Quality Disclosure and Competition (Very Preliminary)

Transcription

1 Quality Disclosure and Competition (Very Preliminary) DanLevin,JamesPeck,andLixinYe August 17, 24 Abstract In this paper we analyze quality disclosure with horizontally differentiated products under duopoly and monopoly market structures. We characterize equilibria and compare the market performances in two cases. We show that the welfare comparison is unambiguously parameterized by the disclosing cost. When the disclosing cost is below a threshold, the welfare is higher under monopoly case, but when the disclosing cost is above the threshold, welfare is higher under duopoly competition. We also show that both market structures result in excessive disclosure compared to the socially optimal level, and imposing tax on disclosure can lead to welfare improvement. 1 Introduction Afreeandefficient flow of information is important for modern, information driven economies. Although prices are the main carriers of information critical for decision making, important information is revealed to consumers directly by firms, either voluntarily or as mandated by law. For example, drug manufacturers must incur substantial costs in order to certify the safety and efficacy of new drugs. A substantial portion of these costs can be attributed to the cost of disclosing the quality, as opposed to the cost of learning the quality. 1 In other industries, disclosure is not regulated, Department of Economics, The Ohio State University. 1 After all, firms sometimes learn that drugs approved to treat one condition are effective in treating a completely different condition. Manufacturers often bypass the costly trials necessary to certify safety and efficacy for the new use of the drug, suggesting that the trials are more about certification of quality than learning about quality. 1

2 yet firms choose to incur expenses in order to disclose quality. Private schools and summer camps produce glossy brochures and quality videos, detailing the quality of the services they provide. What determines firms disclosure behavior? What is the motive for a firm to reveal its private information? In a market with inefficient disclosure, how can a regulatory policy correct the inefficiency? We are motivated by these questions, and in particular, by how the answers depend on the market structure in the relevant industries. A key result in the early literature is that, if disclosure is costlessly credible, a privately informed seller will voluntarily disclose all information. See, for example, Milgrom and Roberts (1986) in the monopoly context and Okuno-Fujiwara, Postlewaite, and Suzamura (199) in the oligopoly context. If we look at the set of qualities that are not disclosed, a firm with the highest quality in the set could increase its perceived quality by disclosing. In such environments, mandatory disclosure rules are unnecessary. 2 However, we often do not observe full disclosure in practice. To reconcile the gap between the theoretical predictions and this empirical observation, other studies utilize one of two approaches. The first approach assumes that disclosing is costly. Viscusi (1978) and Jovanovic (1982) show that if disclosure is costly, sellers will voluntarily disclose only if their quality exceeds some threshold. Introducing a disclosing cost thus leads some sellers to choose not to disclose. However, the analysis does not really justify the motive for a mandatory disclosure rule. In their models, consumers are equally well off with and without disclosure, since the price they pay equals their expected valuation conditional on their information. Sellers may be even worse off with mandatory disclosure, because it eliminates the option to withhold information and save the disclosure cost. Jovanovic shows that, in equilibrium, there is actually too much disclosure. A small subsidy for sales in which quality is not disclosed improves welfare. The second approach employs behavioral models (currently fashionable in economics and finance) that assume that not all consumers are fully rational (Hirshleifer et al. 22, Fishman and 2 Matthews and Postlewaite (1985) turn common intuition on its head, by considering a model in which the firm chooses whether to learn the quality of its product, and then chooses whether to disclose the quality. Both learning and disclosing are costless. Since the firm cannot credibly argue that it did not learn its quality, quality is learned and disclosed. However, a mandatory disclosure law might give an incentive for the firm not to observe quality, because now the firm could credibly argue that it did not learn its quality, because then it would have been disclosed. Thus, mandatory disclosure laws can lead to less disclosure. 2

3 Hagerty 23). It assumes that there are two types of consumers: fully rational consumers who could correctly infer from firms disclosure the relevant quality, and bounded rational consumers who naively neglect to draw inferences from the fact that a firm does not disclose quality, believing instead that the distribution of quality is the prior distribution. The presence of boundedly rational consumers introduces at least some incentive for a seller not to disclose information about quality, even if it is costless. Qualitatively speaking, these behavioral models are similar to the disclosurecost models. In both types of models, unraveling of the decision not to disclose is only partial. For a firm with sufficiently low quality, either separating oneself from the pool of lowest-quality firms is not worth the cost of disclosure, or separating oneself from the pool of lowest-quality firmsisnot worth the foregone earnings from naive consumers. While all the papers mentioned above only consider disclosure in the settings with a monopolistic firm, our approach in this research is different. We are interested in questions about how bringing competition affects the firms disclosure behavior and the market performance. This relates our research to two recent papers. Board (23) examines a duopoly model with differentiated products when disclosing quality is costless. His duopoly example shows that the firms disclosure strategy may exhibit non-monotonic feature, in which only the firms with intermediate level of quality discloses. High quality firm does not disclose since the increasing competition induced by the disclosure may outweigh the gain from the enhanced market perception resulted from quality disclosure. On the other hand, the low quality firm does not disclose as it wants to take advantage of the raised consumer expectation. Our model is different from Board s in that the products are horizontally, rather than vertically differentiated in our model. Our focus of welfare analysis is also very different from Board s. While his interest is in the effect of mandatory disclosure policy on the market performance, we focus on the direct comparison between duopoly and monopoly market structures. Cheong and Kim (24) examine the effect of competition by comparing the incentives to disclose qualities between separate monopolistic markets and an integrated global market with multiple firms. Their major finding is that when the number of firms in the market is getting larger, no firm will disclose quality in the limit. Their result is fairly intuitive. Since firms engage in Bertrand price competition after quality disclosure decisions, the expected rent to the only winning firm goes to zero in the limit. Thus as long as the disclosure cost is positive, however small it is, no firm can afford quality disclosure in the limit. One problem with their results is that as long as there are two 3

