aa r X i v : . [ ec on . T H ] A p r Price competition with uncertain quality and cost
Sander Heinsalu ∗ April 12, 2019
Abstract
Consumers in many markets are uncertain about firms’ qualities and costs,so buy based on both the price and the quality inferred from it. Optimal pricingdepends on consumer heterogeneity only when firms with higher quality havehigher costs, regardless of whether costs and qualities are private or public. Ifbetter quality firms have lower costs, then good quality is sold cheaper thanbad under private costs and qualities, but not under public. However, if higherquality is costlier, then price weakly increases in quality under both informationalenvironments.Keywords: Bertrand competition, price signalling, incomplete information,price dispersion.JEL classification: D82, C72, D43.
In many markets, buyers are uncertain about the qualities and costs of sellers. Inthat case, purchasing decisions depend on both the price and the quality that thebuyers infer from price (as opposed to the actual quality). This paper shows thatthe pricing decisions of the firms differ based on customer heterogeneity when higherquality producers have greater marginal cost, irrespective of whether costs and qualitiesare private or public.If firms privately know their quality and cost, and a higher quality firm has a smallermarginal cost, then it signals its quality by pricing lower than a bad quality rival. Bycontrast, with public qualities and costs, better quality is priced higher. In markets ∗ Research School of Economics, Australian National University, 25a Kingsley St, Acton ACT 2601,Australia. Email: [email protected], website: https://sanderheinsalu.com/ Many industries have higher quality associated with a lower cost, e.g. mutual funds good or bad . The good type has better quality than the bad, and in themain model also higher marginal cost. Consumers have heterogeneous valuations forthe firms’ products, with a greater valuation also implying a weakly larger premiumfor quality. Each player knows her own type, but other players only have a commonprior over the types. The firms simultaneously set prices, which the consumers observe.Then each consumer chooses either to buy from one of the firms or leave the market.Consumers Bayes update their beliefs about the types of the firms based on the prices.The equilibrium notion is perfect Bayesian equilibrium (PBE).In equilibrium, price is above the competitive level, regardless of whether the goodand bad types pool or the good quality firms signal their type by raising price. Pricesin pooling and semipooling equilibria exceed the cost of the good type, thus both typesof at least one firm make positive profit. In separating equilibria, both firms’ bad typesobtain positive profit.By contrast, complete information Bertrand competition between identical firms,whether good or bad quality, leads to zero profit and a lower price than under incompleteinformation. Complete information competition between a good and a bad firm mayraise price compared to private cost and quality, but one of the types still makes zeroprofit. The ex ante expected price may be higher or lower under complete information.The ex ante price dispersion under public types exceeds that under private. If the costand quality differences between the types vanish or the probability of the good typegoes to zero, then the complete and incomplete information environments convergeto the same price: the marginal cost of the bad type. The outcome in this paper isindependent of whether the firms observe each other’s cost or quality, but relies onconsumers not observing these.The equilibrium pricing differs from a privately informed monopolist, and fromcompetition when consumers find it costly to learn prices. A monopolist with goodquality signals its type by a high price. When observing the prices of competing firms (Gil-Bazo and Ruiz-Verd´u, 2009), cotton weaving (Bloom et al., 2013), medical innovations(Nelson et al., 2009). Additional empirical examples and theoretical reasons for negatively correlatedquality and cost are in Heinsalu (2019).
2s costly, the outcome under complete information is monopoly pricing (Diamond, 1971).Incomplete information leads to above-monopoly pricing in costly search, except whenquality and cost are negatively correlated, in which case pricing is competitive (Heinsalu,2019).
Literature.
The signalling literature started from Spence (1973), and price as asignal was studied in Milgrom and Roberts (1986). In the present paper, the consumersare the receivers of the price signal, unlike in limit pricing (Milgrom and Roberts (1982)and the literature following) where the incumbent deters potential entrants from enter-ing the market by signalling its low cost via a low price.Bertrand competition has been combined with price signalling in Janssen and Roy(2015), where consumers are homogeneous and firms may verifiably disclose their types.In Sengupta (2015), consumers may or may not value quality, but are otherwise homo-geneous. Bertrand competing firms publicly invest to obtain a random private qualityimprovement, and signal quality via price.Hertzendorf and Overgaard (2001a) assume that one firm has high and the otherlow quality (firm types are perfectly negatively correlated, thus firms know each other’stype) and that cost does not depend on quality. Firms may signal via price or adver-tising. Full separation requires advertising.Hertzendorf and Overgaard (2001b); Daughety and Reinganum (2007, 2008) con-sider Hotelling competition with quality differences (thus both horizontal and verticaldifferentiation). Daughety and Reinganum (2007, 2008) focus on symmetric separatingequilibria. Hertzendorf and Overgaard (2001b) show the nonexistence of full separa-tion, similarly to the current work.If firm types only differ in their private marginal cost, but not quality, then thehigh cost type prices at its marginal cost, but the low cost type mixes over a range ofprices strictly above its marginal cost and weakly below the price of the high-cost type(Spulber, 1995).The next section sets up the model and Section 2 characterises the equilibrium set,first when cost and quality are positively associated. Negative correlation is examinedin Section 2.3. 3
Model
