitutingp̂bt ðQb

1Þ into the incumbent’s objective function givenin equation (25) and solving for p̂b2, we can getthe incumbent’s profit function p̂b2ðQb

1Þ. Since in stage1, ξ is not resolved, the objective function for theincumbent is therefore given by:

maxl̂b

: E½p̂b2� ¼ E½ðp̂b2 � c2Þðl̂bða2 þ nÞ � p̂b2 þ hp̂bt Þ �sðl̂bÞ22

�( )

;

ð26Þ

which yields the optimal investment decision inmarket preference

l̂bðQb1Þ

¼ð4�3hÞ½ha2ðtþa1þ3a2�2Qb1Þ�c2a2ð4�3h2Þþ3r2h�

8sð4�3h2Þ�ð4�3hÞ2ða22þr2Þ:

ð27Þ


As the entrant’s pre-commit quantity is binding, atstage 2, the entrant will incur production cost c1Qb

1.

6.1.2. Case 2: Pre-Commitment ConstraintEquation (24) is not Activated. We use the notationub to denote the case when the constraint is not acti-vated. By solving the objective function given inEquation (23) without activating the constraint condi-tion given in Equation (24), we can get:

p̂ub1 ðp̂ub2 Þ ¼ a12

and

p̂ubt ðp̂ub2 Þ ¼ tþ a2 þ p̂ub2 h� a2lub þ n� lubn

2:

ð28Þ

Hence, solving backward, we get the entrant’s

expected profit: E½p̂ub1 ðlÞ� and the entrant will chooseits pre-commitment quantity at stage 2 equal to

Qub1 ¼ E½q̂ub1 � þ E½q̂ubt �, which accordingly incurs pro-

duction cost c1Qub1 . Therefore the entrant’s profit is:

p̂ub1 ðlÞ ¼ E½p̂ub1 ðlÞ� � c1Qub1 ðlÞ at stage 2. By using the

same logic as we solve for l̂bðQb1Þ in case 1, we can

get the incumbent’s optimal investment in marketpreference as:

l̂ub ¼ ð2� hÞ hða2ðtþ a2Þ þ r2Þ � c2a2ð2� h2Þ� �4sð2� h2Þ � ð2� hÞ2ða22 þ r2Þ : ð29Þ

On substituting into Qub1 we can get the entrant’s

profit p̂ub1 ðl̂ubÞ. The optimal pre-committed quantityQb

1�is given in the next proposition.

PROPOSITION 7. With demand uncertainty in the incum-bent’s market, an unconstrained entrant’s pre-committedquantity is Qb

1�, where

Qb1

� ¼ 1

4

2ða1 � c1Þ þ c2h

þ t 4� 3h2� �þ a2 4� h2 þ ð4� 2h� h2Þl̂ub� �

2� h2

!:

Intuitively, the entrant faces the risk of not beingable to sell all of its pre-committed quantity whendemand in incumbent’s market is low, it is only will-ing to pre-commit to sell Qb

1�. Therefore, when

k\Qb1�, the entrant will pre-commit its constrained

capacity up to k.

COROLLARY 4. With demand uncertainty in the incum-bent’s market, the entrant’s pre-committed quantity isQb

1 ¼ minfk;Qb1�g.

By pre-committing itself to sell some amount ofquantity in both markets, the entrant can in fact send

a credible signal to the incumbent that it would notsell more quantity into its market and make itself lessthreatening. As a result, the entrant would not sufferthe profit decline as the game switches to the regionof unlimited capacity. Figure 5 shows the profitswhen r = 20 and Qb

1� ¼ 77:5128. As shown in the fig-

ure, there is no profit decline since the entrant wouldnever pre-commit to quantity higher than Qb

1� ¼

77:5128. As a result, the entrant can be better off withquantity pre-commitment when its capacity is suffi-ciently large.

6.2. Capacity Constraint ActivatedIn this section, we consider the case when k�Qb

1�,

that is, the entrant would pre-commit to its entirecapacity k. It can be shown that the incumbent’soptimal strategy is divided into four regions basedon the entrant firm’s capacity. We thus omit thedetails but focus on the new findings under demanduncertainty. We know that if entry does not occur,the objective function for the incumbent firm isgiven by:

maxp̂m2

: p̂m2 ¼ ðp̂m2 � c2Þ½ðl̂m þ hð1� l̂mÞÞða2 þ nÞ

� ð1� h2Þp̂m2 �;ð30Þ

and since (30) is concave, from the first-order condi-tion we get:

p̂m2 ðl̂mÞ ¼c2ð1� h2Þ þ ðhþ l� hlÞða2 þ nÞ

2� 2h2: ð31Þ

Substituting equation (31) into equation (30) andsince demand is not resolved in stage 1, we get:

maxl̂m

:

(E½p̂m2 �¼E

"ðp̂m2 � c2Þððl̂mþhð1� l̂mÞÞ

�ða2þnÞ�ð1�h2Þp̂m2 Þ�sðl̂mÞ2

2

#):

ð32Þ

Figure 5 Entrant’s Profit with Demand Uncertainty


Therefore, we can get the incumbent’s optimalinvestment in market preference and its price

l̂m ¼ a2½ha2 � c2ð1� h2Þ� þ hr2

2ð1þ hÞs� ð1� hÞða22 þ r2Þ;

p̂m2 ¼ 2a2hsþ c2ð1� hÞ½2ð1þ hÞs� ð1� hÞð2a22 þ r2Þ�2ð1� hÞ½2ð1þ hÞs� ð1� hÞða22 þ r2Þ� ;

ð33Þand accordingly its profit in the Blockade Entryregion

p̂m2 ¼ 2sða2h� c2ð1�h2ÞÞ2þ½2sh2� c22ð1�hÞ3ð1þhÞ�r24ð1�hÞ½2sð1þhÞ�ð1�hÞða22þr2Þ� :

ð34Þ

LEMMA 1. With demand uncertainty in the incumbent’smarket, the incumbent can get higher monopolistic profitin the Blockaded Entry region compared to the case whenthere is no demand uncertainty, that is, p̂m2 [ pm2 .

The incumbent is willing to invest more in marketpreference, that is, l̂m [ lm, to increase its marketsize and at the same time able to set a higher mono-poly price and therefore make higher profits.Similar to the discussion in previous sections, based

on the capacity of the entrant, the incumbent can usesome combination of pricing and branding strategy todeter entry. However, due to the impact of demanduncertainty on the incumbent’s profit function in theEffectively Impeded Entry region and in the IneffectivelyImpeded Entry region, the pricing and branding strate-gies became more complicated.

REMARK 1. Based on our numerical results, weobserve that there exists a threshold value r̂ suchthat: (1) If r\ r̂ then the incumbent will adopttwo different combinations of pricing and brandingstrategy to deter entry; (2) If r [ r̂, then it willadopt only one combination of pricing and brandingstrategy.

For example, as shown in the Figure 6, whenr = 5, the incumbent will first invest in market pref-erence according to l̂m

0 ðkÞ and if k[ kc the incum-bent will invest according to l̂m

00 ðkÞ. Proof of Lemma2 in the Appendix S1 contains all the expressions.However, when r = 20, the incumbent will onlyinvest according to l̂m

0 ðkÞ. The reason, as discussedin the proof of Lemma 2, is that when demanduncertainty r is relatively high (for example, r = 20),p̂m

02 ðkÞ is strictly better off than p̂m

002 ðkÞ in the Effec-

tively Impeded region. As a result, only one combina-tion of pricing and branding strategy is sufficient todeter the entrant. Both scenarios lead to the nextlemma.

LEMMA 2. The competition region starts later comparedto the case of demand certainty, that is, K̂

b

2 [ kb2.

Due to demand uncertainty, the entrant’s willing-ness of entry is lower as compared to the case whendemand is certain. As a result, the incumbent is morelikely to hold its monopoly status when market poten-tial is uncertain.

LEMMA 3. With demand uncertainty, the entrant sellsmore in its own market but less in the incumbent’s mar-ket compared to the case without demand uncertainty,that is, qb1 \ q̂b1 and qbt [ q̂bt .

Due to demand uncertainty in the incumbent’smarket, the entrant aims to sell more in its ownmarket but less in the incumbent’s market. As aresult, its profit in the incumbent’s market islower.

LEMMA 4. With demand uncertainty, the incumbent’sprofit are higher when competition starts, that is,p̂b2ðkÞ [ pb2ðkÞ.

Lemma 4 shows that the incumbent can benefitfrom demand uncertainty. Indeed, the incumbent iswilling to invest more in consumer’s preference toincrease its demand (l̂b [ lb) and at the same time is

Figure 6 Market Preference with Demand Uncertainty


able to set a higher price (p̂b2ðkÞ [ pb2ðkÞ) in the Ineffec-tively Impeded Entry region.

REMARK 2. Based on our numerical analysis, weobserve that there exists a threshold value �r suchthat: (1) If r\ �r then there exists a value ~k such thatif K̂

b

2 \ k\~k then p̂b1ðkÞ [ pb1ðkÞ and if ~k\ k\Qb1�

then p̂b1ðkÞ\ pb1ðkÞ; (2) If r [ �r, p̂b1ðkÞ\ pb1ðkÞ.

