Quality and Competition between Public and Private Firms

We study a multi-stage, quality-price game between a public (cid:133)rm and a private (cid:133)rm. The market consists of a set of consumers who have di⁄erent quality valuations. A public (cid:133)rm aims to maximize social surplus, whereas the private (cid:133)rm maximizes pro(cid:133)t. In the (cid:133)rst stage, both (cid:133)rms simultaneously choose qualities. In the second stage, both (cid:133)rms simultaneously choose prices. Consumers(cid:146)quality valuations follow a general distribution. Firms(cid:146)unit production cost is an increasing and convex function of quality. There may be multiple equilibria. In some, the public (cid:133)rm chooses a low quality, and the private (cid:133)rm chooses a high quality. In others, the opposite is true. We characterize subgame-perfect equilibria, and provide conditions on consumer valuation distribution for (cid:133)rst-best equilibrium qualities. Various policy implications are drawn.


Introduction
In many markets such as health care, education, transportation, and utility, public and private …rms jointly serve consumers. Product and service 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'valuations. At a social optimum, the average consumer valuation 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 valuation to the marginal cost of quality. This gives the classic Spence (1975) result: even 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 private …rms interact, may be a mechanism for remedying this ine¢ ciency.
We use a standard model of vertical 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 have 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 private …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 private …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-valuation 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 private …rms, deviations from the …rst best go in tandem. That is, either qualities in public and private …rms are both below the corresponding …rst-best levels, or they are both above. Equilibrium qualities form a rich set, and we have constructed examples with many con…gurations.
Our analysis proceeds in the standard way. Given a subgame de…ned by a pair of qualities, we …nd the equilibrium prices. Then we solve for equilibrium qualities, letting …rms anticipate that their quality choices will result in the corresponding equilibrium prices in the next stage. In the pricing subgame, qualities are given. The public …rm's objective is to maximize social surplus, so its price best response must achieve 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. Given the private …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 private …rm's best response is the typical inverse demand elasticity rule.
When …rms choose qualities, they anticipate the equilibrium prices in the next stage. Given the private …rm's quality, the public …rm chooses its quality to maximize the surplus of consumers that it will serve. 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 private …rm, however, 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 private …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 private …rm would have chosen the quality that would be optimal for the marginal consumer, just as we have stated above (Spence (1975)). A larger quality di¤erence, however, would be preferred because that would raise the price. Because of the price manipulation, the private …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 private …rms.
In the …rst best, the socially e¢ cient qualities are determined by equating average consumer valuations and marginal cost of quality. The surprise is that in contrast to private duopoly, the private …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 private …rm happens to have the average valuation among the private …rm's customers.
The (su¢ cient) conditions for …rst-best equilibria refer to properties of consumers' quality-valuation distribution. In the class of equilibria where the public …rm produces at a low quality, equilibrium qualities are …rst best when the valuation distribution has a linear hazard rate. 1 The linear hazard rate condition is equivalent to the private …rm's marginal revenue function linear in consumer valuation. In the class of equilibria where the public …rm produces at a high quality, equilibrium qualities are …rst best when the valuation distribution has a linear reverse hazard rate. The linear reverse hazard rate condition is equivalent to the private …rm's marginal revenue function being linear in consumer valuation.
Although the hazard and reverse hazard rates have …gured prominently in the information economics literature, we are unaware of any work that imposes linearity on them. We derive all distributions that possess the linearity properties. We wish to note that the uniform distribution, which has been used often to describe consumer valuations, has linear hazard and reverse hazard rates (but it is not the only distribution with this property-see Remark 5 below). Also, because hazard and reverse hazard rates can behave 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 vice versa.
We draw various policy implications from our results. Our use of a social-welfare objective function for the public …rm can be regarded as making a normative 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 private …rms will never be the quality cost di¤erence because the private …rm never prices at marginal cost. The …rst best is never achieved (even when hazard or reverse hazard rates are linear). A social-welfare objective does mean that the public …rm tolerates high prices. However, 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 Thisse (1979, 1986) 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 f , and the reverse hazard rate is F f . By a function being linear, we mean that it has a constant slope and an intercept. This is often called a¢ ne linear in mathematics, but we trust that our abbreviation will not cause any confusion.
public …rm does not. This di¤erence has led to the mixed oligopoly literature, which revolves around the theme that the presence of a public …rm may improve welfare. Grilo (1994) studies a mixed duopoly in the quality, vertical di¤erentiation framework. In her model, consumers'valuations of qualities follow a uniform distribution. The unit cost of production may be convex or concave in quality. The paper derives …rst-best equilibria. In a Hotelling, horizontal di¤erentiation model with quadratic transportation cost,  show that a public …rm improves 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 gives some cost-reduction incentives. In a Cournot model, Cremer et al. (1989) consider replacing some private …rms by public enterprises, and nationalizing some private …rms. In these models, public …rms may discipline private …rms.
For pro…t-maximizing …rms,  show that, under very mild conditions on transportation costs, horizontal di¤erentiation models are actually a special case of vertical product di¤erentiation (see also Champsaur and Rochet (1989)). The isomorphism can be transferred to mixed duopolies. The key in the  proof is that demands in horizontal models can be translated into equivalent demands in vertical models. Firms'objectives are unimportant. Hence, results in horizontal mixed oligopolies do relate to vertical 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 valuations and a quadratic quality cost function in vertical di¤erentiation models.
The …rst-best results in Grilo (1994) are related to the e¢ cient equilibria in the two-…rm case in  because both papers use the uniform distribution for consumer valuations. (Grilo (1994), however, does not use the quadratic transportation cost assumption.) By contrast, we use a general distribution for consumer valuation and a general quality cost function. In this general environment, we fully characterize equilibria. Our results simultaneoulsly reveal the limitation of the uniform distribution and which properties of the uniform distribution (linear hazard and reverse hazard rates) have been the driver of earlier results.
Furthemore, when consumer valuations follow a uniform distribution, the issue of multiple equilibria is moot for a duopoly. As we show below, multiple equilibria are important for general distributions. We derive conditions on general distributions for …rst-best equilibria. Moreover, our equilibrium qualities translate to equilibrium locations under general consumer distributions on the Hotelling line and transportation costs.
Therefore, our conditions on consumer valuation distributions for …rst-best qualities will be corresponding conditions on consumer location distributions on the Hotelling line for horizontal mixed oligopolies.
For private …rms, Anderson et al. (1997) provide 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 valuation distribution and explores consumers'nonpurchase options. Yurko (2011) has worked with lognormal distributions. Our monotone hazard and reverse hazard rate assumptions are valid under the trapezoid distribution, but invalid under lognormal distributions. In any case, our general characterization on the private oligopoly complements these recent advances.
Qualities in mixed provisions are often discussed in the education and health sectors. However, perspectives such as political economy, taxation, and income redistribution are incorporated, so public …rms typically are assumed to have objective functions di¤erent from social welfare. Brunello and Rocco (2008) combine consumers voting and quality choices by public and private schools, and let the public school be a Stackelberg leader. Epple and Romano (1998) consider vouchers and peer e¤ects but have used a competitive model for interaction between public and private schools. Ma (2011, 2012) present models of publicly rationed supply and private …rm price responses under public commitment and noncommitment.
Our results here indicate that commitment may not be necessary, and imperfectly competitive markets may sometimes be e¢ cient.
Privatization 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 vertical differentiation. However, all consumers have the same valuation on quality, and are uniformly distributed on the horizontal product space (as in Ma and Burgess (1993)). They show that the government should manipulate the objective of the public …rm so that it maximizes a weighted sum of pro…t and social welfare, a form of partial privatization. (Using a Cournot model, Matsumura (1998) earlier demonstrates that partial privatization is a valuable policy.) Our model is richer on the vertical dimension, but consists of no horizontal di¤erentiation. Our policy implication has a privatization component to it, but a simple social welfare objective 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 private …rm's, and Section 4 studies the opposite case. In each section, we …rst derive 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 various robustness issues. We consider alternative preferences for the public …rm. We also let cost functions of the …rms be di¤erent. Then we let consumers have outside options. Finally we consider multiple private …rms. Section 6 presents a benchmark private 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