Two firms indexed by i ∈ { X, Y } compete. Each draws an i.i.d. type θ ∈ { G, B } ,interpreted respectively as good and bad, with Pr( G ) = µ ∈ (0 , v ∈ [0 , v ], is distributed according to the strictlypositive continuous pdf f v , with cdf F v . Consumer types are independent of firm types.All players know their own type, but only have a common prior belief over the types ofother players.A type θ firm has marginal cost c θ , with c G > c B >
0. A type G firm has betterquality: a type v consumer values a type B firm’s product at v and G ’s product at h ( v ) ≥ v , with h ′ ≥ h ( v ) > v . For some results, h is restricted to the form h ( v ) = v + ν , with ν >
0. In this case, all consumers are willing to pay the samepremium for quality. Assume v > c G ≥ h (0), so demand for the good type firm ispositive. The previous assumption c B > B under complete information. Firms and consumers are risk-neutral. Consumers haveunit demand.The firms observe their types and simultaneously set prices P X , P Y ∈ [0 , P max ],where P max ∈ ( h ( v ) , ∞ ). A behavioural strategy of firm i maps its type to ∆ R + . Theprobability that firm i ’s type θ assigns to prices below P is denoted σ θi ( P ), so σ θi ( · ) isthe cdf of price. The corresponding pdf is denoted dσ θi ( P ) dP if it exists.After seeing the prices the firms, a consumer decides whether to buy from firm X (denoted b X ), firm Y ( b Y ) or not at all ( n ). The behavioural strategy σ : [0 , v ] × R → ∆ { b X , b Y , n } of a consumer maps his valuation and the prices to a decision.The ex post payoff of a type θ firm if it sets price P and a mass D of consumers buyfrom it is ( P − c θ ) D . Define v µ ( P ) := inf { v : µh ( v ) + (1 − µ ) v ≥ P } . In particular, v ( · ) = h − ( · ) and v ( · ) = id ( · ). Total demand at price P and a fixed posterior belief µ ofthe consumers is D µ ( P ) = 1 − F v ( v µ ( P )). If h ( v ) = v + ν , then D µ ( P ) = 1 − F v ( P − µν ).The monopoly profit of type θ at P, µ is π mθ,µ ( P ) = ( P − c θ ) D µ ( P ). The complete-information monopoly profit π mB, ( P ) is denoted π mB ( P ), and π mG, ( P ) denoted π mG ( P ).The monopoly price is P mθ := arg max P π mθ ( P ). Assume that π mθ ( P ) is single-peaked in P . To avoid trivial separation, assume that P mB ≥ c G or π mB < ( c G − c B ) D ( c G ). Denote by ∆ S the set of probability distributions on the set S . i conditional on price P and the firm’s trategy σ ∗ i is µ i ( P ) := µ ddP σ G ∗ i ( P ) µ ddP σ G ∗ i ( P ) + (1 − µ ) ddP σ B ∗ i ( P ) (1)whenever µ ddP σ G ∗ i ( P ) + (1 − µ ) ddP σ B ∗ i ( P ) >
0. A discontinuity of height h θ in the cdf σ θ ∗ i is interpreted in the pdf as a Dirac δ function times h θ , thus makes the denominatorof (1) positive. A jump in σ G ∗ i ( · ), but not σ B ∗ i ( · ) at P yields µ i ( P ) = 1, and a jump in σ B ∗ i ( · ), but not σ G ∗ i ( · ) results in µ i ( P ) = 0. If each σ θ ∗ i has an atom of respective size h θ at P , then µ i ( P ) = µ h G µ h G +(1 − µ ) h B . Finally, if the denominator of (1) vanishes, thenchoose an arbitrary belief.The equilibrium notion is perfect Bayesian equilibrium (PBE), henceforth simplycalled equilibrium: each player maximises its payoff given its belief about the strategiesof the others, and the beliefs are derived from Bayes’ rule when possible.The minimal and maximal price of firm i ’s type θ in a candidate equilibrium aredenoted P iθ and P iθ , respectively. First the benchmark of complete information is considered, which illustrates some gen-eral features of the framework. After that, the case of private cost and quality isexamined.One general observation is that among the consumers who end up buying, the oneswith a low valuation for good quality relative to bad sort to firms believed to have lowerquality, while the high valuation consumers go to expected high quality. If the qualitydifference between the firm types is large compared to their cost difference and firmsdraw unequal types, then the low quality firm is left with zero demand. Similarly, iffirm types differ and the variation in quality is small and in cost large, then the highquality firm receives no customers.
Symmetric firms with publicly known qualities price at their marginal cost, regardless ofwhether consumers are homogeneous or not and whether their quality premium h ( v ) − v is constant or increasing. 5ith asymmetric firms and a constant quality premium, one type prices at itsmarginal cost and the other higher by just enough to make the consumers indifferent.For example, if c G − c B > ν , then P G = c G and P B = c G − ν , but if c G − c B < ν , then P G = c B + ν and P B = c B . Given the chance, all consumers leave the type with relativelyhigher cost for its quality (if c G − c B > ν , then G , otherwise B ), because otherwisethe type with the lower relative cost would undercut slightly. The outcome of onetype pricing at its marginal cost and receiving zero demand is standard in asymmetricBertrand competition.In asymmetric price competition when the quality premium increases in consumervaluation, define the indifferent consumer as v ∗ ( P G − P B ) := inf { v ≥ h ( v ) − v ≥ P G − P B } . Assume h ′ >
1, which ensures v ∗ is a strictly increasing continuous function.Consumers with valuations above v ∗ ( P G − P B ) buy from G , those below from B ornot at all—the latter if v < v ( P G , P B ) := min { P B , h − ( P G ) } . A good type firmwho expects P ∗ B solves max P G ( P G − c G )[1 − F v ( v ∗ ( P G − P ∗ B ))] and a bad type solvesmax P B ( P B − c B )[ F v ( v ∗ ( P ∗ G − P B )) − F v ( P B )]. Equilibrium exists, because prices mayw.l.o.g. be restricted to the convex compact interval [0 , h ( v )] and payoffs are continuousin the action profiles.If h ( c B ) − c G < c B − c B , then type B obtains positive demand even if G pricesat its marginal cost c G . Symmetrically, if h ( v ) − c G > v − c B , then G can attract somecustomers facing P B = c B . Under these conditions, each type prices above its marginalcost and obtains positive demand, unlike with homogeneous consumers or a constantquality premium. Results for the general case of an increasing quality premium for the good type arepresented first, followed in Section 2.2.1 by derivations that require the additional re-striction of a constant quality premium. The first lemma proves that in any equilibrium,if demand for a firm’s bad type is positive, then the bad type prices below the good andobtains greater demand, strictly so under (partial) separation. The proof combines theincentive constraints (ICs) of the types and is standard.
