Gender-based price discrimination in matching markets

International Journal of Industrial Organization 42 (2015) 34–45

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International Journal of Industrial Organization journal homepage: www.elsevier.com/locate/ijio

Gender-based price discrimination in matching markets☆ Thomas Trégouët Université de Cergy-Pontoise, THEMA, UMR 8184, France

a r t i c l e

i n f o

Article history: Received 16 December 2013 Received in revised form 16 May 2015 Accepted 31 May 2015 Available online 11 July 2015 JEL classification: D80 L10 Keywords: Price discrimination Gender discrimination Matching Signaling Online dating Platform

a b s t r a c t This paper develops a new model to analyze price discrimination in matching markets where agents have private information about their respective qualities. On the basis of signals (car, clothing, club membership, etc.) purchased from profit-maximizing firms, men and women form beliefs about each other's qualities. The matching must then be stable in the following sense: there cannot be a man and woman who are unmatched and who both believe they would be better off if they were matched with one another. The model enables an analysis of the impact of third-degree (or gender-based) price discrimination on welfare. When third-degree price discrimination is not feasible, the cost of eliciting private information is higher but a monopoly intermediary may have stronger incentives to implement an efficient allocation. I show that gender-based price discrimination is more likely to have a positive impact the more symmetric the matching environment is. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Men and women use signals to inform potential partners of their relevant characteristics. Such signals include premarital investment (education, culture, etc.), conspicuous consumption (expensive cars or clothing), membership to selective clubs or dating services, and more. In some of these examples, signals are provided by profit-maximizing firms.1 This raises several questions: Do private firms have an incentive to provide an effective number of signals? What is their impact on the matching of men and women? Do current regulations (like a ban on gender-based price discrimination) have an impact on the provision of signals and, ultimately, on the matching of men and women? In order to answer these questions, I will build a model in which men and women invests in costly signaling prior to matching. Based on their ☆ I would like to thank two anonymous referees, the editor, Jeanne Hagenbach, Frederic Koessler, Andras Niedermayer, Régis Renault, and participants at the 2012 IIOC and 2012 EARIE conferences, and seminar participants at CREST, Ecole Polytechnique, Telecom Paristech and Université de Cergy-Pontoise for comments and suggestions. This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01). E-mail address: [email protected]. 1 For instance, IAC operates three popular dating websites – OkCupid.com, Match.com and Chemistry.com – that can be ranked according to membership fees, $0, $35 and $50 per month respectively: the most expensive offers are supposed to attract people interested in a long-term relationship, while more basic services are designed for casual daters. In addition, there are often different types of membership for a given website: for instance, users of Match.com can pay an additional $5 per month to get their profiles highlighted, $5 to become premium members, etc.

http://dx.doi.org/10.1016/j.ijindorg.2015.05.007 0167-7187/© 2015 Elsevier B.V. All rights reserved.

choice of signals, men and women form beliefs about their potential partners' types. The model departs from the existing literature in that the matching depends explicitly on these beliefs. In particular, I assume that, in equilibrium, the matching must be (pairwise) stable according to the beliefs: there cannot be a man and a woman who both believe they would be better off being matched to one another compared to their current assignment. The advantage of this approach is that there is no need to model precisely how the matching occurs following signaling decisions. Existing models in the literature on price discrimination in matching markets rely on a very precise description of how the matching occurs. Damiano and Li (2007, 2008) consider the following matching mechanism: a profit-maximizing matchmaker opens several meeting places – with different entrance fees – where agents are then matched randomly. Gomes and Pavan (2015) and Johnson (2013) adopt a mechanism design perspective and obtain results on the optimal mechanism. The drawback is that some questions are difficult to assess in these models. In this paper, I will illustrate the benefits of my approach by analyzing the impact of a ban on third-degree (or gender-based) price discrimination. The question is especially relevant in the light of recent debates concerning the legality of gender-based discrimination (see the discussion below). Formally, I analyze a matching model with non-transferable utility and asymmetric information. Agents can be of two types, high or low, with the same ordinal preference: everyone prefers to be matched with a high-type agent. Before they can enjoy any gain from matching,

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agents have to go through the service of a profit-maximizing matchmaker. When gender-based price discrimination is feasible, the matchmaker implements either an exclusive allocation, in which only hightype men and women participate, or a separating allocation, in which all men and women participate, and men and women are matched assortatively. The matchmaker faces a rent-extraction/efficiency tradeoff. On the one hand, in an exclusive allocation, the matchmaker captures the entire high-type surplus, but is unable to capture the low-type surplus. On the other hand, in a separating allocation, the matchmaker captures the entire low-type surplus, but high-type men and women obtain an information rent. When gender-based price discrimination is not feasible, the matchmaker offers the same set of prices on both sides of the market. The cost of information revelation increases: information rents are higher and some users who were left with zero surplus when gender-based price discrimination were feasible now receive an information rent. Suppose that high type men and women were offered pm and pw respectively. Then, when gender-based price discrimination is not feasible, if the matchmaker offers {pm, pw} on both sides, it is possible that both the high-type men and women choose min{pm, pw} if this “signaling” strategy is dominated for low-type agents. In other words, either the high-type men or women now receive an additional information rent: max{pm, pw} − min{pm, pw}. I find sufficient conditions under which this additional “cost” to the matchmaker is higher in an exclusive allocation than in a separating allocation, and conversely. Because total welfare is maximal in a separating allocation, these conditions also indicate in which case gender-based price discrimination has a positive or negative impact on total welfare. My paper is also related to the literature on matching tournaments (see, for example, Bhaskar and Hopkins (2011), Chiappori et al. (2009), Hopkins (2012), Hoppe et al. (2009), Mailath et al. (2011), and Peters and Siow (2002)) which examine how premarital investment or investment before trading shapes the matching of men and women, or buyers and sellers. Given that no restrictions are made on the set of signals, papers in this literature focus on the fully separating equilibrium which consists of the assortative matching of agents based on their signal choices. In most papers, the process by which agents end up being matched assortatively is not explicitly modeled, nor are the beliefs that sustain the equilibrium. My notion of stability provides a theoretical foundation for the separating equilibrium considered in the literature. In addition, my paper is related to the literature on third-degree price discrimination. Economists have long noted that third-degree price discrimination may either reduce or raise social welfare:2 in a monopoly market, moving from non-discrimination to discrimination raises a firm's profits, harms consumers in markets where prices increase, and benefits consumers who enjoy lower prices. I offer an analysis of the problem of a monopolist who can use both seconddegree and third-degree price discrimination in a case where demands are interdependent. Layson (1998) and (Adachi (2002, 2005)) extend the classical analysis of third-degree price discrimination to the case of interdependent demands. The argument is as follows: suppose there are two groups of consumers and the only difference between the two groups is that the first one's participation exerts a positive externality on the second one. The optimal price structure in this case is that individuals of the first group pay a lower price, so that a ban on thirddegree price discrimination is likely to reduce total welfare.3 In a one2 The analysis of third-degree price discrimination goes back to the seminal works of Pigou (1920) and Robinson (1933), later built on by Schmalensee (1981) and Varian (1985). More recently Aguirre et al. (2010) provided general conditions on the curvature of demand functions under which third-degree price discrimination has a positive or a negative impact on total welfare. 3 The analysis of such cross-group network effects is at the heart of the literature on multi-sided platforms (see, Armstrong (2006), Caillaud and Jullien (2003), Rochet and Tirole (2003, 2006)). Wright (2004) discusses informally the impact of a ban on thirddegree price discrimination in these industries.

