Voluntary Quality Disclosure under Price-Signaling Competition 1

Voluntary Quality Disclosure under Price-Signaling Competition 1 Fabio Caldieraro Dongsoo Shin 3 Andrew Stivers 4 008 1 We thank the the co-editor and two anonymous referees for helpful comments. Department of Marketing and International Business, University of Washington, Seattle WA, 98195, Email: CFabiou.washington.edu 3 Department of Economics, Leavey School of Business, Santa Clara University, Santa Clara, CA 95053, E-mail: Dshinscu.edu 4 Center for Food Safety and Applied Nutrition, U.S. Food and Drug Administration, 5100 Paint Branch Parkway, College Park, MD 0740, Email: andrew.stiversfda.hhs.gov

Voluntary Quality Disclosure under Price-Signaling Competition ABSTRACT We analyze an oligopolistic competition with di erentiated products and qualities. The quality of a product is not known to consumers. Each rm can make an imperfect disclosure of its product quality before engaging in price-signaling competition. There are two regimes for separating equilibrium in our model depending on the parameters. Our analysis reveals that, in one of the separating regimes, price signaling leads to intense price competition between the rms under which not only the high-quality rm, but also the low-quality rm has an incentive to disclose its product quality to soften the price competition. JEL Classi cation: D8, L13, L15, M3 Key Words: Oligopoly, Price Competition, Signaling, Voluntary Quality Disclosure 1

1 Introduction Since Akerlof (1970), asymmetric information has long been considered an important factor that causes ine ciencies in markets. It is well understood by now that when consumers cannot observe product qualities, rms can use price as a signaling device for their product quality the low-quality rms have an incentive to pretend that their products are highquality, and hence the high-quality rms need to distort their prices to deter imitating behavior by the low-quality rms. Alternatively, to reduce such distortions associated with asymmetric information, rms may choose to subject their products or services to some process that credibly discloses their qualities, if such a disclosure is feasible. As noted by Daughety and Reinganum (008c), a credible information disclosure would involve using an independent quality evaluation process with public announcement of found quality, or advertisement with potential penalty for misrepresentation in the presence of truth-in-advertising. For instance, some agencies, such as AHAM (Association of Home Appliance Manufacturers) and IAACM (International Association of Air Cleaner Manufacturers), have a voluntary certi cation program that air purifying manufacturers can choose to participate at a cost. make the quality information public. These agencies evaluate the participants products and Another example is the disclosure through special interest magazines a manufacturer may submit its product to such magazine s review process, so that the magazine s assessment of the product can be published in an article, and consequently, made available to readers. 1 An interesting observation is that manufacturers with products that are seemingly inferior in quality sometimes chooses to disclose their product qualities. Moreover, such products sometimes are priced higher than other competing products whose disclosed qualities are higher. According to the AHAM ratings, for example, model 30378 produced by Hunter ($180) outperformed model 01 produced by Blueair ($99) in all quality categories. Also, in a recent LCD evaluation by Maximum PC, 3 Dell s 407WFP ($850) received a rate of 9 (where 10 is the best), while HP s LP465 ($1000) received a rate of 4. Why would a rm with a relatively inferior product with a higher price disclose information on its product quality? This seemingly puzzling phenomena merits a closer examination. 1 Some magazines, such as Consumer Reports, do not accepts submissoin from manufacturers. For some other magazines, such as Maximum PC or PC World, the manufacturers must submit their products for product ratings. See the website, http://cadr.org. Price information is from the internet retailer Amazon. Other websites also provide similar price ranges for these products. 3 Maximum PC (September 006), p76 77.

The objective of this paper is to provide a rational for rms choices of quality disclosure that accommodates the phenomena described above. We look at an oligopolistic industry in which each rm chooses price as well as whether to withhold or advance information about the quality of their products through a disclosure channel. We develop a two-stage game in which two rms compete with products that are di erentiated both horizontally (design or brand preferences) and vertically (quality). To illustrate our point in a parsimonious setting, we assume that one rm produces a high-quality product while the product of the other rm is low-quality. In the rst stage, the rms choose whether or not to disclose their product qualities simultaneously. A rm s quality disclosure allows a fraction of consumers to learn about its product quality, while the remaining fraction of the consumers stay uninformed at the end of the rst stage. At the second stage, the rms engage in price competition, and may use price to signal their product qualities to the uninformed consumers. To be sure, we want to make it clear that quality disclosure in the rst stage, which is our issue of interest, is not a signaling device. On the contrary, it is an instrument that a rm may use to reduce the need of costly distortions to signal in the second stage (when products are nally o ered for sale in the market). We examine two separating equilibrium regimes in our model. First, we nd that the high-quality rm has an incentive to disclose information so as to reduce the price distortion. Second and more interestingly, we nd that the low-quality rm may also have an incentive to disclose the quality of its product. More speci cally, quality disclosure by both highquality and low-quality rms occurs in a regime where the high-quality rm distorts its price downward. In this regime, without quality disclosure the price competition (due to signaling e ort) becomes very intense. Thus, by increasing the proportion of the informed consumers in the rst stage, the low-quality rm can soften price competition in the second stage. That is, by giving up some uninformed consumers in the rst stage (by making them informed), the low-quality rm can reduce the battleground for the remaining uninformed consumers in the second stage. This, together with strategic complementarity of prices, makes the low-quality rm better o by unveiling its quality information to the market. Our result suggests that a rm with relatively lower quality product may have an incentive to voluntarily disclose its quality information, and that such a disclosure takes place with a high price of the product (higher than the price of competing high-quality products). According to our analysis, this can be the case when a high-quality rm can produce at a lower cost. In high-tech industries, for example, producing a higher quality product at a lower cost is not very uncommon. As Gibson, Goldenson and Kost (006) report, 3