Lemma 1.
In any equilibrium, D µ i ( P iB ) ( P iB ) ≥ D µ i ( P iG ) ( P iG ) , and if D µ i ( P iB ) ( P iB ) > ,then P iG ≥ P iB , with equality iff D µ i ( P iB ) ( P iB ) = D µ i ( P iG ) ( P iG ) . If D µ i ( P iθ ) ( P iθ ) > , hen P iθ ≥ c θ .Proof. Denote D µ i ( P iθ ) ( P iθ ) by D θ to simplify notation. In any equilibrium, for any P iθ in the support of σ θ ∗ i , the incentive constraints IC G : ( P iG − c G ) D G ≥ ( P iB − c G ) D B and IC B : ( P iB − c B ) D B ≥ ( P iG − c B ) D G hold. Rewrite the ICs as P iG D G − P iB D B ≥ c G D G − c G D B and c B D G − c B D B ≥ P iG D G − P iB D B . Using c θ >
0, the ICs become P iG D G − P iB D B c G ≥ D G − D B ≥ P iG D G − P iB D B c B . Then from c G > c B , we get D G ≤ D B and P iG D G ≤ P iB D B .In any equilibrium, if D θ >
0, then P iθ ≥ c θ . Thus if D G >
0, then D B > B deviates to P iG ≥ c G > c B to get positive profit. If D B = D G >
0, thenthe ICs imply P iG = P iB . The converse implication is obvious.An implication of Lemma 1 is that the supports of σ B ∗ i and σ G ∗ i have at most oneprice in common. Thus if one type at a firm semipools, i.e. only sets prices that theother type also chooses, then the semipooling type plays a pure strategy.The next lemma rules out some equilibria even when belief threats are possible. Lemma 2.
In any equilibrium, at least one firm’s type B obtains positive demand. Ifboth types of firm i get zero demand, then both types of j set P j > c G and receive positiveprofit. If µ h ( c B ) + (1 − µ ) c B − c G ≤ or type B of a profitable firm partly separatesor firms play symmetric strategies, then both firms’ B types obtain positive profit.Proof. Suppose both types of both firms get zero demand in equilibrium. Then
P > v for all prices. Both types deviate to P ∈ ( c G , v ) to obtain positive demand and profiteven at the worst belief µ i ( P ) = 0. If firm i ’s type G obtains positive demand, thenany P iG that G sets is above c G , otherwise G would deviate to P = c G . Demand andprofit are positive for iB , otherwise iB would deviate to P iG .Suppose both types of firm j receive zero demand. Total demand at P < v is positiveat any belief, so both types of i obtain positive demand and profit at any P i ∈ ( c G , v ).Positive π ∗ iθ implies P iθ > c θ . If iB partly separates, i.e. sets P iB with µ i ( P iB ) = 0, then jB can get positive demand and profit by setting P j ∈ ( c B , P iB ), regardless of µ j ( P j ).If iB (semi)pools with iG , i.e. only sets prices that type G also sets, then there exists P i in the support of σ iG s.t. µ i ( P i ) ≤ µ and P i > c G . If µ h ( c B )+(1 − µ ) c B − c G ≤ ǫ > v ∈ ( c B , c B + ǫ ) strictly prefer to buyfrom j at P j ∈ ( c B , v ) and any belief, rather than from i at P i > c G and µ i ( P i ) ≤ µ .This makes jB deviate to P j ∈ ( c B , v ) to get positive profit.7ymmetric strategies imply that the firms split the total demand on average. Eachfirm’s type G receives positive demand with probability at least µ (when the otherfirm has type G ). Then P G ≥ c G . Type B can imitate P G , thus π ∗ iB > P = c G , which existsif µ h ( c B ) + (1 − µ ) c B > c G and only if a weak inequality holds.Equilibria where one firm obtains zero demand require µ h ( c B ) + (1 − µ ) c B > c G and asymmetric strategies. For example, firm X pools on some P X ∈ [ c G , µ h ( c B ) +(1 − µ ) c B ) and firm Y on some P Y ∈ ( P X , P X + ǫ ) for ǫ > P i , i ∈ { X, Y } is µ i ( P i ) = µ , at other prices zero. Firm Y gets zero demand on or off theequilibrium path from prices P ≥ c B , because if µ h ( c B )+(1 − µ ) c B − P X > c B − c B = 0,then all consumers with v ≥ c B prefer to buy at P X and µ X ( P X ) = µ , rather thanat P = c B and µ X ( P ) = 0. Neither firm wants to deviate, because P = P i results inthe worst belief and zero demand. Weakly dominated strategies are not played in thisequilibrium, provided P X > c G .Some non-pooling equilibria also involve zero demand for one firm—modify thepreceding example so that each type of firm Y (partly) separates, for example B sets P Y B > v and G sets P Y G > h ( v ). To ensure zero demand for Y , any P Y θ receivingpositive probability must satisfy P Y θ − P i ≥ [ µ Y ( P Y θ ) − µ ][ h ( v ) − v ] for all v , whichholds if P Y θ − [ µ Y ( P Y θ ) − µ ][ h ( v ) − v ] ≥ P i for any µ Y ( P Y θ ) ≥ µ . An additionalnecessary condition for equilibria with P Y B > v and P Y G > h ( v ) is that firm X asa monopolist does not deviate: ( P − c θ ) D ( P ) ≤ ( P X − c θ ) D µ ( P X ) for any θ and P < P
