Income Heterogeneity and Product Quality Choice∗ Serene Tan† National University of Singapore April 2018
Abstract This paper proposes a theory of segmentation of the product market when buyers’ incomes are heterogenous. Using a directed search model, ex-ante identical firms choose the product quality of a good to sell, knowing the aggregate income characteristics of potential buyers, though they cannot tell the income level of a buyer who goes to their firm. I show how income heterogeneity of buyers matters in determining the market structure. Segmentation of the product market by buyer income can be obtained as an equilibrium phenomenon, but need not be, and I characterize how changes in income inequality impact the market structure. JEL Codes: D40, E01, L10, L11 Keywords: Buyer Income Heterogeneity; Firm Quality Choice; Segmentation of Markets; Price Dispersion; Directed Search ∗
I would like to thank, for their helpful comments, Jim Albrecht, David Andolfatto, Kosuke Aoki,
Stéphane Auray, Saki Bigio, Pao-li Chang, Davin Chor, Seyed Mohammadreza Davoodalhosseini, Eric Fesselmeyer, Thanasis Geromichalos, Galina Hale, Nicolas Jacquet, Massimiliano Landi, Gea M. Lee, Nicolas LePage-Saucier, Jingfeng Lu, Sephorah Mangin, Pascal Michaillat, Alessandro Pavan, Brennan Platt, Guillaume Rocheteau, Alberto Salvo, Satoru Takahashi, Lawrence Uren, Venky Venkateswaran, Susan Vroman, Liang Wang, Steve Williamson, Randy Wright, Yu Zhu, as well as seminar/conference participants at the NUS Macro Brown Bag, Society for the Advancement of Economic Theory 2017, Workshop of the Australasian Macroeconomics Society 2017, ENSAI Rennes, Fall 2017 West Coast Search and Matching Workshop. The author has not had any other source of financial support beyond that given by her parent institution. † Email:
[email protected]. Department of Economics, National University of Singapore, AS2 #06-02, 1 Arts Link, S117570, Singapore. Tel: +65-65163964.
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1
Introduction
Many, if not all, of us, engage in an activity called shopping, or the purchases of goods from stores. In real life, when a consumer wants to purchase a unit of a particular good, say, a baby stroller, he will face a range of qualities of that good for sale, with some perceived as high quality, and others as low quality. From a consumer’s perspective, the range of qualities of a good offered by firms is exogenous to him since firms decide which qualities to sell. However, firms do not decide in a vacuum the qualities of a good to sell; in fact, firms have in mind who they want to sell to, i.e., their target audience, so what determines the range of qualities offered for sale, and their prices, that a consumer observes? A simple thought experiment would suggest that firms clearly care about who they are selling to: imagine the retail landscape in a neighborhood where the super wealthy reside (imagine some wealthy enclave in the Silicon Valley), and let us suppose the good in question is baby strollers. The range of qualities of baby strollers offered there will be overweight in the upmarket strollers, perceived as high quality, like Bugaboo, and underweight in the cheap and flimsy, presumably low quality, strollers. Compare this product market to that in one of the most impoverished neighborhoods in the country, and the opposite scenario in terms of quality choices offered will prevail. In this paper, a product market is modeled where firms are able to make use of aggregate income characteristics of buyers in their decision on the quality and price of goods to offer for sale. In my model, buyers only differ by income: a fraction of these buyers have high income and the rest have low income. To make the model realistic, I suppose that firms cannot observe the income of each buyer that comes into its store, but know the levels of
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income earned by each type of buyer, as well as the fractions of high and low income buyers, and will make use of this information. The first aim of this paper is to understand how buyers’ incomes influence the product qualities of a good offered for sale in a market, as well as the prices charged by firms. In other words, can who are you and your neighbors are influence what you see for sale in stores? The second aim of this paper is to understand if, and when, segmentation of the product market by buyer income type can take place. This is related more broadly to papers on segmentation and sorting (see Chade et al. (2017) for a recent survey paper on sorting in search and matching models). But it is also an interesting question because it appears to be relevant empirically. As the market structure is determined endogenously in my model, the third aim of my paper is to understand how changes in income inequality can matter for the market structure. To answer these questions, I use a class of models called directed search augmented to allow firms to condition their behavior on buyers’ aggregate income characteristics. In my model firms are allowed to choose to offer either a high or low quality good for sale, and each firm posts a quality-price advertisement which is observable to all buyers before they begin their search. The fact that a firm cannot observe the income of each buyer that steps into its store turns out not to matter, because in choosing its advertisement, and hence its target audience, in equilibrium only the right buyers, i.e., those its advertisement is geared to, will show up at the store; the fact that a particular buyer shows up at a firm gives the firm information about the income of that buyer. I view my model as a good approximation of reality, in that firms know broad income information about the population it is selling to, makes use 3
of this information in deciding what quality of a good to sell, as well as its price, and does not actually have to know each buyer’s income in selling the good. It turns out that one important object in determining the market structure is how the surplus of the high quality good compares to that of the low quality good, and there are three cases, where the former is higher, lower, or equal compared to the latter, denoted by Cases (), (), and () respectively. In Case (), the product market need not feature segmentation, and it turns out that when all buyers are sufficiently rich, even if there are income differences among the buyers, the equilibrium market structure features no segmentation: all firms offer the high quality good for sale; this is a pooling equilibrium. However, when low income buyers are sufficiently poor compared to the high income buyers, then I obtain a perfectly segmented equilibrium (PSE) where low (high) income buyers are in a marketplace with firms selling the low (high) quality good. In other words, the product market is perfectly segmented, and buyers of different income groups consume differently. My model predicts that stores catering to low income buyers are more crowded, i.e., there are more buyers per store, than stores catering to high income buyers, which seems to correspond to reality, where rich buyers do not buy their goods at the same stores as poor buyers, and there are less buyers per store at, say, Louis Vuitton, than Zara. When low income buyers have intermediate levels of income, then the segmentation that obtains in my model is a price dispersion equilibrium (PDE) where buyers of different types are in different marketplaces, firms in both marketplaces are offering the high quality good for sale, but at different prices: low income buyers are offered a lower price, which can be thought of as a discounted price, and high income buyers are offered the full, or regular 4
price. The reason this can be sustained as an equilibrium is because high income buyers are asked to pay more with the tradeoff that there are fewer buyers per store, and they prefer this outcome. It is interesting that there can be a PDE in my model, since all buyers in my model are alike in every way except income, and firms face the same cost of producing the high quality good no matter which marketplace they operate in. An empirical counterpart to a PDE is the coexistence of discount or outlet stores and regular stores, where the same good is sold at a discount in the former compared to the latter. Case () is interesting because buyers agree that the high quality good gives them higher utility, but this good offers a lower surplus to split between the buyer and the firm. Hence, the unique equilibrium is a pooling equilibrium where all firms offer the low quality good for sale, and all buyers are together in one marketplace; segmentation can never take place. Case () is a knife-edge case because a slight change in any of the relevant parameters will bring us to either Case () or (): the market structure is indeterminate, and there are multiple equilibrium market structures. The answer to the question raised earlier, “can who are you and your neighbors are influence what you see in stores?” is an emphatic yes, incomes of buyers in the market matter, but the characteristics of the goods in question also matter. These two, combined, will determine the equilibrium market structure, and whether segmentation is observed. Changes in income inequality may or may not matter for the market structure, and in my paper I can fully characterize how changes in income inequality impact market structure across all the cases. As my paper predicts that changes in income inequality in my model may manifest itself in changes in prices and/or quality offered, it provides a new lens to think about how income inequality affects all of us. 5
This paper is the first to my knowledge that tries to model, in a very parsimonious way, the importance of ex ante identical firms using buyers’ income information (where buyers are otherwise completely identical) in deciding on the quality of a good to offer for sale, thus endogenizing the range of qualities of goods observed for sale in a product market, as well as solving for the equilibrium market structure, without any restriction on where agents are searching, and without any restriction on the number of marketplaces that are open. Segmentation in my paper is by buyers’ income, and is an endogenous outcome. And because I allow the number of marketplaces that are open to be an endogenous object, I propose a new definition of equilibrium to deal with the potential opening up of marketplaces.