Consumers
There is a set of consumers with total mass normalized at 1. Each consumer would like to receive one unit of a good or service. In our context, it is helpful to think of such goods and services as education, transportation, and health care including child care, medical, and nursing home services. The public sector often participates actively 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 positive. Each consumer has a valuation of quality v. This valuation varies among consumers. We let v be a random variable de…ned on the positive support [v; v] with distribution F and strictly positive 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 (v) < 0 and k 0 (v) > 0. The assumptions ensure that pro…t functions, to be de…ned below, are quasi-concave, and are implied by f being logconcave (Anderson et al. (1997)). These 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 reverse hazard rate, although the terminology used by economists varies. 2 Valuation variations among consumers have the usual interpretation of preference diversity due to wealth, taste, or cultural di¤erences. We may call a consumer with valuation v a type-v consumer, or simply consumer v. If a type-v consumer purchases a good with quality q at price p, his utility is vq 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 v, or that the minimum valuation v is su¢ ciently high. The full-market coverage assumption is commonly used in the extant literature of product di¤erentiation (either horizontal or vertical), but Delbono et al. (1996) and Benassi et al. (2016) have explored the implications of consumer outside options, and we defer to Subsection 5.4 to discuss more.

Public and private …rms
There are two …rms, Firm 1 and Firm 2, and they have 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 trivial case of two public …rms. Often a mixed oligopoly is motivated by a more e¢ cient private sector, so in Subsection 5.3 we let …rms have di¤erent technologies, and will explain how our results remain robust.
The variable, unit production cost of the good at quality q is c(q), where c : R + ! R + is a strictly increasing and strictly convex 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 2 In statistics f =(1 F ) is called the hazard rate. Suppose that the random variable x has distribution F and density f . Then f (v)=(1 F (v)) is the conditional density of x = v given that x v. For example, if x denotes the time of failure, the hazard rate measures the density of failure occurring at v given that failure has not occurred before v. We are unable to …nd a common usage for f =F in statistics. However, f =F is the conditional density of x = v given that x v. That is, this is the density of failure occurring at x = v given that failure has occurred by v.
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 extensive forms to be analyzed, both …rms will be active.
Firm 1 is a public …rm, and run by a utilitarian regulator. Firm 1's objective is to maximize social surplus; the discussion of a general objective function for the public …rm is deferred until Subsection 5.2.
Firm 2 is a pro…t-maximizing private …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. Given 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 We sometimes call consumer b v the indi¤erent or marginal consumer.
, 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 v) when its price is su¢ ciently lower than Firm 2's price. Conversely, if Firm 2's price is not too high, then its demand is 1 F (b v). If the two …rms'product qualities are identical, then they must charge the same price if both have positive demands.
In this case, all consumers are indi¤erent between them, and each …rm receives 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.

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 Here, the qualities at the two …rms are q`and q h , q`< q h . Those consumers with valuations between v and v get the good with quality q`, whereas those with valuations between v and v get the good with quality q h .
The …rst best is (q `; q h ; v ) that maximizes (2), and is characterized by the following: The characterization of the …rst best in (3), (4), and (5) is standard. Those consumers with lower valuations should consume the good at a low quality (q `) , and those with higher valuations should consume at a high quality (q h ). Therefore, for the …rst best, divide consumers into two groups: (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 valuations. Finally, the division of consumers into the two groups is achieved by identifying consumer v 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 average valuation should be equal to the marginal cost of quality.
As a result the indi¤erent consumer v actually receives too little surplus from q`because v > c 0 (q`), but too much from q h because v < c 0 (q h ).