Y B . This condition is easier to satisfy for smaller P Y θ . Reducing P Y θ makes therequirement P Y θ − P i ≥ [ µ Y ( P Y θ ) − µ ][ h ( v ) − v ] harder to satisfy.Asymmetric pooling passes the Intuitive Criterion under a constant quality premiumfor some parameter values, as shown in the following lemma. Lemma 3. If h ( v ) − v = ν for all v , then the Intuitive Criterion does not eliminateequilibria in which firm i pools on some P i ∈ ( c G , c B + µ ν ) and firm j on some P j > P i such that [ P j + (1 − µ ) ν − c B ][1 − F v ( P j − µ ν )] > ( P i − c B )[1 − F v ( P i − µ ν )] . The symmetric requirement that if µ Y ( P Y θ ) ≤ µ , then P Y θ + [ µ − µ Y ( P Y θ )][ h ( c B ) − c B ] ≥ P i holds whenever µ h ( c B ) + (1 − µ ) c B > c G , which by Lemma 2 is necessary for one firm to obtain zerodemand. roof. Let P Y > P X w.l.o.g., so that D Y ( P Y ) = 0. Asymmetric pooling equilibriaexist only if µ h ( c B ) + (1 − µ ) c B ≥ c G , by Lemma 2. If setting belief to µ Y ( P Y ) = 1 forsome P Y results in D Y ( P Y ) >
0, then both types of firm Y deviate, but if D Y ( P Y ) = 0,then neither type does. Thus for the firm with zero demand, the Intuitive Criterion hasno effect.Consider the firm X with D X ( P X ) > h ( v ) − v = ν for all v (constantquality premium), in which case µ h ( c B ) + (1 − µ ) c B ≥ c G becomes µ ν ≥ c G − c B .Define P X := P Y + (1 − µ ) ν , which is the price that makes all consumers indifferentbetween firms X and Y at beliefs µ X = 1 and µ Y = µ . Setting belief to µ X ( P X ) = 1for any P X > P X does not attract either type of firm X to deviate, because even atthe best belief, all consumers switch to firm Y .If type G prefers P < P X to P X , then so does type B .Setting belief to µ X ( P X ) = 1 for any P X ∈ ( P X , P X ] makes the bad type of X deviate to P X if ( P X − c B )[1 − F v ( P X − ν )] > ( P X − c B )[1 − F v ( P X − µ ν )]. This isequivalent to ( P Y + (1 − µ ) ν − c B )[1 − F v ( P Y − µ ν )] > ( P X − c B )[1 − F v ( P X − µ ν )],which holds for P Y − P X small enough, because F v is continuous by assumption. Thusthere is no price P such that under the best belief, type G wants to deviate to P , buttype B prefers not to.Symmetric pooling on P ∈ [ c G , c B + (1 − µ ) ν ) exists if c G < c B + (1 − µ ) ν andpasses the Intuitive Criterion by reasoning similar to Lemma 3. Take P X = P Y = P .If the firms set the same price, then assume they split the market equally. The badtype deviates whenever the good type does if ( P + (1 − µ ) ν − c B )[1 − F v ( P − µ ν )] > ( P − c B )[1 − F v ( P − µ ν )], i.e. (1 − µ ) ν > P − c B .In contrast to the constant quality premium case, the Intuitive Criterion eliminatesall pooling equilibria when the quality premium strictly increases in the consumer’svaluation. The idea of the proof relies on profit at a fixed belief being continuous whenthe quality premium is strictly increasing. At the best belief (certainty of the goodtype), the bad type prefers to deviate to a price just above pooling and prefers not todeviate to a high enough price, so by the Mean Value Theorem, there is a price at whichthe bad type is indifferent. At prices just above this indifference and the best belief,the good type still strictly prefers to deviate, which justifies the belief and eliminatesthe candidate equilibrium. 9 emma 4. If h ′ ( v ) > , then the Intuitive Criterion rules out any equilibria where thesupport of σ ∗ iθ includes P i with µ i ( P i ) ∈ (0 , and D i ( P i ) > and the support of σ ∗ jG does not include P j s.t. µ j ( P j ) = 1 .Proof. Recall that v µ ( P ) := inf { v : µh ( v ) + (1 − µ ) v ≥ P } . For any P j , firm i ’s demand D µi ( P ) = Z vv µ ( P ) { µh ( v ) + (1 − µ ) v − P > µ j ( P j ) h ( v ) + (1 − µ j ( P j )) v − P j } + 12 { µh ( v ) + (1 − µ ) v − P = µ j ( P j ) h ( v ) + (1 − µ j ( P j )) v − P j } dF v ( v )decreases in P and increases in µ , strictly if D µi ( P ) ∈ (0 , µ σ ∗ jG + (1 − µ ) σ ∗ jB over P j , firm i ’s expected demand decreases in P and increases in µ , strictly if E D µi ( P ) ∈ (0 , h ′ >
1, then v µ ( P ) is continuous in µ, P , and strictly increases in P and strictlydecreases in µ for P ∈ [ µh (0) , µh ( v ) + (1 − µ ) v ]. If h ′ >
1, then the inverse of h ( v ) − v is continuous and on [ h (0) , h ( v ) − v ], strictly increases in its argument. Further, if h ′ > µ = µ j ( P j ), then the indifference µh ( v ) + (1 − µ ) v − P = µ j ( P j ) h ( v ) + (1 − µ j ( P j )) v − P j holds for at most one v ∈ [0 , v ]. Therefore if h ′ > µ = µ j ( P j ), then D µi ( P ) = Z vv µ ( P ) { [ µ − µ j ( P j )][ h ( v ) − v ] > P − P j } dF v ( v )is continuous in P, µ . In that case, for any distribution over P j , the expected demandand profit are continuous in P, µ .Suppose w.l.o.g. that firm X does not fully separate and gets positive demand, i.e.there exists P X chosen by both types of X , with D X ( P X ) >