1.1
Related Literature
Related Literature. This paper takes as a starting point the canonical directed search model of Burdett et al. (2001) (for a recent survey on directed search models, see Wright et al., 2017). but it does two things differently: it explicitly incorporates a budget or income constraint on the buyer’s side, which the firm takes into account, and it explicitly allows firms to choose between different qualities of a good to offer for sale, and it is the interaction of these two elements that is important in my paper. In this paper, I make a distinction between a buyer’s valuation of the good (which measures his willingness to pay) and his ability to pay for the good (via the income constraint), and I would argue that “willingness” and “ability” are two very different concepts - as a buyer I may like Ferrari cars, and if I had the money I would consider buying one, but I cannot afford to. I also make explicit budget constraints buyers have, which firms can
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incorporate into their decision making process. In Burdett et al. (2001), as well as in almost all papers in the micro literature, buyers are said to be able to pay up to their “reservation price” which is normalized to some constant, but what exactly this amount is, whether buyers are willing to pay up to their valuation, or whether buyers are only able to pay up to a certain income they have, is left unsaid. My paper makes it very explicit that these two concepts are not the same, and this distinction matters. My paper is also related to a large literature in search and matching which is interested in sorting and segmentation, like Eeckhout and Kircher (2010), and Menzio and Shi’s (2010) notion of block-recursive equilibria, but buyers in my model have the same preferences and only differ in income, and firms in my model are ex ante identical, so sorting in my paper, should it occur, is by buyer income type, which is novel compared to the existing literature. My paper is also related to the literature examining price dispersion, like Aguiar and Hurst (2007) and Kaplan and Menzio (2015), as well as consumption inequality like Attanasio and Pistaferri (2016), and my contribution is to propose a potential avenue, income heterogeneity of buyers, which can have implications for firms’ quality choice offerings and prices. My paper is also related to Guerrieri et al. (2010): in that paper agents are ex ante heterogeneous, an agent’s type is private information, and uninformed principals use the posted terms of trade to attract certain types and screen out others. My paper also has the flavor of sellers potentially (but not necessarily) posting advertisements to screen out some types of buyers, but Guerrieri et al. (2010) consider an adverse selection problem and consider separating equilibria only, whereas my paper is not about adverse selection and I do not restrict equilibria to be separating or pooling. This paper is also related to a large literature on second-degree price discrimination or 7
non-linear pricing (survey papers include Stole (2007) and Armstrong (2015)), but firms in this literature are modeled as either oligopolistic or monopolistic, and firms tend to create different product lines to segment the product market in the presence of unobserved buyer heterogeneity. I have a continuum of firms in my model, I do not have buyer heterogeneity beyond income levels, and firms segment the market in my model, potentially, by buyers’ incomes. The PDE in my model actually has some flavor of third-degree price discrimination. This paper is also related to the literature on trade which takes seriously firm heterogeneity and the quality choices made by firms, like Fajgelbaum, Grossman, Helpman (2011, JPE); Faber and Fally (2015), but these papers have frictionless product markets with monopolistically competitive firms, whereas I model a frictional product market where atomistic firms “direct” buyers’ search. Lastly, my paper is related to the literature in macroeconomics which discusses measurement issues in GDP and how GDP accounting does not account properly for consumption changes which reflect changes in the qualities of goods consumed. In my paper, when the incomes of buyers change, the market structure can change totally as firms offer different qualities for sale or keep the qualities offered the same but change the prices charged. Hence, measuring consumption through just consumption expenditure does miss out quite a bit. Moreover, as my paper is able to examine the impact of income inequality on market structure, it suggests that it may be fruitful to consider examining frictional product markets in macroeconomics, rather than assume them as frictionless as is the norm currently. Here is the plan for the rest of my paper. In section 2, I set up the model, define equilibrium, and discuss modeling choices; Case () equilibrium is characterized in section
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3; sections 4 and 5 deal with Cases () and () respectively; and section 6 concludes. All proofs not in the main text are in the Appendix, along with related literature.
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A Model of the Product Market
Participants There is a continuum of ex ante identical risk neutral firms of measure Firms can sell one unit of a good, and they face a product quality choice: they can choose to either offer for sale a unit of a high quality good at cost , or a low quality good at cost , . There is a continuum of risk neutral buyers of measure who each want to purchase one unit of an indivisible good. Buyers differ only in incomes: a fraction are high income buyers with income, and a fraction (1 − ) are low income buyers with income, . Both types of buyers value the good at and the good at , 0.1 Each buyer’s income is known only to himself, and is unobserved by firms, nor other buyers. However, the firm is aware that in the aggregate the income characteristics of the buyers are ( ), though the firm does not know which buyer has what income.2 Quality-Price Posting Game This game is modeled as a static game, like in Burdett et al. (2001).3 In stage 1, each 1
It is possible to let buyers of different income groups have different valuations, e.g., buyers have a higher valuation for the quality good than buyers, which provides an impetus for firms to segment the market by income groups. I prefer to keep the model parsimonious and let buyers differ only in incomes, and not valuations. 2 Instead of assuming unobserved buyer heterogeneity, it is possible to assume that the income type of each buyer is perfectly observable, but that firms are not allowed to price discriminate based on the characteristics of an individual, including income, i.e., first degree price discrimination (FDPD) is ruled out by assumption. In the real world, FDPD is often not illegal, but as consumers seem to think it unfair that firms carry out FDPD, many firms end up not doing it. I prefer the intepretation I gave in the main body of the paper, that the income type of individual buyers is unobserved, so firms cannot carry out FDPD. 3 It is possible to model this as a dynamic game, where a firm exits the market upon selling the good and
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firm decides the quality ∈ { } of the good to offer for sale as well as a price to charge, and this firm will post an advertisement, or announcement, { } ∈ { }. Let the set of all firm advertisements be F. In Stage 2, F is publicly observed. All buyers then decide which firm(s) to buy the good from, and each buyer can make one visit to a firm. We assume anonymity. From a seller’s point of view, buyers are anonymous to it, in that if more buyers show up than it has capacity for the firm will randomly allocate the good it has to a buyer, with the consequence that all other buyers are rationed. From a buyer’s point of view, firms are also anonymous to him in that all firms posting the same advertisements are treated identically. In addition, I also assume that firms offering the same expected utility to a buyer are also treated identically by a buyer. Frictions arise here because buyers are unable to coordinate their visit decisions; these are called coordination frictions, in that there can be more buyers showing up than firms have capacity for, so some buyers are rationed. Assumptions Let me assume the following. (0) : and . The surplus of a good of quality that is produced is ( − ), ∈ { }. If the surplus were negative, the problem is uninteresting because no good will be offered for sale; if the surplus were zero, there is effectively nothing left to split between the firm and the buyer, and hence, also uninteresting. is replaced by a new entrant; buyers, once they purchase a good, also exit the market and are replaced by clones. In this dynamic setup, at any point in time, we are just solving a static game, like that laid out here.
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Another assumption I make is: (1) : ≥ and ≥ . The first (second) inequality implies that the richest (poorest) buyers, the ( ) buyers, can afford to purchase the highest (lowest) quality good (), if offered, as they are willing to pay up to their valuation ( ) for that good.
2.1
Solving for the Equilibrium
I first solve for symmetric equilibria where buyers of the same type use identical visit strategies and firms within a marketplace offering quality = are posting the same advertisement { }. In Section 3, I show there does not exist equilibria where there is a distribution of prices posted within a marketplace for a particular quality of good sold. For firm that posts { } = , the realized number of buyers that show up at this firm is a Poisson random variable with the Poisson parameter the average number of buyers at the firm, which is the expected queue length, denoted as and termed the queue length at firm which is selling quality . Given the advertisements of all other firms −, suppose the highest expected utility that a buyer the last firm is trying to sell to is , and suppose firm produces quality and offers to a buyer an expected utility of . If , then = ∞, implying that the probability of obtaining the good for a buyer who goes to this firm is zero, contradicting , so this cannot be true. Hence, it must be that ⎧ ⎪ ⎪ ⎨ then = 0; and ⎪ ⎪ ⎩ = then = [0 ∞) 11
Firm ’s expected profit when choosing quality , , is given by ( 1), where is the price charged by firm when posting quality : ( 1)
:
¡ ¢ − = (1 − ) − max
−
1−
¡ ¢ − = and
(1)
≤ ∈ { }
(2)
This firm will choose the quality to max { }. For firm , given a quality it has decided to sell, it takes as given ( is an endogenous object that will be determined in
equilibrium). The probability to firm of having no buyer show up is − , so (1 − − ) is the probability that at least one buyer shows up, and conditional on that happening, firm earns ( − ). Firm wants to maximize its expected profit, but it faces two constraints. The first constraint, equation (1), states that its choice of , and the implied , is such that the buyer that comes to it has an expected utility of , where a buyer’s expected utility
is the probability that he is served, which is (1 − − ) , multiplied by his payoff in the event he is served, which is ( − ). The second constraint is a budget constraint: the price charged by firm has to be such that the targeted audience of firm can afford that good: in this directed search setup, firms, when deciding what quality to sell, will be trying to influence how buyers come to it, and hence, will take the budget constraint of the buyer it intends to sell to in its optimization problem. Clearly, if firm wants to sell to all buyers or buyers only, the relevant constraint in (2) is ≤ , whereas if firm wants to sell to buyers only, the relevant constraint in (2) is ≤ 4 4
More precisely, in ( 1), depending on which buyers firm wants to sell to, which pins down = { } in equation (2), the expected utility of a buyer, in equation (1), refers to the expected utility of a buyer the firm is trying to sell to, which depends on which buyers the firm is targeting. But this complicates notation, so I chose to write ( 1) as in the above, but it should be clear from the context what means.
