Quality and Competition between. Public and Private Firms

Transcription

1 Quality and Competition between Public and Priate Firms Liisa Laine* Ching-to Albert Ma** March 2016 Abstract We study a multi-stage, quality-price game between a public rm and a priate rm. The market consists of a set of consumers who hae di erent quality aluations. A public rm aims to maximize social surplus, whereas the priate rm maximizes pro t. In the rst stage, both rms simultaneously choose qualities. In the second stage, both rms simultaneously choose prices. Consumers quality aluations follow a general distribution. Firms unit production cost is an increasing and conex function of quality. There are multiple equilibria. In some, the public rm chooses a low quality, and the priate rm chooses a high quality. In others, the opposite is true. We characterize subgame-perfect equilibria, and proide conditions on consumer aluation distribution for rst-best equilibrium qualities. Various policy implications are drawn. Keywords: price-quality competition, quality, public rm, priate rm JEL Classi cations: D4, L1, L2, L3 Acknowledgement: The rst author receied nancial support from the Yrjö Jahnsson Foundation and the KAUTE Foundation. We receied aluable comments and discussions from seminar participants at the Uniersity of Bologna, Boston Uniersity, the 16th European Health Economics Workshop in Toulouse, and the Helsinki Center of Economic Research. For their suggestions and comments, we thank Francesca Barigozzi, Giacomo Calzolari, Helmuth Cremer, Vincenzo Denicolo, and Katharina Huesmann. Much of this research was done while Laine was isiting the Economics Department at Boston Uniersity; she is indebted to their hospitality. *School of Business and Economics, Uniersity of Jyäskylä; liisa.t.laine@jyu. **Department of Economics, Boston Uniersity; ma@bu.edu

2 1 Introduction In many markets such as health care, education, transportation, and utility, public and priate rms jointly sere consumers. Product and serice qualities are major concerns in these markets. These concerns stem from a fundamental point made by Spence (1975). Because a good s quality bene ts all buyers, the social bene t of quality is the sum of consumers aluations. At a social optimum, the aerage consumer aluation of quality should be equal to the marginal cost of quality. Yet, a pro t-maximizing rm is only concerned with the consumer who is indi erent between buying and not. A rm s choice of quality will be one that equates this marginal consumer s aluation to the marginal cost of quality. This gies the classic Spence (1975) result: een when products are priced at marginal costs, their qualities will be ine cient. In this paper, we show that a mixed oligopoly, in which public and priate rms interact, may be a mechanism for remedying this ine ciency. We use a standard model of ertical product di erentiation. In the rst stage, two rms simultaneously choose product qualities. In the second stage, rms simultaneously choose product prices. The two rms hae access to the same technology. The only di erence from the textbook setup is that one rm is a socialsurplus maximizing public rm, whereas the other remains a pro t-maximizing priate rm. Surprisingly, this single di erence has many implications. First, the model exhibits multiple equilibria: in some equilibria, the public rm s product quality is higher than the priate rm s, but in others, the opposite is true. More important, equilibrium qualities may be socially e cient. In fact, we present general conditions on consumers quality-aluation distribution in which qualities in low-public-quality equilibria are e cient, as well as general conditions in which qualities in high-public-quality equilibria are e cient. When equilibrium qualities are ine cient, for both public and priate rms, deiations from the rst best go in tandem. That is, either qualities in public and priate rms are both below the corresponding rst-best leels, or they are both aboe. Equilibrium qualities form a rich set, and we hae constructed examples with many con gurations. Our analysis proceeds in the standard way. Gien a subgame de ned by a pair of qualities, we nd the equilibrium prices. Then we sole for equilibrium qualities, letting rms anticipate that their quality choices 1

3 will result in the corresponding equilibrium prices in the next stage. In the pricing subgame, qualities are gien. The public rm s objectie is to maximize social surplus, so its price best response must achiee the e cient allocation of consumers across the two rms. This requires that consumers fully internalize the cost di erence between high and low qualities. Gien the priate rm s price, the public rm sets its price so that the di erence in prices is exactly the di erence in quality costs. The priate rm s best response is the typical inerse demand elasticity rule. When rms choose qualities, they anticipate the equilibrium prices in the next stage. Gien the priate rm s quality, the public rm chooses its quality to maximize the surplus of consumers that it will sere. It anticipates the e cient assignment of consumers across rms in the next stage, so it chooses quality for the best welfare of its own customers. The priate rm, howeer, will try to manipulate the equilibrium prices through its quality. Because the equilibrium price di erence will be the quality cost di erence, when the priate rm chooses a quality di erent from the public rm s, it implements a larger price di erence. Without any price response from the public rm, the priate rm would hae chosen the quality that would be optimal for the marginal consumer, just as we hae stated aboe (Spence (1975)). A larger quality di erence, howeer, would be preferred because that would raise the price. Because of the price manipulation, the priate rm s equilibrium quality is one that maximizes the utility of an inframarginal consumer, not the utility of the marginal consumer who would just be indi erent between buying from the public and priate rms. In the rst best, the socially e cient qualities are determined by equating aerage consumer aluations and marginal cost of quality. The surprise is that in contrast to priate duopoly, the priate rm s equilibrium quality choice may coincide with the rst-best quality. In other words, the inframarginal consumer whose utility is being maximized by the priate rm happens to hae the aerage aluation among the priate rm s customers. The (su cient) conditions for rst-best equilibria refer to properties of consumers quality-aluation distribution. In the class of equilibria where the public rm produces at a low quality, equilibrium qualities 2