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to-one matching market, the demands of men and women are positively interdependent, but intra-group participation externalities also occur: men compete to attract women, and conversely.4 Accordingly, some consumers may benefit from a price increase, and it is unclear whether one effect necessarily dominates the other, i.e. whether thirddegree price discrimination is harmful or not. Problems with both second- and third-degree price discrimination arise naturally in insurance markets. The literature on risk classification (Crocker and Snow, 2000) discusses the implication of third-degree price discrimination on efficiency and equity in insurance markets. The focus is on finding the insurance contracts that maximize consumers' welfare under the constraint that insurers make non-negative profits. Therefore, in some sense, I will consider the “dual problem” of the situation. 1.1. Debates on gender-based price discrimination Gender-based pricing is prevalent in many industries (insurance, dry-cleaning, hairdressing, nightclubs and dating services, bars, clothing, etc.). The legality of gender-based price discrimination in matching markets – and of gender-based pricing in general – has been debated in the US since the 90s and in the EU more recently.5 The debate is not whether gender-based pricing may harm competition – a standard concern for competition authorities – but rather whether gender-based pricing is a form of gender discrimination. Gender-based pricing has indeed been criticized for conveying and reinforcing negative stereotypes about both women and men, especially in matching markets.6 In the US, gender discrimination in shops and services is an issue left to the states.7 This resulted in considerable variation in court judgments on gender-based pricing: whereas some courts have systematically ruled against gender-based pricing (California, Florida, Pennsylvania, Iowa and Maryland), others have adopted a case-by-case approach (Illinois, Washington and Michigan). This division among the courts has been debated by American jurists since the 90's. Opponents invoke a de minimis argument: courts should dismiss cases involving genderbased pricing because the plaintiff generally suffers very little damage.8 The economic efficiency of gender-based pricing is sometimes invoked, 4 The same intra-group externalities are examined by Rayo (2013), who analyzes the problem of a discriminating monopolist serving a population of consumers who use the goods as a signaling device (conspicuous goods). 5 For the US, see Joyce L. McClements and Cheryl J. Thomas (1986): “Public Accommodation Statutes: Is Ladies' Night Out?”, Mercer Law Review, 37; Heidi C. Paulson (1991): “Ladies' Night Discounts: Should We Bar Them or Promote Them”, Boston College Law Review, 32; “Civil Rights. Gender Discrimination. California Prohibits Gender-Based Pricing. Cal. Civ. Code §51.6 (West Supp 1996)”, Harvard Law Review, 109 (1996); Jessica Rank (2005): “Is Ladies' Night Really Sex Discrimination?: Public Accommodation Laws, De Minimis Exceptions, and Stigmatic Injury”, Seton Hall Review, 36; Mark A. Herzberg (2010): “Girls Get in Free: A Legal Analysis of the Gender-Based Door Entry Policies”, South California Review of Law and Social Justice, 19; Shana S. Brouwers (2011): “A Guy Walks Into a Bar: Gender Discriminatory Pricing and Admission Policies in Las Vegas Establishments”, Nevada Law Journal, 11. For the EU, see the European Commission's reports “Sex Discrimination in the Access to and Supply of Goods and Services and the Transposition of Directive 2004/113/EC” and “Sex-Segregated Services” (2009), and Aileen McColgan (2009): “The Goods and Services Directive: a curate's egg or an imperfect blessing?”, European Gender Equality Law Review, 1. 6 The idea is that the fact that women usually pay less for matching services replicates a dominance/submission stereotype. More generally, sociologists and psychologists noticed that the organization of romantic relationships has shifted only slowly toward more gender equality. See Paula England (2010): “The Gender Revolution: Uneven and Stalled”; Janet Lever, David A. Frederick and Rosanna Hertz (2013): “Who pays for dates? Following versus Challenging Conventional Gender Norms”. 7 This is due to the fact that the word gender is not mentioned in the section of the Civil Rights Act of 1964 related to “places of public accommodation”: “All persons shall be entitled to the full and equal enjoyment of […] any place of public accommodation […] without discrimination or segregation on the ground of race, color, religion, or national origin”. Consequently, most states have enacted “local” legislations to ensure equal access to accommodations. 8 Rank, supra note.

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notably in matching markets.9 Others argue that the economic defensibility of gender-based pricing does not make it socially justifiable.10 In the EU, Directive 2004/113/EC provides a framework for assessing “sex discrimination in the access to and the supply of goods and services”. The Directive had a limited impact on gender-based price discrimination: very few cases have been brought in connection with the Directive. There are several explanations for this: few people are aware of the existence of the Directive; the Directive is ambiguous (see below), thereby complicating its transposition in domestic laws; and in most cases, the damage incurred by the plaintiff is less than the cost of litigation.11 Experts underline in particular the ambiguity of Article 4.5.12 Under this article, gender-based pricing is considered legal if it aims “primarily” to attract men or women who, otherwise, would not have joined or used the service. It implies that it is legal to subsidize men for attending a course on parenting, for example. It is, however, illegal to subsidize women for joining a club or a dating website. The paper is organized as follows: Section 2 describes the matching model; Section 3 explores the monopoly outcome when gender-based price discrimination is allowed; Section 4 examines the impact of a ban on gender-based price discrimination on total welfare in the case of a monopoly; and Section 5 discusses situations where firms specialized in the provision of gender-specific signals, and offers a conclusion.

2. The model 2.1. Men and women There are two populations of agents: men and women. The two populations have the same size, normalized to one. There are two types of men and women: a proportion 0 b λ b 1 of men (or women) are of type h and the others are of type l. Each agent's type is their private information. A match between a type-i man and a type-j woman, (i, j) ∈ {l, h}2, creates a surplus uij(≥ 0) for the man and vij(≥0) for the woman. sij = uij + vij denotes the total surplus from a match between a type-i man and a type-j woman. I assume homogeneous preferences, uih ≥ uil and vhi ≥ vli, ∀ i ∈ {l, h}, and increasing differences (or complementarities in matching): uhh − uhl ≥ ulh − ull and vhh − vlh ≥ vhl − vll :

This last assumption is the equivalent of the single-crossing condition in screening problems. It ensures that information revelation will be possible. Agents are risk neutral and have quasi-linear preferences. They only care about the difference between the expected value of the match and the cost of signaling (see below). An unmatched agent gets zero payoff, regardless of his/her type. Under these assumptions, the total matching surplus is greatest if (i) all men and women are matched, and (ii) men and women are matched assortatively, i.e. h-types together and l-types together. Condition (i) stems from the fact that uij ≥ 0 and vij ≥ 0 for all (i, j). Condition (ii) is due to the increasing differences of the matching surplus.

9 In 2011, the state of Nevada enacted a law in favor of gender-based pricing. It was explicitly written to protect the bar, club and dating industries. See Brouwers, supra note, and “Ladies' night remains legal, despite anti-discrimination law”, Las Vegas Sun, Jul. 13, 2011. 10 The argument is a moral one: gender should not be used as a proxy for other characteristics, and particularly when it is based on stereotypes. See Harvard Law Review, supra note, and Larry Alexander (1992): “What Makes Wrongful Discrimination Wrong? Biases, Preferences, Stereotypes, and Proxies”, Penn Law Review, 141. 11 One issue in practice is that, contrary to, for example, racial discrimination, associations or groups of plaintiffs cannot bring a case to court. 12 “This Directive shall not preclude differences in treatment, if the provision of the goods and services exclusively or primarily to members of one sex is justified by a legitimate aim and the means of achieving that aim are appropriate and necessary.”

2.2. The matchmaker A monopoly matchmaker, unable to observe types of men and women, offers a pair of fee schedules Σm and Σw, where Σm and Σw are closed subsets of ℝ+.13 In order to join the matchmaker, a man (a woman) must pick one price in Σm(Σw). The agents who join the matchmaker observe the prices chosen by each participant and a matching occurs. I will describe the matching in the presentation of my equilibrium concept (see below). It will become apparent that the only role of the matchmaker is to provide men and women with a signaling device: once users have paid the entrance fees, the matching occurs without the help of the matchmaker.14 The objective of the matchmaker is to maximize the sum of subscription fees collected from men and women. 2.3. Timing and equilibrium The timing of the game is as follows: 1. The matchmaker announces prices Σm and Σw. 2. Men and women pick signals in Σm and Σw respectively, or do not participate.15 3. Men and women who participate in stage 2 observe each other choices. The matching occurs. In stage 3, men and women form beliefs about each other's types. These beliefs, or reputations, depend only on the prices chosen by participants in stage 2. Intuitively, if a man chooses a very high price, women should reasonably believe that he is a high-type man. On the other hand, women should believe that a man who chooses a low price is a low-type man. The matching should depend on these beliefs/reputations. For instance, if a man is believed to be of type l while there are plenty of men who are believed to be of type h, then, a woman should reject the l-man and try to match with an h-man. Formally, M and W denote the sets of men and women who participate in stage 2. For all i ∈ M and j ∈ W, let m(i) and w(j) be man i's and woman j's reputations: participants believe that man i (woman j) has type h with probability m(i) (w(j)). I require the matching to be stable in the following sense: there can be no pairs of matched men and women (i, j) and (i′, j′), such that

 mðiÞ N m i0 and wð jÞ b w j0 :16

In words, in equilibrium, there cannot be a man and a woman who both believe they would be better off being matched to one another compared to their current assignment. Stability has a straightforward implication in my framework: in equilibrium, men and women are matched assortatively in terms of their reputations. In practical terms, suppose that two men (i and i′) and two women (j and j′) are participating with reputations m(i) = 1, m(i′) = 0 and w(j) = 1 and w(j′) = 0 respectively. Stability would then require that man i is matched with woman j, and i′ with j′. A matching should also be feasible: if, say, there are more men than women, then some men must remain single. A precise definition of a feasible matching is rather complex and not very informative. For expositional clarity, let us adopt the following loose definition: a feasible matching is a measure-preserving function from M to W. Notice that 13