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2.2
Definition of Equilibrium
There are two types of equilibria in this paper: a pooling equilibrium where all buyers are together in one marketplace; and a separating, or segmented equilibrium, where buyers of different income levels are in different marketplaces. Without loss of generality, in a segmented equilibrium let and buyers be in marketplaces 1 and 2 respectively. I first define equilibrium in the usual Nash sense of being robust to individual deviations. Definition 1
A pooling equilibrium in quality ∈ { }, consists of {∗ ∗ ∗ } s.t.:
() each firm solves its profit maximization problem ( 1); () ( ∗ ) ≥ ( e ) for e 6= ∗ ;
() ( ∗ ) ≥ (0 0 ) for 0 6= and some 0 ; () ∗ is the solution to (1) ; and () firms and buyers weakly prefer to participate, i.e., ( ∗ ) ≥ 0 ∗ ≥ 0. In a pooling equilibrium only one type of quality is offered for sale, and all buyers can be thought of as being in the same marketplace as all firms. For a pooling equilibrium to be robust to individual deviations, a firm cannot deviate to offer a different price for that quality offered and be made strictly better off (part ()), nor can a firm offer a different quality good at some price and be made strictly better off (part ()). On the buyer’s side, the only individual deviation the buyer can undertake is not to participate in the marketplace, so part () states that in equilibrium a buyer has to weakly prefer to participate than not, and this is also true for a firm. In a segmented equilibrium let = be the quality offered in marketplace = 1 2. ∗ ∗ Definition 2 A segmented equilibrium consists of { ∗ ; ∗ }, = 1 2 ∈
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2 2∗ { }, {1 1∗ 1 } 6= { 2 }, s.t.: 2 2 2∗ () ∗ , the fraction of firms in marketplace 1, solves 1 (1 1∗ 1 ) = ( 2 );
() each firm solves its profit maximization problem ( 1); () ( ∗ e ) for e 6= ∗ ) ≥ ( = 1 2;
e e 6 and some , = 1 2; and () ( ∗ ) ≥ ( ) for =
() ∗ is the solution to (1) for marketplace = 1 2;
() for a buyer in marketplace = 1 2 with income level and expected utility ∗ , either 0
0
0
∗ ≥ 0∗ for 0 6= or if ∗ 0∗ that 0∗ ; and ∗ () firms and buyers weakly prefer to participate, i.e., ( ∗ ) ≥ 0 ≥ 0, = 1 2.
In a segmented equilibrium, buyers of different incomes are in different marketplaces, and 2 2∗ because {1 1∗ 1 } 6= { 2 }, one marketplace is not simply a duplicate of another. In this
equilibrium, firms are indifferent between being in either marketplace, and the fractions of firms operating in marketplaces 1 and 2, ( ∗ 1 − ∗ ), are solved such that this indifference holds, as in part () of the definition above. An equilibrium is defined to be robust to individual deviations in that a firm in a marketplace cannot be made strictly better off offering a different price (part ()), or by offering a different quality in that marketplace for some price (part ()). A segmented equilibrium is also robust to individual deviations on the buyer’s side, in that buyers of an income level in marketplace are either doing 0
the best they can (∗ ≥ 0∗ for 0 6= as in part ()), or that if they are getting a lower 0
expected utility level where they are at (∗ 0∗ ) that they cannot afford to purchase the 0
good in the other marketplace ( 0∗ ). Lastly, both firms and buyers have to be weakly better off participating than not, as stated in part ().
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In a segmented equilibrium, it can be that 1 6= 2 , where different qualities are offered to buyers of different incomes, which we call a perfectly segmented equilibrium (PSE); or 1 = 2 , where the same qualities are offered in both marketplaces, but the prices posted across the two marketplaces are different, which we call a price dispersion equilibrium (PDE). In a segmented equilibrium where buyers of different income levels sort into different marketplaces, a marketplace can denote some physical space, so two marketplaces will correspond to two different physical spaces. However, a marketplace can simply be an abstract concept, and both types of buyers are physically inhabiting the same space, everyone sees all advertisements posted by all firms, but the two types of buyers effectively partition the advertisements into two sets: one set of advertisements meant for people like him with his income type, and another set of advertisements meant for others, and he will ignore the latter set as it does not concern him. The notion of a pooling or segmented equilibrium being robust to individual deviations as discussed above takes the market structure as given and asks if, given this market structure, any individual agent would be better off doing something else. In my model, as I allow firms to choose the product quality to offer, the market structure is endogenous, so as to completely characterize equilibrium, it is important to ask how agents would have chosen among the equilibria with different market structures that were not offered. In this offequilibrium path consideration, since I have a continuum of firms and a continuum of buyers, it would make sense that the marketplaces that were potentially opened could be populated by a measure of agents in such a way that the queue lengths that would result in these marketplaces are well defined. Hence, I allow firms to deviate in a group. The question is 15
what members of the group are doing in deviating, so structure has to be put on what these deviations are. The second part of the definition of equilibrium is in terms of the equilibrium being robust to group, or coalitional, deviations, so let me first define what it means to deviate profitably in a group.
Definition 3 A group deviation is said to be profitable if there exists a mass 0 of firms that can deviate to set up their own marketplace by posting an advertisement { , },
¡ ¢ where = is the quality offered, are the deviation price and promised deviation queue length which all firms commit to, and ∈ { buyers, buyers, all buyers} is the targeted income group these deviating firms would like to attract, such that: () deviating firms are no worse off; () deviating buyers who are targeted and who go to the new marketplace are no worse off; () at least one type of deviating agents is made strictly better off; () if a buyer of a targeted income group is made no worse off in this new marketplace then all buyers of that type would go to the new marketplace; and () a buyer from a non-targeted group will not strictly prefer to go to this new marketplace, and will not go to the new marketplace.
