Horizontal Product Differentiation: Disclosure and Competition

Transcription

1 Horizonal Produc Differeniaion: Disclosure and Compeiion Mariya Teeryanikova Universiy of Vienna, Ausria. Maaren C.W. Janssen Universiy of Vienna, Ausria. January 31, 2012 This Version Absrac The unravelling argumen in he disclosure lieraure esablished ha when firms produce differen qualiies ha are unknown o consumers, firms have an incenive o disclose his privae informaion. Recen lieraure has esablished ha his argumen does no carry over o an environmen where a monopoly firm produces a variey of horizonally differeniaed producs. This paper argues ha he resuls of he recen lieraure are due o he assumpion of a monopoly firm. We consider a horizonally differeniaed duopoly marke and show ha all equilibria of he disclosure game have firms fully disclosing he variey hey produce. JEL Classificaion: D43, D82, D83, M37 Keywords: Informaion disclosure, horizonal differeniaion, price compeiion, asymmeric informaion Deparmen of Economics, Universiy of Vienna, Hohensaufengasse 9, 1010 Vienna, Ausria. Tel: ( ; mariya.eeryanikova@univie.ac.a. Deparmen of Economics, Universiy of Vienna, Hohensaufengasse 9, 1010 Vienna, Ausria. Tel: ( ; maaren.janssen@univie.ac.a.

2 1 Inroducion In a large number of markes, sellers have imporan informaion abou produc aribues ha are no publicly observable. In many insances, however, firms have he opion of volunarily disclosing his informaion in a credible and verifiable manner hrough a variey of means such as independen hird pary cerificaion, labeling, raing by indusry associaions (or governmen agencies and hrough informaive adverising. There is a large lieraure dealing wih he quesion wheher firms have appropriae incenives o disclose informaion abou he produc hey produce. Mos of his lieraure deals wih his issue in he conex of verical produc differeniaion, where differen firms sell differen qualiies. In his conex, he well-known unraveling argumen, 1 esablishes ha a firm whose produc is acually beer han he average has a posiive incenive o volunarily disclose he qualiy of is produc o buyers. This hen induces every firm whose qualiy is above he average undisclosed qualiy o also disclose. The unraveling argumen resuls in a siuaion where all privae informaion abou qualiy should be revealed hrough volunary disclosure. Observed nondisclosure is hen explained in erms of disclosure fricions, such as disclosure coss, consumers no undersanding he informaion ha is disclosed, ec. Alernaively, Janssen and Roy (2010 show ha nondisclosure can also be explained by a combinaion of marke compeiion and he availabiliy of signaling as an alernaive means (o disclosure of communicaing privae informaion. Recenly, Sun (2011 and Celik (2011 have analyzed he incenives for firms o disclose heir produc characerisics when horizonal produc differeniaion is he only or main dimension of differeniaion. Boh papers are se in a monopoly conex. Sun (2011 shows ha seller ypes wih unfavorable horizonal aribues (owards he exreme poins of he produc line do no have an incenive o disclose. In combinaion wih verical differeniaion, her resuls imply ha if eiher full disclosure of boh aribues or no disclosure a all are he only possible reporing sraegies, a seller wih privae informaion abou boh horizonal and verical aribues may no wan o disclose qualiy even if i is high. Celik (2011 shows ha he amoun of informaion disclosure criically depends on he srengh of he buyer s preference for her ideal aribue. If buyers have very srong preferences for paricular produc varieies, hen here exiss an equilibrium in which he seller fully reveals variey. Oherwise, he seller only parially reveals he variey he produces. Moreover, he se of fully revealed locaions monoonically shrinks from all o (almos none as he buyer s preference for her ideal ase becomes weaker. 1 See, Viscusi (1978, Grossman (1981, Milgrom (1981, Jovanovic (