4 are rst best when the aluation distribution has a linear hazard rate. 1 The linear hazard rate condition is equialent to the priate rm s marginal reenue function linear in consumer aluation. In the class of equilibria where the public rm produces at a high quality, equilibrium qualities are rst best when the aluation distribution has a linear reerse hazard rate. The linear reerse hazard rate condition is equialent to the priate rm s marginal reenue function being linear in consumer aluation. Although the hazard and reerse hazard rates hae gured prominently in the information economics literature, we are unaware of any work that imposes linearity on them. We derie all distributions that possess the linearity properties. We wish to note that the uniform distribution, which has been used often to describe consumer aluations, has linear hazard and reerse hazard rates (but it is not the only distribution with this property see Remark 5 below). Also, because hazard and reerse hazard rates can behae quite di erently, for some distributions equilibria for one class (say, low quality at public) can be the rst best but not equilibria in the other (say high quality at public), and ice ersa. We draw arious policy implications from our results. Our use of a social-welfare objectie function for the public rm can be regarded as making a normatie point. If the public rm aims to maximize only consumer surplus, then it will subscribe to marginal-cost pricing. Then the equilibrium price di erence between public and priate rms will neer be the quality cost di erence because the priate rm neer prices at marginal cost. The rst best is neer achieed (een when hazard or reerse hazard rates are linear). A social-welfare objectie does mean that the public rm tolerates high prices. Howeer, our policy recommendation is that undesirable e ects from high prices should be remedied by a tax credit or subsidy to consumers regardless of from where they purchase. This ensures that consumers face a price di erence equal to the quality di erence, a necessary condition for the rst best. Our research contributes to the literature of mixed oligopolies. We use the classical model of qualityprice competition in Gabszewicz and Thisse (1979, 1986) and Shaked and Sutton (1982, 1983). Whereas pro t-maximizing rms use quality di erentiation to relax price competition, a social-surplus maximizing 1 See Lemmas 3 and 6 below. If F denotes the distribution, and f the density, then the hazard rate is 1 F, and f the reerse hazard rate is F. By a function being linear, we mean that it has a constant slope and an intercept. This f is often called a ne linear in mathematics, but we trust that our abbreiation will not cause any confusion. 3

5 public rm does not. This di erence has led to the mixed oligopoly literature, which reoles around the theme that the presence of a public rm may improe welfare. Grilo (1994) studies a mixed duopoly in the quality, ertical di erentiation framework. In her model, consumers aluations of qualities follow a uniform distribution. The unit cost of production may be conex or concae in quality. The paper deries rst-best equilibria. In a Hotelling, horizontal di erentiation model with quadratic transportation cost, Cremer et al. (1991) show that a public rm improes welfare when the total number of rms is either two, or more than six. Also using a Hotelling model, Matsumura and Matsushima (2004) show that mixed oligopoly gies some cost-reduction incenties. In a Cournot model, Cremer et al. (1989) consider replacing some priate rms by public enterprises, and nationalizing some priate rms. In these models, public rms may discipline priate rms. For pro t-maximizing rms, Cremer and Thisse (1991) show that, under ery mild conditions on transportation costs, horizontal di erentiation models are actually a special case of ertical product di erentiation (see also Champsaur and Rochet (1989)). The isomorphism can be transferred to mixed duopolies. The key in the Cremer-Thisse (1991) proof is that demands in horizontal models can be translated into equialent demands in ertical models. Firms objecties are unimportant. Hence, results in horizontal mixed oligopolies do relate to ertical mixed oligopolies. In most horizontal di erentiation models, consumers are assumed to be uniformly distributed on the product space, and the transportation or mismatch costs are quadratic. These assumptions translate to a uniform distribution of consumer quality aluations and a quadratic quality cost function in ertical di erentiation models. The rst-best results in Grilo (1994) are related to the e cient equilibria in the two- rm case in Cremer et al. (1991) because both papers use the uniform distribution for consumer aluations. (Grilo (1994), howeer, does not use the quadratic transportation cost assumption.) By contrast, we use a general distribution for consumer aluation and a general quality cost function. In this general enironment, we fully characterize equilibria. Our results simultaneoulsly reeal the limitation of the uniform distribution and which properties of the uniform distribution (linear hazard and reerse hazard rates) hae been the drier of earlier results. Furthemore, when consumer aluations follow a uniform distribution, the issue of multiple equilibria is moot 4

6 for a duopoly. As we show below, multiple equilibria are important for general distributions. We derie conditions on general distributions for rst-best equilibria. Moreoer, our equilibrium qualities translate to equilibrium locations under general consumer distributions on the Hotelling line and transportation costs. Therefore, our conditions on consumer aluation distributions for rst-best qualities will be corresponding conditions on consumer location distributions on the Hotelling line for horizontal mixed oligopolies. For priate rms, Anderson et al. (1997) proide the rst characterization for a general location distribution with quadratic transportation costs. Our techniques are consistent with those in Anderson et al. (1997), but we use a general cost function. A recent paper by Benassi et al. (2006) uses a symmetric trapezoid aluation distribution and explores consumers nonpurchase options. Yurko (2011) has worked with lognormal distributions. Our monotone hazard and reerse hazard rate assumptions are alid under the trapezoid distribution, but inalid under lognormal distributions. In any case, our general characterization on the priate oligopoly complements these recent adances. Qualities in mixed proisions are often discussed in the education and health sectors. Howeer, perspecties such as political economy, taxation, and income redistribution are incorporated. Brunello and Rocco (2008) combine consumers oting and quality choices by public and priate schools, and let the public school be a Stackelberg leader. Epple and Romano (1998) consider ouchers and peer e ects but hae used a competitie model for interaction between public and priate schools. Grassi and Ma (2011, 2012) present models of publicly rationed supply and priate rm price responses under public commitment and noncommitment. Our results here indicate that commitment may not be necessary, and imperfectly competitie markets may sometimes be e cient. Priatization has been a policy topic in mixed oligopolies. Ishibashi and Kaneko (2008) set up a mixed duopoly with price and quality competition. The model has both horizontal (Hotelling) and ertical differentiation. Howeer, all consumers hae the same aluation on quality, and are uniformly distributed on the horizontal product space (as in Ma and Burgess (1993)). They show that the goernment should manipulate the objectie of the public rm so that it maximizes a weighted sum of pro t and social welfare, a form of partial priatization. (Using a Cournot model, Matsumura (1998) earlier demonstrates that partial 5