The ‘closed set’ assumption ensures that inf Σi ∈ Σi. Damiano and Li (2007) assume that the matchmaker opens different meeting places that are mutually exclusive. We will see in Section 4 that, under this assumption, some allocations can no longer be implemented when gender-based price discrimination is not feasible. 15 If 0 is not an available signal, participation is not guaranteed. 16 This notion of stability is a convenient adaptation of the notion of pairwise stability defined in complete information two-sided matching model (see Chapter 2 in Roth and Sotomayor (1992)). 14

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a matching where a measure zero of agents are left unmatched is consistent with this definition. It has the important implication that a single agent who deviates from the equilibrium strategy can be “punished” by being left unmatched. We will use this property when such a punishment makes sense. In the end, stages 2 and 3 boil down to a strategic form game where the payoffs are obtained by mapping the choices of men and women into public beliefs about their types and then matching men and women assortatively with respect to those public beliefs. An equilibrium of the game therefore specifies prices Σm and Σw, men and women participation M and W, and beliefs such that: (i) prices Σm and Σw maximize the matchmaker's profit; (ii) the choices of men and women must be individually rational given that the subsequent matching is positive assortative in terms of reputations; (iii) beliefs are determined by stage 2 choices using Bayes' rule. There can be multiple equilibria for two reasons: first, men and women play a coordination game in stage 2; second, multiple equilibria can be sustained in stage 3 by choosing appropriate out-of-equilibrium beliefs. To eliminate these potential issues, I focus on the equilibria with maximum participation in stage 2, and I require beliefs to be intuitive in the sense of Cho and Kreps (1987).17 3. Monopoly matchmaker In this Section, I will describe the matching implemented by the matchmaker when gender-based price discrimination is feasible. When only few prices (or signals) are available, an assortative matching may not obtain in equilibrium. For instance, if Σ m = Σw = {0}, all agents participate, no information is revealed and, accordingly, men and women are matched randomly with one another. In this case, we will say that a pooling allocation is implemented. Although various allocations can be implemented, some allocations are better candidates than others: in an exclusive allocation, only hightype men and women participate; in a separating allocation, all men and women participate, and men and women are matched assortatively. In the following section, I will examine the matchmaker's profits where it implements an exclusive or a separating allocation. I will then argue that the matchmaker cannot obtain more profits by implementing other allocations. Lastly, I will provide the conditions under which an exclusive or a separating allocation is implemented. 3.1. Implementation of an exclusive allocation The matchmaker can implement an exclusive allocation by proposing singletons Σm = {pm} and Σw = {pw} since only high-type men and women participate in an exclusive allocation. Let us examine the conditions that must be met by pm and pw. First, high-type men and women must be willing to participate: uhh −pm ≥ 0 and vhh −pw ≥ 0:

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should “signal” a high-type woman. Clearly, maximum profits are obtained by capturing the entire high-type surplus, i.e. pm = uhh and pw = vhh. We conclude: Lemma 1. The maximum profit in an exclusive allocation is ΠEx = λshh. It is achieved for instance with Σm = {uhh} and Σw = {vhh}. 3.2. Implementation of a separating allocation The matchmaker can implement a separating allocation by offerw m w w m m ing pairs of prices Σm = {pm l , ph } and Σ = {pl , ph }. Prices pl and ph 18 must satisfy a set of participation and incentive constraints: uhh −pm ≥ 0; h uhh −pm ≥ uhl −pm h l ; ull −pm ≥ 0; l m m ull −pl ≥ ulh −ph : Increasing differences of the matching surplus ensures that the above inequalities define a non-empty set of ℝ2. We can observe that m m the last inequality implies that pm h ≥ pl . Price pl “signals” a low-type man since this is the lowest price in Σm and it satisfies the low-type men participation constraint. Price pm h “signals” a high-type man, since it is dominated for low-type men and compatible with the incentive and participation constraints for high-type men. Furthermore, standard arguments show that maximum profits are obtained by capturing the entire low-type surplus and the entire high-type surplus, minus an m information rent: pm l = ull and ph = uhh − (uhl − ull). Similarly, on the women's side of the market, profits are maximized when pw l = vll and pw h = vhh − (vlh − vll). We conclude: Lemma 2. The maximum profit in a separating allocation is: ΠSep ¼ λshh þ ð1−λÞsll −λðuhl þ vlh −sll Þ: It is achieved for instance with Σm = {ull, uhh − (uhl − ull)} and Σw = {vll, vhh − (vlh − vll)}. 3.3. Other allocations? In this paragraph, I argue that maximum profits are obtained either with an exclusive or a separating allocation. This can be established formally by writing the problem of the matchmaker and solving for the optimal prices. The proof, however, is rather lenghty and not very informative. Accordingly, I will only present the two reasons why the result is obtained: (i) there are only two types of agents on both sides, and the incentive problems for men and women can basically be solved separately. With more than two types, conflict between local and global incentive constraints may lead to the implementation of pooling allocations.19 (ii) the matchmaker's problem is linear so that “corner” allocations are implemented: in order to reduce the high-type information rent, the matchmaker stops serving low-type men and women.

ð1Þ 3.4. Conclusion

Second, low-type men and women must not be willing to participate: Lemma 1 and 2 together give: ulh −pm b 0 and vhl −pw b 0:

ð2Þ

If the matchmaker offers Σm = {pm} that meets conditions (1) and (2), participants should believe that a man who chooses pm in stage 2 is a high type, since pm is dominated for low-type men. Similarly, pw 17 The intuitive criterion selects a unique equilibrium in my model and gives extreme out-of-equilibrium beliefs, given that there are only two types of agents on both sides. This property has been discussed extensively in the literature on signaling games (see, e.g., Riley (2001)).

Proposition 1. If sll b λ(uhl + vlh), the matchmaker implements an exclusive allocation and makes profits Π = λshh; If sll ≥ λ(uhl + vlh), the matchmaker implements a separating allocation and makes profits Π = λshh + (1 − λ)sll − λ(uhl + vlh − sll).

18 19

w Conditions on pw l and ph can be derived similarly. See Chapter 3 in Laffont and Martimort (2001).

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A standard rent extraction/efficiency trade-off is at stake in Proposition 1. On the one hand, in an exclusive allocation, the matchmaker captures the entire high-type surplus, λshh, but is unable to capture the low-type surplus. On the other hand, in a separating allocation, the matchmaker captures the entire matching surplus, λshh + (1 − λ)sll, minus information rents left to high type men and women, λ(uhl − ull + vlh − vll)). 4. Welfare effects of gender-based price discrimination In order to better understand the effect of gender-based price discrimination on welfare, assume in this section that the matchmaker offers a pair of fee schedules Σm and Σw under the constraints that the two sets must be the same: Σm = Σw = Σ.20 The model can be solved without facing any major technical difficulties, but the analysis may be tedious on account of the many different cases to be discussed. Assumption 1 below reduces the number of parameters from eight (uij and vij) to five, and allows me to obtain more clear-cut results. 2