Extensive form
We study subgame-perfect equilibria of the following game.
Stage 0: Nature draws consumers'valuations v and these are known to consumers only.
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 observe price-quality o¤ers from both …rms, and pick a …rm for purchase.
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 given that prices are given 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 private …rm chooses high quality, and in Stage 2, the public …rm sets a low price, and the private …rm chooses a high price. In the other class, the roles of the …rms, in terms of their ranking of qualities and prices, are reversed. However, because the two …rms have di¤erent objectives, equilibria in these two classes yield di¤erent allocations.

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 private Firm 2's quality q 2 .

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 have a positive demand only if p 1 < p 2 , and there is where we have emphasized that e v, 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: The expression in (7) is social surplus when consumers with valuations in [v; e v] 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.
Firm 1 chooses its price p 1 to maximize (7) given the demand (6) and price p 2 . Firm 2 chooses price p 2 to maximize (8) given the demand (6) and price p 1 . Equilibrium prices, (b p 1 ; b p 2 ), are best responses against each other.
Lemma 1 In subgames (q 1 ; q 2 ) with q 1 < q 2 , and v < c(q 2 ) c(q 1 ) Lemma 1 says that the equilibrium price di¤erence across …rms is the same as the cost di¤erence: b . 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 v, to maximize social surplus (7). This is achieved by getting consumers to fully internalize the cost di¤erence between the high and low qualities. Therefore, given b p 2 , Firm 1 sets b p 1 so that the price di¤erential b p 2 b p 1 is equal to the cost di¤erential c(q 2 ) c(q 1 ). In equilibrium, the indi¤erent consumer is , which indicates an e¢ cient allocation in the quality subgame (q 1 ; q 2 ).
Firm 2 seeks to maximize pro…t. Given Firm 1's price b p 1 , Firm 2's optimal price follows the marginal- . (This is also the standard inverse elasticity rule for the determination of Firm 2's price-cost margin. 3 ) Putting …rms'best responses together, we have Lemma 1.
The key point in Lemma 1 is that equilibrium market shares and prices can be determined separately.
Once qualities are given, Firm 1 will aim for the socially e¢ cient allocation, and it adjusts its price, given Firm 2's price, to achieve 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 5.2.
To complete the characterization of price equilibria, we consider subgames (q 1 ; q 2 ) with q 1 < q 2 , and 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 values of b p 1 when all consumers go to Firm 2, but low values of b p 1 in the other. In any case, equilibria in the game must have two active …rms, so these subgames cannot arise. 4 The equilibrium prices (b p 1 ; b p 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 . Applying the Implicit Function Theorem, we derive how equilibrium prices and market share change with qualities. As it turns out, we will only need to use the information of how b p 1 (q 1 ; q 2 ) and b p 2 (q 1 ; q 2 ) change with q 2 : Lemma 2 From the de…nition of (b p 1 ; b p 2 ) and b v in (9) and (10), we have b v increasing in q 1 and q 2 , and 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 v. We have drawn the utility function of the marginal consumer b v, whose utilities are b vq 1 p 1 = b vq 2 p 2 . Suppose that q 1 increases. From Figure 1, consumer b v strictly prefers to buy from Firm 1, as does consumer b v + for a small and positive value of . Next, suppose that q 2 increases, consumer also b v strictly prefers to buy from Firm 1. The point is that quality q 1 is too low for consumer b v but quality q 2 is too high. An increase in q 1 makes Firm 1 more attractive to consumer b v, but an increase in q 2 makes Firm 2 less attractive to him. If Firm 2 increases its quality, it expects to lose market share. However, 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 v, 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 1's: @ b p 2 =@q 2 @ b p 1 =@q 2 = c 0 (q 2 ) (see (11) for @ b p 1 =@q 2 and (12) for

Subgame-perfect equilibrium qualities
At qualities q 1 and q 2 , the continuation equilibrium payo¤s for Firms 1 and 2 are, respectively, where b p 2 is Firm 2's equilibrium price and b v is the indi¤erent consumer from Lemma 1. Let (b q 1 ; b q 2 ) be the equilibrium qualities. They are mutual best responses, given continuation equilibrium prices: A change in quality q 1 has two e¤ects on social surplus (13). First, it directly changes vq 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 v (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 derivative of (13) with respect 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 b vq 2 c(q 2 ). Second, it changes the equilibrium prices and the marginal consumer. We rewrite (16) as which gives 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) via b v(q 1 ; q 2 ) has a second-order e¤ect. Therefore, the …rst-order derivative of (17) with respect to quality We set the …rst-order derivatives 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 (b q 1 , b q 2 ), and the marginal consumer b v solve the following three equa- vq 2 c(q 2 ) = vq 1 c(q 1 ): As we have explained, given Firm 2's quality and the continuation equilibrium prices, Firm 1's return to quality q 1 consists of its own consumers. Hence b q 1 equates the conditional average valuation of consumers , and the marginal cost c 0 (q 1 ). This is the …rst equation.
Firm 2's quality will a¤ect Firm 1's price in Stage 2. If this were not the case (imagine that @ b p 1 =@q 2 were 0), the pro…t-maximizing quality would be the optimal level for the marginal consumer: b v = c 0 (q), reminiscent of the basic property of quality in Spence (1975). By raising quality from one satisfying b v = 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 . We use (11) to simplify, and show that. Firm 2 sets its quality to be e¢ cient for a consumer with valuation . This is the second equation.
According to Proposition 1, the only di¤erence between equilibrium qualities and those in the …rst best stems from how Firm 2 chooses its quality. Firm 2's consumers have average valuation should be set to the marginal cost of Firm 2's quality for social e¢ ciency. Can this average valuation be equal ? Our next result gives a class of valuation distributions for which the answer is a¢ rmative.
First, we present a mathematical lemma, which, through a simple application of integration by parts, allows us to write the conditional expectation of valuations in terms of hazard rate and the density.
Lemma 3 For any distribution F (and its corresponding density f and hazard rate h Proposition 2 Suppose that the hazard rate h is linear; that is, , for some and Equilibrium qualities and market shares are …rst best.
Proposition 2 exhibits a set of consumer-valuation distributions for which the quality-price competition game yields …rst-best equilibrium qualities. We have managed to write the conditional average in terms of the hazard rate in Lemma 3, and this is . When the hazard rate is linear, , Firm 2's pro…t maximization incentive aligns with the social incentive. The following remark gives the economic interpretation for the linear hazard rate.
Remark 1 When Firm 2 sells to high-valuation consumers, its marginal revenue is linear in consumer valuation if and only if h(v) is linear.
As far as we know, the linearity of the hazard rate has never been used in the theoretical literature such as auction design, regulation, and screening and pricing under asymmetric information. The Myerson virtual cost and La¤ont-Tirole information rent adjustments almost always involve 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 very 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.
For the uniform distribution, we have h(x) = v x (so = v, and = 1).
Next, we show that …rms'equilibrium qualities must be either simultaneously excessive 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 (b q 1 ; b q 2 ; b v), corresponding to Firm 1's quality, Firm 2's quality, and the marginal consumer. If the equilibrium is not …rst best, either That is, when equilibrium qualities are not …rst best, either both …rms have equilibrium qualities lower than the corresponding …rst-best levels, or both have 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 v(q 1 ; q 2 ), the e¢ cient allocation of consumers across the two …rms. Quality q 1 is too low for consumer b v, whereas quality q 2 is too high. If q 2 increases, consumer b v would become worse o¤ buying from Firm 2, so actually b v increases. This also means that Firm 1 should raise its quality because it now serves consumers with higher valuations. 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 `i f and only if Firm 2's quality is higher than the …rst best q h .