7 priatization is a aluable policy.) Our model is richer on the ertical dimension, but consists of no horizontal di erentiation. Our policy implication has a priatization component to it, but a simple social welfare objectie for the public rm is su cient. Section 2 presents the model. Section 3 studies equilibria in which the public rm s quality is lower than the priate rm s, and Section 4 studies the opposite case. In each section, we rst derie subgame-perfect equilibrium prices, and then equilibrium qualities. We present a characterization of equilibrium qualities, and conditions for equilibrium qualities to be rst best. Section 5 considers policies, and arious robustness issues. We consider alternatie preferences for the public rm. We also let cost functions of the rms be di erent. Then we let consumers hae outside options. Finally we consider multiple priate rms. Section 6 presents a benchmark priate duopoly model. The last section draws some concluding remarks. Proofs are collected in the Appendix. Details of numerical computation are in the Supplement. 2 The model 2.1 Consumers There is a set of consumers with total mass normalized at 1. Each consumer would like to receie one unit of a good or serice. In our context, it is helpful to think of such goods and serices as education, transportation, and health care including child care, medical, and nursing home serices. The public sector often participates actiely in these markets. In fact, in the literature, many papers are written for these speci c markets; see, for example, Epple and Romano (1998) and Brunello and Rocco (2008). A good has a quality, denoted by q, which is assumed to be positie. Each consumer has a aluation of quality. This aluation aries among consumers. We let be a random ariable de ned on the positie support [; ] with distribution F and strictly positie density f. We also assume that f is continuously di erentiable We will use two properties of the distribution, namely [1 F ]=f h, and F=f k. We assume that h is decreasing, and that k is increasing, so h 0 () < 0 and k 0 () > 0. The assumptions ensure that pro t functions, to be de ned below, are quasi-concae, and are implied by f being logconcae (Anderson et al. (1997)). These 6

8 monotonicity assumptions are satis ed by many common distributions such as the uniform, the exponential, the beta, etc. (Bagnoli and Bergstrom (2004)). We will call h the hazard rate, and k the reerse hazard rate, although the terminology used by economists aries. 2 Valuation ariations among consumers hae the usual interpretation of preference diersity due to wealth, taste, or cultural di erences. We may call a consumer with aluation a type- consumer, or simply consumer. If a type- consumer purchases a good with quality q at price p, his utility is q p. The quasi-linear utility function is commonly adopted in the literature (see, for example, the standard texts Tirole (1988) and Anderson et al. (1992)). We assume that each consumer will buy a unit of the good. This can be made explicit by postulating that each good o ers a su ciently high bene t which is independent of, or that the minimum aluation is su ciently high. We defer to Subsection 5.4 to discuss the possibility that a consumer does not purchase anything at all. 2.2 Public and priate rms There are two rms, Firm 1 and Firm 2, and they hae access to the same technology. Production requires a xed cost. The implicit assumption is that the xed cost is so high that entries by many rms cannot be sustained. We focus on the case of a mixed oligopoly so we do not consider the rather triial case of two public rms. Often a mixed oligopoly is motiated by a more e cient priate sector, so in Subsection 5.3 we let rms hae di erent technologies, and will explain how our results remain robust. The ariable, unit production cost of the good at quality q is c(q), where c : R +! R + is a strictly increasing and strictly conex function. A higher quality requires a higher marginal cost, and this marginal cost also increases in quality. We also assume that c is twice di erentiable, and that it satis es the usual Inada conditions: lim q!0 + c(q) = lim q!0 + c 0 (q) = 0, so in both the rst best and in the equilibria of the extensie forms to be analyzed, both rms will be actie. Firm 1 is a public rm, and run by a utilitarian regulator. Firm 1 s objectie is to maximize social 2 In statistics f=(1 F ) is called the hazard rate. Suppose that the random ariable x has distribution F and density f. Then f()=(1 F ()) is the conditional density of x = gien that x. For example, if x denotes the time of failure, the hazard rate measures the density of failure occurring at gien that failure has not occurred before. We are unable to nd a common usage for f=f in statistics. Howeer, f=f is the conditional density of x = gien that x. That is, this is the density of failure occurring at x = gien that failure has occurred by. 7

9 surplus; the discussion of a general objectie function for the public rm is deferred until Subsection 5.2. Firm 2 is a pro t-maximizing priate rm. Each rm chooses its product quality and price. We let p 1 and q 1 denote Firm 1 s price and quality; similarly, p 2 and q 2 denote Firm 2 s price and quality. Gien these prices and qualities, each consumer buys from the rm that o ers the higher utility. A consumer chooses a rm with a probability equal to a half if he is indi erent between them. Consider any (p 1 ; q 1 ) and (p 2 ; q 2 ), and de ne b by bq 1 p 1 = bq 2 p 2. Consumer b is supposed to be just indi erent between purchasing from Firm 1 and Firm 2. If b 2 [; ], then the demands for the two rms are as follows: Demand for Firm 1 Demand for Firm 2 F (b) 1 F (b) if q 1 < q 2 1 F (b) F (b) if q 1 > q 2 (1) 1=2 1=2 if q 1 = q 2 We sometimes call consumer b the indi erent or marginal consumer. (Otherwise, if b =2 [; ], or fails to exist, one rm will be unable to sell to any consumer.) If Firm 1 s product quality is lower than Firm 2 s, its demand is F (b) when its price is su ciently lower than Firm 2 s price. Conersely, if Firm 2 s price is not too high, then its demand is 1 F (b). If the two rms product qualities are identical, then they must charge the same price if both hae positie demands. In this case, all consumers are indi erent between them, and each rm receies half of the market. The demand functions exhibit discontinuity when rms o er products with identical qualities: any small price di erence will cause demand to shift completely to the rm that o ers the lower price. 2.3 Allocation, social surplus, and rst best An allocation consists of a pair of product qualities, one at each rm, and an assignment of consumers across the rms. The social surplus from an allocation is Z [xq` c(q`)]f(x)dx + Z [xq h c(q h )]f(x)dx: (2) Here, the qualities at the two rms are q` and q h, q` < q h. Those consumers with aluations between and get the good with quality q`, whereas those with aluations between and get the good with quality q h. 8

10 The rst best is (q ` ; q h ; ) that maximizes (2), and is characterized by the following: Z Z xf(x)dx F ( ) xf(x)dx 1 F ( ) = c 0 (q ` ) (3) = c 0 (q h) (4) q ` c(q ` ) = q h c(q h): (5) The characterization of the rst best in (3), (4), and (5) is standard. Those consumers with lower aluations should consume the good at a low quality (q` ), and those with higher aluations should consume at a high quality (q h ). Therefore, for the rst best, diide consumers into two groups: those with 2 [; ] and those with 2 [ ; ]. The (conditional) aerage aluation of consumers in [; ] is in the left-hand side of (3), and, in the rst best, this is equal to the marginal cost of the lower rst-best quality, the right-hand side of (3). A similar interpretation applies to (4) for those consumers with higher aluations. Finally, the diision of consumers into the two groups is achieed by identifying consumer who enjoys the same surplus from both qualities, and this yields (5). As Spence (1975) has shown, quality is like a public good, so the total social bene t is the aggregate consumer bene t, and in the rst best, the aerage aluation should be equal to the marginal cost of quality. As a result the indi erent consumer actually receies too little surplus from q` because > c 0 (q`), but too much from q h because < c 0 (q h ). 2.4 Extensie form We study subgame-perfect equilibria of the following game. Stage 0: Nature draws consumers aluations and these are known to consumers only. Stage 1: Firm 1 chooses a quality q 1 ; simultaneously, Firm 2 chooses a quality q 2. Stage 2: Qualities in Stage 1 are common knowledge. Firm 1 chooses a price p 1 ; simultaneously, Firm 2 chooses a price p 2. Stage 3: Consumers obsere price-quality o ers from both rms, and pick a rm for purchase. 9