The definition of group deviations allows precisely for the creation of new, and competing marketplaces by agents. However, the structure put on a group deviation is that when a group of firms deviate they post the same (competing) advertisement which they commit to, and the assumption of anonymity prevails. The reason for the structure imposed is that since a new marketplace can potentially be open, it would seem reasonable to ask that this 16
new marketplace, when open, will operate like any other marketplace that would prevail in an equilibrium, so we assume anonymity, commitment, and look for a symmetric outcome. In this group deviation, deviating firms post four objects { , }. The third object, the deviation queue length, is somewhat redundant, since
© ª conveys sufficient
information. But to make clear that group deviations have the structure imposed above
about anonymity, commitment, etc., I include it in the competing advertisement. Note that this definition rules out duplications of marketplaces because part () of definition 3 has that at least one type of agents deviating is made strictly better off. Definition 4 A pooling equilibrium or segmented equilibrium of the two-stage game as defined in definitions 1 and 2 respectively is said to be stable if there does not exist a profitable group deviation. That is, an equilibrium has to be robust to both individual and group deviations. In this paper, I first solve for equilibria that are robust to individual deviations, and then check if those equilibria are robust to group deviations. Some comments about the definitions of equilibrium The literature has dealt with “off-equilibrium path” advertisements in several ways. The closest two to this paper are Moen (1997) and Jacquet and Tan (2007). In Moen (1997), marketmakers can post competing advertisements “off the equilibrium path,” but the restriction is that the new marketplace has the same queue length, and hence the same matching probability as that in the candidate equilibrium, which is essentially a “trembling hand” type of argument. Because of this, it is hard to use Moen’s definition to think about firms deviating and creating a new marketplace. In Jacquet and Tan (2007) one interpretation 17
of their off the equilibrium path advertisements is that a marketmaker can create a new marketplace with a mass of types of agents (they have heterogeneous types in their model), and ask if deviating agents can be made better off; hence, their definition of equilibrium is such that it is robust to the creation of these marketplaces. But that model has agents searching randomly to form a match, and does not allow firms to post advertisements to direct buyers’ search. In this paper, the way I handle off the equilibrium path beliefs has elements of these two papers. I want to allow a new marketplace to be opened potentially, but I allow the competing advertisements to have any queue length potentially, unlike in Moen (1997). In what I do, deviating firms can choose the quality they wish to sell as well as its price, and compute whether it is in their best interest to sell to buyers only, or sell to buyers only, or sell to everyone, and by choosing the mass of deviating firms, effectively pins down the queue length. And, to discipline this deviation, I do not force anyone to go to this new marketplace. In fact, in part () of definition 3 deviating firms cannot be made worse off with this deviation, and in part () deviating buyers are also made not worse off, with at least one type of deviating agents made strictly better off in part () of the definition (the strict inequality rules out duplicate marketplaces). Very importantly, this deviation is optimal because the buyers who are not meant to go there, i.e., the advertisement is not for them, will optimally choose not to go, as in part () of definition 3.56 5
Heuristically, the way I think about trembles in papers like Moen (1997) is that they are checking for equilibrium “locally,” whereas since marketplaces can be opened potentially in my paper I need a more “global” argument to check for equilibrium. 6 When considering group, or coalitional, formation in this paper, one way to map games of cooperation to games of non-cooperation is to consider the notion of coalition-proof equilibrium in non-cooperative settings, which is in turn related to work on “coalitional rationalizability,” as in Ambrus (2006) and Luo and Yang (2009), to understand how, in non-cooperative settings, groups or coalitions of agents can act in their collective interest. Alternatively, and more straightforwardly, one can interpret firms coordinating
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In definition 3, I allow group deviations of this form: suppose the candidate equilibrium is a pooling equilibrium where all buyers enjoy expected utility ∗ but a group of firms would like to deviate which target only one group of buyers, say the buyers, who are offered a deviation expected utility = ∗ . By part () of this definition, all buyers will go to the newly created marketplace, but notice that buyers are assumed by part () of this definition not to go to this new marketplace, even though = ∗ . In this example I have constructed, a group deviation is profitable so long as deviating firms are strictly better off, which would render this candidate equilibrium not stable. My assumption about how buyers choose to move (or not) is not as unreasonable or restrictive as it first appears, because a profitable group deviation is simply that which takes place in order to exploit some gain in surplus, so in this example, deviating firms are the ones reaping the entire gain in surplus, but I could have assumed that these firms give a small amount 0 to each deviating buyer, so the targeted income group is now strictly better off and they would strictly prefer to move to this new marketplace, whereas the non-targeted income group, the buyers, are indifferent, and part () of the definition states that in the case of indifference they do not move. Some Comments about the Setup Coordination frictions play an important role in all directed search papers. In the baseline model of Burdett et al. (2001), firms have one unit of a good to sell; and they also considered firms with two units of a good. This assumption can be relaxed to allow firms to offer a general number of goods for sale, as was done in Watanabe (2010). It is not important deviations in groups in a stage 0 that opens prior to the game laid out above as pre-play communication, with all results in my paper going through.
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what the capacity constraint exactly is, but that there is a capacity constraint at all. The whole point of coordination frictions is to capture the idea that buyers can be rationed, or that stockouts can occur, and this can occur whether the capacity constraint is 1, 10, or 100. Bils (2016) examined stockout probabilities in micro-CPI data for 1988-2009, across 20 goods categories and he finds that stockouts occur frequently, at an average of 4.6% over the sample period and across all goods. He also finds that stockouts occur whether the economy is booming or in a recession; and the average stockout rate can be as low as 1.6% over the sample period in “tires & vehicle parts” and “alcohol & tobacco,” or as high as 9% in “jewelry, watches, luggage.” The point is that stockouts are frequently observed in reality, which is an empirical fact that directed search models can generate through coordination frictions. In this paper I have adopted a directed search model à la Burdett et al. (2001) where the number or measure of firms is a constant, and asked how firms strategically choose which marketplace to go to, and even allow them the possibility of creating new marketplaces. In what I do, firms have to be indifferent across being located in the different marketplaces. A competing but parallel narrative, called competitive search, also stresses the importance of coordination frictions, but has taken a different approach, with the existence of a marketmaker which decides on the number of marketplaces to open, and the measure of firms in each marketplace is pinned down by a free entry condition, i.e., firms will enter a market till the expected profit net of a constant fixed cost that has to be paid to be active, is driven down to zero. In this latter narrative, firms are also indifferent in equilibrium across where to locate themselves, just like in the earlier narrative, but this fixed cost is an important consideration, which is something purely exogenous. I prefer the narrative implied by directed 20
search models, where firms can strategically choose where to locate themselves, which is not dependent on an exogenous cost parameter. I also view the directed search approach as more internally consistent with my consideration of how agents behave off the equilibrium path, since I wish to fully endogenize the market structure, including the number of marketplaces. Comment #1 in the Appendix’s discusses other modeling choices.
2.3
Preliminaries: Equilibrium Characterization
It turns out that it is important to compare the surplus of the high quality good to the low quality good, where the surplus is how much of a pie there is to split, and since, conditional on a match, a firm has a payoff of price minus cost, and a buyer has a payoff of valuation minus price, the surplus of the match is just valuation minus cost. There are three cases:7 () : − − ; () : − − ; and () : − = − .
Case (): − −
3
The first thing to note is that in an equilibrium, it cannot be that a group of buyers are inactive, in the sense that the advertisements by firms exclude them from participation in a market (see lemma A.0 in the Appendix). Hence, we know that in any equilibrium, all buyers are served, so they are active. To start, define the average measure of buyers to sellers as = , and let 7
( − ) − = − for ∈ { } 1 − −
See Comment #2 of the Appendix for a discussion on which case fits reality better.
21
(3)
Let us first solve for stable pooling equilibria (PE). There are only two outcomes for PE: one where all firms are selling to all buyers in one marketplace; and another where all firms are selling to all buyers in one marketplace. In a pooling equilibrium, given that all other firms are posting { }, the last firm is solving his optimization problem in ( 1). The budget constraint for buyers in equation (2) in a PE is ≤ . Rewriting equation (1) as = −
1 − −
(4)
and substituting into the firm’s objective function, the firm’s problem is: ∙ ´ ³ − ( − ) − + − + L= 1− max
¸ 1 − −
where is the multiplier on equation (2). The first order conditions (F.O.C.s) to the above problem (as shown in Lemma A.1 in the Appendix, this problem is strictly concave in ) are both necessary and sufficient, and − ∗ then using the fact that in a symmetric equilibrium ∗ = = − , = = = , so
−
∗ (1 − − − − ) ( − ) − + = 0; (1 − − )2 ≤ ; − 1 − − µ ¶ ∗ = 0; ∗ ≥ 0 − + 1 − −
(5) (6) (7)
If the budget constraint does not bind, ∗ = 0, substituting (5) into (4): ∗ = −
( − ) − = ; 1 − −
∗ = = ( − ) − ; and ¡ ¢ ∗ = ( − ) 1 − − − − . 22
(8)
And if the budget constraint binds, ∗ ≥ 0, from (7), we have that − +(1−− ) = 0, or that ∗ = , which satisfies all F.O.C.s (5) − (7) above.