Assumption 1. There exists β ∈ [0, 1] such that, for all (i, j) ∈ {l, h} , uij = β ⋅ sij. Assumption 1 provides that the sharing rule of the matching surplus does not depend on the type of man and woman. Hereinafter, βm = β and βw = 1 − β shall denote the share of men and women in the matching surplus, respectively. Our goal in this section is to find conditions under which the total welfare is higher or lower when gender-based price discrimination is feasible. To this end, a possible approach would be to derive the equivalent of Proposition 1 when gender-based price discrimination is unfeasible. However, this approach is unfruitful as it involves discussing numerous cases. The analysis in this Section will indeed reveal that when gender-based price discrimination is unfeasible, there may be a link between incentive problems for men and women. Loosely speaking, we move from two 2-types screening problems to a single 4-types screening problem. It is well known that a pooling or partially pooling allocation may be optimal with more than two types and the same phenomenon is true in my model: the matchmaker implements a pooling allocation for some parameter values. While interesting, this makes the comparison with Section 3 difficult and unclear. Accordingly, I will not characterize these situations in the following sections, but focus instead on situations where the matchmaker's profit is maximized when an exclusive or a separating allocation is implemented. First, I will show how to characterize the profit-maximizing allocations in both the classes of exclusive and separating allocations (Section 4.1). Second, I will deduce sufficient conditions under which gender-based price discrimination improves welfare (Section 4.2). 4.1. Implementation of exclusive and separating allocations 4.1.1. Exclusive allocation In the following, assume w.l.o.g. that βm ≥ βw. Suppose first that the matchmaker offers the same prices as if gender-based price discrimination were feasible: Σ = {βmshh, βwshh} and high-type men and women are supposed to choose βmshh and βwshh respectively. Let us find conditions under which this can be sustained in equilibrium. A high-type woman never chooses βmshh (≥βwshh) since this would violate her participation constraint. Things are more complicated on the other side of the market: suppose that a high-type man deviates and chooses the lower price βwshh. Since women can “punish” him by leaving him

20 Another interpretation could be that the matchmaker is a firm that offers goods which appeal to both men and women (phones, membership to a club, etc.).

unmatched if they believe that the man is low-type,21 this deviation is profitable only if women believe that only a high-type man could have chosen βwshh. Given our assumption that beliefs should be intuitive in the sense of Cho and Kreps (1987), this is the case only if paying βwshh is a dominated strategy for low-type men, i.e. if βmslh − βwshh b 0. Put differently, there is a potential conflict between providing low-type men with incentives not to participate and providing high-type men with incentives not to choose the high-type woman's signal. This conflict between an upward and a downward incentive constraints explains why, in some cases, the matchmaker finds it optimal to “pool” high-type men and women, that is, to propose a unique signal for both low-type men and women. Lemma 3 below discusses the different cases: Lemma 3. Assume that βm ≥ βw. When gender-based price discrimination is not feasible, the profit-maximizing allocation in the class of exclusive allocations is such that • if βwshh ≤ βmslh,   Σ ¼ βw shh ; βm shh and ΠEx ¼ λshh ; • if βm slh b βw shh ≤ βm shh 2þslh ,   Σ ¼ βm slh ; βm shh and ΠEx ¼ λβm ðshh þ slh Þ; • if βm shh 2þslh b βw shh ; X

  ¼ βw shh and ∏Ex ¼ 2λβw shh :

The case where βw N βm can be stated similarly. 5

Proof. See Appendix A.1. m

The first case in Lemma 3 states that, if the difference between β shh and βwshh is great enough, a high-type man will never consider choosing the price βwshh since this would clearly signal that he is a low-type man. Accordingly, the matchmaker is still able to capture the entire high-type surplus. In the second and third case, the difference between βmshh and βwshh is small enough that a high-type man may now consider choosing βwshh. It follows that, if the matchmaker is willing to capture the entire high-type men surplus, it has to reduce pw down to a price that signals a low-type if it is chosen by a high-type man, that is pw = βmslh. Alternatively, the matchmaker may prefer to capture the entire high-type women surplus by offering pw = βwshh, knowing that pw will also be chosen by high-type men. Therefore the matchmaker trades off between two options: either it captures the entire high-type men surplus and leaves an information rent to high-type women (second case), or it captures the entire high-type women surplus and leaves an information rent to high-type men (third case). To summarize, when βm ≥ βw, the “cost” of unfeasible gender-based price discrimination is that hightype men or women now receive an information rent if βmslh b βwshh. What conclusions can be drawn from Lemma 3? First, if the two sides are sufficiently asymmetric (first case in Lemma 3), a “ban” on gender-based price discrimination has no cost. Indeed, in this case, the maximum price on the low-profit side of the market is low enough that agents on the other side are not tempted to use it as a signal. 21 This punishment is consistent with the definition of a matching because a single agent has measure zero. One possible interpretation is that a high type woman matched with the “deviating” man can find an unmatched man who has chosen the “expected” membership fee βmshh at a small cost. This would be for instance the case in a dynamic matching market where a large number of new agents join the matching service at each period. With this interpretation my model can be understood as the stationary equilibrium of a dynamic matching market. If deviating agents cannot be punished by being left unmatched, it is not possible to implement an exclusive allocation in some cases (first case in Lemma 3). This would accordingly lead the matchmaker to implement other allocations “more often”, thereby making a separating allocation more attractive.

T. Trégouët / International Journal of Industrial Organization 42 (2015) 34–45

39

Second, if the two sides are roughly symmetric, some agents may receive an information rent following a “ban” on gender-based price discrimination, but this has little effect on the matchmaker's profits: when βw is close to βm, there is little difference between 2βwλshh and (βm + βw)λshh. It would follow that the “cost” of implementing an exclusive allocation is the highest when asymmetries between men and women are neither too strong nor too weak. Last, the above analysis illustrates a key difference between my matching model and the existing literature on price discrimination in matching markets. If the matchmaker had to open different meeting places that are mutually exclusive (as in Damiano and Li, 2007), the common membership in an exclusive allocation should satisfy inequalities p N βmslh and p ≤ βwshh to ensure exclusion of low-type men and participation of high-type women. It follows that an exclusive allocation could not be implemented when βwshh ≤ βmslh. Strikingly, this corresponds to Case 1 in Lemma 3 in which the outcome Lemma 1 obtains since there is no conflict between downward and upward incentive constraints.

to finding conditions under which the cost of implementing an exclusive allocation is higher than the cost of implementing a separating allocation. Conditions under which a “ban” on gender-based price discrimination has no cost are different in a separating and in an exclusive allocations. First, in a separating allocation, the matchmaker has to leave an information rent to low-type men or women if βm ≠ βw. Obviously, this is not the case if the matchmaker implements an exclusive allocation. Second, the conflict between upward and downward incentive constraints may occur in one case but not in the other. Assume, for instance, that more profits can be made from men than from women. On the one hand, in an exclusive allocation, there is no conflict between upward and downward incentive constraints when βmslh N βwshh.23 On the other hand, in a separating allocation, there is no conflict between upward and downward incentive constraints when βm(slh − sll) N βw(shh − slh).24 Given that none of these conditions implies the other, it may be less costly to implement a separating allocation than an exclusive allocation, for example.

4.1.2. Separating allocation Implementing a separating allocation is more complicated. The following section provides an overview of the argument. Details are provided in Appendix A.2. Let Σ denote the set of prices offered by the matchmaker to implement a separating allocation. In a separating equilibrium, low-type men and women choose prices that “signal” that they have low types. Accordingly, low-type men and women choose the lowest price in Σ, since in the worst case this signals a low type. This has an important implication: in a separating allocation, low-type men or women receive an information rent. Let i ≠ j such that βi ≥ βj. Since price inf Σ is chosen by low type j–agents, we have inf Σ ≤ βjsll ≤ βisll: low-type i–agents receive an information rent. Furthermore, we can observe that the lowtype i–agents information rent is passed to the high-type i–agents. In other words, high-type i–agents should receive higher information rents. These are the first effects of a “ban” on gender-based price discrimination when a separating allocation is implemented. The second effect is similar to the one described in the case of an exclusive allocation. Let pm and pw denote the signals of high-type men and women respectively. Assume that there are more profits to be made on high-type men than on high-type women, so that pm is higher than pw in principle. If pm and pw are too close, high-type men may be tempted to choose the price pw. Accordingly, there is a potential conflict between providing low-type men with incentives not to choose the high-type woman's signal (pw high enough) and providing hightype men with incentives not to choose the high-type woman's signal (pw low enough). Again, in some cases, this conflict between an upward and a downward incentive constraints may lead to some “pooling” between high-type agents.22 The logic is the same as in Lemma 3: depending on the situation, this conflict benefits either to high-type men or to high-type women. To summarize, the “cost” of unfeasible gender-based price discrimination is that low type men or women now receive an information rent, high type men or women receive a higher information rent and, possibly, both high type men and women receive a higher information rent.

4.2.1. Positive effect on total welfare In the following section, I present conditions under which genderbased price discrimination has a positive effect on total welfare. To begin with, I observe that gender-based price discrimination has a positive effect if the sharing rule of the matching surplus is symmetric.