3 In his paper, we show ha hese resuls for horizonal produc differeniaion do no exend o a more compeiive environmen. In paricular, we show in a duopoly se-up ha full disclosure is always an equilibrium, and moreover, ha here does no exis any equilibrium where firms do no fully disclose heir produc informaion. As here can be many messages wih which firms fully disclose heir informaion, he equilibrium sraegies are no unique, bu he equilibrium oucome of full disclosure is. The model we consider has wo firms locaed on a Hoelling line, where each paricular locaion represens he variey of he produc. Locaion is known o boh firms, bu no o consumers. 2 The wo firms firs simulaneously choose a message abou heir locaion. We assume ha firms canno lie. Tha is, he rue locaion should be consisen wih he message ha is chosen. One way o hink abou his grain of ruh assumpion is ha informaion is verifiable and ha here is a large fine for providing informaion ha urns ou o be false. The assumpion is in line wih regulaions concerning adverisemen or oher disclosure mechanisms requiring ha firms provide ruhful informaion. Firms can eiher send a raher vague message, indicaing ha heir locaion is somewhere on he produc line, as one exreme, or a much more precise message, indicaing he precise locaion, as he oher exreme, or anyhing in beween. Afer firms have sen heir messages, hey boh simulaneously choose prices. Consumers have quadraic ransporaion coss and decide where o buy he produc afer observing he messages and he prices. Given he informaion hey receive, consumers updae heir beliefs abou he locaion of he wo firms and buy from he firm where he sum of ransporaion coss and price is he lowes. The reason why in his environmen all equilibria mus be fully revealing is inimaely relaed o he reason why in he sandard Hoelling model wih locaion choice, firms wan o maximally differeniae from each oher. Suppose ha a firm would no choose a fully revealing sraegy and would choose he same message for differen locaions. As he price canno be used o signal he locaion, consumers will hen be uncerain abou he rue locaion of he firm. In his case, he updaed beliefs of consumers will be such ha hey do no assign full probabiliy mass o he exreme locaions ha send his paricular message. A leas one of hese exreme locaions have hen an incenive o deviae for wo reasons. Firs, by fully revealing is locaion, a firm can reduce he uncerainy concerning he locaion for consumers and wih convex ransporaion coss any 2 The compeiive disclosure lieraure can be divided ino a sream ha considers markes where firms know each ohers ype (see, e.g., Daughey and Reinganum (2007, Caldieraro, Shin and Sivers (2008 and Janssen and Roy (2011 and anoher lieraure where hey do no (see, e.g., Board (2009 and Hoz and Xiao (2011. The firs ype of lieraure, and hus his paper, is relevan for markes where firms are acive for some ime and have he abiliy (and due o he frequen ineracion also he incenives o find ou he produc produced by a compeior. 2

4 reducion in uncerainy increases demand ceeris paribus. 3 Second, by having a perceived locaion ha is furher away from he compeior, firms charge higher price in he pricing game and his price effec ouperforms he direc demand effec. The res of he paper is organized as follows. Secion 2 presens he model. Secion 3 describes he resuls and Secion 4 concludes. 2 Model Consider a horizonally differeniaed duopoly, where he variey produced by each firm is represened by a paricular locaion on he uni inerval. Le x i denoe he variey produced by firm i and x i [0, 1], i = 1, 2. We focus on he disclosure policy of he firms and consider hese varieies o be given for he firms. In he following, x 1 and x 2 will be referred o as locaions or ypes of firms 1 and 2, respecively. We consider markes where firms know each ohers s locaion, bu consumers are unaware of he specific locaion of firms. One way o hink abou his is ha i requires resources o research he produc characerisics of a firm and ha rival firms are beer equipped or have more incenives o do his han consumers. Producion coss do no depend on firms locaions and wihou loss of generaliy are se o be equal o zero. The demand side of he economy is represened by a coninuum of consumers. Each consumer has a preference for he ideal variey of he good ha she would like o buy, denoed by λ. The value of λ, or consumers locaion on [0, 1], follows a uniform disribuion. 4 A consumer s ne uiliy from buying variey x i a price P i, i = 1, 2, is v (λ x i 2 P i, where v is he gross uiliy of a consumer when he variey of he good, x i, maches wih her ideal variey, λ, perfecly (i.e., when x i = λ and measures he degree of disuiliy a consumer incurs when x i and λ differ from each oher. We assume ha v is sufficienly large so ha he marke is fully covered. Each consumer hen chooses o buy he good from he firm where her expeced uiliy is maximized. The consumer has uni demand and if she buys from firm i, hen firm i s payoff from he ransacion is P i ; oherwise, he payoff of firm i is zero. The iming of he game is as follows. A sage 0, Naure independenly selecs locaions x 1 for firm 1 and x 2 for firm 2 from a sricly posiive densiy funcion f(x. The locaions are known o boh firms, bu no o consumers. A sage 1, firms send a cosless message M i [0, 1], i = 1, 2, abou 3 One way o inerpre his is ha convex ransporaion cos inroduces an elemen of risk aversion in consumers preferences: ceeris paribus a consumer raher buys a a known locaion han a an unknown locaion wih he same expeced value. 4 This specificaion wih a coninuum of consumers whose preferences for variey is uniformly disribued over he uni inerval, is idenical o he specificaion wih a single consumer who has a privaely known ase for a variey drawn from he uniform densiy funcion defined over [0, 1]. 3