quality improvement and cost reduction are often paired with each other in such industries. In fact, according to their report using Capability Matuarity Model Integration standard, companies quality improvement by 48% are accompanied by cost reduction by 34%. There are not many papers that deal with signaling in oligopoly models. The papers most closely related to ours are Hertzendorf and Overgaard (001, 00) and Fluet and Garella (00). These studies construct duopoly models in which rms use both price and advertising to signal their qualities. Unlike ours, the main focus of these models is on the existence of signaling versus pooling equilibria as a function of parameter ranges and assumptions regarding out-of-equilibrium beliefs. Yehezkel (008) extends Hertzendorf and Overgaard (001) by introducing a fraction of informed consumers. This paper is di erent from ours in two ways. First, in his study, the fraction of informed consumers are exogenous, while in our paper, the fraction of informed consumers are determined endogenously by the rms disclosure strategies. Second, as in Hertzendorf and Overgaard (001), the products are only di erentiated vertically in his paper. In our paper, the rms products are di erentiated both vertically and horizontally, raising situations in which the low-quality rm discloses its quality information if the disclosure cost is not too large. Daughety and Reinganum (007, 008a) also consider signaling models in oligopoly setting. In their models, the rms are not competing signal senders, because the rms types are stochastically independent and a rm does not know the other rm s type. Therefore, consumers beliefs about one rm s quality depend only on that rm s signaling instrument. In the current paper, the rms types are correlated, and therefore consumer beliefs depend on both rms prices. Barigozzi, Garella and Peitz (008) also study a duopoly setting with signaling advertisement, but in their model, one rm s type is known to the consumer, and hence there is no direct issue associated with multiple signal senders. The bulk of the signaling literature, unlike our paper, considers monopolistic industries. In this branch of research, the paper by Moraga-Gonzales (000) is more connected to ours. The author considers a model in which rms could endogenously inform a fraction of the consumers through some form of informative advertising. This feature is similar to credible information disclosure in our paper. However, the paper does not consider the e ect of information revelation on equilibrium prices. Therefore, in his model, information disclosure only occurs in a pooling equilibrium, and only by the high-quality rm. We contribute to the literature by identifying that, when there is competition, information disclosure occurs also in a signaling equilibrium, and that both the high-quality and the low-quality rms may have incentives to do so. Fishman and Hagerty (003) and Daughety 4

and Reinganum (008b and 008c) also study both disclosure and signaling in one model. In Fishman and Hagerty (003), disclosure and signaling are complements, whereas in our paper, they are substitutes. In Daughety and Reinganum (008b and 008c), unlike in ours, if disclosure takes place there is no need for signaling since disclosure informs all consumers. Other studies in the signaling literature in monopolistic environments include Wolinsky (1983), Bagwell and Riordan (1991), Bagwell (199), Shieh (1993) and Daughety and Reinganum (1995). The rst three papers studied monopoly models in which high quality is associated with high marginal costs. They show that the low quality rm chooses its full information price, while the high quality rm distorts its price upward to signal its quality. Shieh (1993) considers a monopolistic model in which a rm could rst realize cost-reducing investments and then make pricing decisions. The research shows that it is optimal for the high quality rm to signal by distorting its price upward (downward) when high quality is associated with high (low) marginal costs. Daughety and Reinganum (1995) nd very similar results by analyzing a model in which safety and liability costs add to marginal costs. Their study demonstrates that upward or downward price-signaling (distortion) is optimal depending on the net marginal cost. 4 The rest of the paper is organizaed as follows. We presents the model in the next section. The game in the second stage (price-signaling competition between rms) is analyzed in Section 3. In Section 4, we analyze the game in the rst stage (choice of information disclosure) using the results from Section 3. Some extensions to our model are discussed in Section 5. We gather concluding remarks in Section 6. Model setup We consider a two-stage game with two single-product rms indexed by i I = f1; g : For notational purposes we employ j I (j 6= i) to designate the other rm. The products are di erentiated both horizontally and vertically. In the rst stage the rms choose whether to disclose their vertical quality, and in the second stage they choose prices. We focus on competition between rms with di erent qualities and assume that rm 1 produces high-quality products and rm produces low-quality products. This 4 Other papers in monopolistic environments focus on the possibility of signaling through price and other instruments. For example, the aforementioned paper by Milgrom and Roberts (1986) considering price and advertising as signaling devices; Hertzendorf (1993) analyzing noisy advertising; Lutz (1989) considering product warranties; Moorthy and Srinivasan (1995) studying money-back guarantees as signals of quality; and Zhao (000) considering advertising that a ects demand. 5