4.2. Welfare comparison We aim to find conditions on βi, sij and/or λ such that if a separating allocation is implemented when gender-based price discrimination is feasible, it is also implemented when it is not. Under such conditions, the total welfare would be (weakly) higher when gender-based price discrimination is unfeasible, since the total welfare is maximized in a separating allocation. As the above analysis suggests, this is equivalent 22

See Lemma 4 in Appendix A.2.

Proposition 2. If βw = βm, then, the total welfare is (weakly) higher when gender-based price discrimination is feasible. 5

Proof. See Appendix A.3. m

w

The intuition is as follows: if β = β , then, the matchmaker can capture the entire high-type surplus at no cost in an exclusive allocation when gender-based price discrimination is unfeasible (see Lemma 3). Accordingly, if an exclusive allocation is optimal when gender-based price discrimination is feasible, it is also optimal when it is not. However, there is still an additional cost to implement a separating allocation: if shl ≠ slh, high-type men and women' incentives to misrepresent their types are different even when βm = βw. In the end, a separating allocation is implemented less often when gender-based price discrimination is unfeasible if βm = βw. The next proposition shows that gender-based price discrimination has a positive effect if the distribution of the matching surplus is symmetric and log-supermodular. Assume that shl = slh. If sll/slh ≥ slh/shh, then, the total welfare is (weakly) higher when gender-based price discrimination is feasible. Proposition 3. Assume that shl = slh. If su/slh ≥ slh/shh, then, the total welfare is (weakly)higher when gender-based price discrimination is feasible. Proof. See Appendix A.4.

5

The proof of Proposition 3 is rather lengthy and complicated: first, I show that the matchmaker implements either an exclusive or a separating allocation when gender-based price discrimination is unfeasible; second, I show that, if a separating allocation is implemented when gender-based price is unfeasible, then it is also implemented when it is feasible (i.e., a separating allocation is “more likely” to be implemented when gender-based price discrimination is feasible). The second part of the proof involves the comparison of the matchmaker's profits in exclusive and separating allocations in several cases. Under the assumptions of Proposition, the conflict between downward and upward incentive constraints is more pronounced in a separating allocation than in an exclusive allocation. To understand why, 23

See Lemma 3. This condition ensures that the highest price compatible with women incentive constraints is lower than the lowest price compatible with men incentive constraints, or, equivalently, that the high-type woman's signal can only signal a low-type on the men side of the market. See Lemma 4 in Appendix A.2. 24

40

T. Trégouët / International Journal of Industrial Organization 42 (2015) 34–45

recall that, both in an exclusive and a separative allocation, there is no conflict between downward and upward incentive constraints when the highest price admissible for high-type women signals a low type if it is chosen by a man. Formally, in an exclusive allocation, it is the case when βmslh ≥ βwshh and, in a separating allocation, when βm(slh − sll) ≥ βw(shh − slh). Notice that the latter inequality rewrites:     s s βw shh 1− lh ≤βm slh 1− ll : shh slh It is then apparent that, in both cases, the ratio slh/shh must be large enough. However, as shown by the latter inequality, in a separating allocation, this ratio must not be too large compared to sll/slh. 4.2.2. Negative effect on total welfare In the following section, I will show that asymmetry in both the allocation and the distribution of the matching surplus is sufficient to find a negative effect of gender-based price discrimination. Definition 1. The matching market is said to be dominated by men (women) if

degree of asymmetry between the two sides of the market is key to understanding the impact of a ban on gender-based price discrimination. It is worth noting that most papers in the matching literature assume a symmetric allocation and/or distribution of the matching surplus when utility is non-transferable.26 The matching model is also well suited to analyze situations where companies are specialized in the provision of gender-specific signals. In this regard, I refer to situations where men and women buy conspicuous goods from different companies (e.g. cars, suits, dresses and perfume).27 In such an environment, one can show that allocations requiring rich sets of signals to be implemented (i.e., the separating allocation in our specific model) are less likely to be implemented.28 In my model I made the implicit assumption that any information volunteered by a participant that stretches beyond what is signaled by his or her choice of price is not credible and, as such, cannot be used by the matchmaker or the other participants. However dating website users (for example) spend a huge amount of time polishing their online profile, thereby suggesting that it conveys information. The same users also often complain that profiles may not reflect reality. The conditions under which costless signaling (or cheap talk) may improve matching efficiency is an interesting question that I explore in another paper.29 Appendix A

βm N ðbÞ βw and βm ðshh −shl Þ N ðbÞ βw ðshh −slh Þ:

A.1. Proof of Lemma 3 Proposition 4. In a matching market dominated by men (women), the total welfare is higher when gender-based price discrimination is not feasible if βwshl N (b)βmslh. Proof. See Appendix A.5.

5

Proposition 4 states that in a matching market dominated by men, i.e. where more profits can be made on the men's side of the market, gender-based price discrimination has a negative effect if a low-type woman benefits more from being matched with a hightype man than a low-type man benefits from being matched with a high-type woman.25 More specifically, Proposition describes situations where the “cost” to implement an exclusive allocation is higher than the “cost” to implement a separating allocation when genderbased price discrimination is unfeasible. On the one hand, information rents are high if the matchmaker implements an exclusive allocation because every price that satisfies the high-type woman's participation constraint also signals a high type if it is chosen by a high-type man. Hence, high-type men and women choose the same price even if the allocation of the matching surplus is highly asymmetric. On the other hand, if the matchmaker implements a separating allocation, the additional information rents are low because the deviation payoffs for high-type men and women are similar: whereas high-type men benefits more than high-type women from being matched with a high type, they also benefit less from being matched with a low type. 5. Conclusion This paper offers a theory of third-degree price discrimination in matching markets. It proposes a plausible matching model that does not rely on a very precise description of how the matching occurs. It also provides sufficient conditions on the allocation and distribution of the matching surplus under which third-degree price discrimination has a positive impact on total welfare. The analysis reveals that the 25

These situations arise typically when the population of women is less differentiated than the population of men. Indeed, asymmetric gains from matching can be explained by different degrees of heterogeneity among men and women. Intuitively, competition for attracting the more desirable partner should be more intense among the more homogeneous, i.e. less differentiated, group and should therefore result in asymmetric gains from matching. This argument explains, for instance, the transition from bride prices to dowries (i.e. “groom” prices) in pre-industrial societies (Anderson, 2007).

The problem of the matchmaker writes: maxΠ ¼ λðpm þ pw Þ m w ðp ;p Þ

s:t βm shh −pm ≥ 0;

ð3aÞ

βm shh −pm ≥ μ  βm shh þ ð1−μ Þ  0−pw ;

ð3bÞ

βw shh −pw ≥ 0;

ð3cÞ

βw shh −pw ≥ v  βw shh þ ð1−vÞ  0−pm ;

ð3dÞ

βm slh −pm b 0;

ð3eÞ

μ  slh þ ð1−μ Þ  0−pw b 0;

ð3fÞ

βw shl −pw b 0;

ð3gÞ

v  shl þ ð1−vÞ  0−pm b 0:

ð3hÞ

where μ (ν) is the probability that a type-h man (woman) matches with a type-h woman (man) if he (she) deviates and chooses price pw (pm). Inequalities (3a) and (3b), for instance, are respectively type-h men participation and incentive constraints. Precisely, constraint (3b) says that type-h must prefer price pm to price pw. Note that μ coincides with women' beliefs that a man who chooses pw has type h. Let μ ′ = Pr{h − man|pw} and notice that μ ′ ∈ {0, 1} since the intuitive criterion gives extreme out-of-equilbrium beliefs. If μ ′ = 1, assortative matching implies that a man who deviates and chooses pw is still matched with a type-h woman. However, if μ ′ = 0, our definition of an equilibrium matching implies that he can be punished by being left unmatched: he gets a payoff of 0 (see inequality (3b)). Since we have continuum populations on both sides of the 26 For instance, Damiano and Li (2007) assume that a match between a type x man and a type y woman creates an xy surplus for both the man and the woman. 27 There is extensive literature showing that conspicuous consumption is part of a mating-oriented signaling strategy. See Sundie et al. (2011) and the many references therein. 28 This comes from the fact that starting from a separating allocation a firm can always unilaterally exclude low-type agents in order to implement an exclusive allocation. 29 See (Hagenbach et al., 2014).