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 have di¤erent objectives, equilibria in this class are not isomorphic to those in the previous section. However, many de…nitions and proof procedures that have been used previously can be applied analogously, so where appropriate we will omit proofs.

Subgame-perfect equilibrium prices
When q 1 > q 2 , the …rms have positive 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 v: e vq 1 p 1 = e vq 2 p 2 or e v(p 1 ; p 2 ; q 1 :q 2 ) = p 1 p 2 q 1 q 2 : The …rms'payo¤s are respectively: 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 valuations buy from the low-quality-low-price private …rm, whereas consumers with high valuations buy from the high-quality-high-price public …rm.
Firm 1 chooses price p 1 to maximize social surplus (22) given the demand function (21) and price p 2 . Firm 2 chooses price p 2 to maximize pro…t (23) given the demand function (21) and price p 1 . Equilibrium prices, , 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 v < 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: b p 1 b p 2 = c(q 1 ) c(q 2 ). Firm 2's pro…t maximization follows the usual marginal-revenue-marginal-cost tradeo¤. We use the reverse 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 < v or v < c(q 1 ) c(q 2 ) q 1 q 2 one …rm will be inactive, so these subgames are irrelevant.
The equilibrium prices and allocation in (24) and (25) depend on the qualities, so we write them as b p 1 (q 1 ; q 2 ), b p 2 (q 1 ; q 2 ), and b v(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 (b p 1 ; b p 2 ) in (24) and (25), we have b v increasing in q 1 and q 2 , 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 . However, the e¤ect of a higher quality q 2 on prices may be ambiguous, but the e¤ect of q 2 on b p 2 is larger than that on b p 1 by c 0 (q 2 ).
Finally, we consider subgames where both …rms have 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. However, Firm 1's objective is social surplus, which, for q 1 = q 2 , is R v v [vq 1 c(q 1 )]dv, irrespective 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.

Subgame-perfect equilibrium qualities
Given qualities q 1 and q 2 , Firm 1 and Firm 2 have, respectively, the continuation equilibrium payo¤s where b p 2 is Firm 2's equilibrium price and b v is the indi¤erent consumer (see Lemma 4). Let (b q 1 ; b q 2 ) be the equilibrium qualities. They are mutual best responses, given continuation equilibrium prices: 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 via the …rst integral is second order by the Envelope Theorem. To study the e¤ect of changing q 2 on Firm 2's payo¤, we use the de…nition of b v to rewrite pro…t in (31) as Hence, changing q 2 has only two e¤ects: the direct e¤ect on the surplus of the marginal consumer b vq 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 Envelope Theorem. We obtain the …rst-order conditions Then we apply Lemma 5, and use k(b v) + k 0 (b v) [b v c 0 (q 2 )] to substitute for @ b p 1 =@q 2 . To sum up, we present the characterization in the next Proposition (proof omitted).
Proposition 4 Equilibrium qualities (b q 1 , b q 2 ), and the marginal consumer b v solve the following three equa- vq 2 c(q 2 ) = vq 1 c(q 1 ): The intuitions behind Proposition 4 are similar to those in Proposition 1 in the previous section. Firm 1 chooses q 1 to maximize the surplus of those consumers with valuations higher than b v. The marginal consumer is b v but Firm 2 chooses the quality that is e¢ cient for a lower type b . Firm 2's lower quality serves 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 incentive aligns with the social incentive. First, we present a mathematical result that relates the reverse hazard rate and conditional expectations.
Lemma 6 For any distribution F (and its corresponding density f and reverse hazard rate k F=f ), we Proposition 5 Suppose that the reverse hazard rate k is linear; that is, k( Equilibrium qualities and market shares are …rst best.
We also present the following relationship between the linear reverse hazard rate and the private …rm's marginal revenue.

Remark 3 When Firm 2 sells to low-valuation consumers, its marginal revenue is linear in consumer valuation if and only if k(v) is linear.
The linear reverse 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 reverse hazard rate can behave rather di¤erently. For example, the exponential distribution has a constant hazard rate (see Remark 2), but the reverse hazard rate is nonlinear. 6 As another example, a "triangular" distribution has a linear reverse hazard rate, but its hazard rate is nonlinear (see Example 1 below). We present all distributions that have linear reverse rates in the following. For the uniform distribution, both hazard and reverse hazard rates are linear, but this is not the only one. The following characterizes those distributions whose hazard and reverse hazard rates are both linear.

Remark 5 Finally
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 verbatim 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.