perfect negative correlation is employed for expositional purpose and our qualitative results hold as long as there is some negative correlation between the types of the rms (we will discuss more about this issue in Section 5). We assume that the rms have better knowledge about the industry than the consumers, and hence they know their competitor s product quality. Consumers cannot observe the quality of a product, but they have a prior belief regarding a rm s product quality that is P r( rm i = high-quality) = 1=: In the rst stage, each rm can credibly disclose information about its product quality at a cost : The choice of disclosure is denoted by x i f0; 1g; with x i = 0 (withhold the information) and x i = 1 (disclose the information at the cost ). Disclosures by the rm are imperfect we denote by (X) the probability that a consumer becomes informed of the rms product qualities when such information is disclosed, where X = x 1 + x ; with (0) = 0; 0 (X) > 0; and () < 1: The situation we have in mind is that there are numerous disclosure channels, through which rms can make their product qualities public information. For simplicity, we are implicitly assuming that a rm can a ord information disclosure through only one channel. Likewise, it is prohibitively costly for a consumer to check all such channels for information. Hence, the probability that a consumer remains uninformed is strictly positive, but as X (= x 1 + x ) increases, a consumer becomes more likely to be informed. 5 Throughout this paper, we interpret (X) as the fraction of informed consumers at the end of the rst stage. In the second stage, rm 1 (high-quality) and rm (low-quality) choose p 1 and p respectively, with horizontally and vertically di erentiated demand D i (p 1 ; p ; a i ). The parameter a i fa L ; a H g, with a H = a L + a and a > 0: For the fraction of consumers (informed at the end of the rst stage), a 1 = a H and a = a L : For the fraction 1 of consumers (uninformed at the end of the rst stage), a i are replaced by the expected quality E(a i ) = i (p 1 ; p ) a H + (1 i (p 1 ; p ))a L, where i (p 1 ; p ) is the posterior belief about the probability that rm i is selling a high quality good. Without price-signaling, only the fraction of consumers will be able to distinguish the two rms. The remaining 1 of consumers can only use their priors and the information from the pair of prices to update their posteriors about quality. Demand for rm i s product is described below: D i (p 1 ; p ; a i ) = E(a i ) p i + bp j = a i + (1 ) i a H + (1 i )a L p i + bp j ; where b < 1 is the cross-price e ect on demand. The own-price e ect is normalized to 1. 5 See Grossman (1981) and Hirshleifer and Teoh (003) for a similar setup. 6

Note that D i p i < 0, D i p j > 0, and D i a i > 0 (again, the index j I denotes the other rm ). Expected pro ts for the rms are then given by the expression: i (p 1 ; p ; i ) = (p i c i )D i (p 1 ; p ; E(a i )) x i ; where c i is the marginal production cost of rm i: To guarantee a strictly positive demand, we assume that a L > c i : We assume that these speci cations are common knowledge. We close this section by summarizing the timing of the game. At the rst stage, the rm 1 and rm decide whether or not to disclose their quality information (choice of x 1 f0; 1g and x f0; 1g). Accordingly, the proportion of informed consumers () is realized at the end of the rst stage. At the second stage, the rms engage in price-signaling competition. In the subsequent sections, we analyze the game by backward induction. 3 Analysis of the second stage choice of prices In the second stage we look at the rms pricing strategies. The results in this section will be used in Section 4 where we discuss information disclosure by each rm before pricesignaling competition. Recall that 1 of consumers remain uninformed at the end of the rst stage, and thus, if pro table, a rm will price so as to signal its product quality to the uninformed consumers. The building block of our analysis is as follows. First, assuming that rm 1 s signaling equilibrium price, denoted by p H, exists (later, we show that p H exists), we de ne the consumer belief structure. Second, we identify two separating regimes, and construct a set of candidates for p H in each regime. Third, we show that such set in each regime is not empty, depending on parameters. Finally, we show that, in each regime, there is a unique p H (among the candidates) that satis es all conditions for a separating equilibrium. De nition 1 Consumers have the following system of beliefs: If p i = p H and p j 6= p H then i (p 1 ; p ) = 1 and j (p 1 ; p ) = 0; If p i = p H and p j = p H then i (p 1 ; p ) = j (p 1 ; p ) = 1 ; If p i 6= p H and p j 6= p H then i (p 1 ; p ) = j (p 1 ; p ) = 1 : The interpretation of this system of beliefs is as follows. The rst bullet point: If rm i is pricing according to the high-quality product and rm j is not, then consumers believe 7

that rm i is the high-quality rm and rm j is the low-quality rm. The last two bullet points: If both rms are pricing according to the high-quality product, or both rms choose di erent prices from the one that the high-quality rm would choose, then consumers cannot tell a rm s product quality and revert to their priors (and believe that both rms have the same probability of having high-quality). We want to characterize the Perfect Bayesian Equilibrium for this game, in which each rm maximizes its expected pro t, given its rival s pricing and given the consumers beliefs that are consistent with the equilibrium pricing strategy played by the rms. Below, we de ne a separating equilibrium (denoting rm s price in an equilibrium as p L ). De nition A separating equilibrium is a pair of prices fp 1 ; p g = fph ; p L g and a system of beliefs 1 ph ; p L ; ph ; p L such that: (i) p H arg max p1 1 p 1 ; p L ; 1 (p 1 ; p L ) ; (ii) p L arg max p p H ; p ; (p H ; p ) ; (iii) 1 p H ; p L ; 1 1 (bp 1 ; p L ; 1 ) 8bp 1 6= p H ; (iv) p H ; p L ; 0 (p H ; p H ; 1 ); (v) 1 ph ; p L = 1 and ph ; p L = 0: Conditions (i) and (ii) specify that p H is the optimal price of a high-quality rm when it is perceived to be high-quality and p L is the optimal price of a low-quality rm. These are consistent with the Intuitive Criterion of Cho and Kreps (1987). The next two conditions follow the standard argument in signaling models. Condition (iii) require that rm 1 (high-quality) prefers the signaling price p H to some other price that allow mimicry by rm (low-quality). Condition (iv) requires that rm s pro t from being truthfully perceived as the low-quality is higher than its pro t from pretending to be the high-quality. Finally, condition (v) says that consumers beliefs are consistent with the separating outcome the consumers believe that the product is high-quality when the rm charges p H and that the the product is low-quality when the rm charges p L : With these de nitions, we characterize each rm s optimal price and payo. As often occurs in signaling games, both separating and pooling equilibria may emerge. Furthermore, it is also possible that the high-quality rm s best response price automatically allows separation without distortion. Since such outcomes are trivial situations, we focus on separating equilibria with price distortions to make our point. 8