5 heir locaion, where M i = [0, 1] can be inerpreed as no message a all, or full non-disclosure of informaion by firm i. Messages have o conain a grain of ruh in he sense ha x i M i for i = 1, 2. Tha is, firms canno lie abou heir locaion. In he following we will refer o his assumpion as he grain of ruh assumpion. A sage 2, firms simulaneously se prices. Finally, a sage 3, consumers observe he messages and he prices of he wo firms and decide where o buy. A he end of he game, he payoffs of all players firms and consumers are realized. All aspecs of he game are common knowledge. Two imporan observaions are in order a his poin. Firs, he quadraic erm in he uiliy funcion of consumers (he ransporaion coss implies risk aversion wih respec o x i. Tha is, a consumer dislikes uncerainy abou he variey of he good and given wo messages wih he same condiional mean, favors he one wih he smaller variance. Second, even hough consumers are assumed o have uni demand for he good, he probabiliy of a purchase from a given firm declines wih is price so ha he expeced demand funcion faced by each firm is downward sloping. To proceed wih he more formal analysis, we define he sraegy spaces as follows. The reporing sraegy of firm i is denoed by m i (x i, x j. The image of m i belongs o all subses of [0, 1] such ha x i m i. The pricing sraegy of firm i is denoed by p i (x i, x j M i, M j where he messages sen by he wo firms are M i and M j, respecively. Similarly, le he vecor b(λ, M i, M j, P i, P j describe he buying sraegy of a consumer wih preferred variey λ, where b = (1, 0 if he consumer buys he good from firm 1 and b = (0, 1 if he consumer buys he good from firm 2. Finally, le µ i (z M i, M j, P i, P j be he probabiliy densiy ha consumers assign o x i = z when he firms send messages M i, M j and se prices P i, P j. Noe ha a he momen when consumers have o decide from which firm o buy, hey form beliefs µ i no only on he basis of he observed messages and prices, bu also on he basis of he equilibrium sraegies, ha is, equilibrium messages and prices, m i (x i, x j and p i (x i, x j M i, M j. All consumers process he informaion received in he same way and herefore have symmeric beliefs. Before providing he deails of he equilibrium noion which we will use o analyze he game, we consider he decision making of a consumer. To do so, we firs find he ideal variey, λ, of he indifferen consumer, who obains he same expeced ne uiliy of buying from eiher of he wo firms, given he observed se of messages and prices. Then all consumers wih ideal varieies below λ buy from he firm wih he mos lef perceived locaion and all ohers buy from he oher firm. Therefore, λ deermines he expeced demand faced by each firm and allows describing opimal prices and messages chosen by he firms a he previous wo sages of he game. 4

6 Given he updaed beliefs, he ideal variey λ of he indifferen consumer is defined by he equaliy beween he expeced ne uiliy of buying from firm 1 and he expeced ne uiliy of buying from firm 2: v E (( λ x 1 2 µ 1 P 1 = v E (( λ x 2 2 µ 2 P 2 (2.1 In his expression, E (( λ x i 2 µ i, i = 1, 2, is he expecaion of he ransporaion coss of he indifferen consumer associaed wih buying from firm i, condiional on consumers beliefs. We solve his equaliy for λ. Noice ha E (( λ x i 2 µ i = λ 2 + E ( x 2 i µ i 2 λe (xi µ i = λ 2 + var (x i µ i + E 2 (x i µ i 2 λe (x i µ i so ha (2.1 becomes: λ 2 + var (x 1 µ 1 + E 2 (x 1 µ 1 2 λe (x 1 µ 1 + P 1 = λ 2 + var (x 2 µ 2 + E 2 (x 2 µ 2 2 λe (x 2 µ 2 + P 2. Thus, he ideal variey of he indifferen consumer is equal o: λ = 1 2 P 2 P 1 + var (x 2 µ 2 var (x 1 µ E (x 2 µ 2 E (x 1 µ 1 2 (E (x 1 µ 1 + E (x 2 µ 2 (2.2 This resul has an immediae implicaion for he form of he expeced demand funcions of firms 1 and 2. In fac, since consumers wih preferred variey λ < λ (λ > λ buy from he firm wih he mos lef (righ perceived locaion and since he value of consumer s bes-preferred variey is disribued uniformly over [0, 1], λ is also he value of he expeced demand faced by he firm wih he mos lef perceived locaion, given he prices P 1 and P 2 and he messages M 1 and M 2. Accordingly, 1 λ, he remaining share of he marke, is he expeced demand of he oher firm. Wihou loss of generaliy, hroughou he paper we consider ha E (x 1 µ 1 E (x 2 µ 2. In case of sric inequaliy, λ is he expeced demand of firm 1 and 1 λ is he expeced demand of firm 2. The case when E (x 1 µ 1 = E (x 2 µ 2 will be addressed separaely laer. The derivaion of expeced demand helps o define he equilibrium noion we use. From (2.2 i follows ha apar from he price difference, a firm s expeced demand and, hence, is profi only depends on expeced locaions and on he precision of he messages abou hese locaions, bu (and his is imporan no on acual locaions. This is rue boh on he equilibrium and off he equilibrium pah. Tha is, any ype of firm ha sends he same equilibrium message concerning locaion has equal incenives o se any ou-of-equilibrium price. Given his fac, price canno reasonably ac as = 5