3.1
Suppose ≥
Referring to equation (8), let us suppose that = . And let us suppose that the buyers are sufficiently rich, in that ≥ , where this threshold, , is the price posted in equilibrium if all buyers and sellers were together in the same marketplace when firms are all selling quality , and poor buyers can afford to purchase good. It may be tempting to think the stable equilibrium when ≥ is where all firms and buyers are indeed in one marketplace but that firms post ∗ = . That is, firms extract all surplus from trade. Lemma 1 When ≥ , an equilibrium where all firms post { ∗ = } is not stable. The intuition why an equilibrium where firms extract all the surplus is not stable is because a buyer’s expected utility from participating is zero; hence, a group of firms can propose to create a new marketplace offering deviating buyers a strictly positive expected utility 0 which is arbitrarily small, and deviating firms can choose a deviation queue length and price combination, ( ) such that each deviating firm makes strictly higher expected profit. It is also interesting to note the following. Lemma 2 When ≥ , a segmented equilibrium is not stable. The following corollary is a direct implication of lemma 2:
23
Corollary 1 When ≥ , an equilibrium where one type of buyers is in a marketplace with firms posting { ∗ = } and another type of buyers is in a marketplace with firms posting { ∗ = } is not stable. The intuition why a segmented equilibrium is not stable when is sufficiently high is the following. Since quality has a higher surplus than quality , firms would like to offer if they can, simply because there is more surplus to split between the firm and the buyer. When buyers are sufficiently wealthy, in that ≥ , they are wealthy enough that they could afford to pay for this quality . Hence, firms want to be in a pooling equilibrium with all other firms and all other buyers where only is produced and consumed. An equilibrium where buyers are charged their maximum willingness to pay, as measured by their valuation of the good, is not an equilibrium because it is not stable: a group of firms can get together and offer a slightly positive (and higher) expected utility to the deviating buyers they are trying to attract, and because these firms when they create a new marketplace can post a deviating queue length, can effectively choose a price and queue length pair such that they are maximizing their expected profit when deviating. From the above, one would expect that when ≥ a stable equilibrium is a pooling equilibrium where is sold, and that is correct. Proposition 1 () When ≥ , there exists a unique stable equilibrium which is a pooling
© ª equilibrium where all firms face a queue length of and post ∗ ( ) = , ( − ) − = ; 1 − − ¡ ¢ ∗ ( ) = 1 − − − − ( − ) ; and ∗ ( ) = −
∗ ( ) = − ( − ) 24
(9) (10) (11)
() When , there does not exist any stable pooling equilibrium. Part () of Proposition (1) is very intuitive. As we are in Case (), quality good has a higher surplus than quality , and since all buyers prefer quality (as ) firms will choose to offer quality if they can, and firms can indeed do so if the poorest buyers in the economy are rich enough and are able to afford to buy that good. From an ex ante point of view, a firm does not always manage to form a match since a firm could have no one showing up, which is why a firm’s expected profit is his share of the surplus ¡ ¢ (∗ ( ) − ) multiplied by the probability the match is indeed formed, which is 1 − − ,
and on rewriting, it works out to be equation (10), which states that a firm captures a share ¡ ¢ 1 − − − − of the total surplus ( − ). As for buyers, their expected utility is their
share of the surplus ( − ∗ ( )) multiplied by the probability they can form a match
¡ ¢ (i.e., they are not rationed), which is [ 1 − − ], and on rewriting, it works out to be equation (11) where a buyer captures a share − of the total surplus ( − ).
What is less obvious is part () of proposition 1. It is tempting to think that if the budget constraint of the buyers “binds,” in that the buyers cannot afford to pay the price that would prevail in this equilibrium, that the stable equilibrium in this economy is a pooling equilibrium with all firms selling the quality good and charging a price up to . Corollary 2 When , there does not exist a stable pooling equilibrium where all firms post { ∗ = }. This result follows directly from part () of proposition 1. The intuition is: since is generically not an “optimal” price for a firm, if a firm had to offer an expected utility corresponding to that which would be obtained by buyers in this candidate equilibrium, 25
there is a profitable deviation for a mass of firms to offer an “optimal” price to buyers, who can afford to purchase this good, and earn more. More generally, when there does not exist any stable pooling equilibrium where firms are posting { e ≤ }.
One can think that if , then surely the stable pooling equilibrium for this economy
© ª is for all firms to post ∗ = since all buyers can afford to pay for the good, and as the price posted is optimal? It turns out that this candidate equilibrium is not stable, for
the reason that the good has a lower surplus than the good, so it is possible to find a mass of firms to deviate to create their own marketplace offering the good to buyers, who can afford the good, and that will satisfy conditions () − () of definition 3, i.e., there exists a profitable group deviation.
Corollary 3 In the stable pooling equilibrium of part () of Proposition 1, changes in or in such a way that ≥ will not affect the equilibrium. Neither will changes in affect the equilibrium.
This follows directly from the proof of proposition 1. In this pooling equilibrium, the poorest agents in the economy are sufficiently rich, or that ≥ . In this model, firms do not know the identity of any of the individuals; all firms know is that any buyer who steps into its store can afford the good, for otherwise the buyer will not be there. Hence, if there are changes in or in such a way that ≥ or if there are changes in , firms know that a buyer who steps into its store can still afford to buy the good, which is why the equilibrium outcomes are not affected.
26
3.2
Suppose
Now suppose that buyers have incomes that are not high enough, that is, . From part () of proposition 1 we know that any equilibrium, if it exists, must be a segmented equilibrium. It is tempting to think that there is an equilibrium where firms segment the market and charge the maximum willingness to pay, or the valuation of each individual, but this is not true. Lemma 3 When there does not exist a stable segmented equilibrium where buyers are in marketplace 1 with firms posting { ∗ = } and buyers are in marketplace 2 with firms posting { ∗ = } The intuition for the result is the following: if each individual pays his valuation for the good, his net payoff is zero, which means that it is easy to find a profitable group deviation where deviating firms attract some, or all, buyers with a small, but positive, net payoff, and the deviating queue length is such that deviating firms lose a bit by sharing some surplus with the deviating buyers, but gain by an appropriate choice of deviating queue length,8 so overall, deviating firms are made no worse off, and actually can be made better off. Now let us suppose that buyers are so poor that they are unable to pay for quality , so any equilibrium, if it exists, would have firms sell quality () to ( ) buyers in one marketplace, which I call a perfectly segmented equilibrium (PSE). To begin, suppose is low, and to be precise, let us suppose ≤ , and define condition (0 ) as follows: (0 ) : 8
¡ ¢ − 1 − −1 − 1 −1 |1 = −
In other words, it is possible for an appropriate choice of the mass of deviating firms to ensure that this latter effect dominates the former effect.
27
Assume (0 ) holds henceforth. Define the queue length of a firm in marketplace as , = 1 2. If were the fraction of firms operating in marketplace 1 where buyers are, it must be that 1 = , and 2 = [(1 − ) ] (1 − ). Proposition 2 Suppose ≤ . There exists a unique stable PSE where a fraction ∗ of firms are selling the quality good to buyers, and (1 − ∗ ) fraction of firms are selling the quality good to buyers, with ∗ the unique solution to: ¡ ¡ ∗ ∗ ¢ ∗ ∗¢ ∗ − 1 − − − ( − ) = 1 − − − ∗ − ( − )
(12)
∗ ∗ where = ( ) = ∗ , ∗ = ∗ ( ) = [(1 − ) ] (1 − ∗ ) firms selling quality
post { ∗ ( )}, ∈ { }, with ∗
( − ) ∗ − ( ) = − ; ∗ 1 − − ¡ ∗ ∗¢ ∗ ( ) = 1 − − − ∗ − ( − ) ; ∗
∗
∗ ( ) = − ( − )
(13) (14) (15)
The idea is that if the poorest agents in the economy are too poor to afford the good, which is true if ≤ ,9 then the only equilibrium in this economy is a separating equilibrium where buyers are in their own marketplace with firms selling , and buyers are in their own marketplace with firms selling . In a PSE, firms have to be indifferent between offering quality or , and the fraction of firms offering in equilibrium, ∗ , is such that a firm’s expected profit in equilibrium from offering or is the same, which is just equation (12). Having solved for ∗ , this then pins down the equilibrium queue lengths ∗ and ∗ . Firms offering quality in a PSE are posting the optimal prices as given by (13),
and earn expected profits as given in (14). Buyers then receive expected utility in (15). 9
This is a sufficient, not necessary, condition for a PSE, as will be discussed shortly.