11 An outcome of this game consists of rms prices and qualities, (p 1 ; q 1 ) and (p 2 ; q 2 ), and the allocations of consumers across the two rms. Subgames at Stage 2 are de ned by the rms quality pair (q 1 ; q 2 ). Subgame-perfect equilibrium prices in Stage 2 are those that are best responses in subgames de ned by (q 1 ; q 2 ). Finally, equilibrium qualities in Stage 1 are those that are best responses gien that prices are gien by a subgame-perfect equilibrium in Stage 2. There are multiple equilibria. In one class of equilibria, in Stage 1 the public rm chooses low quality, whereas the priate rm chooses high quality, and in Stage 2, the public rm sets a low price, and the priate rm chooses a high price. In the other class, the roles of the rms, in terms of their ranking of qualities and prices, are reersed. Howeer, because the two rms hae di erent objecties, equilibria in these two classes yield di erent allocations. 3 Equilibria with low quality at public rm In this section, we study equilibria when the public Firm 1 s quality q 1 is lower than the priate Firm 2 s quality q Subgame-perfect equilibrium prices Consider subgames in Stage 2 de ned by (q 1 ; q 2 ) with q 1 < q 2. According to (1), each rm will hae a positie demand only if p 1 < p 2, and there is e 2 [; ] with eq 1 p 1 = eq 2 p 2 or e(p 1 ; p 2 ; q 1 :q 2 ) = p 2 p 1 ; (6) where we hae emphasized that e, the consumer indi erent between buying from Firm 1 and Firm 2, depends on qualities and prices. Expression (6) characterizes rms demand functions. Firms payo s are: Z e [xq 1 c(q 1 )]f(x)dx + Z e [xq 2 c(q 2 )]f(x)dx (7) [1 F (e)][p 2 c(q 2 )]: (8) The expression in (7) is social surplus when consumers with aluations in [; e] buy from Firm 1, whereas others buy from Firm 2. The prices that consumers pay to rms are transfers, so do not a ect social surplus. The expression in (8) is Firm 2 s pro t. 10

12 Firm 1 chooses its price p 1 to maximize (7) gien the demand (6) and price p 2. Firm 2 chooses price p 2 to maximize (8) gien the demand (6) and price p 1. Equilibrium prices, (bp 1 ; bp 2 ), are best responses against each other. Lemma 1 In subgames (q 1 ; q 2 ) with q 1 < q 2, and < c(q 2) c(q 1 ) <, equilibrium prices (bp 1 ; bp 2 ) are: bp 1 c(q 1 ) = bp 2 c(q 2 ) = ( ) 1 F (b) f(b) ( )h(b); (9) where b = c(q 2) c(q 1 ) : (10) Lemma 1 says that the equilibrium price di erence across rms is the same as the cost di erence: bp 2 bp 1 = c(q 2 ) c(q 1 ). Second, it says that Firm 2 makes a pro t, and its price-cost margin is proportional to the quality di erential and the hazard rate h. We explain the result as follows. Firm 1 s payo is social surplus, so it seeks the consumer assignment to the two rms, e, to maximize social surplus (7). This is achieed by getting consumers to fully internalize the cost di erence between the high and low qualities. Therefore, gien bp 2, Firm 1 sets bp 1 so that the price di erential bp 2 bp 1 is equal to the cost di erential c(q 2 ) c(q 1 ). In equilibrium, the indi erent consumer is gien by bq 1 c(q 1 ) = bq 2 c(q 2 ), which indicates an e cient allocation in the quality subgame (q 1 ; q 2 ). Firm 2 seeks to maximize pro t. Gien Firm 1 s price bp 1, Firm 2 s optimal price follows the marginalreenue-marginal-cost calculus. For a unit increase in p 2, the marginal loss is [p 2 c(q 2 )]f(e)=( ), whereas the marginal gain is [1 F (e)]. Therefore, pro t maximization yields bp 2 c(q 2 ) = ( ) 1 F (b). (This is also the standard inerse elasticity rule for the determination of f(b) Firm 2 s price-cost margin. 3 ) Putting rms best responses together, we hae Lemma 1. The key point in Lemma 1 is that equilibrium market shares and prices can be determined separately. Once qualities are gien, Firm 1 will aim for the socially e cient allocation, and it adjusts its price, gien Firm 2 s price, to achiee that. Firm 2, on the other hand, aims to maximize pro t so its best response depends on Firm 1 s price as well as the elasticity of demand. Firm 1 does make a pro t, and we will return to this issue in Subsection Firm 2 s demand is 1 F (e). Hence, elasticity is d(1 F (e)) p 2 q2 q1 = dp 2 1 F (e) h(e) p2. 11