7 a signal of locaion. A similar argumen applies o he inabiliy of firms o signal he locaion of heir compeior. Noe ha as firms know heir own locaion and he locaion of heir compeior, a firm s ype is, in principle, a wo-dimensional objec (x 1, x 2. Therefore, consumers could, in principle, make some inference on he locaion of he compeior upon observing a firm s message. Given he above, his, however, would no be a reasonable inference. Consider ha, all ypes of a firm ha have sen he same equilibrium message concerning locaion and ha herefore are believed o have an idenical locaion along he equilibrium pah have equal incenives o make consumers believe ha heir compeior has a cerain locaion. Moreover, equilibrium pay-offs of all ypes ha could given he grain of ruh assumpion have sen he same equilibrium message, should receive he same equilibrium pay-off as oherwise one of he ypes should have an incenive o send anoher message. Therefore, if consumers would infer a cerain locaion of he compeior afer observing a paricular ou-of-equilibrium message, eiher all ypes (x 1, x 2 of he firm wih he same x 1 componen would wan o deviae o ha ou-of-equilibrium message or no ype would. Bu hen i would be unreasonable for consumers o discriminae beween firm ypes ha differ only in he locaion of he compeior. In principle, he same could apply o a firm rying o signal is own locaion. Bu here is where he grain of ruh assumpion becomes relevan. If a firm would deviae o a very precise message, hen because of he grain of ruh assumpion only few (or in he limi, no oher ypes wih differen own locaion can imiae ha signal. Thus, he grain of ruh assumpion makes i possible for ou-of-equilibrium messages o signal some informaion abou own locaion. These consideraions imply ha in he conex of our model where profis are only governed by prices and expeced locaions (and no by real locaions, i is reasonable o confine aenion o equilibria where consumer beliefs concerning a firm s locaion only depend on is own message and no on pricing decisions. Also, due o he grain of ruh assumpion consumers inerpre he one dimensional message M i of firm i as being uninformaive abou he locaion of he compeior. We call such a perfec Bayesian equilibrium a sable belief equilibrium and i is defined as follows. Definiion A sable belief equilibrium is a se of reporing and pricing sraegies m 1, m 2, p 1, p 2 of he wo firms, sraegy b of a consumer, and he probabiliy densiy funcions π 1, π 2 which saisfy he following condiions: 6

8 (1 For all M 1, M 2, P 1 and P 2, b is a consumer s bes buying decision as defined below: b(λ, M 1, M 2, P 1, P 2 = (1, 0 if 1 0 (v (λ x 1 2 P 1 µ 1 (x 1 M 1, M 2, P 1, P 2 dx (v (λ x 2 2 P 2 µ 2 (x 2 M 1, M 2, P 1, P 2 dx 2 = (0, 1 if 1 0 (v (λ x 2 2 P 2 µ 2 (x 2 M 1, M 2, P 1, P 2 dx 2 ( (v (λ x 1 2 P 1 µ 1 (x 1 M 1, M 2, P 1, P 2 dx 1 (2 Given (1 and given he messages sen by he wo firms and he price se by he compeior, p i is he price ha maximizes he expeced profi of firm i, i = 1, 2. (3 Given (1, (2 and given he message sen by he compeior, m i is he message ha maximizes he expeced profi of firm i, i = 1, 2,, subjec o he consrain ha x i m i. (4 For all M 1, M 2, P 1 and P 2, a consumer updaes his or her beliefs in he following way: 5 µ i (z M 1, M 2, P 1, P 2 = f(z z M i f(zdz on he equilibrium pah any beliefs saisfying he propery µ i (z M i, M j, P 1, P 2 = µ i (z M i, M j, P 1, P 2 for any P 1, P 2, P 1, P 2, M j, M j off he equilibrium pah Par (1 of he definiion saes ha for any observed messages and prices, a consumer buys a uni of he produc from he firm, where her expeced ne uiliy, given he updaed beliefs, is maximized. Each firm raionally anicipaes he bes response of consumers o any given messages and prices, and chooses he price and message ha maximize is expeced profi. This is saed in pars (2 and (3. Finally, par (4 claims ha consumers updae beliefs abou he locaions using Bayes rule for any M 1, M 2, P 1 and P 2 ha occur wih posiive densiy along he equilibrium pah and ha, as discussed above, off-he-equilibrium pah beliefs abou firm i s locaion canno depend on prices and he message sen by he oher firms as any ype of firm has equal incenives o choose any price. Given his independence of beliefs from prices and he message of he rival firm (on and off he equilibrium pah, prices and he message of he oher firm can be omied from he noaion for beliefs, leading o µ i (z M i. 5 Noe ha Bayes rule canno be applied when M i or a subse of M i is discree. For example, if M i = {y, z}, hen boh evens, x i = y and x i = z have ex-ane zero probabiliy. In his case, updaing proceeds as follows: Using l Hôpial s rule, F (z + ε F (z µ i(z M 1, M 2, P 1, P 2 = lim ε 0 F (z + ε F (z + F (y + ε F (y f(z + ε µ i(z M 1, M 2, P 1, P 2 = lim ε 0 f(z + ε + f(y + ε = f(z f(z + f(y 7