T. Trégouët / International Journal of Industrial Organization 42 (2015) 34–45

market, this does not leave any participating women to remain single, and the resulting allocation is feasible, stable and measure preserving. We have:  μ¼

1 if pw N βm slh ; and ν ¼ 0 otherwise;



1 if pm N βw shl ; 0 otherwise:

Assume in the following that βm ≥ βw. We solve the relaxed problem: max Π ¼ λðpm þ pw Þ

ðpm ;pw Þ

s:t ð3aÞ; ð3bÞ; ð3cÞ; ð3eÞ; ð3gÞ:

Constraint (3b) rewrites: pm ≤ pw + (1 − μ) ⋅ βmshh. There are three cases to consider. Case 1. βwshh ≤ βmslh. Then all prices pw compatible with inequalities (3c) and (3 g) are below βmslh so that μ = 0. In particular, inequality (3b) is satisfied if inequality (3a) and (3 g) are satisfied. Therefore, the solution to the matchmaker problem are the maximum prices pm and pw that satisfy (3a) and (3c) respectively: pm = βmshh and pw = βwshh. Case 2. βmslh ≤ βwshl. Then all prices pw compatible with inequalities (3c) and (3 g) are above βmslh so that μ = 1. Inequality (3b) is therefore binding: pm = pw. The solution to the matchmaker problem is the maximum price pw that satisfies (3c): pw(=pm) = βwshh. w

m

w

w

w

m

Case 3. β shl b β slh ≤ β shh. On the one hand, if β shl b p ≤ β slh, μ = 0 so that the maximum price that satisfies (3a) and (3e) is pm = βmshh. On the other hand, if βmslh ≤ pw ≤ βwshh, μ = 1 so that the maximum price that satisfies (3a) and (3e) is pm = pw. The matchmaker therefore chooses between the two following options: either Σ = {pm, pw} = {βmshh, βmslh}, in which case Π = λβm(shh + slh); or Σ = {pw(=pm)} = {βmshh}, in which case Π = 2λβwshh. To conclude, notice that, if the latter option is chosen, Σ is the same as in Case 2 above. A.2. Implementation of a separating allocation when gender-based price discrimination is not feasible Let Σ the set of prices offered by the matchmaker to implement a separating allocation. In a separating equilibrium, low type men and women choose prices that “signal” they have low types. Therefore low type men and women choose the lowest price in Σ since in the worst case this signals a low type. This has an important implication: in a separating allocation, low type men or women receive an information rent. Let i ≠ j, (i, j) ∈ {m, w}, such that βi ≥ β j. Since price inf Σ is chosen by low type j–agents, we have infΣ ≤ β jsll ≤ βisll: low type i–agents receive an information rent. Notice also that the low type i–agents information rent passes to the high type i–agents. In other words, high type i–agents should receive higher information rents. These are the first effects of a “ban” on gender-based price discrimination when a separating allocation is implemented. Notice that w.l.o.g we can assume that Σ contains only three elements: Σ ¼ fp; pm ; pw g, where pm, pw and p ¼ inf Σ are chosen by h–type men, h–type women and l–type men and women respectively. Prices pm, pw and p must satisfy participation constraints: • High type men participation constraint: m

m

β shh −p ≥ 0;

• Low type men and women participation constraint:   min βm ; βw sll −p ≥ 0:

Prices p , pm and pw must also satisfy a total of eight incentive constraints, two for each gender/type combination: • High type men incentive constraints: βm shh −pm ≥ βm shl −p;

ð7Þ

βm shh −pm ≥ μ 0  βm shh þ μ ″  βm shl −pw ;

ð8Þ

where μ′ (resp. μ″) is the probability that a type-h man will match with a type-h (resp. l) woman if he deviates and chooses price pw. Note that μ′ coincides with μ = Pr{h − man|pw}, women' beliefs that a man who chooses pw has type h. Recall that the intuitive criterion implies extreme out-of-equilibrium belief μ ∈ {0, 1}. If μ = 1, assortative matching implies that he will match with a type h woman. If μ = 0, assortative matching implies that he is “punished” by being matched with a type l woman. It follows in particular that μ″ = 1 − μ. Inequality (8) therefore rewrites: βm shh −pm ≥ μ  βm shh þ ð1−μ Þ  βm shl −pw ;

ð9Þ

• High type women incentive constraints: βw shh −pw ≥ βw slh −p;

ð10Þ

βw shh −pw ≥ ν  βw shh þ ð1−νÞ  βw slh −pm ;

ð11Þ

where ν = Pr{h − woman|pm}(∈{0, 1}) is men' beliefs that a woman who chooses pm has type h (see the explanation before inequality (9)). • Low type men incentive constraints: βm sll −p ≥ βm slh −pm ;

ð12Þ

βm sll −p ≥ μ  βm slh þ ð1−μ Þ  βm sll −pw :

ð13Þ

• Low type women incentive constraints: βw sll −p ≥ βw shl −pw ;

ð14Þ

βw sll −p ≥ ν  βw shl þ ð1−νÞ  βw sll −pm :

ð15Þ

In the end, the matchmaker's problem writes: max Π ¼ λðpm þ pw Þ þ 2ð1−λÞp ðpm ;pw ;p Þ s:t: ð4Þ to ð15Þ; p ≤pm ; pw : In order to simplify the problem, we would like to identify a priori the binding constraints. The standard approach suggests to order agents according to their incentives to misrepresent their true type and, then, to ignore the “upward” incentive constraints. Unfortunately there is no such natural ordering in our context. In the following, I describe the case where high type men are more likely to pretend having a low type than high type women, i.e. where constraint (7) is “above” constraint (10). The opposite case can be treated similarly. Assumption 2. βm(shh − shl) ≥ βw(shh − slh).

ð4Þ

• High type women participation constraint: βw shh −pw ≥ 0;

41

ð5Þ

ð6Þ

Basically, Assumption 2 implies that pm will be higher that pw, therefore allowing us to ignore women' upward incentive constraints (11) and (15). Then, following the standard analysis of price discrimination, we ignore the high type men and women participation constraints (4) and (5). In the end, we consider the relaxed problem: max Π ¼ λðpm þ pw Þ þ 2ð1−λÞp ðpm ;pw ;p Þ s:t: ð6Þ; ð7Þ; ð9Þ; ð10Þ; ð12Þ; 14:

42

T. Trégouët / International Journal of Industrial Organization 42 (2015) 34–45

Clearly the low type men and women participation constraint (6) is binding so that p ¼ minfβm ; βw gsll . We are still left with five constraints. Notice that if constraint (9), that describes the incentives of high type men to choose pw, were absent, we could treat the men and women incentive problems separately, as in Section 3. Constraint (9) says that the price for high type men should not be too high compared with the price for high type women. In other words, there is now a link between the men and women incentive problems. Constraint (9) rewrites: pm ≤ pw þ ð1−μ Þβm ðshh −shl Þ; where μ = 1 if women believe that a man who chooses pw has type h. In words, if pw “signals” a high type (μ = 1), the matchmaker cannot charge a higher price pm for high type men. This happens if pw is sufficiently high so that choosing pw is (equilibrium) dominated for low type workers (i.e. if βm slh −pw b βm sll −p):  μ¼

1 if pw N p þ βm ðslh −sll Þ; 0 otherwise:

There are three cases to consider depending on whether μ is constant or not for all pw compatible with women incentive constraints (10) and (14): • when βm(slh − sll) N βw(shh − slh), the highest price compatible with women incentive constraints is lower than the lowest price compatible with men incentive constraints. Hence, pw can only signal a low type (μ = 0) so that the men and women incentive problems can be treated separately. Incentive constraints (7) and (10) are binding: pm ¼ p þ βm ðshh −shl Þ and pw ¼ p þ βw ðshh −slh Þ. • when βw(shl − sll) N βm(slh − sll), the lowest price compatible with women incentive constraints is higher than the lowest price compatible with men incentive constraints. Hence, pw can only signal a high type (μ = 1) so that pm = pw. Incentive constraints (9) and (10) are binding: pm ¼ pw ¼ p þ βw ðshh −slh Þ. • when βw(shh − slh) N βm(slh − sll) N βw(shl − sll), the highest (lowest) price compatible with women incentive constraints is higher (lower) than the lowest price compatible with men incentive constraints. Hence, a low pw signals a low type, while a high pw signals a high type. There are two candidate prices in this case (see Fig. 1 below): either the matchmaker offers the same price for high type men and women or it sets different prices. We conclude: Lemma 4. Assume that βm(shh − shl) ≥ βw(shh − slh). Let β ¼ minfβm ; βw g. When gender-based price discrimination is not feasible, the matchmaker implements the separating allocation with: 1. if βm(slh − sll) ≥ βw(shh − slh), Σ ¼ fp; pm ; pw g ¼ fβsll ; βsll þ βm ðshh − shl Þ; βsll þ βw ðshh −slh Þg and ΠSep ¼ 2βsll þ λðshh −βm shl −βw slh Þ; 2. if β w (s hl − s ll ) ≥ β m (s lh − s ll ), Σ ¼ fp; pm ð¼ pw Þg ¼ fβsll ; βsll þ βw ðshh −slh Þg and ΠSep ¼ 2βðsll þ λðshh −slh ÞÞ; 3. if βw(shh − slh) ≥ βm(slh − sll) ≥ βw(shl − sll), (a) if βm(shh − shl + slh − sll ) ≥ 2βw(shh − slh), Σ ¼ fp; pm ; pw g ¼ fβsll ; βsll þ βm ðshh −shl Þ; βsll þ βm ðslh −sll Þg and ΠSep ¼ 2βsll þ λ βm ðshh −shl þ slh −sll Þ; (b) if βm(shh − shl + slh − sll) b 2βw(shh − slh), Σ ¼ fp; pm ð¼ pm Þg ¼ fβsll ; βsll þ βw ðshh −slh Þg and ΠSep ¼ 2βðsll þ λðshh −slh ÞÞ. The case where βm(shh − shl) b βw(shh − slh) can be stated similarly. Proof. Cases 1 and 2 in Lemma 4 have already been established. Case 3 is similar to case 3 in the proof of Lemma 3. I therefore rely on a graphical proof (see Fig. 1). When βw(shh − slh) ≥ βm(slh − sl) ≥ βw(shl − sll)

Fig. 1. The set of prices pm and pw compatible with incentive and participation constraints (gray area) when βm(shh − shl) N βw(shh − slh) N βm(slh − sll) N βw(shl − sll). Thick line: constraint pm ≤ pw + (1 − μ)βm(shh − shl); black circles: candidate prices.

the optimization constraints define a non-convex polyhedron the extreme points of which are: ðpw ; pm Þ ¼ ðp þ βw ðshh −slh Þ; p þ βw ðshh − slh ÞÞ and ðpw ; pm Þ ¼ ðp þ βm ðslh −sll Þ; p þ βm ðshh −shl ÞÞ. Profits are Π = 2βw(sll + λ(shh − slh)) at the former, and Π = 2βwsll + λβm(shh − shl + slh − sll) at the latter. Comparison of these two profits yields the announced result. □ A.3. Proof of Proposition 2 The proof proceeds in two steps: first, I derive the matchmaker's profits in pooling allocations and I show that the matchmaker can achieve higher profits with an exclusive or a separating allocation if βm = βw; second, I show that, if βm = βw, if an exclusive allocation is implemented when gender-based price discrimination is feasible, then, it is also implemented when it is not. A.3.1. Step 1: Profits in pooling allocations There are three different pooling allocations to consider depending on whether all men and/or women join the matchmaker. A.3.1.1. All men and women participate. The matchmaker offers Σ = {pm, pw} such that all men and women participate, and men and women are randomly matched. Prices pm and pw must satisfy the low type men and women participation constraints: βm ðλslh þ ð1−λÞsll Þ−pm ≥0 and βw ðλshl þ ð1−λÞsll Þ−pw ≥0: If, for instance, pm N pw, all men (and women) should choose pw. Indeed, since pw is acceptable for both the high type and the low type men, a woman must believe a man who chooses pw have type l with probability 1 − λ and type h with probability λ. The matchmaker's profits in a pooling allocation are therefore given by:   ΠPool ¼ 2 min βw ðsll þ λðshl −sll ÞÞ; βm ðsll þ λðslh −sll ÞÞ :

T. Trégouët / International Journal of Industrial Organization 42 (2015) 34–45

Assume now that βw = βm and, for instance, shl ≥ slh so that ΠPool = sll + λ(slh − sll). There are two cases to consider.

43

discrimination is (not) allowed. We already noted that ΠgEx = ΠEx when βm = βw. Since we always have ΠSep ≤ ΠgSep, we conclude:

• If shl − sll ≤ shh − shl, then, we have: ΠgEx −ΠgSep ≥ 0⇒ΠEx −ΠSep ≥ 0:

ΠSep ≥ sll þ λðshh −shl Þ ≥ sll þ λðslh −sll Þ ¼ ΠPool : The first inequality comes from the fact that the matchmaker can always implement a separating allocation with only two prices. When shl ≥ slh (and βm = βw), this is done with Σ = {1/2sll, 1/ 2sll + 1/2(shh − shl)}. In other words, the matchmaker's profits are at least equal to sll + λ(shh − shl). The second inequality comes from increasing differences of the matching surplus. • If shl − sll N shh − shl, then, ΠSep = sll + λ/2(2shh − shl − slh) (case 1. in Lemma 4, Appendix A.2). Hence λ sll þ ð2shh −shl −slh −ðsll þ ðslh −sll ÞÞÞ; 2 λ ¼ ð2ðshh þ sll Þ−ð3slh þ shl ÞÞ; 2 λ ≥ ð2ðslh þ shl Þ−ð3slh þ shl ÞÞ ¼ λðshl −slh Þ ≥ 0; 2

ΠSep −ΠPool

¼

A.4. Proof of Proposition 3 The proof follows the same steps as the proof of Proposition 2. First, I show that the matchmaker can achieve higher profits with an exclusive or a separating allocation if shl = slh; second, I show that, when shl = slh and sll/slh ≥ slh/shh, if an exclusive allocation is implemented without the ban, then, it is also implemented under the ban. A.4.1. Step 1. Profits in pooling allocations

where the first inequality comes from increasing differences of the matching surplus. A.3.1.2. All men participate, only the high type women participate. We call this allocation a pooling M — exclusive W allocation. The lemma below gives an upper bound of the matchmakers' profits if it implements a pooling M — exclusive W allocation: Lemma 5. When gender-based price discrimination is not allowed, the matchmaker's profits in a pooling M — exclusive W allocation satisfies:
 Π ≤ λ  βm slh þ βw ðλshh þ ð1−λÞslh Þ :

In words, if an exclusive allocation is implemented without the ban then it is also implemented under the ban. This concludes the proof.

ð16Þ

A.4.1.1. All men and women participate. When shl = slh, the matchmaker's profits in a pooling allocation is given by (see the proof of Proposition 2 above):   ΠPool ¼ 2 min βm ; βw ðsll þ λðshl −sll ÞÞ: In the following, assume for instance that βw ≤ βm. There are two cases to consider. • if βw(shh − slh) ≥ βm(shl − sll), then, we are either in case 2 or 3 of Lemma 4: ΠSep

¼

8 9 0 1 < = max 2βw sll þ λβm @shh −shl þ slh −sll A; 2βw sll þ 2λβ w ðshh −slh Þ ; |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} : ; ¼0

Proof. Notice first that the matchmaker's profits in a pooling M — exclusive W allocation is lower when gender-based price discrimination is not feasible compared with the case where it is feasible. Then notice that the matchmaker's profits in a pooling M — exclusive W allocation when gender-based price discrimination is allowed is given by:

max

ðpm ;pw Þ

m

w

p þ λp

9 > > > =

≥ 2βw sll þ 2λβ w ðshh −slh Þ; ≥ 2βw sll þ 2λβ w ðshl −sll Þ ¼ ΠPool

where the latter inequality comes from increasing differences of the matching surplus. • if βw(shh − slh) b βm(shl − sll), then, case 1 in Lemma 4 yields ΠSep = 2βwsll + λ(shh − shl). Therefore ΠSep −ΠPool ¼ 2β
w sll þ λðshh −shl Þ−2βw ðsll þ λðshl −sll ÞÞ; ¼ λ ðshh −shl Þ−2βw ðshl −sll Þ ≥0;






λβm slh −pm ≥0 ¼ λ  βm slh þ βw ðλshh þ ð1−λÞslh Þ : > βw ðλshh þ ð1−λÞslh Þ−pw ≥ 0 > > ; βw ðλshl þ ð1−λÞsll Þ−pw b 0

where the inequality is obtained by noticing that shh − shl ≥ shl − sll (increasing differences) and βm ≥ βw ⇔ 1 ≥ 2βw.