13 To complete the characterization of price equilibria, we consider subgames (q 1 ; q 2 ) with q 1 < q 2, and either c(q 2) c(q 1 ) < or < c(q 2) c(q 1 ). In the former case, Firm 1 would like to allocate all consumers to Firm 2, whereas in the other case, Firm 1 would like to allocate all consumers to itself. In both cases, there are multiple equilibrium prices. They take the form of high alues of bp 1 when all consumers go to Firm 2, but low alues of bp 1 in the other. In any case, equilibria in the game must hae two actie rms, so these subgames cannot arise. 4 The equilibrium prices (bp 1 ; bp 2 ) in (9) and (10) formally establish three functional relationships, those that relate any qualities to equilibrium prices and allocation of consumers across rms. We can write them as bp 1 (q 1 ; q 2 ), bp 2 (q 1 ; q 2 ), and b(q 1 ; q 2 ) e(bp 1 (q 1 ; q 2 ); bp 2 (q 1 ; q 2 ); q 1 :q 2 ). Applying the Implicit Function Theorem, we derie how equilibrium prices and market share change with qualities. As it turns out, we will only need to use the information of how bp 1 (q 1 ; q 2 ) and bp 2 (q 1 ; q 2 ) change with q 2 : Lemma 2 From the de nition of (bp 1 ; bp 2 ) and b in (9) and (10), we hae b increasing in q 1 and q 2, 1 (q 1 ; q 2 ) = h(b) + h 0 (b) [c 0 (q 2 ) b] 2 (q 1 ; q 2 ) = c 0 (q 2 ) + h(b) + h 0 (b) [c 0 (q 2 ) b] : (12) Lemma 2 describes how the equilibrium consumer changes with qualities, and the strategic e ect of Firm 2 s quality on Firm 1 s price. Consider a subgame (q 1 ; q 2 ). Figure 1 shows the determination of b. We hae drawn the utility function of the marginal consumer b, whose utilities are bq 1 p 1 = bq 2 p 2. Suppose that q 1 increases. From Figure 1, consumer b strictly prefers to buy from Firm 1, as does consumer b + for a small and positie alue of. Next, suppose that q 2 increases, consumer also b strictly prefers to buy from Firm 1. The point is that quality q 1 is too low for consumer b but quality q 2 is too high. An increase in q 1 makes Firm 1 more attractie to consumer b, but an increase in q 2 makes Firm 2 less attractie to him. 4 The discussion of equilibria of rms haing identical qualities is in Subsection

14 Figure 1: Qualities and the marginal consumer s utility If Firm 2 increases its quality, it expects to lose market share. Howeer, it does not mean that its pro t must decrease. From (8), Firm 2 s pro t is increasing in Firm 1 s price. 5 Hence if in fact Firm 1 raises its price against a higher q 2, Firm 2 may earn a higher pro t. In any case, because h is decreasing, and c 0 (q 2 ) > b, according to Lemma 2, an increase in q 2 may result in higher or lower equilibrium prices. The point is simply that Firm 2 can in uence Firm 1 s price response. Also, Firm 2 s equilibrium price always increases at a higher rate than Firm = = c 0 (q 2 ) (see (11) 1 = and (12) 2 = ). 3.2 Subgame-perfect equilibrium qualities At qualities q 1 and q 2, the continuation equilibrium payo s for Firms 1 and 2 are, respectiely, Z b(q1;q 2) Z [xq 1 c(q 1 )]f(x)dx + [xq 2 c(q 2 )]f(x)dx (13) b(q 1;q 2) [1 F (b(q 1 ; q 2 ))][bp 2 (q 1 ; q 2 ) c(q 2 )]; (14) 5 The partial deriatie of (8) with respect to p 1 is f(e)[p2 c(q2)] > 0. 13

15 where bp 2 is Firm 2 s equilibrium price and b is the indi erent consumer from Lemma 1. Let (bq 1 ; bq 2 ) be the equilibrium qualities. They are mutual best responses, gien continuation equilibrium prices: bq 1 = argmax q 1 Z b(q1;bq 2) Z [xq 1 c(q 1 )]f(x)dx + [xbq 2 c(bq 2 )]f(x)dx (15) b(q 1;bq 2) bq 2 = argmax q 2 [1 F (b(bq 1 ; q 2 ))][bp 2 (bq 1 ; q 2 ) c(q 2 )]: (16) A change in quality q 1 has two e ects on social surplus (13). First, it directly changes q 1 c(q 1 ), the surplus of consumers who purchase the good at quality q 1. Second, it changes the equilibrium prices and the marginal consumer b (hence market shares) in Stage 3. This second e ect is second order because the equilibrium prices in Stage 3 maximize social surplus. Hence, the rst-order deriatie of (13) with respect to q 1 is R b(q 1;q 2) [x c 0 (q 1 )]f(x)dx. Similarly, a change in quality q 2 has two e ects on Firm 2 s pro t. First, it directly changes the marginal consumer s surplus bq 2 c(q 2 ). Second, it changes the equilibrium prices and the marginal consumer. We rewrite (16) as [1 F (b(q 1 ; q 2 ))] [b(q 1 ; q 2 )q 2 c(q 2 ) b(q 1 ; q 2 )q 1 + bp 1 (q 1 ; q 2 )] (17) because b(q 1 ; q 2 ) = e(bp 1 (q 1 ; q 2 ); bp 2 (q 1 ; q 2 ); q 1 :q 2 ) bp 1(q 1 ; q 2 ) bp 2 (q 1 ; q 2 ) q 1 q 2 (18) which gies the channels for the in uence of q 2 on prices. Firm 2 s equilibrium price in Stage 3 maximizes pro t, so the e ect of q 2 on pro t in (17) ia b(q 1 ; q 2 ) has a second-order e ect. Therefore, the rst-order deriatie of (17) with respect to quality q 2 is b(q 1 ; q 2 ) c 0 (q 2 ) 1(q 1 ; q 2 ) factor [1 F (b(q 1 ; q 2 ))]). (where we hae omitted the We set the rst-order deriaties of social surplus with respect to q 1 and of pro t with respect to q 2 to zero. Then we apply (11) in Lemma 2 to obtain the following. Proposition 1 Equilibrium qualities (bq 1, bq 2 ), and the marginal consumer b sole the following three equa- 14