0. Being chosen byboth types implies µ X ( P X ) ∈ (0 ,
1) and by Lemma 1, P X ≥ c G and is unique. Theassumption c G > h (0) implies D X ( P X ) < σ ∗ Y θ , µ Y ( · ), there exists P X large enough s.t. all consumers v ≥ c B preferfirm Y to X even at belief µ X = 1. Then firm X , in particular type B , does not deviatefrom P X to P X ≥ P X , because D µX ( P X ) ≤ D X ( P X ) = 0 ∀ µ .Due to D X ( P X ) ∈ (0 ,
1) and µ X ( P X ) <
1, we have D X ( P X ) > D X ( P X ). Conti-nuity of D X ( · ) is ensured, because the support of σ ∗ jG does not include P j s.t. µ j ( P j ) = 1.Thus for ǫ > D X ( P X + ǫ ) > D X ( P X ).Using the Mean Value Theorem and ( P X + ǫ − c B ) D i ( P X + ǫ ) > ( P X − c B ) D µ X ( P X ) i ( P X ) > ( P X − c B ) D i ( P X ), there exists P ∗ ∈ ( P X , P X ) such that ( P ∗ − c B ) D i ( P ∗ ) = ( P X − B ) D µ X ( P X ) i ( P X ). Type B strictly prefers the equilibrium price P X to any P > P ∗ .For ǫ > G strictly prefers P ∗ + ǫ to P X , because the indifference of B between P X and P ∗ implies D i ( P ∗ ) < D µ X ( P X ) i ( P X ), thus ( P ∗ − c B ) D i ( P ∗ ) − ( c G − c B ) D i ( P ∗ ) = ( P ∗ − c B ) D µ X ( P X ) i ( P X ) − ( c G − c B ) D i ( P ∗ ) > ( P X − c B ) D µ X ( P X ) i ( P X ) − ( c G − c B ) D µ X ( P X ) i ( P X ).An immediate corollary of Lemma 4 is that equilibria where both firms pool areremoved by the Intuitive Criterion. In such equilibria, at least one firm obtains positivedemand without fully separating and the other firm’s good type does not separate.Lemma 4 stands in contrast to the constant quality premium case in Lemma 3, whereall consumers switch firms at the same price, which makes profit discontinuous. Thenthere exist parameter values such that for any price and belief combination, either bothtypes deviate to it or neither does. Simultaneous deviations mean that the IntuitiveCriterion cannot eliminate the equilibrium. This section restricts attention to consumers who all have the same difference h ( v ) − v = ν in their valuations for good and bad quality.The following lemma shows that in any separating equilibrium, the bad type pricesstrictly above its marginal cost. The intuition is that otherwise the bad type wouldimitate G to make positive profit. Lemma 5.
In any separating equilibrium, P iB > c B , π ∗ iB > and P iG > P iB + ν , andif c G − c B ≤ ν , then P iG > c G and π ∗ iG > for i ∈ { X, Y } .Proof. Assume P iG ≤ P jG w.l.o.g., so D i ( P iG ) ≥ µ [1 − F v ( P iG − ν )] >
0. Firm i ’stype B has the option to set P iG and make positive profit, so π ∗ iB >
0. Then π ∗ jB > jB can set P ∈ ( c B , P iB ) and attract all customers from iB . Positive profitfor B implies P iB > c B .Suppose P iG ≤ P iB + ν , then any consumer who buys at P iB also buys at P iG . Thisimplies D i ( P iG ) ≥ D i ( P iB ), which motivates B to imitate G .If c G − c B ≤ ν , then P iG > P iB + ν > c B + ν ≥ c G , therefore π ∗ iG > Lemma 6.
In any separating equilibrium, P iB = P jB ≤ P mB for i ∈ { X, Y } . roof. The result P iG > P iB + ν in Lemma 5 implies that if P iG − P jB ≥ P jG − P iB ,then all consumers at iG who do not leave the market switch to jB given the chance.If all customers at iG prefer to switch to P jB , then P jB ≥ min { P mB , P iB } , because atprices below P iB , firm j ’s type B is a monopolist and deviating only improves belieffor B in a separating equilibrium. Thus demand after raising price from P jB is at leastas great as for a monopolist known to be type B .By Lemma 5, P jG > P jB + ν and by the previous paragraph, P jB ≥ P iB , so due to P jB ≥ P jB , all consumers at jG strictly prefer P iB . Thus P iB ≥ min (cid:8) P mB , P jB (cid:9) .Suppose P iB > P mB ≤ P jB . Deviating from P iB to P mB results in demand D µ i ( P mB ) i ( P mB )weakly above the monopoly level D i ( P mB ). In addition, some consumers may switch to j when facing P iB , but none switch at P mB ≤ P jB ≤ P jB < P jG − ν . Thus demandfor iB increases relatively more than in a complete information monopoly environment,where cutting price from P iB to P mB is profitable. The implication P iB ≤ P mB ≤ P jB contradicts P iB > P mB ≤ P jB , therefore P iB = P jB ≤ P mB .Separation implies positive profit by Lemma 5, so P iB > c B and a small enoughprice cut does not lead to negative profit. Both firms’ B types choose an atomless pricedistribution in any separating equilibrium, because belief threats do not deter the badtype from undercutting an atom of the rival.At P iB ≥ P jB , demand is only positive if the other firm is type G . Deviat-ing from P iB > P mB to P mB increases demand