9 3 Resuls Given ha we have already derived consumer demand in he previous secion, we sar our analysis by sudying he pricing decision of firms. Each firm anicipaes he opimal behavior of consumers and chooses price so as o maximize is expeced profi, for any given messages of he firms sen a he previous sage. Expression (2.2 for λ implies ha he profis of firms 1 and 2 are given by ( P 1 2 P 1 + var (x 2 µ 2 var (x 1 µ 1 π 1 = P E (x 2 µ 2 E (x 1 µ 1 2 (E (x 1 µ 1 + E (x 2 µ 2 ( π 2 = P P 2 P 1 + var (x 2 µ 2 var (x 1 µ E (x 2 µ 2 E (x 1 µ 1 2 (E (x 1 µ 1 + E (x 2 µ 2. Funcion π i, i = 1, 2, is a sricly concave, quadraic funcion of P i. Hence, as prices do no effec consumers beliefs abou locaion he profi-maximizaion problem of each firm is well-defined and he firs-order condiions yield he price a which π i is maximized: 6 1 P 1 (E (x 2 µ 2 E (x 1 µ P 2 (E (x 2 µ 2 E (x 1 µ ( P2 + var (x 2 µ 2 var (x 1 µ 1 E (x 2 µ 2 E (x 1 µ 1 + E (x 1 µ 1 + E (x 2 µ 2 ( P 1 + var (x 2 µ 2 var (x 1 µ 1 + E (x 1 µ 1 + E (x 2 µ 2 E (x 2 µ 2 E (x 1 µ 1 = 0 = 0. The firs equaion above is he firs-order condiion for firm 1, π 1 P 1 is he firs-order condiion for firm 2, π 2 P 2 sraegy of firm 2: = 0, while he second equaion = 0. The second equaion yields he bes-response pricing P 2 = (E (x 2 µ 2 E (x 1 µ 1 1 2( P1 + (var (x 2 µ 2 var (x 1 µ ( E 2 (x 2 µ 2 E 2 (x 1 µ 1 (3.1 Summing up he wo firs-order condiions leads o he bes-response pricing sraegy of firm 1: P 1 = 2 (E (x 2 µ 2 E (x 1 µ 1 P 2 (3.2 6 This derivaion makes use of he fac ha we are resricing our aenion o equilibria where prices do no affec consumers beliefs regarding locaion. 8

10 Collecing (3.1 and (3.2 resuls in he soluion of he price seing sage of he game: P 1 = ( 2 3 (E (x 2 µ 2 E (x 1 x 1 Ω ( E 2 (x 2 x 2 Ω 2 E 2 (x 1 µ (var (x 2 µ 2 var (x 1 µ 1 (3.3 P 2 = ( 4 3 (E (x 2 µ 2 E (x 1 µ 1 1 ( E 2 (x 2 µ 2 E 2 (x 1 µ (var (x 2 µ 2 var (x 1 µ 1 (3.4 Plugging expressions (3.3 (3.4 for prices ino he profi funcions of he wo firms, yields reducedform profi funcions ha are expressed in erms of he condiional expecaions and variances of x 1 and x 2 : π 1 = π 2 = 18 (E (x 2 µ 2 E (x 1 µ 1 (2 (E (x 2 µ 2 E (x 1 µ 1 + (3.5 + ( E 2 (x 2 µ 2 E 2 (x 1 µ 1 + (var (x 2 µ 2 var (x 1 µ (E (x 2 µ 2 E (x 1 µ 1 (4 (E (x 2 µ 2 E (x 1 µ 1 (3.6 ( E 2 (x 2 µ 2 E 2 (x 1 µ 1 (var (x 2 µ 2 var (x 1 µ 1 2 We can now consider sage a which firms decide on he messages hey will send. As a special case, consider firs he siuaion where locaions of boh firms are fully revealed. This means ha in all expressions above E (x i µ i = x i, var (x i µ i = 0 and profis of firm 1 and 2 are funcions of exac locaions x 1, x 2. In paricular, following he assumpion ha x 1 x 2 (he analogue of E (x 1 µ 1 E (x 2 µ 2, he profis of firms 1 and 2 are given by: π 1 (x 1, x 2 = 18 (x 2 x 1 (2 + x 1 + x 2 2 (3.7 π 2 (x 1, x 2 = 18 (x 2 x 1 (4 x 1 x 2 2 (3.8 Boh hese expressions are sricly posiive as long as x 1 < x 2. If x 1 = x 2, hen consumers buy from he firm wih he lowes price. The usual Berrand-ype argumen hen esablishes (cf., 3.3 and 3.4 ha P 1 = P 2 = 0 and so, π 1 = π 2 = 0. Noe ha he profi of firm 1 in (3.7 is decreasing in x 1, while he profi of firm 2 in (3.8 is increasing in x 2. Indeed, ( π 1 = (2 + x 1 + x 2 x 1 18 x x 1 < 0 9 ( π 2 = (4 x 1 x 2 x 2 18 x x > 0, 9 where he signs of he derivaives are implied by he fac ha 0 x 1, x 2 1. This finding is 9