16 tions in q 1, q 2, and R xf(x)dx F () + h() 1 h 0 () = c 0 (q 1 ) = c 0 (q 2 ) q 2 c(q 2 ) = q 1 c(q 1 ): As we hae explained, gien Firm 2 s quality and the continuation equilibrium prices, Firm 1 s return to quality q 1 consists of its own consumers. Hence bq 1 equates the conditional aerage aluation of consumers R b in [; b], xf(x)dx, and the marginal cost c 0 (q 1 ). This is the rst equation. F (b) Firm 2 s quality will a ect Firm 1 s price in Stage 2. If this were not the case (imagine 1 = were 0), the pro t-maximizing quality would be the optimal leel for the marginal consumer: b = c 0 (q), reminiscent of the basic property of quality in Spence (1975). By raising quality from one satisfying b = c 0 (q), Firm 2 may also raise Firm 1 s price, hence its own pro t. This is a rst-order gain. The optimal tradeo is now gien by b 1(bq 1 ; bq 2 ) = c 0 (bq 2 ). We use (11) to simplify, and show that. Firm 2 sets its quality to be e cient for a consumer with aluation b + h(b) 1 h 0. This is the second equation. (b) According to Proposition 1, the only di erence between equilibrium qualities and those in the rst best R b stems from how Firm 2 chooses its quality. Firm 2 s consumers hae aerage aluation xf(x)dx 1 F (b), which should be set to the marginal cost of Firm 2 s quality for social e ciency. Can this aerage aluation be equal to b + h(b) 1 h 0? Our next result gies a class of aluation distributions for which the answer is a rmatie. (b) First, we present a mathematical lemma, which, through a simple application of integration by parts, allows us to write the conditional expectation of aluations in terms of hazard rate and the density. Lemma 3 For any distribution F (and its corresponding density f and hazard rate h (1 F )=f), we hae Z xf(x)dx 1 F () + Z f(x)h(x)dx f()h() : (19) Proposition 2 Suppose that the hazard rate h is linear; that is, h(x) = x, x 2 [; ], for some and 0. Then for any + h() R 1 h 0 () = xf(x)dx R + f(x)h(x)dx : (20) 1 F () f()h() 15

17 Equilibrium qualities and market shares are rst best. Proposition 2 exhibits a set of consumer-aluation distributions for which the quality-price competition game yields rst-best equilibrium qualities. We hae managed to write the conditional aerage in terms of R b the hazard rate in Lemma 3, and this is b + f(x)h(x)dx h(b). When the hazard rate is linear, f(b)h(b) 1 h 0 (b) R b f(x)h(x)dx, Firm 2 s pro t maximization incentie aligns with the social incentie. The following remark f(b)h(b) gies the economic interpretation for the linear hazard rate. Remark 1 When Firm 2 sells to high-aluation consumers, its marginal reenue is linear in consumer aluation if and only if h() is linear. As far as we know, the linearity of the hazard rate has neer been used in the theoretical literature such as auction design, regulation, and screening and pricing under asymmetric information. The Myerson irtual cost and La ont-tirole information rent adjustments almost always inole the hazard rate (see, for instance, Myerson (1997), La ont and Tirole (1993)), but no linearity assumption has been used before. By contrast, in the empirical literature (such as labor economics), the linear hazard model has been ery popular, although our assumption has no direct bearing on the estimation of, and inference from, such models. Many distributions satisfy the linear hazard rate assumption. They include the popular uniform and exponential distributions. Essentially, the condition [1 F ()]=f() = requires that all distributions conditional on the random ariable exceeding to be quite similar. For example, if the lower parts of the uniform or exponential distributions are remoed, the remaining distributions still are uniform or exponential. In any case, [1 F ()]=f() = de nes a di erential equation, and we sole for all distributions that hae linear hazard rates. Remark 2 Suppose that h(x) = x. Then if = 0, f is the exponential distribution f(x) = A exp( x ), with = 1, and A = exp( ), so when = 0, f(x) = 1 exp( x ) for x 2 R +. If > 0, then f(x) = ( x) (1 ) 1, with = 0. For the uniform distribution, we hae h(x) = x (so =, and = 1). ( ) 16

18 Next, we show that rms equilibrium qualities must be either simultaneously excessie or de cient. It cannot be an equilibrium for one rm s quality higher than the rst best while the other rm s quality lower than the rst best. Proposition 3 Let an equilibrium be written as (bq 1 ; bq 2 ; b), corresponding to Firm 1 s quality, Firm 2 s quality, and the marginal consumer. If the equilibrium is not rst best, either (bq 1 ; bq 2 ; b) < (q ` ; q h; ) or (bq 1 ; bq 2 ; b) > (q ` ; q h; ): That is, when equilibrium qualities are not rst best, either both rms hae equilibrium qualities lower than the corresponding rst-best leels, or both hae equilibrium qualities correspondingly higher. The proposition can be explained as follows. Firm 1 aims to maximize social surplus. If Firm 2 chooses q 2 = q h, Firm 1 s best response is to pick q 1 = q `. Next, Firm 1 s best response is increasing in q 2. This stems from the properties of b(q 1 ; q 2 ), the e cient allocation of consumers across the two rms. Quality q 1 is too low for consumer b, whereas quality q 2 is too high. If q 2 increases, consumer b would become worse o buying from Firm 2, so actually b increases. This also means that Firm 1 should raise its quality because it now seres consumers with higher aluations. In other words, if Firm 2 raises its quality, Firm 1 s best response is to raise quality. Therefore, Firm 1 s quality is higher than the rst best q ` 2 s quality is higher than the rst best q h. if and only if Firm We hae constructed a number of examples to erify that equilibrium qualities can be either below or aboe the rst best. Howeer, it is more e ectie if we discuss these examples after we hae presented the other class of equilibria in which the public rm chooses a higher quality than the priate rm. The examples are presented in Subsection 4.3. Also, at this point we already can draw arious policy implications from the results, but will defer those discussions until after we hae presented the other class of equilibria. See Subsections 5.1 and

19 4 Equilibria with high quality at public rm In this class of equilibria Firm 1 s quality is higher than Firm 2 s, q 1 > q 2. Because the two rms hae di erent objecties, equilibria in this class are not isomorphic to those in the preious section. Howeer, many de nitions and proof procedures that hae been used preiously can be applied analogously, so where appropriate we will omit proofs. 4.1 Subgame-perfect equilibrium prices When q 1 > q 2, the rms hae positie demand only if p 1 > p 2. The de nition for demand in (1) continues to apply. We rewrite the de nition of the indi erent consumer e: The rms payo s are respectiely: eq 1 p 1 = eq 2 p 2 or e(p 1 ; p 2 ; q 1 :q 2 ) = p 1 p 2 q 1 q 2 : (21) Z e [xq 2 c(q 2 )]f(x)dx + Z e [xq 1 c(q 1 )]f(x)dx (22) F (e)[p 2 c(q 2 )]: (23) The expressions in (22) and (23) are social surplus and Firm 2 s pro t, and are similar to those in (7) and (8). Here, consumers with low aluations buy from the low-quality-low-price priate rm, whereas consumers with high aluations buy from the high-quality-high-price public rm. Firm 1 chooses price p 1 to maximize social surplus (22) gien the demand function (21) and price p 2. Firm 2 chooses price p 2 to maximize pro t (23) gien the demand function (21) and price p 1. Equilibrium prices, (bp 1 ; bp 2 ), are best responses against each other. The following lemma is the characterization of equilibrium prices and consumer allocation. Its proof is similar to that of Lemma 1, and omitted. Lemma 4 In subgames (q 1 ; q 2 ) with q 1 > q 2, and < c(q 1) c(q 2 ) q 1 q 2 <, equilibrium prices (bp 1 ; bp 2 ) are: bp 1 c(q 1 ) = bp 2 c(q 2 ) = (q 1 q 2 ) F (b) f(b) (q 1 q 2 )k(b); (24) where b = c(q 1) c(q 2 ) q 1 q 2 : (25) 18