28
Segmentation is an equilibrium outcome, where firms earn the same expected profit in either marketplace, and buyers choose to go to the “right” marketplaces. There are no barriers to entry of marketplaces, nor do these marketplaces charge an entrance fee; all agents are free to go between marketplaces, and agents behave optimally. The following corollary follows directly from proposition 2
Corollary 4 In the PSE as laid out in Proposition 2: ∗ () ∗ ( ) ( );
() ∗ ( ) ; () ∗ ( ) ; and () ∗ ( ) ∗ ( ) First notice that buyers in their marketplace enjoy a higher expected utility than buyers who are in their own marketplace: ∗ ( ) ∗ ( ), as stated in part () of corollary 4. Then why does a buyer not deviate and shop in the other marketplace? A buyer could always switch marketplaces because he is not prevented from doing so, but he would choose not to, because he cannot afford to buy the good, as stated in part () of corollary 4. The model also predicts, as stated in part () of corollary 4 that the queue length at a firm is higher than the queue length at a firm, which does seem to fit the retail landscape, where, for example, firms that sell more expensive clothes, say Chanel, will have fewer buyers per store than firms that sell more mass-market clothing, like Zara (or that a Ferrari showroom would see less buyers compared to a Toyota or Ford showroom). Part () of corollary 4 implies that if we were to compare two economies, one where is sufficiently high, ≥ , and another where ≤ , keeping the fraction of buyers, 29
, the same as well as the level of income , the former economy with more equal incomes will be in a stable pooling equilibrium where all buyers pay for good , as we know from proposition 1. However, in the latter economy where the poorest agents are very poor, or that incomes are very unequal, we know from proposition 2 that the stable equilibrium is a PSE, and the richest agents in the economy are paying ∗ ( ) when they purchase a good. In other words, buyers benefit from the presence of low income agents in their economy: as firms face a choice of supplying or , when the poorest agents in an economy cannot afford to spend too much on , this reduces the expected profit that firms make when supplying the quality good, which in turn lowers the expected profit a firm supplying can make. In fact, it is possible to compare the expected utilities of agents in the stable pooling equilibrium when ≥ to that of the stable PSE when ≤ : it is simple to show that ∗ ( ) ∗ ( ) ∗ ( ). Proposition 3
In a PSE , as the fraction of buyers, , goes up, ∗ ( ) and
∗ ( ) increase; all buyers are strictly worse off; and firms are strictly better off. It is true that as increases the average income in the economy goes up, but this is not the reason why both ∗ ( ) and ∗ ( ) go up. The intuition why ∗ ( ) and ∗ ( ) increase is that as goes up, more firms will enter the marketplace where the buyers are, but by proportionately less, so the equilibrium queue length at this marketplace ∗ ( ) = ∗ also goes up, i.e., it is harder for a buyer to be served at a seller,
which is what enables sellers to charge a higher price ∗ ( ). What is more surprising is that ∗ ( ) also goes up. The reason why this is so is because firms choose the quality to offer, so if firms in the marketplace selling are earning a higher expected profit when 30
goes up, to maintain the equal expected profit condition, firms in the marketplace selling also have to earn more, and they achieve this by raising ∗ ( ).10 Corollary 5 Comparing two economies featuring PSE with different fractions of buyers, , the economy with a higher fraction of buyers has both types of buyers pay more for their goods and they have lower levels of expected utilities, and all firms make higher expected profit.
Because of assumption (1), buyers can always afford to buy the good, so what matters in determining the market structure is . Proposition 2 states that when ≤ holds, a PSE is the unique stable equilibrium, but this is a sufficient condition only. More generally, we have: ¡ ¢ Proposition 4 There exists a unique ∈ such that for ≤ , a PSE as described in proposition 2 is stable. In fact, ∗ ( )
In a PSE, the properties of the PSE as stated in corollary 4 hold, and the result in proposition 3 where both ∗ ( ) and ∗ ( ) are increasing in , the fraction of buyers, also holds.11 When ≤ , there is no way for firms to offer a good of quality to these buyers since the price charged has to weakly exceed its production cost , which is why ≤ is a sufficient condition ensuring that the quality good is unaffordable to buyers, so as to support a PSE. However, this condition ≤ is not a necessary condition for the 10
Naturally, with this change in , even though the equilibrium queue lengths at both marketplaces go ∗ ∗ up, it is still true that ( ) ( ). 11 Both corollary 4 and proposition 3 apply to values of ∈ ( ] because the proofs take as given the existence of a PSE, and ≤ did not matter for the proof.
31
existence of a stable PSE. When is marginally higher than , buyers still have such low incomes that, because it constrains what firms serving these buyers can charge if they were to sell , firms still find it optimal to offer to buyers. However, as gets higher, since these buyers are now richer, firms have more reason to offer instead to these buyers at a price they can afford, since quality good has the higher surplus. Proposition 4 states the ¡ ¢ existence of a threshold level of , which I denote ∈ , such that for incomes
lower than a PSE is a stable equilibrium. Intuitively, as becomes “too high” in the
sense of exceeding , then a group of firms can profitably deviate by offering buyers the quality good instead at a price that they can afford. In the proof of proposition 4, I show that ∗ ( ). One may reason that when ≥ ∗ ( ), a PSE is not stable since buyers can afford to purchase the good, and since in a PSE, ∗ ( ) ∗ ( ) that any buyer will unilaterally deviate, and this is correct, but one may then be tempted to think that so long as ∗ ( ), then a PSE is stable. However, this is not correct. The idea is that for ∈ ( ∗ ( )), buyers are sufficiently rich in that some firms could deviate and offer them the quality good and charge them a deviation price equals to , and all buyers would go to the newly created marketplace, and none of the buyers, who are not targeted in the first place, would go.12 It is only for ≤ that it is not possible to find a profitable group deviation. Given that the equilibrium market structure is a pooling equilibrium in when ¡ ¢ is sufficiently high ≥ and the equilibrium market structure is a PSE when is below a threshold ( ≤ ), where , this then begs the question as to what the 12
In the proof of proposition 4, I show that in a group deviation it does not matter how this surplus from deviating is split between the deviating firms and the deviating buyers.
32
equilibrium market structure is for intermediate values of . Suppose the level of is just slightly below but arbitrarily close. We know from part () of proposition 1 that the moment drops below that there is no stable pooling equilibrium. Any equilibrium, if it exists, is a segmented equilibrium, and from the explanation to proposition 4, a PSE is not stable. However, there is another type of segmented equilibrium to consider which is a Price Dispersion Equilibrium (PDE): both types of buyers are in different marketplaces, firms in both marketplaces offer the same quality good for sale, but at different prices, and firms are indifferent between being in either marketplace as they earn the same expected profit. First note that a PDE where quality good is offered cannot be an equilibrium.
Lemma 4 In a PDE, firms cannot be offering the low quality good for sale.
This lemma is very intuitive. In Case (), ( − ) ( − ), firms want to offer the good for sale; as buyers can afford the good given assumption (1), that is what firms would sell them. Hence, there is no way to have an equilibrium where all buyers are in a marketplace with firms offering as there are many ways to construct profitable group deviations. Hence, a PDE, if it exists, will fit the description contained in the following lemma. Lemma 5 In a PDE where buyers are in marketplace 1 with a fraction ∗ of firms ∗ posting { 1∗ }, and buyers are in marketplace 2 with a fraction (1 − ) of firms posting ∗ 1∗ 2∗ { 2∗ }, 6= is the solution to
¢ 1∗ ¡ 2∗ − ) ( − ) (1 − − ) 1∗ − = (1 − | {z } | {z } = 2∗
= 1∗
33
(16)
1∗ 2∗ = ∗ , = (1 − ) (1 − ∗ ), and the following properties must hold: 1∗ 2∗ () 2∗ , with = and
1∗
1∗
1∗ − (1 − − )]; 1∗ = − [( − )
(17)
1∗ 2∗ ; and ()
() 1∗ 2∗ , where ∗ = in marketplace = 1 2.
£¡ ¤ ∗ ¢ ∗ 1 − − ( − ∗ ) is the expected utility of a buyer
A PDE truly features price dispersion as it features two marketplaces with the same quality good sold at different prices. In particular, buyers are offered the sale of the good at a higher price 1∗ , which corresponds to firms’ “optimal” or profit-maximizing price, whereas 1∗ buyers are offered the sale of the good at a lower price 2∗ , which corresponds to
firms’ “constrained” price, as in these buyers are charged the maximum they can afford to pay since their budget constraint becomes binding, i.e., 2∗ = . The question to ask is: why would buyers not deviate to marketplace 2 to enjoy the lower price? The answer is contained in part () of this lemma: buyers would not deviate to marketplace 2 to enjoy the lower price because they would enjoy a lower level of expected utility. And why 1∗ 2∗ is that so? Part () of the lemma tells us that , so buyers enjoy a lower
queue length in marketplace 1, so the chance of being rationed is lower, and since a buyer’s expected utility is the product of the probability of being able to buy the good as well as the payoff conditional on doing so, the lower queue length compensates for the higher price paid, and in such a way that buyers get a higher 1∗ so none of these buyers will want to deviate to marketplace 2. In fact, buyers actually would like to go to marketplace 1 even though the price posted there is higher, precisely because 1∗ 2∗ ; however, they would 34
not do so because they cannot afford to pay the higher price, so they would choose not to unilaterally deviate either. This price dispersion outcome has the flavor of outlet stores coexisting with full-priced stores, like we observe in reality in the retail landscape, where essentially the same items are sold at different prices, and this model would say that this is because the outlet stores are actually targeting a different set of buyers (the lower income group), since the higher income group would prefer to pay the “full price” and shop at a regular store instead.13 Given the description above about how a PDE would look like, if it existed, the natural question to ask is if there actually exists a PDE in my model. To start, let us suppose that is very high, but just slightly below , so we know from part () of proposition 1 that there is no pooling equilibrium, and we also know that there does not exist any PSE from proposition 4. So could there be a PDE instead? Lemma 6 When is just slightly below but arbitrarily close, the unique stable equilibrium is a PDE where ∗ solves (16), firms in marketplace 1 with buyers are posting 1∗ { 1∗ }, with given in equation (17) and firms in marketplace 2 with buyers are
posting { 2∗ = } When exceeds the equilibrium is a pooling equilibrium, but once drops below , the market structure changes, and the equilibrium now features buyers of different income types sorted into different marketplaces, but with firms offering the same quality across the two marketplaces. 13
It is true that in reality, higher-income people shop at outlet stores too, perhaps because people enjoy the “thrill” of getting a good bargain, but I do not have this “thrill” element in my model, so retail segmentation occurs very cleanly in my model by buyer income types, which provides a way to think about segmentation in retail markets.