11 consisen wih he argumen in he sandard Hoelling model of locaion choice. Firms wan o be locaed maximally far from each oher as differeniaion allows hem o charge higher prices, which urns ou o ouweigh he adverse effec of a decline in demand. This funcional dependence of profis on locaions plays a key role in he proof of he firs heorem: Theorem 3.1. There exiss a sable belief equilibrium where firms fully disclose heir locaion. Theorem 3.1 claims ha full disclosure is always an equilibrium of he game. Clearly, he fully revealing equilibrium is no unique since here are many ses of messages wih which firms are able o fully disclose heir locaion. In he proof we use sraegies where firms disclose heir locaion precisely. This faciliaes he proof in he sense ha i is impossible for ypes o imiae each ohers message due o he resricion ha messages mus be ruhful. For oher fully revealing equilibria, we should verify in addiion ha his ype of imiaion is no profiable. The proof also uses specific ou-of-equilibrium beliefs ha discourage firms o deviae from heir equilibrium sraegies. These ou-of-equilibrium beliefs are somewha exreme in he sense ha consumers believe ha an-ou-of-equilibrium message is sen by one of he ypes wih he lowes equilibrium profi of ypes in he se of ypes ha is consisen wih he message, even hough all oher ypes ha are consisen wih he message could also have ruhfully send such a message. However, one can show ha hese ou-of-equilibrium beliefs are reasonable in he sense ha hese exreme ou-of-equilibrium beliefs are consisen wih he logic of he D1 crierion. 7 The nex resul is probably even more imporan for he general message of he paper han Theorem 3.1 saing he exisence of fully disclosing equilibrium. Theorem 3.2 shows ha when here is compeiion beween firms, here canno be a sable belief equilibrium where firms do no perfecly disclose he variey hey produce. Thus, even hough he fully revealing equilibrium sraegies are hemselves no unique, he equilibrium oucome of full disclosure is. Theorem 3.2. There does no exis a sable belief equilibrium where firms do no fully disclose heir locaion. The inuiion behind he resul of Theorem 3.2 is relaed o he argumen in he sandard Hoelling model wih locaion choice, where firms have an incenive o maximally differeniae from each oher. 7 Inuiively, he D1 crierion requires ha for a given observed deviaion from he equilibrium sraegy, consumers believe ha such deviaion was chosen by he ype of a firm ha has mos incenives o deviae. To define which ype has mos incenives o deviae, observe ha he profis are compleely deermined by consumer beliefs abou locaion and no by locaion iself. Thus, afer sending an ou-of-equilibrium message, he profis of a deviaing firm are independen of is ype. Therefore, he incenive o deviae is larges for he ype (or ypes in M i wih he smalles equilibrium payoff. This way o modify he D1 crierion o his game is suggesed by a similar adapaion in Janssen and Roy (

12 If a firm does no follow a fully revealing sraegy, here are locaions wihin is non-fully revealing message ha are furher away from he perceived locaion of he rival firm han he own perceived locaion. These locaions have an incenive o deviae by fully disclosing hemselves. The reason for his is wofold. Firs, by fully revealing is locaion, a firm reduces he uncerainy associaed wih he locaion for consumers and given he quadraic ransporaion coss, any reducion in uncerainy increases he demand ceeris paribus. Second, by having a perceived locaion ha is furher away from he compeior, a firm can charge higher price and his price effec ouperforms he direc effec of a decline in demand. 4 Conclusion In his paper we developed a duopoly model of horizonal produc differeniaion. We sudied he incenives of a firm o disclose is horizonal produc characerisic when his characerisic is known o boh firms bu no o consumers. Firms firs simulaneously choose a message abou heir locaion, such ha his message is ruhful, ha is, he rue locaion of a firm is consisen wih he message. The messages can range from being very precise (indicaing he exac locaion o very vague. Afer firms have sen heir messages, hey simulaneously choose prices. Given he messages and prices, consumers updae heir beliefs abou firms locaions and decide where o buy. As profis in his environmen only depend on expeced locaions and prices, bu no on real locaions or ypes, and as we insis ha messages have o conain a grain of ruh, we define a sable belief equilibrium where consumers beliefs concerning a firm s locaion afer observing some ouof-equilibrium message or prices only depend on a firm s own message concerning is locaion. We argue ha his is he naural equilibrium noion in his environmen. We find ha all sable belief equilibria of he game are such ha boh firms fully reveal heir locaions. In oher words, here always exiss an equilibrium where firms fully disclose heir locaion and here does no exis a sable belief equilibrium where firms do no disclose. This full-disclosure resul conrass wih he finding of possible non-disclosure in he lieraure on horizonal produc differeniaion in a monopolisic se-up, suggesing ha compeiion plays a key role in deermining incenives for firms o disclose. Inuiively, he reason why in he compeiive environmen all equilibria mus be fully revealing is relaed o he reason why in he sandard Hoelling model wih locaion choice, firms wan o maximally differeniae from each oher. Suppose ha a firm would no fully reveal is locaion, choosing he same message for differen locaions. Consumers will hen be uncerain abou he rue locaion of he firm and hence, will form beliefs such ha he exreme locaions, sending ha 11