4 or more firms disclose, only the firm with the highest quality can survive with positive profit. This result is the joint effect of Bertrand competition and homogeneous consumers. With the assumption of heterogeneous consumers, our result allows for the coexistence of multiple firms surviving in the market with simultaneous disclosure. More specifically we consider a Hotelling model of price competition with horizontally differentiated products. Consumers are located along the unit interval, interpreted to be the most preferred product characteristic, and the two firms are located at either endpoint. Each firm s product is also defined along a second dimension, interpreted to be product quality. Product quality is privately observed by the firm, and the two qualities are independent. In the symmetric equilibrium we characterize, firms whose quality is below a certain threshold, q D, choose not to disclose, while those with quality higher than q D disclose. In the monopoly case, everything remains the same except that the monopolist produces both products and is responsible for the disclosure decision for both products. 3 Compared to the duopoly case, the monopolist possesses an additional coordination instrument in quality disclosure, as the disclosure decision can be made contingent on the realization of both qualities. Although qualities are independent in our model, the strategic coordination can create correlations which may be desirable for the monopolist. Indeed, we show that in equilibrium, the monopolist follows disclosure strategies characterized by a pair of cutoffs (q M,G( )). The monopolist discloses the higher quality (q H )aslongasitishigherthanq M. As for the lower quality, themonopolistwillonlydiscloseitwhenitishigherthang(q H ). Since G(q H ) varies with q H,the strategic coordination thus has positive value for the monopolist profit maximization. Our welfare analysis shows that the comparison is nicely parameterized by the disclosure cost (δ). When δ is lower than some cutoff point (δ ), the welfare is lower under the duopoly market structure, while when δ > δ, the welfare is higher under the duopoly market structure. Thus introducing competition improves the market performance as long as the disclosing cost is above some threshold. The intuition is clear. In our model inefficiency comes from two sources, one due to product misallocation and the other due to disclosure costs. When disclosure is costless, standard literature implies complete unraveling in both market structures. It can be shown that in that case duopoly competition results in more distortion from the first-best allocation compared to the monopoly case, thus the expected total surplus starts lower for duopoly. However, when δ increases, 3 The monopoly in our model can thus be re-interpreted as a collusive duopoly. 4

5 it is shown that monopoly always discloses more on average, which incurs more disclosing cost. When δ is sufficiently large, the effect due to disclosure cost dominates the effect of misallocation, which results in higher welfare in duopoly case. Although disclosing quality has the beneficial effect of inducing some consumers to switch to the higher quality product, we show that the equilibrium level of disclosure is too high in both market structures. Thus our model does not provide support for mandatory disclosure policy. Instead, we show that imposing a small tax on disclosure can always improve welfare in both market structures. In Section 2 we lay down the model for duopoly competition with quality disclosure and characterize the equilibrium. In Section 3 we characterize the equilibrium for monopoly benchmark. We present the welfare comparisons in Section 4 and Section 5 concludes. 2 The Duopoly Model There are two firms producing a differentiated product, represented by firm and firm 1 located at either endpoint of the unit interval. Products and 1 differ on two dimensions: the taste dimension and the quality dimension. Consumers all have the same preference for high quality, but different consumers have different preferences along the taste dimension. We model the taste dimension as the location of a consumer within the unit interval, representing the ideal product characteristic for that consumer. The distance from a consumer s location to the product location is the loss in utility from consuming a product that is less than ideal. Thus, consumers located near have a strong preference for product over product 1, consumers located near 1 have a strong preference for product 1 over, and a consumer located at 1/2 is indifferent between the two products (holding quality constant). A consumer is thus characterized by the distance, x, from the location of firm (x is the type of the consumer). We assume that consumers are uniformly distributed along the unit interval. For i =, 1, the quality of the good produced by firm i is ψ +q i,whereψ is a common value component exogenously given in our model and q i s are independent draws from the uniform distribution over [, 1]. A consumer purchases either or 1 unit of output, where the utility of not purchasing is normalized to zero. Given q and q 1, the values of product and 1 toatype-x consumer are given by ψ + q x and ψ + q 1 (1 x), respectively. Thus, given prices p and p 1 charged by two firms, 5