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 q 2 , 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(v) = , and k(v) = h exp( v ) 1 i . 7 The proof of Proposition 3 can just be repeated here. The only di¤erence is that the cross partial derivative of Firm 1's objective function now becomes [b v c 0 (q1)]@b v=@q2. This is positive because now in an equilibrium b v < c 0 (q1) whereas @b v=@q2 remains positive. computations are in the Supplement.
Example 1 illustrates Proposition 5 and considers distributions for which either the hazard rate or the reverse hazard rate is linear. Examples 2 and 3 consider distributions for which neither the hazard nor the reverse hazard rates are linear and therefore represent the equilibrium outcomes in which the qualities can be higher or lower than the …rst best.
In the …rst triangular distribution, we have : so the hazard rate is not linear, but the reverse hazard rate is. Proposition 5 says that when the public …rm's quality is higher than the private …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: 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 have so the hazard rate is linear but the reverse 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:
Example 2 Two exponential distributions: In the …rst exponential, we have ( In both equilibria, …rms'qualities are higher than the …rst best. Moreover, the equilibrium with the public …rm producing a lower quality has a higher equilibrium welfare.
In the second exponential distribution, we have Again, neither the hazard rate nor the reverse hazard rate are linear. We use the same values of and v, In both equilibria, …rms'qualities are lower than the …rst best. However, the equilibrium in which the public …rm produces a higher quality yields a higher welfare.
The Beta distribution with parameters and (as in the above expression) constitutes a big class. For some values of and , its hazard or reverse hazard rates are linear (for example a Beta distribution with = = 1 is the uniform distribution). We have computed the equilibria two di¤erent sets of parameters: = 5, = 2, and = 2, = 5: The densities are illustrated in the following diagram. Equilibrium qualities are not …rst best, and now the deviations from the …rst best are di¤erent from the examples above. If the public …rm produces a lower quality than the private …rm, both …rms produce equilibrium qualities below the …rst best. If the public …rm produces a higher quality, then both …rms produce equilibrium qualities above the …rst best. The equilibrium welfare is higher when the public …rm produces high quality.
For = 2 and = 5, we have Also in this example, the equilibrium qualities are not …rst best. If the public …rm produces a lower quality than the private …rm, both …rms produce equilibrium qualities below the …rst best. If the public …rm produces a higher quality, then both …rms produce equilibrium qualities above the …rst best. However, now the equilibrium welfare when the public …rm produces low quality is higher than the equilibrium welfare when the public …rm produces high quality.

Competition and regulatory policies
The analysis in the previous two sections points to various policy implications. The regulation literature has commonly adopted a mechanism-design approach. In our setting, this would take the form of a regulator …rst committing to the quality and price of the product of a public …rm, and then the private …rm reacts. Instead, we use a conventional simultaneous-move, quality-price competition model. In fact, the commitment-Stackelberg model, as we will argue, adds few conceptual advantages.
First, Propositions 2 and 5 present conditions for the …rst best (linear hazard and reverse hazard rates).
These propositions have a direct implication for competition policy. Suppose that the market initially consists of private duopolists (as in Section 6 ). If a regulator would like to improve quality e¢ ciency, taking over a private …rm and adopting an objective of social-surplus maximization may be all it takes. Propositions 2 and 5 also indicate whether a public …rm should take over a …rm producing a low quality or a high quality.
Second, equilibrium qualities are …rst best in the simultaneous-move games if and only if they are …rst best in the Stackelberg game (when the public …rm can commit to quality or price). The reason is this. Suppose that Stackelberg equilibrium qualities are …rst best. If public …rm chooses the (…rst-best) low quality, the private …rm must choose the (…rst-best) high quality as a best response. Because the public …rm's payo¤ is social surplus, the (…rst-best) low quality is a best response against the (…rst-best) high quality, so the …rst best is an equilibrium in the simultaneous-move game. Improvement due to commitment is inadequate for the …rst best.
Proposition 3 implies that the improvement in welfare from a predetermined public quality must come from the public …rm choosing a quality closer to the …rst best. For example, if in an equilibrium, qualities are lower than the …rst best (as in the reverse truncated exponential distribution case in Example 2), a higher public quality leads to a higher best response by the private …rm, so both qualities will become closer to the …rst best.

General objective for the public …rm and subsidies
So far our focus has been on quality e¢ ciency. The public …rm's objective function has been social welfare, so prices are transfers between consumers and …rms, so they do not a¤ect social welfare. A more general objective function for a public …rm can be a weighted sum of consumer surplus, and pro…ts, also a common assumption in the literature. In this case, we can rewrite Firm 1's objective function as Here, consumers are paying for the lower quality q 1 at price p 1 , and the higher quality q 2 at price p 2 . The weight on consumer surplus is > 1 2 , whereas the weight on pro…ts is 1 , so pro…ts are unattractive from a social perspective. We can rewrite (34) as which always decreases in Firm 1's price. If we impose a balanced-budget constraint, then the public …rm must set price p 1 at marginal cost c(q 1 ) to break even.
In this speci…cation, Lemmas 1 and 4 would not apply. Price di¤erentials no longer equal cost di¤erentials.
In fact, in any price equilibrium, we have p 2 p 1 > c(q 2 ) c(q 1 ) due to Firm 2's pro…t-maximizing price-cost margin: p 2 > c(q 2 ). The incremental price for purchasing the good at a higher quality exceeds the true cost di¤erence, so fewer consumers will use the private …rm. The …rst best cannot be an equilibrium because consumers will never bear the full incremental cost between high and low qualities.
The concern for distribution naturally suggests a subsidy policy. Consider an equilibrium in which Firm 1 chooses a low quality and Firm 2 chooses a higher quality. Each …rm's price is given by Lemma 1, so each …rm earns a pro…t. Firm 1's pro…t can be set aside for distribution. Firm 2's pro…t can be taxed as a lump sum. The total collection now can be given as a subsidy to consumers who purchase from either the public or the private …rms. This subsidy policy is often implemented as a voucher or tax credit. In cases where conditions in Propositions 2 or 5 are satis…ed, this would allow the …rst best to become an equilibrium.
From a normative perspective, a government instructing an administrator of a public …rm to adopt a goal of social-surplus maximization may allow the implementation of e¢ cient qualities.