13 paricular message, will no obain full probabiliy mass. These exreme locaions have hen incenive o deviae o full disclosure for wo reasons. On one hand, by fully revealing is locaion, a firm reduces he uncerainy concerning he locaion for consumers and wih quadraic ransporaion coss, his increases heir demand. On he oher hand, by indicaing a locaion ha is perceived by consumers as being furher away from he compeior, price compeiion is sofened and his price effec ouweighs he direc effec of a decline in demand. In he presen model we have considered a simple framework where consumers are uniformly disribued over he uni inerval and have quadraic ransporaion coss. Moreover, disclosure is compleely cosless and firms know no only heir own locaion, bu also he locaion of heir compeiors. We have considered his simple framework o focus on he role of compeiion in providing incenives for firms o fully disclose. In fuure work, we inend o relax some of hese assumpions. For example, he case where firms have purely privae informaion abou heir produc characerisics could be of considerable ineres. Appendix Proof of Theorem 3.1. Suppose ha firms fully reveal heir locaion by ruhfully announcing i, i.e., every firm wih locaion x i sends message M i = {x i }. Since firms canno lie, he firm of any given ype x i canno imiae he message of anoher ype. Any deviaing message is herefore an ou-of-equilibrium message and he proof of an equilibrium hen requires o consruc a se of ouof-equilibrium beliefs such ha given hese beliefs, no firm has an incenive o deviae. Le us consider he following ou-of-equilibrium beliefs. For any ou-of-equilibrium message M i sen by firm i, consumers assign probabiliy one o firm i being of ype x(m i where x(m i arg min x i M i π i (x 1, x 2 i.e., x(m i is any selecion from he se of minimizers of he funcion π i (x 1, x 2 on he se M i. Observe ha given hese ou-of-equilibrium beliefs, no ype of any firm wishes o deviae from he candidae equilibrium sraegies. If firm i of ype x i deviaes and sends some admissible message M i {x i }, hen he subsequen choice of consumers will be as if he rue ype of firm i is x(m i for sure, and since all ha maers for he payoff of firm i is her perceived ype (and no her rue ype, he expeced coninuaion payoff afer his deviaion is exacly equal o π i ( x(m i, x j. As x i M i i follows ha π i ( x(m i, x j π i (x i, x j. Therefore, he deviaion is no gainful. 12