6 a type-x consumer s utilities obtained from purchasing one unit of products and 1, are given by ψ + q p x,andψ + q 1 p 1 (1 x), respectively. For convenience in our model we assume ψ> 5 4. Since the value of ψ is common knowledge to all parties in the model, henceforth we simply refer the private value part of the quality q i as the quality of the product i, i =, 1. The consumer will either not purchase, purchase from firm, or purchase from firm 1, depending on which decision induces the most expected utility conditional on the consumer s available information. The timeline of the duopoly game is as follows. First, each firm privately observes its quality q i, and decides on whether or not to disclose their quality. Thus each firm s disclosure strategy can be represented by a function D i :[, 1] {, 1}, mapping a private quality to a disclosure choice, where D i =means do not disclose and D i =1means disclose. If firm i discloses its quality, consumers are told q i, so we assume that certifying a false quality is impossible. After the disclosure choices have been made by both firms, the firms engage in a duopoly competition by simultaneously choosing prices. Since at this point quality disclosure decisions have been made, consumers have formed market perceptions about each product s quality. Let the induced market perceptions be q and q 1, respectively. It is easily seen that the firms profits only depend on the prices and the market perceptions of the qualities, instead of the true qualities. 4 Thus each firm s pricing (reduced) strategy can be represented by a function P i :[, 1] [, 1] R +, mapping the pair of perceived qualities to a positive price. Finally, after observing firms disclosure and pricing decisions, consumers choose to purchase from one of the firms, or choose not to purchase. We assume that the cost of disclosing is denoted by δ, and the marginal cost of production is normalized to zero. Our solution concept is Perfect Bayesian Equilibrium (PBE). 5 Because types are uncorrelated and quality does not affect production cost, the optimal price for a firm that does not disclose its quality does not depend on the value of that quality. Thus, we feel justified in restricting attention 4 Thereisnoconflict between the market perception of the quality and the quality itself, if the quality is disclosed. 5 Throughout the paper, we ignore technical issues involving a continuum of qualities and a continuum of consumer types. Sequential rationality should hold "almost everywhere," and statements about the uniqueness of equilibrium should be similarly qualified. 6

7 to PBE in which all types that do not disclose choose the same price, and consumers beliefs (about the quality of a firm that does not disclose) do not depend on the rival s disclosure choice or either price. Consumers simply update their priors and assign a uniform distribution over qualities that are not disclosed on the equilibrium path. Also, in view of the symmetry of the model, we restrict attention to symmetric PBE. We will use the term duopoly equilibrium to denote a PBE satisfying the restrictions imposed here. 6 Lemma 1 In any duopoly equilibrium, there is a threshold, q D >, such that firms with quality below q D do not disclose, and firms with quality above q D disclose. Proof. Suppose, in a duopoly equilibrium, firm i discloses quality q, but does not disclose quality q,whereq >q holds. Since consumers cannot distinguish qualities of firms that do not disclose, the expected profit ofafirm that does not disclose is a constant, π. Since firm i discloses quality q, its expected profit,whichwedenotebyπ,mustsatisfyπ π. However, if firm i were to disclose quality q, it could choose the same price it would have chosen with quality q, and receive strictly more customers. 7 Thus, this deviation would yield firm i strictly higher profits than π,and therefore strictly higher profits than π, a contradiction. Finally, note that q D must be positive, because a firm that discloses with quality, q i =, could increase profits by δ if it decided not to disclose. ItfollowsfromLemma1andthedefinition of a duopoly equilibrium that, when a firm does not disclose, consumers believe that its quality is uniformly distributed on the interval, [,q D ]. Hence the perception of the quality is given by E(q q q D )=q D /2. Our assumption, ψ> 5 4, ensures an interior solution, with all consumers purchasing one of the two products. In equilibrium, there will be a cutoff type, x, where consumers with types x<x purchase from firm, consumerswithtypes x>x purchase from firm 1, and consumers with type x = x are indifferent between purchasing 6 Without these restrictions, there would be a class of symmetric PBE, indexed by the price chosen by firms that do not disclose. The reason for the multiplicity is that firms must choose the designated price in order to be considered within the pool of types that do not disclose. A deviation to a different price is taken to be from a firm with zero quality. 7 That is, it receives strictly more customers whenever it receives any customers with quality q. We know quality q must sometimes receive customers, or else it would not incur the disclosure cost. 7

8 from firm or firm 1. Given such cutoff x, we simply say that the market share for product is x. Given the prices (p,p 1 ) and consumers percepted qualities ( q, q 1 ), x is determined by the following equation: which gives ψ + q p x = ψ + q 1 p 1 (1 x ) The firms profits are given by: x = ( q q 1 + p 1 p ) (1) π (p,p 1 ) = p x = p ( ( q q 1 + p 1 p )) π 1 (p,p 1 ) = p 1 (1 x )=p 1 ( ( q q 1 + p 1 p )) To maximize π (p,p 1 ) and π 1 (p,p 1 ),itcanbeverified that the optimal prices are as follows: p = ( q q 1 ) (2) p 1 = ( q 1 q ) (3) Substituting the optimal prices above to the expressions for market share and profits, we have x ( q, q 1 ) = ( q q 1 ) (4) π ( q, q 1 ) = 1 18 (3 + q q 1 ) 2 (5) π 1 ( q, q 1 ) = 1 18 (3 + q 1 q ) 2 (6) Proposition 1 There is a unique duopoly equilibrium characterized by the disclosure threshold q d. Each firm discloses q i if q i q D, and does not disclose if q i <q D. q D 5 3 = δ if δ [, 13/72] 1 if δ>13/72 (7) Proof. Suppose firm 1 follows a disclosure strategy characterized by a cutoff quality q. If firm discloses its quality q, its expected profit from the following duopoly competition is given by 8