20 Lemma 4 presents the equilibrium prices and consumer allocations in subgames with q 1 > q 2. Their properties parallel those in Lemma 1. Firm 1 implements the socially e cient consumer allocation by setting a price di erential equal to the cost di erential: bp 1 bp 2 = c(q 1 ) c(q 2 ). Firm 2 s pro t maximization follows the usual marginal-reenue-marginal-cost tradeo. We use the reerse hazard rate, k = F=f, to obtain (24). Finally, for subgames (q 1 ; q 2 ) with q 1 > q 2, and either c(q 1) c(q 2 ) q 1 q 2 < or < c(q 1) c(q 2 ) q 1 q 2 one rm will be inactie, so these subgames are irreleant. The equilibrium prices and allocation in (24) and (25) depend on the qualities, so we write them as bp 1 (q 1 ; q 2 ), bp 2 (q 1 ; q 2 ), and b(q 1 ; q 2 ). We totally di erentiate these three functions to obtain how prices and allocation change with Firm 2 s quality. The following lemma presents these results. The proof follows the same steps as those in Lemma 2, and is omitted. Lemma 5 From the de nition of (bp 1 ; bp 2 ) in (24) and (25), we hae b increasing in q 1 and q 1 (q 1 ; q 2 ) = k(b) + k 0 (b) [b c 0 (q 2 )] 2 (q 1 ; q 2 ) = c 0 (q 2 ) k(b) + k 0 (b) [b c 0 (q 2 )] : (27) Lemma 5 shows how Firm 2 s quality will alter equilibrium prices and allocation. Unlike subgames where Firm 2 s quality is higher than Firm 1 s, Firm 2 s market share increases with both q 1 and q 2. Howeer, the e ect of a higher quality q 2 on prices may be ambiguous, but the e ect of q 2 on bp 2 is larger than that on bp 1 by c 0 (q 2 ). Finally, we consider subgames where both rms hae chosen the same qualities, q 1 = q 2. According to (1), the rms share the market equally if they charge the same price; otherwise, the rm that charges the lower price gets all consumers. Howeer, Firm 1 s objectie is social surplus, which, for q 1 = q 2, is R [q 1 c(q 1 )]d, irrespectie of prices. Any price can be a best response for Firm 1. Clearly, Firm 2 prefers its price (and Firm 1 s price) to be as high as possible. Here, we select the equilibrium in which the price is at marginal cost c(q 1 ). Our reason for this selection is to main continuity. In both (9) and (24), the price-cost margin tends to zero as q 1 and q 2 tend to each other. 19

21 4.2 Subgame-perfect equilibrium qualities Gien qualities q 1 and q 2, Firm 1 and Firm 2 hae, respectiely, the continuation equilibrium payo s Z b(q1;q 2) Z [xq 2 c(q 2 )]f(x)dx + [xq 1 c(q 1 )]f(x)dx (28) b(q 1;q 2) F (b(q 1 ; q 2 ))[bp 2 (q 1 ; q 2 ) c(q 2 )]; (29) where bp 2 is Firm 2 s equilibrium price and b is the indi erent consumer (see Lemma 4). Let (bq 1 ; bq 2 ) be the equilibrium qualities. They are mutual best responses, gien continuation equilibrium prices: bq 1 = argmax q 1 Z b(q1;bq 2) Z [xbq 2 c(bq 2 )]f(x)dx + [xq 1 c(q 1 )]f(x)dx (30) b(q 1;bq 2) bq 2 = argmax q 2 F (b(bq 1 ; q 2 ))[bp 2 (bq 1 ; q 2 ) c(q 2 )]: (31) We apply the same method to characterize equilibrium qualities. Changing q 1 in Firm 1 s payo in (30) only a ects the second integral there because the e ect ia the rst integral is second order by the Enelope Theorem. To study the e ect of changing q 2 on Firm 2 s payo, we use the de nition of b to rewrite pro t in (31) as F (b(bq 1 ; q 2 ))[b(bq 1 ; q 2 )q 2 c(q 2 ) b(bq 1 ; q 2 )bq 1 + bp 1 (bq 1 ; q 2 )]: Hence, changing q 2 has only two e ects: the direct e ect on the surplus of the marginal consumer bq c(q 2 ), and the e ect on Firm 1 s equilibrium price, because any e ect on the marginal consumer is second order according to the Enelope Theorem. We obtain the rst-order conditions Z b(q 1;q 2) [x c 0 (q 1 )]f(x)dx = 0 b(bq 1 ; q 2 ) c 0 (q 2 ) 1(bq 1 ; q 2 ) = 0: Then we apply Lemma 5, and use k(b)+k 0 (b) [b c 0 (q 2 )] to substitute 1 =. To sum up, we present the characterization in the next Proposition (proof omitted). Proposition 4 Equilibrium qualities (bq 1, bq 2 ), and the marginal consumer b sole the following three equa- 20