Di¤erent cost functions for public and private …rms
We now let …rms have di¤erent cost functions. Let c 1 (q) and c 2 (q) be Firm 1's and Firm 2's unit cost at product quality q, and these functions are increasing and convex. Often the public …rm is assumed to be less e¢ cient, so we can assume c 1 (q) > c 2 (q) and c 0 1 (q) > c 0 2 (q), so both unit and marginal unit costs are higher at the public …rm. Our formal model, however, does not require this particular comparative advantage.
The analysis in Sections 3 and 4 remains exactly the same. Simply replace every c(q 1 ) by c 1 (q 1 ) and every c(q 2 ) by c 2 (q 2 ). In the price subgame, the equilibrium still has price di¤erence equal to cost di¤erence: . The equilibrium qualities continue to satisfy their respective conditions after …rst-order conditions are simpli…ed.
Propositions 2 and 5 have to be adjusted. This is because the …rst best in Subsection 2.3 has to be rede…ned. There are now two ways to assign technology. In one, low quality for low-valuation consumers incurs the cost c 1 (q), and high quality incurs the cost c 2 (q) . In the other, it is the opposite. One of these technology assignments will yield a higher social welfare. However, our abstract model does not allow us to determine which technology should be used for low quality. 8 Suppose that the …rst best has the low quality produced by the public …rm. Equilibria in Section 4 can never achieve the …rst best because the low quality is produced by the private …rm. Hence, the last statement in Proposition 5 has to be dropped. The same reasoning applies to equilibria in Section 3 and Proposition 2 when the low quality is produced by the private …rm in the …rst best. These quali…cations do not seem to pose any conceptual problem. Misallocation is due to a kind of miscoordination on equilibria.
Our policy implication in Subsection 5.1 is actually strengthened. If the government takes over a private …rm, its decision should be guided by both strategic and technological considerations. It may decide to take over a …rm with cost c 1 and produces a low quality because that is what is called for by the …rst best, and because of the potential for quality e¢ ciency in the mixed oligopoly.

Consumer outside option and many private …rms
Formally, the case of the consumer having an outside option is modeled by a …ctitious …rm o¤ering a product at zero quality and zero price. The …rst best may assign null consumption to some consumers whose valuations of quality are below a threshold. A price set by the public …rm may a¤ect two margins: whether a consumer should choose between the low-quality good and the high-quality good, as well as whether a consumer should choose between the low-quality good and non-consumption.
In fact, Delbono et al. (1996) show that under a uniform valuation distribution, the …rst best is not an equilibrium. E¢ cient allocation requires that all consumers face price di¤erentials that correspond to cost di¤erentials. Hence, if Firm 1 produces a low quality q 1 and Firm 2 produces a high quality q 2 , then e¢ ciency requires p 2 p 1 = c(q 2 ) c(q 1 ). When p 2 > c(q 2 ) due to Firm 2's market power, p 1 > c (q 1 ). However, to induce consumers to make e¢ cient nonconsumption decisions, p 1 should be set at c(q 1 ).
Furthermore, Benassi et al. (2016) show that even the existence of a price equilibrium (under the as-8 As an illustration, let c1(q) = (1 + s)c(q), and c2(q) = (1 s)c(q). The social welfare from using c1 to produce the low quality is At s = 0, this is the model in Subsection 2.1. From the Envelope Theorem, the derivative of the maximized welfare with respect to s evaluated at s = 0 is the partial derivative of welfare with respect to s: c(q `) F (v ) + c(q h ) [1 F (v )]. Properties of q `, q h , and v from (3), (4), and (5) do not indicate whether this derivative is positive or negative. sumption of identical costs among …rms and …xed qualities) would require more structure on the valuation distribution (see their Assumption 1 on p88, and Proposition 1 on p89). The usual logconcave distribution assumption (say, in Caplin and Nalebu¤ (1991)) is inconsistent with the su¢ cient conditions proposed by Benassi et al. The case of many private …rms is formally very similar. When a public …rm has to interact with, say, two private …rms, it does not have enough instruments to induce e¢ cient decisions. Suppose that there are three …rms, and they produce low, medium, and high qualities. Suppose that the medium quality is produced by a public …rm, whereas the other qualities are produced by private …rms. Private …rms exploit their market power, but the public …rm cannot simultaneously use one price to induce two e¢ cient margins, so consumers can choose between medium and low qualities e¢ ciently, and at the same time choose between high and medium qualities e¢ ciently. The lack of tractable analysis seems pervasive in the literature of horizontal and vertical di¤erentiation with multiple …rms.

Private Duopoly
We now analyze a duopoly model with two private …rms under the same extensive form in Subsection 2.4.
Firm 1 now maximizes pro…t, so this is a standard model in which product di¤erentiation is used to relax price competition.