9 π D (q )=q [ 1 Z 1 18 (3 + q q /2) 2 1 ]+ q 18 (3 + q q) 2 dq δ (8) If firm does not disclose its quality, its expected profit isgivenby π ND (q )=q 1 Z q 18 (3 + q /2 q) 2 dq (9) Note that given q, π ND (q ) is a constant and π D (q ) is strictly increasing in q. Therefore, for q to characterize a cutoff disclosure equilibrium, we must have π D (q )=π ND (q ) For δ 13 72, the solution is qd = δ. For δ> 13 72, by playing the strategy characterized by qd =1,eachfirm receives profits of 1 2.It is immediate to check that a deviation to a price other than p i =1leads to lower profits. The best deviation involving disclosure is for a firm with quality, q i =1, to choose the price, p i = 7 6, yielding profits of δ. Since we have δ>13 72, this deviation is not profitable. All consumers are willing to purchase, so (7) characterizes a duopoly equilibrium. We omit the argument that there is no duopoly equilibrium in which not all consumers are served. 3 The Monopoly Model In order to evaluate how the degree of competition affects disclosure and welfare, we now consider a monopoly model that is otherwise identical to the model of Section 2. The locations and utility functions of consumers is the same, the location of products is the same, and the joint distribution of qualities is the same. The only difference is that now there is a single firm that produces both products and observes both qualities. Based on the realizations of both qualities, the monopolist chooses to disclose either none, one, or both of the qualities. Thus the disclosure strategy D : [, 1] [, 1] {(, ), (1, ), (, 1), (1, 1)} maps a pair of qualities to a disclosure choice. After disclosure decision, the firm sets prices for both products. Thus a (reduced) strategy of pricing P :[, 1] [, 1] R 2 + maps the market perceptions about two products qualities to two (positive) prices. As before, production cost is normalized to zero, and a disclosure cost, δ, is incurred for each product whose quality is disclosed. 9

10 Because types are uncorrelated and quality does not affect production cost, the profit maximizing prices, (p,p 1 ), for a monopolist do not depend on the value of any undisclosed quality. Thus, we feel justified in restricting attention to PBE in which prices depend only on disclosed qualities, and consumers beliefs about undisclosed qualities do not depend on either price. However, beliefs about, say, q, could well depend on the disclosed value of q 1. On the one hand, the incentive to disclose q depends on consumers willingness to pay once quality is disclosed, which in turn depends on the disclosed value of q 1. On the other hand, we want to avoid undesirable equilibria in which specific high quality values are not disclosed, supported by the belief that the undisclosed quality is zero. To resolve this issue, we restrict attention to the following class of symmetric PBE. Let q H =max{q,q 1 } denote the higher of the two qualities and let q L =min{q,q 1 } denote the lower of the two qualities. Also, let p H denote the price chosen for the good with quality q H,andletp L denote the price chosen for the good with quality q L. Definition 1: A monopoly equilibrium is a PBE satisfying (i) There is a threshold, q M, such that q H is not disclosed if we have q H <q M,andq H is disclosed if we have q H q M, (ii) Beliefs about undisclosed quality are independent of prices, (iii) If, for i =or 1, the monopolist discloses q i = q, wherewehaveq<q M,consumersbelieve that q H = q holds with probability one, and that q L is uniformly distributed over [,q]. Lemma 2 In any monopoly equilibrium, beliefs are as follows. If nothing is disclosed, consumers believe that both qualities are independently and uniformly distributed over [,q M ]. If only q i = q< q M is disclosed, consumers believe that q H = q holds with probability one, and that q L is uniformly distributed over [,q]. If only q i = q q M is disclosed, consumers believe that q H = q holds with probability one, and that q L is uniformly distributed over [,G(q)], for some function G(q), satisfying <G(q) q. The monopolist s disclosure policy is to disclose nothing if q H <q M holds, only to disclose q H if q M q H and q L G(q H ) hold, and to disclose both qualities if q M q H and q L >G(q H ) hold. Proof. If both qualities are near zero, then certainly nothing will be disclosed in any monopoly equilibrium. Therefore, no disclosure is on the equilibrium path, and the beliefs following no disclosure are determined by Bayes rule. If only q i = q < q M is disclosed, then beliefs are 1

11 determined by Definition 1 (iii). If only q i = q q M is disclosed, then we are on the equilibrium path, based on Definition 1 (i), and the fact that q L near zero would not be disclosed no matter what consumers believe. Therefore, Bayes rule implies that q H = q holds with probability one. Also, if the monopolist discloses q H = q and q L = q, then sequential rationality requires that the monopolist discloses q H = q and q L = q for all q >q. The reason is that the monopolist s profit from disclosing only q H = q does not depend on q L,whichwecanwriteasπ M (q). If disclosing q H = q and q L = q yields profits greater than π M (q), then disclosing q H = q and q L = q must yield higher profits still, because the monopolist can raise the price it charges for the lower quality good and continue to sell to all consumers. This establishes the interval property for disclosure of q L when we have q M q H. Consumers must believe that q L is uniformly distributed over [,G(q)], which now follows from Bayes rule. Proposition 2 There is a unique monopoly equilibrium. When δ [, 7/32], the disclosure policy is characterized by q M =4 1+2δ 4 (1) G(q H )= where ˆq = δ 2q H p (qh ) 2 4q H +4+24δ if δ [ˆq, 1] q H if δ [q M, ˆq] (11) When δ [7/32, 9/32], q M =4 1+2δ 4 and G(q H )=q H (the lower quality is never disclosed). When δ>9/32, q M =1,andG(q H )=q H, that is, the monopolist never discloses any quality. Proof: See the appendix. Proposition 2 shows that unlike in the duopoly case, the monopolistic disclosure exhibits some coordination feature, as the disclosure of q L can be contingent on the disclosure of q H.Proposition 2 suggests that this coordination mechanism has value for the monopolist in profit maximization, as G( ) does vary with q H. Is this coordination instrument also beneficial to welfare? In the next section,wewilllookattheeffect of market structures on welfare. 11