This concludes the proof. □ Assume now that βm = βw. By Lemma 3, the matchmaker's profits in an exclusive allocation when gender-based price discrimination is not allowed is

A.4.1.2. All men participate, only the high type women participate. Let us show that the matchmakers' profits in an exclusive allocation is higher that the upper bound for profits in a pooling M — exclusive W allocation obtained in Lemma 5 in Appendix A.3, namely λ(βmslh + βw(λshh + (1 − λ)slh)). There are two cases to consider.

s:t:



ΠEx ¼ 2λβ w shh ¼ λ βm shh þ βw shh ≥λ βm slh þ βw ðλshh þ ð1−λÞslh Þ :

• Case 1. βw ≤ βm. By Lemma 3, if βwshh ≤ βmslh, then, 

ΠEx ¼ λshh ¼ λ βm shh þ βw shh ≥λ βm slh þ βw ðλshh þ ð1−λÞslh Þ :

A.3.1.3. Only the high type men participate, all women participate. Similar to case (b). To summarize, we have shown that, when βm = βw, either an exclusive or a separating allocation is implemented when gender-based price discrimination is not allowed.

Assume then that βw shh N βmslh . We are either in the second or third cases of Lemma 3, so that Π Ex = λ max {2βwshh, βm(shh + slh)} ≥ λβm(shh + slh). Notice then that

A.3.2. Step 2: Welfare comparison Let ΠgSep and ΠgEx (ΠSep and ΠEx) denote the matchmakers' profits in an exclusive and separating allocations when gender-based price


 β m ðshh þ slh Þ− βm slh þ β w ðλshh þ ð1−λÞslh Þ ¼ β m shh −βw ðλshh þ ð1−λÞslh Þ; ≥β m ðshh −ðλshh þ ð1−λÞslh ÞÞ; ≥0;

44

T. Trégouët / International Journal of Industrial Organization 42 (2015) 34–45

which shows that the matchmaker achieves higher profits in an exclusive allocation than in a pooling M — exclusive W allocation. • Case 2. βw N βm. By case 1 in Lemma 3, if βmshh ≤ βwshl, then, ΠEx = λshh ≥ λ(βmslh + βw(λshh + (1 − λ)slh)). Assume then that βmshh N βwshl. We are either in the second or third cases of Lemma 3, so that ΠEx = max{2βmshh, βw(shh + shl)}. Notice then that

2β w s ll . Assume that Π gEx − Π gSep ≥ 0, which is equivalent to λ ≥ s ll / shl by Proposition 1. Then, ΠEx −ΠSep ≥ ≥ ≥


 β ðshh þ shl Þ− βm slh þ βw ðλshh þ ð1−λÞslh Þ ¼ β
w ðshh −shlÞð1−λÞ þ βw −βm shl ≥0; w

≥ ≥

which shows that the matchmaker achieves higher profits in an exclusive allocation than in a pooling M — exclusive W allocation.

≥

sll m β ðs þ sll Þ−2βw sll ; shl  hl  s þ sll βw βm sll hh −2 m ; β  shl  s þ sll shh −sll − ; βm sll hh shl shh −shl m β sll ððs þ sll Þðshh −shl Þ−ðshh −sll Þshl Þ; shl ðshh −hl Þ hl m
 β sll s s −s2 ; shl ðshh −shl Þ hh ll hl 0:

In other words, we have shown that ΠgEx − ΠgSep ≥ 0 ⇒ ΠEx − ΠSep ≥ 0. A.4.1.3. Only the high type men participate, all women participate. Similar to case (b). To summarize, we have shown that, when shl = slh, either an exclusive or a separating allocation is implemented when gender-based price discrimination is not allowed.

w

g g shh þshl β ll Case 3. 2ðsshhhh−s −shl Þ b βm ≤ 2shh . Again, assume that ΠEx − ΠSep ≥ 0, i.e. m w λshl ≥ sll. We have ΠEx = λβ (shh + shl) and ΠSep = 2β (sll + λ(shh − shl)). Hence,

ΠEx −ΠSep A.4.2. Step 2. Welfare comparison g g Let ΠSep and ΠEx (ΠSep and ΠEx) denote the matchmakers' profits in an exclusive and separating allocations when gender-based price discrimination is (not) allowed. We show that: ΠgEx −ΠgSep ≥ 0 ⇒ ΠEx −ΠSep ≥0: Hereafter assume for instance that βw ≤ βm. By Lemma 3 and 4, we have:

¼

  βw βm λðshh þ shl Þ−2 m ðsll þ λðshl −shl ÞÞ β   s þ shl ðsll þ λðshl −shl ÞÞ ; ≥ βm λðshh þ shl Þ− hh shh s þ shl ≥ βm hh ðλshh −ðsll þ λðshl −shl ÞÞÞ; shh s þ shl ðλshl −sll Þ; ≥ βm hh shh ≥ 0: g

g

In other words, we have shown that ΠEx − ΠSep ≥ 0 ⇒ ΠEx − ΠSep ≥ 0. w

ΠEx

8 > > > > shh > > > < ¼ λ  βm ðshh þ shl Þ > > > > > > w > : 2β shh

hl b ββm . Notice that ΠEx − ΠSep = 2βw(λshl − sll). Therefore Case 4. shh2sþs g g hh ΠEx − ΠSep ≥ 0 ⇔ ΠEx − ΠSep ≥ 0. This concludes the proof.

βw s ≤ hl ; βm shh s βw s þ shl if hl b m ≤ hh ; shh β 2shh w s þ shl β if hh b m: 2shh β if

A.5. Proof of Proposition 4 Assume in the following that the matching market is dominated by men. A.5.1. Step 1: Profits in pooling allocation

ΠSep

8 > > > > shh −shl > > > < ¼ 2βw sll þ λ  βm ðshh −sll Þ > > > > > > w > : 2β ðshh −shl Þ

Now notice that since

sll shl

≥

shl shh ,

βw s −s ≤ hl ll ; βm shh −shl s −s βw shh −sll ; if hl ll b m ≤ shh −shl β 2ðshh −shl Þ w s −s β if hh ll b m : 2ðshh −shl Þ β if

the cutoffs in the two previous equa-

tions rank as follow:

Therefore, depending on the value of βw/βm, there are four cases to consider. Case 1. ββm ≤ sshhhl . Notice that ΠEx = ΠEx. Then, since ΠSep ≤ ΠSep, we have: ΠgEx − ΠgSep ≥ 0 ⇒ ΠEx − ΠSep ≥ 0. g

w

  ΠPool ¼ 2 min βw ðsll þ λðshl −sll ÞÞ; βm ðsll þ λðslh −sll ÞÞ : Then notice that, when the market is dominated by men, ΠSep ¼ 2βw ðsll þ λðshh −slh ÞÞ; ≥ 2βw ðsll þ λðshl −slh ÞÞ;  ≥ 2 min βw ðsll þ λðshl −sll ÞÞ; βm ðsll þ λðslh −sll ÞÞ ¼ ΠPool ; where the first inequality comes from increasing differences of the matching surplus.

shl −sll shl s −s s þ shl ≤ ≤ hh ll ≤ hh : shh −shl shh 2ðshh −shl Þ 2shh

w

A.5.1.1. All men and women participate. Recall that the matchmaker's profits in a pooling allocation is given by (see the proof of Lemma 2):

g

m ll Case 2. sshhhl b ββm ≤ 2ðsshhhh−s −shl Þ. We have Π Ex = λβ (shh + s hl ) and Π Sep = 2βw sll + λβ m (shh − sll ). Therefore, ΠEx − Π Sep = λβm (shl + sll ) −

A.5.1.2. All men participate, only the high type women participate. Recall that an upper bound for the matchmaker's profits in a pooling M — exclusive W allocation is λ(βmslh + βw(λshh + (1 − λ)slh)) (see Lemma 5 in the proof of Lemma 2). Then notice that, when the market is dominated by men ΠEx = 2λβwshh and

 2β w shh − β m slh þ βw ðλshh þ ð1−λÞslh Þ ¼ ð1−λÞβ w ðshh −slh Þ þ β w shh −β m slh :

To conclude, notice that the second term in the rhm of the above equation is positive since βwshh ≥ βwshl N βmslh.

T. Trégouët / International Journal of Industrial Organization 42 (2015) 34–45

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