22 tions in q 1, q 2, and Z xf(x)dx 1 F () k() 1 + k 0 () = c 0 (q 1 ) = c 0 (q 2 ) q 2 c(q 2 ) = q 1 c(q 1 ): The intuitions behind Proposition 4 are similar to those in Proposition 1 in the preious section. Firm 1 chooses q 1 to maximize the surplus of those consumers with aluations higher than b. The marginal consumer is b but Firm 2 chooses the quality that is e cient for a lower type b k(b) 1 + k 0. Firm 2 s lower quality (b) seres to use product di erentiation to create a bigger cost di erential, and hence a bigger price di erential between the two rms. Again, the di erence between the equilibrium qualities and the rst best stems from Firm 2 s quality choice. We can identify a class of distributions for which Firm 2 s pro t incentie aligns with the social incentie. First, we present a mathematical result that relates the reerse hazard rate and conditional expectations. Lemma 6 For any distribution F (and its corresponding density f and reerse hazard rate k F=f), we hae Z xf(x)dx F () Z f(x)k(x)dx f()k() : (32) Proposition 5 Suppose that the reerse hazard rate k is linear; that is, k(x) = + x, x 2 [; ], for some and 0. Then for any k() 1 + k 0 () = Z xf(x)dx F () + Z f(x)k(x)dx f()k() : (33) Equilibrium qualities and market shares are rst best. We also present the following relationship between the linear reerse hazard rate and the priate rm s marginal reenue. 21

23 Remark 3 When Firm 2 sells to low-aluation consumers, its marginal reenue is linear in consumer aluation if and only if k() is linear. The linear reerse hazard rate in Proposition 5 may look similar to the earlier condition for the rst best in Proposition 2, but in fact, hazard rate and the reerse hazard rate can behae rather di erently. For example, the exponential distribution has a constant hazard rate (see Remark 2), but the reerse hazard rate is nonlinear. 6 As another example, a triangular distribution has a linear reerse hazard rate, but its hazard rate is nonlinear (see Example 1 below). We present all distributions that hae linear reerse rates in the following. ( + x) 1 Remark 4 Suppose that k(x) = + x. Then > 0, and f(x) = ( + ) uniform distribution, = and = 1. 1 with + = 0. For the For the uniform distribution, both hazard and reerse hazard rates are linear, but this is not the only one. The following characterizes those distributions whose hazard and reerse hazard rates are both linear. Remark 5 Finally, if h() = and k() = + for the same distribution, we hae f() = [( ) + ( ] 1 and F () = ( ) [( ) + ( ] 1. When the equilibrium is not rst best, the distortion in equilibria with higher public qualities exhibits the same pattern as in equilibria with lower public qualities: Proposition 3 holds erbatim for the class of high-public-quality equilibria. (The proof parallels that for Proposition 3, and is omitted. 7 ) Either both rms simultaneously produce qualities higher than rst best, or both simultaneously produce qualities lower. 4.3 Examples and comparison between equilibrium and rst-best qualities We now present three sets of examples to illustrate the di erent types of equilibria. All examples use the same quadratic cost function c = 1 2 q2, but di erent distributions. The Mathematica programs for the 6 Suppose that x has the exponential density 1 exp( x ) on R+, > 0, then h() =, and k() = h exp( ) 1 i. 7 The proof of Proposition 3 can just be repeated here. The only di erence is that the cross partial deriatie of Firm 1 s objectie function now becomes [b c 0 (q 1)]@b=. This is positie because now in an equilibrium b < c 0 (q 1) remains positie. 22

24 computations are in the Supplement. Example 1 illustrates Proposition 5 and considers distributions for which either the hazard rate or the reerse hazard rate is linear. Examples 2 and 3 consider distributions for which neither the hazard nor the reerse hazard rates are linear and therefore represent the equilibrium outcomes in which the qualities can be higher or lower than the rst best. Example 1 Two triangular distributions: f() = 2 and its reerse f() = 2(1 ), for 2 [0; 1]. In the rst triangular distribution, we hae : f() = 2 F () = 2 h() = k() = 2 ; so the hazard rate is not linear, but the reerse hazard rate is. Proposition 5 says that when the public rm s quality is higher than the priate rm s, equilibrium qualities are rst best, but this may not be true when the public rm s quality is lower. The following presents the rst best and the equilibria: Welfare First best q l = 0:4120 q h = 0:8241 = 0:6180 0:2423 Low public quality bq 1 = 0:3849 bq 2 = 0:7698 b = 0:5774 0:2416 High public quality bq 1 = q h bq 2 = q l b = 0:2423 When the public Firm 1 chooses a low quality, equilibrium qualities are all below the rst best, and there is a small welfare loss. In the second triangular distribution, we hae f() = 2 2 F () = 2 2 h() = 1 k() = (1 ) ; so the hazard rate is linear but the reerse hazard rate is not. Equilibrium qualities are rst best when the public rm chooses a low quality (Proposition 2). The following presents the rst best and equilibria: Welfare First best ql = 0:1760 qh = 0:5880 = 0:3820 0:0756 Low public quality bq 1 = ql bq 2 = qh b = 0:0756 High public quality bq 1 = 0:6151 bq 2 = 0:2302 b = 0:4227 0:

25 In this example, when the public Firm 1 chooses a high quality, equilibrium qualities are all aboe the rst best. Example 2 Two exponential distributions: f() = [exp ( =)]= 1 exp ( =) for > 0, and 2 [0; ]. [exp ( ( )=)]= and its reerse f() =, 1 exp ( =) In the rst exponential, we hae f() = [exp ( =)]= 1 exp ( =) F () = 1 exp ( =) 1 exp ( =) h() = [1 exp ( ( )=)] k() = [exp (=) 1] ; so neither the hazard rate nor the reerse hazard rate are linear. We hae computed the rst best and equilibria for = 20 and = 100: Welfare First best ql = 11:1172 qh = 46:9151 = 29: :857 Low public quality bq 1 = 11:4546 bq 2 = 49:0791 b = 30: :617 High public quality bq 1 = 61:4085 bq 2 = 28:9712 b = 45: :835 In both equilibria, rms qualities are higher than the rst best. Moreoer, the equilibrium with the public rm producing a lower quality has a higher equilibrium welfare. In the second exponential distribution, we hae f() = [exp ( ( )=)]= 1 exp ( =) F () = exp ( ( )=) exp ( =) 1 exp ( =) h() = [exp (( )=) 1] k() = [1 exp(=)] : Again, neither the hazard rate nor the reerse hazard rate are linear. We use the same alues of and, and compute the rst best and equilibria: Welfare First best ql = 53:0849 qh = 88:8828 = 70: :69 Low public quality bq 1 = 38:5915 bq 2 = 71:0288 b = 54: :67 High public quality bq 1 = 88:5454 bq 2 = 50:9209 b = 69: :45 In both equilibria, rms qualities are lower than the rst best. Howeer, the equilibrium in which the public rm produces a higher quality yields a higher welfare. 24