Separating equilibrium To nd the separating equilibrium in the model, rst we obtain the strategy of rm, the low-quality rm (Lemma 1 below), then we determine the set of candidates for p H, the separating equilibrium price for rm 1, the high-quality rm (Lemma below). show that such set is non-empty, depending on parameters (Lemma 3 below). Then, we Lastly, we obtain the separating equilibrium price p H from the set of candidates (Lemma 4 below). Lemma 1 In a separating equilibrium pro le fp 1 ; p g = fph ; p L g, the strategy of rm is p L = p (p H ) = al +bp H +c. Proof. See Appendix A. We proceed by nding candidates for rm 1 s separating price in equilibrium (candidates for p H ): Recalling De nition, condition (iii) requires that rm 1 wants to separate itself from rm (low-quality). We use the superscript S in S i to denote separating outcomes and the superscript N in N i to denote non-separating outcomes. In a separating outcome, the pro t for rm 1 (the high quality) is given by: S 1 p H ; p L ; 1 = p H c 1 a H p H + bp L x 1 : When rm is truthfully recognized as the low-quality rm, its price is as stated in Lemma 1: p L = al +bp H +c : Thus, the pro t for rm 1 in a separating equilibrium is: S 1 p H ; p L (p H ); 1 = p H c 1 a H p H + b al + bp H + c x 1 : (1) If rm 1 deviates to a price bp 1 6= p H while p = p L ; then consumers believe with 1 = 1 that rm 1 is the low-quality. In such a case, its pro t would be: N 1 (bp 1 ; p L p H ; 1 1 ) = (bp 1 c 1 ) a H + (1 ) ah + 1 al bp 1 + b al + bp H + c Firm 1 s best outcome in this case is obtained by maximizing the above expression with respect to bp 1. Such maximization yields: x 1 : N 1 (bp 1 (p L p H ); p L p H ; 1 ( + b) a L ) = c 1 + bc + b p H + (1 + ) a x 1 : () 4 9

In order to determine the set of candidates for p H that satis es condition (iii) in Definition, we de ne h(p H ) to be the di erence S 1 p H N 1 p H using the expressions in (1) and () : We verify whether h(p H ) is positive or negative: h(p H ) = p H c 1 a H p H + b al + bp H + c Function h(p H ) is concave and quadratic in p H with two real roots. ( + b) a L c 1 + bc + b p H + (1 + ) a : 4 Let these roots be p 1 and p 1, where p 1 < p 1. These roots satisfy condition (iii) in De nition with equality. Let H = fp H : h(p H ) 0g = fp H : p 1 p H p 1 g: If p H = H, then Condition (iii) in De nition is violated, and rm 1 would prefer not to separate itself from rm. The next condition (condition (iv)) in De nition, requires that, with rm 1 s signaling price p H, rm would rather choose p = p L and be truthfully identi ed as the low quality than attempt to mimic rm 1 by choosing p = p H : In a separating outcome, rm s pro t can be written as: S p H ; p L ; 0 = p L c a L p L + bp H x : Again, the optimal price for rm in a separating equilibrium is given by Lemma 1: p L = al +bp H +c : Substituting for this expression in rm s pro t function we obtain: S p H ; p L p H ; 0 = al + bp H + c 4 x : (3) If rm mimics rm 1, while rm 1 is playing p H ; then rm s pro t would be: N (p H ; p H ; 1 ) = 1 ph c a L + (1 ) ah + 1 al p H + bp H x : (4) Using the expressions in (3) and (4) ; we determine another set of candidates for p H that satisfy condition (iv) in De nition. As before, we de ne l(p H ) to be the di erence S p H N p H ; and verify whether the expression is positive or negative: l(p H ) = al + bp H + c 4 p H 1 c a L + (1 ) ah + 1 al (1 b)p H : Function l(p H ) is convex and quadratic in p H with two real roots. Let these roots be p and p, where p < p. Let L = fp H : l(p H ) 0g = fp H : p H p These roots satisfy condition (iv) in De nition with equality. De nition is violated, and rm would prefer to mimic rm 1. or p H p g: If p H = L; then condition (iv) in Since both (iii) and (iv) in De nition need to be satis ed, our discussion indicates that all candidates for p H must be in the intersection H\L: Thus we have the next lemma. 10