35
Lemma 7 As falls from towards , in a PDE, if it exists, ∗ , the fraction of firms in 1∗ 2∗ marketplace 1, goes up, which results in decreasing and increasing, and ∗ ( ) = 1∗ ∗ , = 1 2 decreases monotonically and increases monotonically. The buyers are 2∗ enjoying a lower price as falls, but face a higher queue length which decreases their
probability of obtaining the good, so their overall expected utility 2∗ could rise or fall. As falls from towards in a PDE, ∗ ( ) falls, when firms decide whether to deviate in a group and offer a competing advertisement, the lower the , the more attractive alternative advertisements appear. ¡ ¢ Proposition 5 There exists ∈ such that for ∈ [ ) a Price Dispersion
Equilibrium (PDE) is stable.
When buyers earn intermediate levels of income, a PDE is the stable equilibrium. Proposition 5 suggests that for ∈ ( ) the stable PSE need not be the unique stable equilibrium, and this is indeed intended. In numerous numerical exercises, is uniquely pinned down, and , so when ∈ [ ] there are two stable segmented equilibria: PSE and PDE; for the unique stable equilibrium is a PSE; for ∈ ( ) the unique stable equilibrium is a PDE; and for ≥ the unique stable equilibrium is a
36
PE in .14 This is illustrated below.
2∗ Proposition 6 In a PDE, as the fraction of buyers, rises, 1∗ rises, but is unaf-
fected; both types of buyers are strictly worse off, and firms are strictly better off.
3.3
How Differences and Changes in Buyers’ Incomes Affect the Market Structure
Changes in buyers’ income, so long as still remains weakly higher than , i.e., assumption (1) is satisfied, do not matter at all in determining the equilibrium market structure. What matters is the level of . ¡ ¢ When buyers’ incomes are sufficiently high ≥ , so the economy is not so unequal
in incomes, the equilibrium market structure features pooling where all buyers are in the © ª same marketplace and all firms post . When there are changes in , the fraction of
buyers, or in the levels of or , so long as they remain above and respectively, the market structure is not impacted, and the equilibrium outcomes remain unchanged. However, the moment drops below , the equilibrium market structure no longer supports pooling, but instead features segmentation, both types of buyers are in their own 14
Let us consider a numerical example. Let = 3 = 1 = 1 = 0 = 1 = 13 This implies that = 184 = 134 ∈ (113 114) .
37
1∗ 2∗ marketplace, with firms offering { ∗ } in marketplace = 1 2 respectively, with ,
so one interpretation is that buyers are paying the “full” price for quality , but buyers are charged a “discounted” price. What is interesting is that all buyers observe the entire set of firms’ advertisements, but a buyer of a particular income type will ignore the advertisements that are the “wrong” ones, and only search among firms posting the correct (to him), or relevant advertisements. In the region of PDE, if the incomes of buyers drop, the price charged falls one for one, and this also impacts the price charged to buyers: buyers are strictly better off the lower the level of is, and all firms are strictly worse off. For a buyer, income changes of buyers matter in affecting the price he is charged, even though his own income has not changed. In a PDE, changes in matter, and if the economy has a larger share of buyers, all buyers are made strictly worse off, and firms, strictly better off. When falls further, there is a region where there is potential multiplicity of equilibria in that both PDE and PSE exist. When buyers’ incomes are sufficiently low, there is too much income heterogeneity in the economy, there is no way for firms to offer buyers the quality good, and the equilibrium features perfect segmentation, in that ( ) buyers are in a marketplace with firms offering quality (). In this economy a buyer of one income group does not meet buyers of another income group in his marketplace. In a PSE, changes in affect equilibrium outcomes: a larger share of buyers makes every buyer worse off, but changes in , so long as they imply the economy is still in a PSE, do not affect the market structure. Comparing outcomes across different market structures, we know that ∗ ( ) ∗ ( ), i.e., buyers pay less in the economy with more heterogeneous incomes. Not 38
only that, ∗ ( ) ∗ ( ) ∗ ( ), which means that buyers are better off in an economy with more heterogenous incomes, so they benefit from the presence of very poor buyers. As for firms, it is easy to show that ∗ ( ) ∗ ( ), so firms in a PSE make lower expected profits than in firms in a pooling equilibrium. Comparing PDE to PE, buyers are better off in a PDE than in a pooling equilibrium; again, they benefit from the presence of poor buyers. The expected utility of buyers can be either higher or lower in a PDE than PE. Firms in a PDE also earn strictly lower expected profit than if they were in a pooling equilibrium. The intuition why firms would prefer to be in a pooling equilibrium than any other equilibrium is because in a pooling equilibrium, they are all offering the highest surplus good for sale, and they are able to charge the unconstrained optimal price, since no buyer’s income constraint binds. In my model, because the market structure is truly endogenous, all marketplaces that are open and populated are intricately linked, and changes that affect buyers in one marketplace can affect buyers in the other marketplace.
3.4
No Price Dispersion within a Marketplace
Proposition 7 There does not exist an equilibrium featuring price dispersion within a marketplace for a particular quality of good sold. In the directed search or competitive search literature, it is quite standard to solve for symmetric equilibria, but hardly any attention is paid to non-symmetric outcomes. In Moen (1997), for example, he defines a competitive search equilibrium where within a marketplace all firms are assumed to post the same price.15 To the best of my knowledge I have not 15
In Moen (1997), a labor market is analyzed, but translating the framework to a goods market setting,
39
seen a paper in this literature which allows ex ante identical firms within a marketplace to potentially post different prices (for a given quality) and then rule out such equilibria. Because there is no equilibrium that supports price dispersion within a marketplace for a particular quality of good offered, what was done earlier in restricting firms within a marketplace offering a particular quality to post the same price in their advertisements is without loss of generality.
3.5
Social Planner’s Problem
The social planner faces the same constraints as agents in the decentralized equilibrium in that there are frictions, so the planner cannot simply assign a buyer to a firm, and has to treat all agents of the same type the same way. The planner has to: () decide how many marketplaces to open - he can choose to open one marketplace where all buyers are asked to search in, or two marketplaces, one for each type of buyer; () instruct buyers of each type which marketplace to go to; () in the event more than one marketplace is open, to choose the measure of firms to operate in each marketplace; and () instruct firms on the quality of the good to produce. As all agents are risk neutral, the planner seeks to maximize the expected surplus, subject to a pair-wise budget constraint, in that the resource of the buyer has to be able to support the production choice of the firm. If ≥ , every buyer in the economy has sufficient resources to consume the quality good, which is the good with the higher surplus, so the social planner will create one marketplace for all buyers and all firms, and instruct all firms the wage can be recast as a price.