12 4 Welfare Analysis 4.1 Duopoly vs. Monopoly In this section we consider the welfare implications from the equilibria characterized in both market structures. First, following Proposition 1, we can compute the expected total surplus as a function of the disclosure cost δ in the duopoly case. For δ [, 13/72], the expected total surplus is given by For δ>13/72, ETS D = ψ +1/4. ETS D = ψ δ ( δ) δ (12) In monopoly case, to save on notation we define the following: A = q 5 2δ 4 1 2δ, B = 1 2δ, C = 1+2δ, and D = 1+24δ. (13) For δ [, 7/32], the expected total surplus is given by ETS M = ψ δ ( ln2)δ2 + +( δ δ )B + 16(3 + δ)c +( δ)d +16δ2 ln AB +( δ)a D 1 1+A 2B (14) where A, B, C and D are definedin(13). For δ (7/32, 9/32], wehave For δ>9/32, we have ETS M = ψ δ +8δ2 + 16(3 + δ) 1+2δ (15) ETS M = ψ (16) To compare the welfare in both market structures, we depict the curves of expected surplus against the disclosing cost in Figure 1. Proposition 3 There exists a unique δ (, 13/72), such that for δ [,δ ),theexpectedtotal surplus is higher in monopoly case, and for δ (δ, 9/32), the expected total surplus is higher in duopoly case. For δ 9/32, the expected surplus is the same under two market structures. 12

13 Expected total surplus Disclosing cost Figure 1: Proof: See the appendix. To understand this welfare comparison result, we need to identify the sources of inefficiency. In fact, there are two potential sources for inefficiency in our model. One is the misallocation of the products, and the other is the disclosure costs. When δ =, there is no disclosure cost, and hence no efficiency loss due to the disclosure costs. In this case the unraveling result established in the literature implies that both qualities will be disclosed under both market structures. The expected total surplus is completely determined by the market shares for both products. Let x D and x M denote the market shares for product under duopoly and monopoly cases. Then x D = (q q 1 ) (17) x M = (q q 1 ) (18) To understand which market structure induces more distortion from the first-best solution, we consider the social planner s problem. The social planner decides on a market share x to maximize 13

14 the expected total surplus. That is, the social planner will choose x to maximize the following: The solution is given by Z x (ψ + q x) dx + Z 1 x (ψ + q 1 x) dx x = (q q 1 ) (19) As long as q 6= q 1, x M is always closer to x than x D. Therefore we expect that the inefficiency due to misallocation in the case of duopoly is larger than in the case of monopoly. This explains the expected surplus curve in monopoly case starts strictly higher than the expected surplus curve in duopoly case. By continuity, we should expect that the expected total surplus will remain higher in the case of monopoly when δ is sufficiently small. To understand why there is a cutoff δ which parameterizes the welfare comparison, we compare the expected disclosures in equilibrium. Proposition 4 In equilibrium, the expected number of disclosed products in duopoly case is strictly lower for δ (, 9/32). Beyond this range, either both products are disclosed (δ =)ornoneofthe products is disclosed (δ 9/32) under both market structures. Proof: See the appendix. ThecomparisonisdepictedinFigure2. Our computation (in appendix) reveals that the probability of disclosing both products is always lower, and the probability of disclosing neither product is always higher in duopoly case. For the probability of disclosing exactly one product, it is higher in duopoly case only when δ is low. As a result, the expected number of disclosed products is strictly lower in duopoly case for δ (, 9/32). Therefore on average monopoly incurs more expected disclosing costs in equilibrium, which explains why the expected total surplus in duopoly case overtakes the expected total surplus after a certain threshold of disclosing cost. To summarize, when δ [,δ ), the misallocation effect (due to pricing) dominates the disclosing effect so the monopoly welfare dominates the duopoly; when δ (δ, 9/32), the disclosing effect dominates the misallocation effect so duopoly welfare dominates the monopoly; when δ 9/32, no 14

15 Expected number 2 of disclosure Disclosing cost Figure 2: disclosure ever occurs in both market structures, and there is no welfare difference generated from differentmarketstructures. So in our model with horizontally differentiated products, the welfare comparison is unambiguously parameterized by the disclosure cost δ. When disclosing cost is not too small, our result suggests that introducing competition is welfare improving. The expected total surplus curves shown in Figure 1 suggest that both market structures tend to disclose too much in equilibrium. To confirm such impression we next consider the socially optimal disclosure. 4.2 Socially Optimal Disclosure To measure the socially optimal disclosure, we modify the game as follows. We augment the games with one more stage prior to the timeline described before, and one more player, that is, the social planner. The new timeline is as follows. First, the social planner picks the disclosure cutoff (which is q D in duopoly case and ((q M,G ( )) in monopoly case). Then the firms (or the firm) makes disclosure decision according to the cutoff chosen by the social planner. The ensuing game is the same as before. That is, after quality disclosure decision, the firms (or the firm) chooses prices. Finally 15