Lemma A separating equilibrium pro le fp 1 ; p g = fph ; p L g can only be supported by p H S fp : p 1 p p g or p H S fp : p p p 1 g. Proof. The proof follows directly from the fact that p H H = fp H : p 1 p H p 1 g and also p H L = fp H : p H p or p H p g: Lemma says that the region of potential separating prices comes from two distinct ranges, S and S: It can be easily seen that when p H S; rm 1 is sending a signal by setting a high price, and when p H S, rm 1 is sending a signal by setting a low price. In other words when the high quality rm prices too high or too low it is not pro table for the low quality rm to mimic the high quality rm. Lemma 3 If c 1 is su ciently larger (smaller) than c ; then S (S) is non-empty. Proof. See Appendix B. To focus on separating outcomes, we assume that the parameters satisfy the conditions in Lemma 3. Figure 1 below presents the two sets for p H candidates: S and S: In panel (a) of Figure 1, S 6=? and S =?; and thus only a low price can signal high quality. As pointed out in Lemma 3, this is the case when the high-quality rm s production cost is signi cantly lower than the low-quality rm s production cost. Here, the high-quality rm can a ord to distort its price downward to signal, while it is too costly for the low-quality rm to match the lower price. In panel (b) on the other hand, S =? and S 6=?; thus only a high price can signal high quality. This is the case when the high-quality rm s production cost is signi cantly higher than the low-quality rm s production cost. It is not pro table for rm to match rm 1 s upward-distorted price because the own-price has a larger impact on the demand than the cross-price (b < 1). That is, rm must cope with a large decrease in quantity demanded to mimic rm 1 s high price. Consequently, it is less pro table for rm to mimic rm 1. (a) l(p H ) (b) l(p H ) S p S h(p H ) h(p H ) Figure 1. Set of the candidates for p H p 11

From now on, p H S refers to a high-price separating equilibrium, and p H S refers to a low-price separating equilibrium. Having found the price ranges that support separating prices, we apply conditions (i) and (ii) in De nition to select a unique separating equilibrium in each equilibrium regime. These conditions will prescribe that rm 1 distort its price to the minimum extent necessary to prevent mimicry by rm. Lemma 4 In a high-price separating equilibrium p H = inf(s); and in a low-price separating equilibrium p H = sup(s): Proof. See Appendix C. The intuition behind Lemma 4 is rather straightforward. Firm 1 prefers to deviate its price as little as possible from the one that satis es the rst order condition. Therefore, if rm 1 wants to change its price from the rst order condition price to a price in the separating region, the closest price to the rst order condition price will be either the minimum element in S or the maximum element in S. We are now ready to determine the rms outcomes in these two separating regimes. For convenience, we introduce the following notations for each separating equilibrium: p H inf(s) and p H sup(s); S 1 S 1 (p H ; p L ; 1) and S 1 S 1 (p H ; p L ; 1); S S (p H ; p L ; 0) and S S (p H ; p L ; 0); where p L = al +bp H +c and p L = al +bp H +c by Lemma 1. By solving l(p H ) = 0 for p H ; we obtain the exact signaling prices: p H and p H : p H = ( b) al + c + (1 ) a + p (1 ) a f ( b) [a L (1 b) c ] + (1 ) a g ( b) ; p H = ( b) al + c + (1 ) a p (1 ) a f ( b) [a L (1 b) c ] + (1 ) a g ( b) Using these expressions for the equilibrium price in the two separating regimes, we obtain the following pro t expressions. S 1 = p H " c 1 a L + a p H + b al + bp H # + c In a high-price separating equilibrium, 1 and S = al + bp H c : 4

Similarly, in a low-price separating equilibrium, S 1 = p H " c 1 a L + a p H + b al + bp H # + c and S = al + bp H c : 4 Figure illustrates the rms best response (BR) functions when the prevailing outcome is a low-price separating equilibrium. Notice that there are two regions in which rm 1 s BR does not change with the price of rm. These two regions are due to rm 1 s deviation to the two separating prices (p H and p H ). Also, rm s BR function shifts up over the range of non-separating prices (between p H and p H ) because its expected quality will increase. p BR1 Equilibrium BR S p H p H S p 1 Figure. Low-price separating equilibrium In the following section, we proceed to the analysis of the rst stage at which each rm decides whether to disclose it s quality information. 4 Analysis of the rst stage choice of information disclosure Using the pro t expressions at the second stage, we now analyze the rms optimal information disclosure strategies at the rst stage. Recall that is the proportion of informed 13

consumers at the end of the rst stage and 0 (X) > 0; where X = x 1 +x : Thus, we can see each rm s disclosure strategy simply by looking at the e ect of on i : In other words, the sign of S i determines x i f0; 1g; given that is small enough. If S i > 0; then rm i will disclose its product quality at the rst stage (x i = 1 at the cost ): If S i < 0; then rm i will withhold its quality information (x i = 0): If is too large, then it is clear that x i = 0: Below, we proceed to analyze the two separating equilibrium regimes. 6 High-price separating equilibrium In a high-price separating equilibrium, rm 1 (the high-quality) distorts its price upward so that consumers can identify its product quality. The higher the at the end of the rst stage, the lower the need for rm 1 to distort its prices upward. Thus, rm 1 bene ts from disclosing information (x 1 = 1) because this increases, which in turn reduces the upward distortion in p 1 (p H decreases). increases. However, rm (the low-quality) becomes worse o as Again, as becomes larger, rm 1 s price becomes lower as rm 1 can reduce its upward price distortion. Thus, with x = 1; not only more consumers are informed of rm s low quality (at the end of the rst stage), but also rm s price in equilibrium goes down as a response to rm 1 s lower price (at the second stage). Together, these e ects diminish rm s pro t. Consequently, rm has an incentive to withhold the information about its product quality (x = 0): The above discussion is summarized in the following proposition. Proposition 1 In a high-price separating equilibrium; the high-quality rm has an incentive S to disclose information of its true quality 1 > 0 if is small enough, while the lowquality rm has an incentive to withhold information of its true quality S. < 0 Proof. See Appendix D. Low-price separating equilibrium In this regime, rm 1 (the high-quality) distorts its price downward (instead of upward) to signal its product quality. That is, for rm 1, the higher the at the end of the rst stage, the lower the need for downward price distortion. Thus, as in the high-price separating regime, rm 1 bene ts from disclosing information (which increases ). Unlike in the 6 In a non-separating equilibrium, rms do not distort their prices, and therefore it is straightforward to verify that only the high-quality rm discloses (certi es) its quality. 14