Subgame-perfect equilibrium prices
Consider a subgame (q 1 ; q 2 ) in Stage 2. Without loss of generality, let q 1 < q 2 . Firm 1's pro…t is now where the demand e v is given by (6). Given Firm 2's price p 2 , Firm 1 chooses p 1 to maximize its pro…t, and the …rst-order condition is We simplify this …rst-order condition, and combine the …rst-order condition of Firm 2's pro…t maximization (which is derived in the proof of Lemma 1) to obtain the following lemma (whose proof is omitted). (We use the same notation as in the previous sections when Firm 1 is the public …rm, but this should not create any confusion.) Lemma 7 In subgames (q 1 ; q 2 ) with q 1 < q 2 , equilibrium prices (b p 1 ; b p 2 ) are given by the following: Lemma 7 presents the usual price markups. The key observation is that the …rst-best allocation of consumers across the two …rms is generally not an equilibrium. We substract (35) from (36) to obtain which says that the price di¤erence between the two …rms is di¤erent from their cost di¤erence. Compared with either Lemma 1 or Lemma 4, for a given pair of qualities, duopoly prices may be higher or lower than prices when Firm 1 aims to maximize social surplus.
We write equilibrium prices in Stage 2 as b p(q 1 ; q 2 ) and b p(q 1 ; q 2 ). The equilibrium marginal consumer b v(q 1 ; q 2 ) is implicitly de…ned by (37). Pro…ts of Firm 1 and Firm 2 are, respectively, F (b v(q 1 ; q 2 ))[b p 1 (q 1 ; q 2 ) c(q 1 )] and [1 F (b v(q 1 ; q 2 ))][b p 2 (q 1 ; q 2 ) c(q 2 )]. In a subgame-perfect equilibrium, each …rm chooses its quality in Stage 1 to maximize its pro…t, given the rival …rm's quality and the continuation equilibrium prices b p 1 (q 1 ; q 2 ) and b p 2 (q 1 ; q 2 ). The following properties of equilibrium prices will be used for the derivation of the equilibrium qualities.
Lemma 8 From the de…nitions of (b p 1 ; b p 2 ) in (35) and (36), and the marginal consumer b v(q 1 ; q 2 ) implicitly de…ned by (37) we have b v increasing in both q 1 and q 2 , and > 0: Furthermore, Firm 1's equilibrium price increases with Firm 2's quality, but Firm 2's equilibrium price decreases with Firm 1's quality: @b v @q 1 < 0: Lemma 8 reports classical tendency of more intense price competition when products are more similar.
If Firm 1 raises its quality, then the lower quality q 1 gets closer to the higher quality q 2 . As a consequence, Firm 2 will reduce its price in Stage 2. Likewise, if Firm 2 raises its quality, then the higher quality q 2 gets farther away from the lower quality q 1 , so Firm 1 now raises its price. Our characterization in Lemma 8, however, uses no speci…c assumptions such as the uniform distribution on quality valuations and quadratic cost functions. Lemma 8 also contrasts with Lemmas 2 and 5. When Firm 1 aims to maximize social surplus, its price responds to quality di¤erences solely to ensure e¢ cient allocation of consumers.

Subgame-perfect equilibrium qualities
We characterize equilibrium qualities. Pro…ts of Firms 1 and 2 are, respectively, As in the earlier subsections on equilibrium qualities when Firm 1 is a public …rm, we substitute b v for b p 1 , and rewrite Firm 1's pro…t function as Changing q 1 changes the marginal consumer b v, Firm 2's price b p 2 , and the surplus for the indi¤erent consumer b vq 1 c(q 1 ). Now the Envelope Theorem applies, and the e¤ect of q 1 on pro…t through b v is second order.
The …rst-order derivative of Firm 1's pro…t with respect to q 1 is where we have omitted the factor F (b v(q 1 ; b q 2 )) (and the partial derivative with respect to b v).
Similarly, for Firm 2, we substitute b p 2 by b v, and rewrite its pro…t as The e¤ect of q 2 on pro…t through its e¤ect on b v is zero by the Envelope Theorem. The derivative of Firm 2's pro…t with respect to q 2 is where again we have omitted the factor 1 F . We now state our main result for the duopoly model.

Proposition 6
The equilibrium qualities and market share solve the three equations in q 1 , q 2 , and v: Proposition 6 gives a full characterization of equilibrium qualities. It con…rms the product di¤erentiation result: Firm 1 chooses a quality lower than one that is optimal for the indi¤erent consumer, but Firm 2 does the opposite. From the …rst two equations in Proposition 6, we have c 0 (b q 1 ) < b v < c 0 (q 2 ). Lemma 7 already says that for any given …rm qualities, the allocation of consumers across the two …rms is not …rst best. Proposition 6 now says that the equilibrium qualities have very little to do with the …rst best. In fact, for all the examples we have presented above, equilibrium qualities are not …rst best.
Excessive product di¤erentiation can be illustrated by the typical uniform-quadratic example. Let f (v) = 1=10 and F (v) = 1=10(v 10), for v 2 [10; 20], and c(q) = 1 2 q 2 . The …rst best has q l = 12 1 2 , q h = 17 1 2 , and v = 15: The equilibrium qualities and the market shares are given by the solution of the three equations in Proposition 6. The equilibrium qualities are b q 1 = 7 1 2 and b q 2 = 22 1 2 , and the equilibrium indi¤erent consumer