The intuition behind this result is as follows: Whendemand uncertainty is relatively low, the entrant sellsmore in its own market and its total profit is highercompared to the model without demand uncertainty.However, when the entrant’s capacity is greater thansome value, as it sells less in the incumbent’s market,the reduction of profits in the incumbent’s market willbe greater than the gain in its own market, and totalprofit will be lower. When demand uncertainty is rel-atively high, the incumbent is more likely to keep theentrant out of its market. As a result, K̂

b

2 would begreater than ~k and the entrant’s gain from its ownmarket does not cover the shortfall of not makingentry in the incumbent’s market.Another interesting observation is as follows. It is

worthwhile to note that if the branding strategy l isexogenous, then with demand uncertainty in theincumbent’s market, the entrant is in fact always bet-ter off. As such, by investing in market preference, theincumbent may make the entrant worse off. In the sec-ond stage, the entrant’s profit function in the incum-bent’s market is

p̂b1ðlÞ ¼ pb1ðlÞ þ eðlÞ; ð35Þ

where eðlÞ ¼ ½ð8�3h2Þð1�lÞþ4hl�2r232ð4�3h2Þ2 [ 0 and pb1ðlÞ is the

entrant’s profit function when demand is certain. Atfirst glance, the entrant benefits from demand uncer-tainty. However, the incumbent will choose

l̂bðkÞ

¼ð4�3hÞ a2hðtþa1þ3a2�2kÞ�c2a2ð4�3h2Þþ3hr2� �

8sð4�3h2Þ�ð4�3hÞ2ða22þr2Þ;

ð36Þwhich can be proved that l̂bðkÞ [ lbðkÞ (the invest-ment decision when demand is certain). It is theinvestment decision (l) that may lower the entrant’stotal profit. Although the expression for the differ-ence in the profit function can be derived, it is notpossible to analytically examine it. With extensivenumerical analysis, we get the results in Remark 2.

7. Conclusions and Future Research

We studied the entry-deterrence problem of anincumbent firm who faces competition from a capaci-

tated entrant firm. The incumbent must choose itspricing and branding strategies in order to block,deter, or accommodate the entrant. When theentrant’s capacity is low, the threat of entry is low.However, when the entrant’s capacity is higher than acritical threshold, the incumbent has to lower price tostop entry. Finally, allowing entry is optimal strategywhen the entrant’s capacity size is sufficiently high,and beyond another threshold value, the capacitatedproblem switches to the uncapacitated case.The study first considers the pricing competition

models and we show that profits for both firms arehigher in Stackelberg competition than in Nash com-petition. Surprisingly, the entrant, as follower, is alsobetter off in Stackelberg competition because ofreduced price competition resulting in higher pricesand higher quantities sold. We show that in Stackel-berg competition, the capacitated entrant sells fewerproducts in its own market but more in the incum-bent’s market than an uncapacitated entrant firm. Ouranalysis also indicates the surprising result that profitsfor the entrant firm are not monotonically increasingin its capacity. While higher capacity for the entrantincreases potential for competition for the incumbentand lower profits for it, as the entrant cannot crediblycommit to selling less quantity in the incumbent’s mar-ket, the price competition becomes intense leading tolower profits for the entrant firm as well.In the branding model, in addition to the pricing

strategy, the incumbent has an additional mechanismto better compete with the entrant. Interestingly, ourresults show that in order to deter the potential entrant,the incumbent would invest in improving consumer’spreference for its product while also increasing theprice. This is in contrast to the results when only thepricing strategy is available. Finally, we incorporatedemand uncertainty into our model and show that theincumbent benefits from demand uncertainty whilethe entrant may be worse off depending on the magni-tude of demand uncertainty and its capacity.In future studies, researchers can extend our model

to a broader setting where the incumbent also has lim-ited capacity. Such a setting requires a two-dimen-sional analysis of capacity thresholds for both firms.Another extension to our work is the case when theentrant also has the option of investing in consumers’preference in the incumbent’s market. In this case,consumer’s preference is a decision variable for bothfirms. Finally, the problem when the entrant firm’scapacity size is private information would be anotherinteresting problem for future research.

Acknowledgments

The authors thank the senior editor and two anonymousreviewers for their excellent suggestions. They also grate-


fully acknowledge Raphael Boleslavsky’s insightful com-ments.

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Supporting InformationAdditional Supporting Information may be found in theonline version of this article:

Appendix S1: Proofs of Results in Stackelberg and NashCompetition in Uncapacitated Case.


entry deterrence of capacitated competition using price ... · we also study the timing difference...

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