previous regime, however, rm (the low-quality) also bene ts from voluntarily disclosing its quality information (contributing to an increase in ). proposition below. have an incentive to disclose information of its true quality rm also has such an incentive > 0 if is small enough. We present our result in the Proposition In a low-price separating equilibrium, not only does the high-quality rm S 1, > 0 but the low-quality Proof. See Appendix E. S Note that, in the low-price separating equilibrium, the signaling price by rm 1 (p H ) is smaller than the low-quality rm s price (p L ). Therefore, information disclosure by the low-quality rm is accompanied by p L > p H : Again, when the high-quality rm can produce at a lower cost, it will aggressively distort its price downward to convince the uninformed consumers at the end of the rst stage. the payo of the low-quality rm in two ways. This leads to a strategic e ect that decreases The lower price by the high-quality rm decreases the demand of the low-quality rm, and it also forces the low-quality- rm to respond with a lower price. reduces the low-quality rm s pro t. As a result, the high-quality rm s downward price distortion By disclosing its quality information to the market (x = 1); rm increases the proportion of informed consumers, which leads to smaller downward distortion in rm 1 s price (p H increases), thus allowing rm to increase its price. These e ects together positively contribute to, and consequently, rm also has an incentive to disclose the information about its product quality. 5 Discussion: correlation between product qualities For expositional purpose, we employed perfect negative correlation between the rms product qualities in our model when one rm is the high-quality (H); the other rm is the low-quality (L): This is, the Nature chooses either state HL or LH, and the rms learn the Nature s choice while the consumers do not. When the Nature can choose all possible combinations of the state, HL, LH, LL, and HH; ex ante, our results still hold, provided that there is some (not necessarily perfect) negative correlation between the rms types. In this section, we rst discuss a case in which the rms type are not perfectly correlated (to make our point, again we focus on the case in which the Nature chose H as rm 1 s type, and L as the rm s type, ex post, although all cases are possible from a consumer s 15

point of view). Then we discuss the circumstances under which some negative correlation between product qualities arises from a consumer s perspective. Imperfect correlation between product qualities With an imperfect negative correlation, all combinations of HL, LH, LL, and HH are possible, but the likelyhood of HL or LH is higher than the other cases. In our framework, as long as there is some degree of negative correlation between the rms types, our results hold. To see this, suppose (x 1 ; x ) = 1 (x 1 )+ (x ); with 0 < 1 (x 1 )+ (x ) < 1: 7 That is, 1 of the consumers are informed by rm 1 s disclosure, and of the consumers are informed by rm s disclosure. With perfect negative correlation (as in our model), i of the consumers automatically learn rm j s quality information. This is not the case when the rm s types are not perfectly correlated. However, with some negative correlation, i of the consumers learn that it is more likely that rm j s quality is the opposite to rm i s. One can model that, for i of the consumers (the consumers who learned rm i s quality), rm j s quality parameter j a i + (1 )(a L + a H a i ); where represents the correlation parameter with [0(perfect negative); 1(perfect positive)]. With = 1=, there is no correlation. With this speci cation, if the state of nature is such that rm 1 is type H and rm is type L; the demand functions for rm 1 and rm are respectively: D 1 = 1 a H + 1 + (1 ) i a H + (1 i )a L p i + bp j ; D = a L + 1 + (1 ) j a H + (1 j )a L p j + bp i ; where 1 a L + (1 )a H and a H + (1 )a L : Notice that these demand functions are essentially the same as the one in our model, except the second term 1 ( 1 ) for rm 1 (): Therefore, for any [0; 1=); our results hold. Also, with imperfect negative correlation discussed above, a consumer s o -the-equilibrium belief can be di erent from the current structure (in our model, 1 = = 1= o the equilibrium path since the consumers know that one rm is type H and the other rm is type L): Because consumers know that all states of nature are possible, instead of 1 = = 1=; one way to model can be just 1 = o the equilibrium. That is, o the equilibrium path, the uninformed consumers believe that rm 1 and rm are equally likely to be high quality, but not necessarily half and half. For example, suppose the consumers believe that both rms are high quality with certainty ( 1 = = 1) when p 1 = p = p H ; and both are low quality with certatinty ( 1 = = 0) when p 1 6= p H and p 6= p H : With such 7 In our model, we let (x 1; x ) = (x 1 + x ) for simplicity. 16