Concluding remarks
In this paper we have studied equilibria in a mixed market in a conventional, two-stage, quality-then-price game. The public …rm maximizes social surplus, and the private …rm maximizes pro…t. We have used a general distribution for consumer's valuations and general cost functions for …rms. We discuss two classes of equilibria. In one class, the public …rm o¤ers low quality and the private o¤ers high quality. In the other class, the opposite is true. It turns out that equilibrium qualities can be …rst best when hazard and the reverse hazard rates are linear. We have related our results to competition policies, and discussed various robustness issues.
Various directions for further research may be of interest. Clearly, duopoly is a limitation. However, a mixed oligopoly with an arbitrary number of …rms is analytically very di¢ cult. In the extant literature, models of product di¤erentiation with many private …rms typically need to impose very strong assumptions on either consumer valuation (equivalently location) distribution or production cost (equivalently mismatch disutility). The contribution here relies on our ability to identify the hazard and reverse hazard rates as the determining factors for properties of equilibrium qualities. It may well be that they also turn out to be useful for a richer model. The unit cost being constant with respect to quantity is a common assumption in the literature. We have used the same "constant-returns"approach. Scale e¤ects may turn out to be important even for the mixed duopoly.
We have taken for granted the existence of equilibria. However, …rms'payo¤ functions are discontinuous at the point when they o¤er the same quality. Proofs of existence in similar duopoly models as in Anderson et al. (1997) or Benassi et al. (2016) have required more assumptions than monotone hazard or reverse hazard rates. Although …nding su¢ cient conditions for existence in the mixed duopoly model is beyond the scope of this paper, further research may turn out to be fruitful.
which are the expressions in the lemma.
Proof of Proposition 1: The …rst-order derivative of (13) with respect to q 1 is @b v @q 1 : By Lemma 1, the term inside the curly brackets is zero. By putting this …rst-order derivative to zero, we obtain the …rst equation in the Proposition. Also, because equilibrium prices b p 1 (q 1 ; q 2 ) and b p 2 (q 1 ; q 2 ) must follow Lemma 1, we have which is the last equation in the Proposition.
Next, we use (17) to obtain the …rst-order derivative of Firm 2's pro…t with respect to q 2 : Again, by Lemma 1, the term inside the curly bracket is zero. After setting the …rst-order derivative to 0, we obtain b v(q 1 ; q 2 ) c 0 (q 2 ) + @ b p 1 (q 1 ; q 2 ) @q 2 = 0: We then use (11) in Lemma 2 to substitute for @ b p 1 (q 1 ; q 2 ) @q 2 , and write the …rst-order condition as the second equation in the Proposition.
Proof of Lemma 3: By de…nition, f (x)h(x) = (1 F (x)). We have where the second equality is due to integration by parts.
Proof of Proposition 2: where the expression in the curly brackets comes from the identity (19). Simplifying, we have v + We have proved (20).
The three equations in Proposition 1 are now exactly those that de…ne the …rst best in (3), (4), and (5).
Equilibrium qualities and consumer allocation must be …rst best.
Proof of Remark 1: When Firm 2 sells to consumers with valuations above v at price p 2 , its revenue where v = p 2 p 1 q 2 q 1 . If we express p 2 as a function of v, we have p 2 (v) = p 1 + v(q 2 q 1 ). The marginal revenue is the derivative of revenue with respect to the …rm's quantity, [1 F (v)]: dp 2 (v) dv = p 2 (v) h(v)(q 2 q 1 ): Because p 2 (v) is linear in v, marginal revenue is linear in v if and only if the hazard rate h(v) is linear.
Proof of Remark 2: De…ne y 1 F , so y 0 = f . We have h(x) = x equivalent to y 0 y = 1 x . First, suppose that = 0. We have y 0 y = 1 , so y(v) = A exp( v ), some A. Therefore, Because we have F (v) = 0, we must have A = exp( v ). We also have F (v) = 1, which requires v = 1.
Second, suppose that > 0. We have 1 . Because F (v) = 1, we must have v = 0, so that and cannot be arbitrary.
Proof of Proposition 3: For any q 2 we consider Firm 1's best response function: [xq 2 c(q 2 )]f (x)dx: First, at q 2 = q h , we have e q 1 (q h ) = q `. Clearly, if Firm 2 chooses q h , from the de…nition of the …rst best, Firm 1's best response is q 1 = q `b ecause Firm 1 aims to maximize social surplus. It follows that the …rst best belongs to the graph of Firm 1's best response function.
Second, we establish that e q 1 (q 2 ) is increasing in q 2 . The sign of the derivative of e q 1 (q 2 ) has the same sign of the cross partial derivative of Firm 1's objective function (13) evaluated at q 1 = e q 1 (q 2 ). The derivative of (13) with respect to q 1 is simply because the partial derivative with respect to b v is zero. The cross partial is then obtained by di¤erentiating the above with respect to q 2 , and this gives [b v(q 1 ; q 2 ) c 0 (q 1 )]f (b v) @b v(q 1 ; q 2 ) @q 2 > 0 where the inequality follows because at q 1 = e q 1 (q 2 ), we have b v(q 1 ; q 2 ) > c 0 (q 1 ) and @b v @q 2 > 0 by (41) in the proof of Lemma 2.
Proof of Lemma 6: By de…nition, f (x)k(x) = F (x). We have where the second equality is due to integration by parts.
The three equations in Proposition 4 are now exactly those that de…ne the …rst best in (3), (4), and (5).
Equilibrium qualities and consumer allocation must be …rst best.
Proof of Remark 3: When Firm 2 sells to consumers with valuations below v at price p 2 , its revenue is F (v)p 2 , where v = p 1 p 2 q 1 q 2 . If we express p 2 as a function of v, we have p 2 (v) = p 1 v(q 1 q 2 ). The marginal revenue is the derivative of revenue with respect to the …rm's quantity, F (v): Because p 2 (v) is linear in v, marginal revenue is linear in v if and only if the reverse hazard rate k(v) is linear.
Proof of Remark 4: De…ne y F , so y 0 = f . We have k(x) = + x equivalent to y 0 y = 1 + x .
These requirements are + v = 0 and A( + v) 1 = 1. We obtain the expression for f in the Remark by di¤erentiation.
Proof of Remark 5: Suppose we have F = 1 ( v)f and F = ( + v)f . We use these two equations to solve for f , and obtain f = [ + + ( )v] 1 . Then we substitute by v and by v, and simplify f to the expression of in the Remark. Finally, we obtain F in the Remark by substituting the solution for f in either of the two equations.
Next, the derivative of b p 1 in (35) with respect to q 2 and the derivative of b p 2 in (36) with respect to q 1 in the lemma are obtained by straightforward computation, and we have kept track of b v(q 1 ; q 2 ) being implicitly de…ned by (37). Again, the inequalities follow from h 0 < 0, k 0 > 0, and the properties of b v derived above.
Proof of Proposition 6: We begin with the derivatives of …rms'pro…ts in (38) and (39), and set them to zero to obtain …rst-order conditions. Equilibrium qualities b q 1 and b q 2 are best responses, so must satisfy the …rst-order conditions simultaneously: The continuation equilibrium in prices must also satisfy Lemma 7, so (37) must also be satis…ed at qualities b q 1 and b which is the third equation in the Proposition.
Next, we use the expressions for @ b p 2 (b q 1 ; b q 2 ) @q 1 and @ b p 1 (b q 1 ; b q 2 ) @q 2 in Lemma 8. After substitution, the …rstorder conditions (42) and (43) Then we apply the expression for @b v @q 1 and @b v @q 2 in Lemma 8 to the above, simplify, and obtain the …rst two equations in the Proposition.