16 the consumers make purchase decisions given firms disclosure decision and pricing decision. The social planner s objective is to maximize the expected total surplus given the constraint of specific market structure (i.e., the way the prices are determined after enforcing the disclosure policy). The socially optimal cutoff is the solution to this problem faced by the social planner. Proposition 5 In duopoly case, the socially optimal cutoff is given by q D =12 p δ/5 for δ [, 5/144], andq D =1for δ>5/144. In monopoly case, the socially optimal cutoff is given by q M = G (q H )=8 p δ/3 for δ [, 3/64] and q M = G (q H )=1for δ>3/64. In either market structure, disclosure is excessive compared to its socially optimal disclosure benchmark. Proof: See the appendix. Thus the socially optimal disclosure rules are quite simple in both market structures. In particular, the socially optimal disclosing rule in monopoly case does not involve coordination between the higher and lower qualities, as G (q H ) does not vary in q H. It appears to be an advantage for the monopolist to be able to coordinate between two quality disclosures through function G( ). However, Proposition 5 implies that such instrument of coordination does not help the social planner with surplus maximization, though it is desirable for profit maximization. By Proposition 5, there is usually too much disclosure in our model, either in duopoly or in monopoly case. So our model does not provide support for mandatory disclosure policy. This suggests that instead of subsidizing to encourage disclosure, imposing tax to discourage disclosure can be welfare improving. In fact we can show the following result: Proposition 6 Introducing a sufficiently small tax on disclosure can always improve welfare in duopoly case with δ (, 13/64), and in monopoly case with δ (, 9/32). Proof: See the appendix. 5 Conclusion In this research we analyze quality disclosure with horizontally differentiated products in both duopoly and monopoly market structures. Our main result is that the welfare comparison is nicely parameterized by the quality disclosing cost. When the cost is lower than some threshold, monopoly 16

17 achieves higher welfare while when the cost is higher than the threshold, duopoly generates higher welfare. While the magnitude of the difference may be sensitive to the parameters chosen in our model, we believe that the qualitative prediction from our result is fairly robust. Intuitively, when disclosing cost is low, the inefficiency is mainly due to the misallocation of the products, thus monopoly will perform better; but when disclosing cost is high, the inefficiency is mainly due to the disclosure cost and hence duopoly will perform better. Introducing competition thus has two effects. One is to increase allocation distortion through more fierce price competition, and the other is to reduce disclosure to help conserve the welfare when disclosure is excessive. One extension of our research is to introduce quality correlation into our model. We conjecture that if qualities are correlated, the incentive to disclose quality will be reduced in both market structures. It would be interesting to investigate the net effect regarding the welfare comparison in equilibrium. 17

18 Appendix ProofofProposition2: Given the disclosure strategy characterized by (q M,G( )), suppose the induced market perceptions about the qualities are given by ( q, q 1 ) according to the belief system prescribed by Lemma 2. Since ψ is sufficiently large, all consumers will be served and the monopolist will optimally extract all rents from the marginal consumer who is just indifferent between purchasing product and product 1. Let this marginal consumer s type be x, then the monopolistic prices are given by p = ψ + q x p 1 = ψ + q 1 (1 x) (2) The profit maximizing market share is thus given by x argmax(ψ + q x)x +(ψ + q 1 (1 x))(1 x) which results in x = ( q q 1 ) (21) Substituting (21) into the revenue function, the monopolistic revenue (gross of any possible disclosure cost) is given by: R( q, q 1 )=ψ 1 2 ( q + q 1 )+ 1 8 ( q q 1 ) 2 (22) We consider three cases in order. 1. Neither product is disclosed. By (22), the monopolistic profit is given by π 1 (q H,q L )=ψ qm (23) 2. Only q H is disclosed. By (22), the monopolistic profit is given by π 2 (q H,q L )=ψ (q H + G(q H )/2) (q H G(q H )/2) 2 δ (24) 18

19 3. Both products are disclosed. By (22), the monopolistic profit is given by π 3 (q H,q L )=ψ (q H + q L )+ 1 8 (q H q L ) 2 2δ (25) It is easily verified that whenever q H or q L is disclosed, the profit is strictly increasing in q H or q L. This justifies the monotonicity of the equilibrium. The equilibrium requires that when q H = q M the monopolist is indifferent between disclosing q H only and disclosing neither products. Substituting q H = G(q H )=q M into (24) and (23) and equating π 2 with π 1,wehave q M =4 1+2δ 4 It remains to solve for G( ). Note that the equilibrium requires that if q H >q M and q L >G(q H ), the monopolist should prefer disclosing both products to disclosing q H only. For q H >q M and q L = G(q H ), the monopolist should weakly prefer disclosing q H only. Letting q L = G(q H ) in (25) and equating π 3 with π 2,wehave where ˆq = δ. G(q H ) = min{q H, 2q H (qh ) 3 3p 2 4q H +4+24δ} 2q H = 3p (qh ) 2 4q H +4+24δ if δ [ˆq, 1] q H if δ [q M, ˆq] For q H = q L (q M, ˆq), itiseasilyverified that disclosing q H only is strictly preferred to disclosing both products by the monopolist. Hence (q M,G( )) given in (1) and (11) indeed characterizes the disclosure equilibrium. ProofofProposition3: We first derive the expected total surplus functions under both market structures in order. 1. Duopoly Case Following the equilibrium characterization in Proposition 1, we consider four regions of the realization of qualities in order: 19