from − σ B ∗ j ( P iB )2 D i ( P iB ) < D i ( P iB )to D i ( P mB ) ≥ D i ( P iB ), which is a greater increase than under complete informa-tion monopoly. By assumption, monopoly profit under complete information is singlepeaked, so the demand increase from P iB > P mB to P mB makes deviation profitable.Therefore P iB ≤ P mB for each firm.Demand is positive on the support of prices, otherwise B would deviate to c B + ǫ .Positive demand and π ∗ G ≥ c G . The assumption v > c G implies that the prices are strictly above c G ,because otherwise G would deviate to P = c G + ǫ even at the worst belief µ i ( c G + ǫ ) = 0.Thus under incomplete information, price is always strictly above marginal cost for bothtypes, which differs from a situation with public cost and quality, as shown in the nextsection. 12 .2.2 Comparison to public positively correlated quality and cost Incomplete information may increase or decrease prices in Bertrand competition, asshown in this section for positive correlation of cost and quality (the case of nega-tive correlation is in Section 2.3.1). This indeterminate effect contrasts with costlysearch (Heinsalu, 2019), where asymmetric information greatly enhances competition(decreases price from monopoly to competitive) under negatively associated quality andcost, but reduces competition under positive correlation.Under complete information Bertrand competition with positively correlated costand quality, if c G − c B > ν , then trade occurs at P G = c G with probability µ , at P B = c G − ν with probability 2 µ (1 − µ ) and at P B = c B with probability (1 − µ ) .If instead c G − c B < ν , then the trading price is P G = c G with probability µ , is P G = c B + ν > c G with probability 2 µ (1 − µ ) and P B = c B with probability (1 − µ ) .By contrast, incomplete information implies that trade occurs at a (semi)poolingprice P > c G (with probability at least 1 − µ ) or a semiseparating price P iGs >P + (1 − µ ) ν . The ex ante expected price under incomplete information is higherthan under complete iff either c G − c B ≥ ν or µ / ∈ ( µ , µ ) ⊂ (0 , If information is complete or cost and quality are negatively related (as in Section 2.3),then the strategies of the firms against homogeneous and heterogeneous consumersare the same. However, under incomplete information and positive correlation of costand quality, Janssen and Roy (2015) Proof of Proposition 2 initially claims the uniquesymmetric D1 equilibrium:(a) If v B > c B + v G − v B , then P G = v G > v B > c B + v G − v B .(b) If v B ≤ c B + v G − v B , then P G = max { c G , c B + 2( v G − v B ) } and all consumers buy.From the second paragraph on, Janssen and Roy (2015) Proof of Proposition 2 says:(a) If v B ≥ c B + v G − v B , then P G = c B + 2( v G − v B ) and type B mixes over P B ∈ [ c B + µ ( v G − v B ) , c B + v G − v B ]. 13b) If v B < c B + v G − v B , then P G = v G and type B mixes over [ c B + µ ( v B − c B ) , v B ].Nonnegative profit for G requires P G = c B + 2( v G − v B ) ≥ c G , so incomplete informa-tion always increases P G , sometimes strictly. If c G − c B ≤ v G − v B , then incompleteinformation strictly increases P B , but if c G − c B > v G − v B and µ < c G − c B − v G + v B v B − c B ,then there is a positive probability that P B is lower under incomplete information. As µ →
0, the probability of trade at P B goes to 1 in both cases. In this section, the only differences from Section 2.2 are that a firm with good qualityhas a lower cost, the gains from trade are positive for a bad quality firm, but not allconsumers buy at the bad type’s cost, and the complete information monopoly profitof the good type increases in price for prices below the bad type’s cost. Formally, c G < c B < v , c B ≥ h (0) and ddP P [1 − F v ( h − ( P ))] > P ∈ [0 , c B ]. Normalise c G = 0 w.l.o.g.Analogously to Lemma 1, demand and price are monotone in type in any equilib-rium, but due to c G < c B , the direction of the monotonicity switches. Denote by D i ( P )the equilibrium demand for firm i at price P . Lemma 7.
In any equilibrium for any P θ in the support of σ θ ∗ i , D i ( P G ) ≥ D i ( P B ) , andif D i ( P B ) > , then P G ≤ P B .Proof. If P B D i ( P B ) − c B D i ( P B ) ≥ P G D i ( P G ) − c B D i ( P G ) and P G D i ( P G ) ≥ P B D i ( P B ),then P B D i ( P B ) − c B D i ( P B ) ≥ P B D i ( P B ) − c B D i ( P G ).In equilibrium, D i ( P B ) > ⇒ P B ≥ c B , otherwise B deviates to P ≥ c B .If ( P B − c B ) D i ( P B ) ≥ ( P G − c B ) D i ( P G ) and D i ( P G ) ≥ D i ( P B ) >
0, then P B − c B ≥ P G − c B .The next lemma, similarly to Lemma 2, rules out some equilibria even when beliefthreats are possible. Lemma 8.