a belief structure, rm 1 will distort its price even further (compared to the case where 1 = = 1= o the equilibrium path) to separate itself from rm. Thus, rm will have a greater incentive to withhold its quality information in the high-price separating regime. On the other hand, rm s incentive to disclose its quality information in the low-price separating regime will be stronger in such a case. Negative correlations in quality driven by consumers preference Here we show that some negative correlation between the rms types can be implied by the fact that a consumer wants a product of relatively better quality. Consider that a consumer derives utility as follows: (the base structure of the utility function is adapted from Daughety and Reinganum (008a)). U(q i ; q j ) = X ii + i + f i > j 1 X q i ii gq i q 1 q ; where captures some base utility for a product (regardless of quality), i is the quality parameter with i f1 (high-quality), 0 (low-quality)g; captures the added absolute utility from a high-quality product, and (> 0) captures the relative utility of buying the best available product. The function f = i j if is true, and f = 0 otherwise. Also, is the degree of substitution between the two products, with 0 < < g: maximizes her utility for consumption. max q 1 ;q U(q 1 ; q ) + B X ii p iq i : A consumer where B is her budget. By taking derivatives with respect to q 1 and q and solving for the rst order condition yields the optimal quantities: Since f j > i = 1 q i = (g ) a + (g i j ) + gf i > j f j > i gpi + p j g : f i > j ; by normalizing g (g )a def = 1 and letting = a L ; g g and (g+) g def = a ; we can rewrite the demand for rm i as: q i = a L + i b j + f i > j a p i + bp j : def = b; g This is the demand expression under full information. Incorporating our model asumptions that a consumer remains uninformed at the end of the rst stage with probability 1 ; the expected demand for rm i is: 17

D i = a L + i b j + f i > j a + (1 ) h i b j + f i > j a i p i + bp j : Notice that for the fraction 1 of consumers, the quality parameter i is replaced with the belief i : This expression for D i is essentially the same as the one in our model, as long as > 0 (i.e., consumers derive utility from relative higher quality). If such relativity does not in uence consumer s utility function ( = 0); then a = 0 (since (1+) 1 = a ) and rms quality become fully independent in the way they in uence consumer utility. On the other hand, if there is only relative efect and no absolute e ect of quality ( = 0); then the negative correlation between the product qualities becomes perfect, followed by the demand: D i = a L + f i > j a + (1 )f i > j a p i + bp j = a i + (1 ) i a H + (1 i )a L p i + bp j ; which is the same as the one in our model. 6 Concluding remarks In this paper we have studied an oligopoly in which two competing rms have the opportunity to reveal information about their product qualities to a fraction of consumers. Firms use their prices as the signaling device for the remaining uninformed consumers. Two separating equilibrium regimes have been identi ed. Our analysis suggests that not only a high-quality rm, but also a low-quality rm may prefer to disclose the true quality of its product. The price distortion for signaling by the high-quality rm can either improve the payo of the low-quality rm (as in the case in which high price signals high-quality) or diminish the payo of the low-quality rm (as in the case in which low price signals high-quality). In the latter case, the low-quality rm has an incentive to advance the true quality of its product to the market, so as to relax price competition. Our result hinges on the following assumptions. First, we adopted Bertrand competition with di erentiated products, i.e., rms products are di erentiated not only vertically, but also horizontally (design, etc.). For example, in Hertzendorf and Overgaard (001), the products are not di erentiated horizontally, and therefore, the quantity demanded for the low-quality rm is zero if the low-quality rm s price is higher than the high-quality rm s price. In such setting (two products are horizontally identical with di erent qualities), our results do not hold. Second, as discussed in Section 5, our result requires some degree of 18

(but not necessarily perfect) negative correlation. Finally, the cost of disclosure needs to be small enough. In the high-price separating regime, there is only one cuto level in ; say ; such that when > the high-quality rm will not choose to disclose its quality information. In the low-price separating regime, there are two cuto levels, say b and e with b < e; such that when < b both rms disclose their information, when b < < e only the high-quality rm discloses its information, and when > b no information is disclosed at the rst stage. Appendix A. Proof of Lemma 1 In a separating equilibrium, rm has no incentive to distort it s price to mimic rm 1, and hence it chooses p to maximize: S = (p c ) a L p + bp H x : The rst order condition with respect to p with a simple rearrangement gives: p p H = al +bp H +c : B. Proof of Lemma 3 We prove Lemma 3 by showing that, rst, p 1 p > 0 when c 1 c is large enough, and then p p 1 > 0 when c c 1 is large enough. Since h(p H ) is a quadratic function with two real roots, solving h(p H ) = 0 for p H gives p 1 and p 1 ; where p 1 > p 1 : Similarly, l(p H ) = 0 gives p and p ; where p > p : Using the values for p 1 and p from h(p H ) = 0 and l(p H ) = 0 respectively, we have the following expression: p 1 p = (c 1 c ) 4 b + (1 + + b b) b a (4 b ) p 8(1 )a f(4 b + ) [( + b)a L ( b )c 1 + bc ] b (1 + ) (3 + ) a g (4 b ) p (1 )a f( b) [a L (1 b)c ] + (1 ) a g (4 b ) : We let X = (c 1 c ) 4 b + (1 + + b b) b a (4 b ) captures the rst term outside the square roots and p 8(1 )a f(4 b ) [( + b)a L ( b )c 1 + bc ] b (1 + ) (3 + ) a g Y = (4 b ) p (1 )a f( b) [a L (1 b)c ] + (1 ) a g (4 b ) 19