20 (a) q,q 1 <q D, both qualities are not disclosed. By (2), in this case the type of the marginal consumer who is indifferent between product and product 1 is given by x = 1 2.Foratype-x consumer, x [,x ], the generated value (surplus) from purchasing product is given by ψ + q x, and similarly, the generated value (surplus) from purchasing product 1 is given by ψ + q 1 (1 x). Integrating over all consumers, we have the total surplus: ts 1 (q,q 1 )= Z 1/2 (ψ + q x) dx + Z 1/2 (ψ + q 1 x) dx Integrating ts 1 (q,q 1 ) over the set of qualities with q,q 1 <q D, we have the expected surplus: ETS 1 = Z q D Z q D ts 1 (q,q 1 ) dq 1 dq (26) (b) q >q D >q 1.Onlyq is disclosed. By (2), the marginal consumer type is given by x = (q q D /2) (27) Given (q,q 1 ), the total surplus generated is given by ts 2 (q,q 1 )= Z x (ψ + q x) dx + Z x (ψ + q 1 x) dx δ Integrating again to get the expected total surplus in this region: ETS 2 = where x is given by (27). Z 1 Z q D q D ts 2 (q,q 1 ) dq 1 dq (28) (c) q 1 >q D >q, q 1 is disclosed but q is not. By symmetry, ETS 3 = ETS 2. (d) q,q 1 >q D.Bothq and q 1 are disclosed. By (2), the marginal consumer type is given by x = (q q 1 ) (29) Given (q,q 1 ), the total surplus generated is given by ts 4 (q,q 1 )= Z x (ψ + q x) dx + 2 Z x (ψ + q 1 x) dx 2δ

21 Integrating again to get the expected total surplus in this region: where x is given by (29). ETS 4 = Z 1 Z 1 q D q D ts 2 (q,q 1 ) dq 1 dq (3) Putting all these together, the expected total surplus in equilibrium is given by: ETS D = ETS 1 +2ETS 2 + ETS 4 = ψ (qd ) 3 2(1 q D )δ Substituting the value of q D,wehaveforδ<13/72, ETS D = ψ δ ( δ) δ For δ 13/72, q D =1, and hence ETS D = ψ +1/4. 2. Monopoly Case Following the equilibrium characterization in Proposition 2, we consider three regions of the realization of qualities in order: (a) q L q H <q M, both qualities are not disclosed. By (21), the market share is given by x =1/2, the total surplus given (q L,q H ) is ts 1 (q H,q L )= Z x (ψ + q H x) dx + Z 1 x (ψ + q L x) dx Integrating over this region, we have the expected total surplus: Z q M Z qh ETS 1 =2 ts 1 (q H,q L ) dq L dq H (31) (b) q H >q M but q L <G(q H ),onlyq H is disclosed. By (21), the market share is given by x =1/2 +(q H G(q H )/2)/4, the total surplus given (q L,q H ) is ts 2 (q H,q L )= Z x (ψ + q H x) dx + Z 1 x Integrating over this region, we have the expected total surplus: ETS 2 =2 Z 1 q M Z G(qH ) (ψ + q L x) dx ts 2 (q H,q L ) dq L dq H (32) 21

22 (c) q H >q M and q L >G(q H ),bothq H and q L are disclosed. By (21), the market share is given by x =1/2 +(q H q L )/4, the total surplus given (q L,q H ) is ts 3 (q H,q L )= Z x (ψ + q H x) dx + Z 1 x (ψ + q L x) dx Integrating over this region, we have the expected total surplus: Z 1 Z 1 ETS 3 =2 ts 3 (q H,q L ) dq L dq H (33) q M G(q H ) The expected total surplus ETS M = ETS 1 + ETS 2 + ETS 3. Substituting the values of q M and the expression of G(q H ) into (31), (32) and (33), we can obtain the expected total surplus under monopoly regime. For δ [, 7/32], ETS M = ψ δ ( ln2)δ2 + +( δ δ )B + 16(3 δ)c +( δ)d +16δ2 ln AB +( δ)a D 1 1+A 2B where A = p 5 2δ 4 1 2δ, B = 1 2δ, C = 1+2δ, andc = 1+24δ as defined in (13). For δ (7/32, 9/32], wehave For δ>9/32, we have ETS M = ψ δ +8δ2 + 16(3 + δ) 1+2δ ETS M = ψ The proof is trivial by inspecting Figure 1, so we skip the detail. Numerical solution for δ is approximately.5. So in this model only for a small range of values of δ that monopoly generates more welfare, and for the rest of the values of δ, duopoly welfare dominates monopoly. ProofofProposition4: 1. Duopoly Case 22

23 Let Pr D (i) denotes the probability of disclosing i products, i =, 1, 2. Thenforδ [, 13/72] we have Pr D () = q D2 = 1 9 ( δ) 2 (34) Pr D (1) = 2q D (1 q D )= 2 9 ( δ 5)( δ) (35) Pr D (2) = (1 q D ) 2 = 1 9 ( δ) 2 (36) For δ 13/72, wehavepr D () = 1 and Pr D (1) = Pr D (2) =. We can now compute the expected number of disclosed products in duopoly market, EN D for δ [, 13/72]: EN D (δ) =Pr D (1) + 2Pr D (2) = δ (37) 3 For δ 13/72, wehaveen D (δ) =. 2. Monopoly Case Let Pr M (i) denotes the probability of disclosing i products, i =, 1, 2. Then for δ [, 7/32] we have Pr M () = (q M ) 2 =(4 1+2δ 4) 2 (38) Z 1 Z G(qH ) Pr M (1) = 2 dq L dq H q M = 1 3 [ δ 8A +16AB +96C 2D]+16δln D 1 2(1 + A 2B) Z 1 Z 1 Pr M (2) = 2 dq L dq H q M G(q H ) = 1 3 [ δ +8A +64B 16AB] 16δ D 1 2(1 + A 2B) For δ (7/32, 9/32], the monopolist never discloses the lower quality. We thus have Pr M (2) =, Pr M () = (q M ) 2 =(4 1+2δ 4) 2,andPr M (1) = 1 Pr M (). For δ>9/32, neither qualities will be disclosed in equilibrium. Based on these, the expected number of disclosed products, EN M can be computed. For δ [, 7/32], EN M (δ) =Pr M (1)+2Pr M (2) = 1 ln 3 [ δ+8A+64B 16AB+96C+2D] 16δ D 1 2(1 + A 2B) (41) For δ (7/32, 9/32), wehaveen M (δ) =1 ( δ) 2.Forδ>9/32, EN M (δ) =. 23 (39) (4)