In any equilibrium, at least one firm’s type G obtains positive profit. Iftype G of firm i gets zero demand, then both types of j set P j > c B and receive positivedemand and profit. If µ h (0) < c B or at least one firm partly separates or firms playsymmetric strategies, then type G of each firm obtains positive profit. roof. Suppose π ∗ iG = 0 for both firms. Then for each firm and any P iG in the supportof σ G ∗ i , either P iG = 0 or D i ( P iG ) = 0. If P iG = 0 ≤ P jG , then D i (0) >
0, so type B offirm i separates and sets P iB >
0. Then jG has probability 1 − µ of facing iB with P iB > µ i ( P iB ) = 0, so jG obtains positive demand and profit from P ∈ (0 , P iB )for any belief µ j ( P ). Thus P jG > π ∗ jG > D i ( P iG ) = 0, then D i ( P ) = 0 for any P >
0, otherwise iG would deviateto P to get positive profit. Total demand D X ( P ) + D Y ( P ) is positive for any P < v for any beliefs µ X ( P ) , µ Y ( P ), so if D i ( P ) = 0, then D j ( P ) >
0. Then both types of j get positive profit from any ˆ P ∈ ( c B , v ), thus jB sets P jB > c B .If jB partly separates, i.e. sets P jB with µ j ( P jB ) = 0, then both types of firm i canget positive demand and profit by setting P i ∈ ( c B , P jB ), regardless of µ i ( P i ).If jB (semi)pools with jG , then there exists P j chosen by both jB and jG with µ j ( P j ) ≤ µ . If µ h (0) < c B , then for ǫ > v ∈ (0 , ǫ )strictly prefer to buy from i at P i ∈ (0 , v ) and any belief, rather than from j at P j ≥ c B and µ j ( P j ) ≤ µ . This makes iG deviate to P i ∈ (0 , v ) to get positive profit.If the firms play symmetric strategies, then consumers’ beliefs are µ X ( P ) = µ Y ( P )for any P chosen in equilibrium, thus D X ( P ) = D Y ( P ). From D X ( P ) + D Y ( P ) > P < v , we get P iB ≥ c B , otherwise B would deviate to P d ≥ c B to obtainnonnegative profit. Type G can imitate P iB ≥ c B and receive D i ( P iB ) >
0, thus π ∗ iG > i set P i > c B and both types of j set P j ∈ ( c B , P i )) ensures zero profitfor both types of j . Symmetric pooling on c B guarantees zero profit for the bad types.Unlike under positive correlation of cost and quality, additional results do not requirea constant quality premium h ( v ) − v . This is because all consumers prefer type G at P G ≤ P B . The Intuitive Criterion of Cho and Kreps (1987) selects a unique equilibrium,as shown below. Theorem 9.
In the unique equilibrium passing the Intuitive Criterion, P B = c B andtype G mixes on [ P G , c B ) , where P G [1 − F v ( h − ( P G ))] = c B (1 − µ )[1 − F v ( h − ( c B ))] and the cdf of price is σ G ∗ i ( P ) = 1 µ − c B (1 − µ )[1 − F v ( h − ( c B ))] µ P [1 − F v ( h − ( P ))] . roof. First, rule out equilibria where some firm’s type G gets zero profit. Lemma 8proved that π ∗ iG = P iG D i ( P iG ) > D i ( P iG ) ≤
1, any P iG in the support of σ G ∗ i is bounded below by π ∗ iG >
0. Denote by P iθ the lowest pricein the support of σ θ ∗ i . To apply the Intuitive Criterion to show π ∗ jG > j = i , set µ j ( P ) = 1 for P ∈ (0 , min { P iG , c B } ). Then all consumers with h ( v ) ≥ P buy fromfirm j at P . Suppose π ∗ jG = 0, then jG strictly prefers P to its equilibrium price, but jB strictly prefers its equilibrium profit π ∗ jB ≥ P < c B .This justifies µ j ( P ) = 1 and removes any equilibrium with π ∗ jG = 0.Second, rule out (partial) pooling. Positive profit for G implies positive demand,so pooling is only possible on P i ≥ c B , otherwise B would deviate to nonnegativeprofit. Define P ∗ := sup { P ≤ P i : ( P − c B ) D i ( P ) < π ∗ iB } . Due to π ∗ iG >
0, we have D i ( P ) >
0, so P ∗ ≥ c B . To apply the Intuitive Criterion to rule out (partial) poolingon P i ≥ c B , set µ i ( P ) = 1 for all P < P ∗ . Because iB puts positive probability on P i ,we have µ i ( P i ) <
1. Then for ǫ > P i − ǫ ) D ( P i − ǫ ) > P i D i ( P i ),so G strictly prefers to deviate to P i − ǫ , but B strictly prefers the equilibrium. Thisjustifies µ i ( P i − ǫ ) = 1 and removes (partial) pooling.Combining separation of types with Lemma 7 shows P iG < P iB . All consumersstrictly prefer P iG at µ i ( P iG ) = 1 to P iB at µ i ( P iB ) = 0, so iB only gets positivedemand if firm j has type B . The bad types Bertrand compete, so undercut each otherto P B = c B .The good types mix atomlessly by the standard reasoning for Bertrand competitionwith captive customers. Atoms invite undercutting. The good types’ price competitioncannot reach P = 0, because with positive probability, the other firm has type B andsets P B = c B . This makes type G strictly prefer P = c B − ǫ to P ≤ ǫ for ǫ > P iθ the supremum of the support of σ θ ∗ i . Combining separation, P iG ≤ P iB and P iB = c B yields P iG ≤ c B for both firms.Suppose P iG ≤ P jG < c B . If σ G ∗ i has an atom at P iG and P iG = P jG , then set µ j ( P jG − ǫ ) = 1 for ǫ > jG will deviate from P jG to undercut P iG . Type B of firm j strictly prefers not to set P jG − ǫ . Thus the Intuitive Criterionjustifies µ j ( P jG − ǫ ) = 1 and eliminates equilibria with an atom at P iG = P jG < c B .If σ G ∗ i has no atom at P iG or P iG < P jG , then demand at P jG is only positive whenfirm i has type B . Set µ j ( P ) = 1 for all P ∈ [ P jG , c B ), then D j ( P ) > i has type16 , in which case jG is a monopolist on P ∈ [ P jG , c B ). By assumption, the completeinformation monopoly profit P D ( P ) strictly increases on [0 , c B ], so G strictly prefersany P ∈ ( P jG , c B ) to P jG . Type B strictly prefers not to set P < c B . Therefore theIntuitive Criterion justifies µ j ( P ) = 1 and eliminates equilibria with P iG ≤ P jG < c B .Suppose P iG < P jG , then set µ i ( P ) = 1 for all P ∈ [ P iG , P jG ]. Then for any P ∈ [ P iG , P jG ), all customers who end up buying buy from i . Because P D ( P ) strictlyincreases on [0 , c B ], type G of firm i will deviate from P iG to any P ∈ [ P iG , P jG ). Thisrules out P iG < P jG .Price P