captures the second and the third term with the square roots. To check whether p 1 p could be positive, we rst inspect Y : The term Y is positive when: 0; where 8(1 ) a 4 b ( + b)a L ( b )c 1 + bc b (1 + ) (3 + ) a and (1 ) a ( b) a L (1 b)c 1 + (1 )a : These are the expressions inside the square root terms of Y : By solving this expression with equality ( that the two square root terms cancel out when: = 0) for c 1 ; we nd c 1 = (7 + 4b)aL + (1 + 3b) 8 4b c + f46 + 18 b [1 + 8b(1 + )]g a 8(8 6b + b : ) This value for c 1 necessarily implies that c 1 > c ; since the rst term is greater than c, and the second term is positive. X > 0: Therefore, when c 1 imply that p 1 p > 0: However, if c 1 > c ; then, it can be easily checked that c (> 0) is large enough, we have X > 0; and Y = 0; which Similarly, using the values for p 1 and p from h(p H ) = 0 and l(p H ) = 0 respectively, we have the following expression: p p 1 = + (c 1 c ) 4 b + (1 + + b b) b a (4 b ) p 8(1 )a f(4 b ) [( + b)a L ( b )c 1 + bc ] b (1 + ) (3 + ) a g (4 b ) p (1 )a f( b) [a L (1 b)c ] + (1 ) a g (4 b ) : Again, we let X = (c 1 c ) 4 b + (1 + + b b) b a (4 b ) captures the rst term outside the square roots and Y = p 8(1 )a f(4 b ) [( + b)a L ( b )c 1 + bc ] b (1 + ) (3 + ) a g (4 b ) p (1 )a f( b) [a L (1 b)c ] + (1 ) a g (4 b ) captures the second and the third term with the square roots. To see that p p 1 = X + Y > 0 when c c 1 (> 0) is large enough, we take the extreme values: c 1 = 0 (the minimum possible c 1 ) and c = a L (the maximum possible c ). With these parametric values, we have: 0

X = al 4 b + (1 + + b b) b a (4 b ) ; Y = p 8(1 )a f(4 b ) [( + b)a L + ba L ] b (1 + ) (3 + ) a g (4 b ) p (1 )a f( b) [a L (1 b)a L ] + (1 ) a g (4 b ) : It can be easily seen that X > 0: For Y ; again, we look at the expressions inside the square root terms of Y : Let! 8(1 ) a 4 b ( + b)a L + ba L b (1 + ) (3 + ) a and (1 ) a ( b) a L (1 b)a L + (1 ) a : From!, we have the expression: (1 ) f46 + 18 b [1 + 8b(1 + )]g a + (8 + 7b) 4 b a L a > 0 () Y > 0), and thus X + Y > 0: C. Proof of Lemma 4 In a high-price separating equilibrium, p H S. Since the rm s pro t is concave in it s own price, the rm s marginal pro t is decreasing for all high signaling prices, i.e., S 1 (p 1;p (p 1 );1) p 1 < 0 for p 1 S; where p (p 1 ) = al +bp 1 +c by Lemma 1. Therefore, rm 1 will choose the lowest price that does not change the consumer beliefs: p H = inf(s): The proof of p H = sup(s) in a low-price separating equilibrium is similar. D. Proof of Proposition 1 We prove Proposition 1 by showing that S 1 enough, x 1 = 1) and S 1 < 0 (implying that x = 0): Since S i > 0 (which implies that when is small = S i p H ph ; we rst determine the sign of S 1 p H : In this regime, rm 1 (high-quality) is setting a price di erent from the rst order condition so as to guarantee that consumers can distinguish between rms. Let p F OC 1 denotes the undistorted First Order Condition price for rm 1. We know that 1 p 1 = 0 when p 1 = p F OC 1 : Because pro ts are a strictly concave function in own prices, it must be the case that S 1 p H distorting its price upward, S 1 p H < 0: < 0 when p H > p F OC 1. Since, in this case, the rm is 1

Next we determine the sign of ph p H = a 8 >< >: 1 : ( b) a L (1 b) c + (1 ) a {z } >= (A1) q a (1 ) ( b) [a L (1 b) c ] + [(1 ) a ] {z } >; (A) ( b) : Notice that expressions (A1) and (A) in the RHS of the above equation are positive, thus the entire expression in the curly brackets is negative, which implies that ph < 0. Therefore S 1 = S 1 ph p H > 0, meaning that rm 1 s pro t increases with. Next we analyze the e ect of quality disclosure on the pro t of rm. By the chain rule, S = S 1 p H ph : S p H ; p L (p H ); 0 p H = b al + bp H c > 0; 4 9 since a L > c. Furthermore, as seen above, ph < 0. Thus, S = S p H ph < 0: E. Proof of Proposition As in the previous proof, we show that S 1 > 0 and S enough, x 1 = x = 1): Again, S i = S i ph p H > 0 (implying that when is small, and we begin the proof by signing the e ect of disclosure on the pro t of rm 1. We rst determine the sign of S 1 p H : In this regime, the rm 1 is playing a price di erent from the rst order condition so as to guarantee that consumers can distinguish between rms. Let p F 1 OC denotes the undistorted First Order Condition price for rm 1. When p 1 = p F 1 OC ; we have 1 p 1 = 0 : Because pro ts are an strictly concave function in own-prices, it must be the case that S 1 p H In this regime, the rm is distorting its price downward, and thus S 1 p H > 0: Next we determine the sign of ph p H = a 8 >< >: : > 0 when p H < p F OC 1. ( b) a L (1 b) c + (1 ) a {z } >= (B1) 1 + q a (1 ) ( b) [a L (1 b) c ] + [(1 ) a ] {z } >; (B) ( + b) : 9

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