14 Proof of Theorem 3.2. Suppose ha a leas one of he wo firms does no follow a fully revealing sraegy and chooses he same message for differen locaions. Le ypes x 1 S 1 send idenical message M 1 and ypes x 2 S 2 send idenical message M 2, where a leas one of he ses S 1, S 2 conains wo or more ypes. 8 Wihou loss of generaliy, assume ha firm 1 does no fully disclose is locaion (while firm 2 may disclose or no disclose. In his case, consumers are uncerain abou he rue locaion of he non-disclosing firm/firms and form expecaions abou his locaion and resuling ransporaion coss. Again, wihou loss of generaliy, we resric he analysis o he case when E (x 1 µ 1 E (x 2 µ 2. If he inequaliy is sric, he profis of firms 1 and 2 are given by (3.5 (3.6. If insead E (x 1 µ 1 = E (x 2 µ 2, hen consider he firm (referred o as firm i whose equilibrium message has he larges variance (referred o as firm j, ha is, var (x i µ i var (x j µ j. The profi of firm i is equal o zero because firm j is a leas as aracive o consumers and hence, can eiher push firm i ou of he marke by seing P j = (var (x i µ i var (x j µ j (if var (x i µ i > var (x j µ j or share he marke wih firm i bu a zero prices (if var (x i µ i = var (x j µ j. Suppose firs ha E (x 1 µ 1 < E (x 2 µ 2. We prove ha if var (x 1 µ 1 var (x 2 µ 2, he deviaion o full disclosure is profiable for any ype y of firm 1 such ha y < E (x 1 µ 1. If he opposie inequaliy for variances holds, he deviaion o full disclosure is profiable for any ype z of firm 2 such ha z > E (x 2 µ 2. 9 The proof of his claim relies on he following wo observaions. 10 Firs, profi funcions π 1 and π 2 are monoonically decreasing in var (x 1 µ 1 and var (x 2 µ 2, respecively. This is an immediae implicaion of (3.5 and (3.6. Second, profi funcion π 1 in (3.5 is monoonically decreasing in E (x 1 µ 1 when var (x 1 µ 1 var (x 2 µ 2, and profi funcion π 2 in (3.6 is monoonically increasing in E (x 2 µ 2 when he opposie inequaliy is rue, i.e., var (x 2 µ 2 > var (x 1 µ 1. To demonsrae his second observaion, consider he derivaive of π 1 wih respec o E(x 1 µ 1 and he derivaive of π 2 wih respec o E(x 2 µ 2 and evaluae heir signs. Sraighforward calculaions lead o π 1 E (x 1 µ 1 π 2 E (x 2 µ 2 = = p 1 6 (E (x 2 µ 2 E (x 1 µ 1 2 ( (E (x2 µ 2 E (x 1 µ 1 (E (x 2 µ 2 3E (x 1 µ var (x 2 µ 2 var (x 1 µ 1 p 2 6 (E (x 2 µ 2 E (x 1 µ 1 2 ( (E (x2 µ 2 E (x 1 µ 1 (4 3E (x 2 µ 2 + E (x 1 µ 1 + +var (x 2 µ 2 var (x 1 µ 1 8 If boh ses, S 1 and S 2, conain only one ype, hen reporing sraegies of boh firms are fully revealing. 9 Type y of firm 1 such ha y < E (x 1 µ 1 exiss because a by assumpion, firm 1 does no fully disclose is locaion, so ha S 1 is no a singleon, and b he probabiliy densiy funcion f(x is sricly posiive. For he same reason, when var (x 2 µ 2 > var (x 1 µ 1, ype z of firm 2 such ha z > E (x 2 µ 2 exiss. 10 As he deviaion is such ha is effec on he variance and he effec on he expeced locaion of consumers boh increase profis, we can ac as if hese wo effecs can be achieved independenly of each oher. 13

15 Given ha 0 E(x 1 µ 1, E(x 2 µ 2 1, he firs expression is sricly negaive when var (x 1 µ 1 var (x 2 µ 2 and he second expression is sricly posiive when var (x 2 µ 2 > var (x 1 µ 1. Now, suppose ha E (x 1 µ 1 = E (x 2 µ 2. Then if var (x 1 µ 1 var (x 2 µ 2, he deviaion by ype y of firm 1 o he fully revealing message is beneficial simply because before he deviaion is profi is zero and afer he deviaion i is posiive: π D 1 = ( 2 (E (x2 µ 2 y + ( E 2 (x 2 µ 2 y 2 + var (x 2 µ (E (x 2 µ 2 y Similarly, if var (x 2 µ 2 > var (x 1 µ 1, hen he deviaion by ype z of firm 2 o he fully revealing message is beneficial. Thus, a leas one firm can always benefi by deviaing. Therefore, an equilibrium where firms do no fully disclose heir locaion does no exis. References [1] Board, O. [2003], Compeiion and Disclosure, Journal of Indusrial Economics, 67, [2] Caldieraro, F., D. S. Shin, and A. E. Sivers [2087], Volunary Qualiy Disclosure under Price- Signaling Compeiion, Working Paper, Deparmen of Economics, Sana Clara Universiy. [3] Celik, L. [2011], Informaion Unraveling Revisied: Disclosure of Horizonal Aribues, working paper CERGE-EI (Prague. [4] Daughey, A. F., and J. F. Reinganum [1995], Produc Safey: Liabiliy, R&D and Signaling, American Economic Review, 85, [5] Daughey, A. F., and J. F. Reinganum [2007], Imperfec Compeiion and Qualiy Signaling, RAND Journal of Economics, 39, [6] Grossman, S. [1981], The Informaional Role of Warranies and Privae Disclosure abou Produc Qualiy, Journal of Law & Economics, 24, [7] Hoelling, H. [1929], Sabiliy in Compeiion, Economic Journal, 39, [8] Hoz, V. J. and M. Xiao [2011], Sraegic Informaion Disclosure: The Case of Muli-Aribue Producs wih Heerogenous Consumers, Economic Inquiry, forhcoming. [9] Janssen, M. and S. Roy [2010], Signaling Qualiy Through Prices under Oligopoly, Games and Economic Behavior, 68,

16 [10] Jovanovic [1982], Truhful Disclosure of Informaion, The Bell Journal of Economics, 13, [11] Milgrom, P. [1981], Good News and Bad News: Represenaion Theorems and Applicaions, Bell Journal of Economics, 12, [12] Sun, M. J. [2011], Disclosing Muliple Produc Aribues, Journal of Economics and Managemen Sraegy 20, [13] Viscusi, W. K. [1978], A Noe on Lemons Markes wih Qualiy Cerificaion, Bell Journal of Economics, 9,