iG ≤ P jG attracts all customers with h ( v ) ≥ P iG . Profit from P iG is P iG [1 − F v ( h − ( P iG ))], which must equal the profit c B (1 − µ )[1 − F v ( h − ( c B ))] from P iG = c B . This determines P iG . Type G must be indifferent between all P ∈ [ P iG , c B )in the support of σ G ∗ i , which determines firm j ’s mixing cdf σ G ∗ j by P [1 − µ σ G ∗ j ( P )][1 − F v ( h − ( P ))] = c B (1 − µ )[1 − F v ( h − ( c B ))]. The same indifference condition for j deter-mines σ G ∗ i , so σ G ∗ X = σ G ∗ Y . Solving for σ G ∗ i yields σ G ∗ i ( P ) = µ − c B (1 − µ )[1 − F v ( h − ( c B ))] µ P [1 − F v ( h − ( P ))] .The equilibrium selected by the Intuitive Criterion in Theorem 9 is similar to thesymmetric equilibrium in the homogeneous consumer case studied in Janssen and Roy(2015) Lemma A.1 in which P B = c B and type G mixes atomlessly over [(1 − µ ) c B + µ c G , c B ]. Thus trade occurs at c B with probability (1 − µ ) , otherwise at some P G < c B .The ex ante expected price under incomplete information is lower than under completeiff µ / ∈ ( µ, µ ) ⊂ (0 , P = c B is an equilibrium when c B is below a cutoff. Similarly, with heterogeneous consumers, pooling on c B is an equi-librium for low c B . However, it does not pass the Intuitive Criterion, regardless ofwhether valuations differ among customers or not. If belief is set to 1 on prices below c B , then the good types will deviate to c B − ǫ and at least double their demand. This section compares competition under incomplete and complete information whencost and quality are negatively associated. When both firms have the same type, bothprice at their marginal cost, as usual in symmetric Bertrand competition. Thus tradeoccurs at price c G = 0 with probability µ (when both firms have type G ), but at c B − µ ) . For asymmetric firms and a constant quality premium, tradeoccurs at c B + ν . The probability of unequal types is 2 µ (1 − µ ).By contrast, incomplete information results in P B = c B and type G mixing atom-lessly over [ P G , c B ) for some P G >
0. Therefore trade occurs at c B with probability(1 − µ ) , otherwise at some P G < c B . The ex ante expected price under incompleteinformation is lower than under complete iff µ / ∈ ( µ, µ ) ⊂ (0 , h ( v ) − v increases in v , strictly for some v . Focus on asymmetricfirms. Type B does not set any P < c B attracting positive demand, thus type G isa monopolist on P < h ( c B ). If the complete information monopoly price P mG of thegood type is below h ( c B ), then G sets P mG , type B gets zero demand and may set any P ≥ h − ( P mG ). This outcome is similar to the case of a constant quality premium.The more interesting case is P mG > h ( c B ), where type G raises price until theconsumers with the lowest valuations above c B prefer to switch to B charging P B = c B .These switchers are captive for type B , inducing it to raise price above c B , which inturn loosens the incentive constraint of type G , allowing it to raise price. The good typeends up pricing between h ( c B ) and P mG , and the bad type strictly above c B . This resultis similar to the above-competitive pricing found in Section 2.1 for positively correlatedpublic cost and quality, but differs from privately known cost and quality.Unlike with public types, private information about negatively correlated cost andquality leads to P G < P B = c B , with P G > c G = 0, and if the firms have different types,then demand for the bad type is zero (Theorem 9). The reason is that the good typesignals its private quality by reducing price. Overall, price may end up higher or lowerthan under public cost and quality. Customer heterogeneity turns out to be important for optimal pricing decisions whenbetter quality producers have higher costs, independently of private information aboutcost and quality. When higher quality costs less, firms price similarly when facingheterogeneous consumers as when homogeneous. Because information is valuable onlyif it changes decisions, a firm may estimate the value of gathering information onconsumer preferences using its own and competitors’ cost and quality data, if available. Sufficient for P mG > h ( c B ) is ddP P [1 − F v ( h − ( P ))] > P ∈ [0 , h ( c B )]. ex ante when firms’ types are unknown if the cost difference between firms isexpected to be large relative to the quality difference or consumers are nearly certainof the firms’ types. Thus in mature industries, certification is likely to increase surplus.If the regulator knows the firms’ costs and qualities, then welfare-maximisationsuggests that with symmetric firms, this information should be revealed. If the firmsdiffer and the better quality producer has a lower cost, then revelation reduces surplus.With asymmetric firms where the higher quality one has a greater cost, the price effectof making information public depends on the relative qualities and costs.The comparison between positively and negatively associated cost and quality sug-gests other policy implications. To maximise welfare under privately known cost andquality, the correlation between these should be made negative. Two ways to do thisare to reward good quality (industry prizes for the best product) and punish for flawedproducts (fines, lawsuits). The quality of larger firms should be checked the most fre-quently to give them the greatest motive to improve it, because they are likely the lowcost producers. Firms whose quality and cost are uncertain to consumers (e.g. start-ups) should receive targeted assistance with cost reduction if their quality is high, andwith quality improvement if their costs are low.19f qualities and costs are negatively associated and private, then a merger to duopolyneed not increase prices by much. Thus the optimal antitrust policy depends on thecorrelation of cost and quality in an industry. References
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