24 2. The comparison shows that EN D (δ) >EN M (δ) for δ (, 9/32), which is illustrated by Figure ProofofProposition5: 1. Duopoly Case The social planner announces and enforces the disclosure policy characterized by q.let( q, q 1 ) be the induced market perceptions about the qualities. Also let x be cutoff consumer type that divides the market shares between purchasing two products. Then under duopoly competition, we have x = ( q q 1 ) 2 Based on this, we can compute the expected total surplus following exactly the same steps as in the derivation of the equilibrium expected total surplus in duopoly case above. Given any disclosure cutoff q that chosen by the social planner, it can be verified that the expected total surplusisgivenby ETS = ψ δ +2δq q3 Therefore the socially optimal cutoff is given by q D = 12 p δ/5 if δ [, q D = 1 if δ ] Obviously q D >q D for all δ (, 13/72) and q D = q D for δ =and δ>13/72. Therefore in duopoly equilibrium there is too much disclosure compared to the social optimal disclosure. 2. Monopoly Case The social planner announces and enforces the disclosure policy characterized by (q,g( )). Let ( q H, q L ) be the market perceptions about (q H,q L ) induced by the monopolist s disclosure decision. Also let x denote the cutoff consumer type that divides the market shares between purchasing the higher quality product and lower quality product. Then x = ( q H q L ) 2 24

25 (a) If neither quality is disclosed, x =1/2. Given(q H,q L ),thesurplusisgivenby ts 1 (q H,q L )= (b) If only q H is disclosed, ts 2 (q H,q L )= Z 1/2 where x =1/2+1/4(q H G(q H )/2). (c) If both q H and q L are disclosed, ts 3 (q H,q L )= where x =1/2+1/4(q H q L ). (ψ + q H x) dx + Z 1/2 Z x Z 1 x (ψ + q H x) dx + Z x Z 1 x (ψ + q H x) dx + (ψ + q L x) dx (ψ + q L x) dx (ψ + q L x) dx The expected total surplus given disclosure rule (q,g( )) is thus Z q ETS =2 Z qh Z 1 (ts 1 (q H,q L ) dq L,dq H +2 Z G(qH ) [ q Z 1 ts 2 (q H,q L ) dq L + G(q H ) ts 3 (q H,q L ) dq L ] dq H {z } =Ω(G) (42) So the optimal G( ) maximizes Ω(G) for any given q H.Differentiating gives G (q H ) = q H if q H < 8 p δ/3 G (q H ) = 8 p δ/3 if q H 8 p δ/3 (43) Substituting G( ) =G ( ) into(42)andthendifferentiating with respect to q we have The optimal q is thus given by det S dq = 3 32 (q ) 3 +2δq q = 8 p δ/3 if δ [, 3/64] q = 1 if δ>3/64 (44) Combining (43) and (44), the social planner s disclosure policy is given below: 25

26 q M 8 p δ/3 if δ [, 3/64] = 1 if δ>3/64 and G (q H )=8 p δ/3 for q H (q M, 1]. Since G ( q H ) does not vary with q H,ineffect the socially planner sets a single cutoff point as in duopoly case. In other words, the potential coordination instrument provided by G( ) does not help to improve social welfare. q M >q M for all δ (, 9/32) and q M = q M for δ =and δ>9/32. It can also be verified that G (q H ) G(q H ) for any q H. Therefore in monopoly equilibrium there is too much disclosure compared to the social optimal disclosure. ProofofProposition6: In duopoly case, suppose a tax t is imposed for each quality disclosure. Then by (12) and (37), the expected total surplus for δ (, 13/72) is given by: ETS D (δ + t) = ψ (δ t) ( (δ t)) p (δ t) EN D (δ + t) t = ψ δ t +[ δ 5 27 t]p (δ + t) Differentiating with respect to t and evaluating at t =,wehave det S D t= = δ + 18δ dt δ > for all δ (, 13/72), which means that introducing a small tax can always improve welfare. For the case with monopoly, the computation is tedious but the result remains the same, thus we skip the detail here. 26

27 References [1] Board, O. (23), Competition and the impact of mandatory disclosure laws, Working Paper. [2] Cheong, I. and Kim, J. (24), Costly information disclosure in oligopoly, Journal of Industrial Economics, 1, [3] Fishman, M.J. and K.M. Hagerty (23), Mandatory versus voluntary disclosure in markets with informed and uninformed customers, Journal of Law, Economics, and Organization, 19, [4] Hirshleifer, D., S.S. Lim, and S. H. Teoh (22), Disclosure to a credulous audience: the role of limited attention, Ohio State University Working Paper. [5] Jovanovic, B. (1982), Truthful disclosure of information, Bell Journal of Economics, 13, [6] Milgrom, P.R. and J. Roberts (1986), Price and advertising as signals of product quality, ournal of Political Economy 66, [7] Okuno-Fujiwara, M., Postlewaite, A. and Suzumura, K. (199), Strategic information revelation, Review of Economic Studies, 57, [8] Viscusi, W.K.(1978), A note on Lemons markets with quality certification, Bell Journal of Economics, 9,