40
to produce . The expected surplus per buyer, denoted as , is thus =
¢ 1¡ 1 − − ( − )
(18)
In other words, all the pooling equilibria in the decentralized economy are efficient, but the planner is going to ask for what corresponds to the pooling outcome for a larger range of values of (for ≥ ) than compared to the decentralized equilibrium ( ≥ , where ). From the planner’s perspective, so long as a budget constraint is satisfied, the planner wants these matches to go ahead. In a PDE, for all ≥ , where , even though all buyers are consuming the good, there are two marketplaces open, whereas the planner would have asked that only one marketplace is open. Hence, all PDE are inefficient. We had also seen that for ∈ [ ] that a stable PSE exists in the decentralized economy, so all these PSE are also inefficient. Now suppose , so only buyers have enough resources to consume the high quality good . The planner will create two marketplaces, one for buyers (marketplace 1), one for buyers (marketplace 2), and choose the fraction of firms operating in marketplace 1, ∗ , to be the solution to max =
¡ ª 1© ¡ ¢ ¢ 1 − − ( − ) + (1 − ) 1 − − ( − )
(19)
where = and = (1 − ) (1 − ) The F.O.C. to the above is
¡ ¡ ∗ ∗ ¢ ∗ ∗ ¢ ∗ − 1 − − − ( − ) = 1 − − − ∗ − ( − )
(20)
which means that ∗ , the solution to the planner’s problem is such that ∗ = ∗ , where ∗ is the fraction of firms offering quality in a PSE, which is immediate on comparing (20) to (12), or that the planner’s solution coincides with the allocation in the decentralized 41
equilibrium. Recall that in the decentralized economy when , the unique stable equilibrium is a PSE; hence, all PSE in this range of values are efficient. To summarize the above we have the following proposition. Proposition 8 When ≥ , the unique stable equilibrium in the decentralized economy, which is a pooling equilibrium, is efficient. When the unique stable equilibrium in the decentralized economy, which is a PSE, is efficient. When ∈ [ ), none of the equilibria in the decentralized economy are efficient.
4
Case (): − −
Like in Case (), we pay attention first to symmetric equilibria.
Proposition 9 There can never exist a segmented equilibrium in Case ().
From the buyers’ point of view, they have a higher valuation of the good to the good. However, as the surplus of the good is higher, both buyers and sellers prefer the good: since each party gets a share of the surplus, the higher the surplus the more there is to split, even though buyers have a strictly lower valuation of good compared to good ( ). Suppose the market was segmented and only one type of buyers was offered good . If buyers were offered good they are indeed happy, but a buyer will find it profitable to deviate to the marketplace with buyers, and a buyer can afford to pay for this good by assumption (1) If the buyers offered good were buyers, then a buyer will strictly prefer to deviate to go to the marketplace where the good is offered, since he 42
will be made strictly better off. This good, because it has a higher surplus than the good, is basically “cheap and good,” so there is an incentive for firms to offer that good to buyers, who will end up enjoying a higher surplus, and hence, firms do not want to segment the market. Proposition 10 When − − , there exists a unique stable equilibrium which is a pooling equilibrium where all firms face a queue length of , and post { ∗ ( ) = }, where ( − ) − = 1 − − ¡ ¢ with ∗ ( ) = 1 − − − − ( − ) ; and ∗ ( ) = −
∗ ( ) = − ( − )
(21) (22) (23)
Although the above considers symmetric equilibria where firms within a marketplace posted the same advertisement, it is easy to verify16 that there does not exist any equilibrium where there is a distribution of prices posted by firms within a marketplace for a given quality offered, just like in Case (), and hence, the equilibrium described in proposition 10 is the only equilibrium. Now consider the Social Planner’s problem. Since quality yields the higher surplus, and since all buyers can afford to purchase this good by assumption (1), the planner will simply open one marketplace, instruct everyone to be there, and ask firms to produce the c, is quality good for sale. The expected surplus per buyer, 16
¡ ¢ c = 1 1 − − ( − ) ,
The proof of proposition 7 does not rely on comparing the surpluses of goods of quality to so the analysis can also be applied to case () to show the non-existence of a distribution of prices for a given quality within a marketplace.
43
and the allocation of the equilibrium as described in proposition 10 is efficient
5
Case (): − = −
Proposition 11 The market structure is indeterminate in Case ().
There is indeterminacy of market structure because there is no good that is “preferred” by firms or buyers, in the sense of having the larger surplus, unlike in Cases () and (). When all goods yield the same surplus, everyone is indifferent across the goods, so the market structures is indeterminate. What can be equilibria depends on whether the allocations are affordable to the buyers being targeted. This case is a knife-edge case, because any slight change in or will result in the economy being in either Case () or (), and hence, is uninteresting.
6
Conclusion
This paper has considered how the market structure is determined when buyers are heterogeneous in income and firms can choose the qualities of a good to offer for sale. I have used a parsimonious directed search model with minimal assumptions to understand if, and when, segmentation of the product market by income groups can occur. I view my model as a good approximation of reality, in that when many firms independently decide what to sell, they make use of aggregate buyer income information before choosing what to sell, which is a novel consideration that has not been explored in the literature, which has hitherto appealed to inherent cost or productivity advantages of firms in deciding the quality of good
44
to offer. As a consumer, is what I see in stores influenced by the composition of buyers in my economy? Absolutely, but it is not the only consideration. And given the same underlying parameters across two economies, except that one economy has a different income inequality than another, can what consumers see in stores be different? Absolutely! One interesting example illustrating that firms do take into account the income information of consumers in their choice of product offering is Singapore Airlines’ choice of seat types in their then record-setting longest non-stop flight of around 16,000 kilometers in approximately 18 hours between Singapore to Newark (all flights between Singapore and the US up to that point required at least one stop). When this non-stop route was first introduced in 2004, the configuration of the planes was a mixture of 64 business class and 117 executive economy seats. However, in March 2008, all the planes serving this route were configured to 100 “all business class seats,” which was a first on a commercial airliner; this change happened in the same month that Bear Stearns went bust, and six months before the collapse of Lehman Brothers. As is well known, the world economy then, including Singapore, was experiencing a massive run up in prices across all asset classes. Singapore Airlines could be thought of as reconfiguring their planes to all business class seats to cater to the boom times in business travel when businesses were willing to splash out on their spending. In retrospect, it was the worst possible time to launch all business class flights as business travel dried up with the implosion of the financial sector, and in November 2013, Singapore Airlines discontinued this all business class non-stop route.
Serene Tan, Department of Economics, National University of Singapore. Email:
[email protected]. Address: National University of Singapore, AS2 #06-02, 1 Arts Link, S117570, Singapore. Tel: +65-65163964.
45
Appendix −
+
Henceforth I denote → as the left-hand limit, and → as the right-hand limit. Lemma A.0. In a stable equilibrium, all buyers are active. Proof. Suppose a candidate equilibrium features a group of buyers that are inactive. Can it be that buyers are inactive? This is not possible, because they can always afford to purchase the good that buyers are buying by assumptions (0) and (1). Can it be that buyers are inactive? There are two ways to think of this. One can imagine that all buyers and all firms are in one marketplace, but the price posted for quality , e , is such that e so
buyers are effectively “shut out” of that marketplace even though they are there, i.e., they are inactive. Or that buyers are in their marketplace with some pricing advertisement there which does not “make
sense,” in that they cannot afford it, and buyers and all firms are in their own marketplace with some price posting for a quality such that e . In either interpretation, effectively only buyers are
served, so buyers face a queue length e = and have expected utility ( ); buyers have expected
e , where utility ( ) = 0; and all firms post { e }, ∈ { }, with corresponding expected profit ¡ ¢ − e = 1 − (e − ) It has to be that e , for otherwise buyers would not remain inactive. Is
this candidate equilibrium { e } where only buyers are served stable?
It is easy to construct a profitable deviation to show that this candidate equilibrium is not stable. © ª Suppose a group of firms posted instead, with ≤ e , so deviation expected profit = ¢ ¡ (1−− ) − = e , which means that e, and since buyers are promised expected utility of = ¡ ¢ [(1−− ) ] − 0, so long as ≤ ( ), this constitutes a profitable deviation. If this deviation
implies ( ), which means buyers would join even though the advertisement is not meant for ª © them, then construct this group deviation instead, where a group of firms offered all buyers , with ¡ ¢ ≤ e , and for = (1 − − ) − = e , which means that e (and it is possible to find a measure of firms deviating to serve all buyers since the measure of firms can be made small enough to ensure ¡ ¢ that e), and = [(1 − − ) ] − 0 = ( ) and here ( ). Nothing in this proof relied on comparing the surplus of the good to the good, so this result that
all buyers have to be active in equilibrium is true for all three cases.¥
Lemma A.1. The firm’s expected profit function as defined in ( 1) is strictly concave, so the F.O.C.s to the firm’s problem are both necessary and sufficient. Proof. For ease of notation in this proof, restate ( 1) by dropping the superscript and subscript that denotes a firm’s identity and quality ∈ { } produced; and ignoring for now the buyer’s budget constraint respectively, we have
max
=
¡ ¢ 1 − − ( − )
(1 − − ) ( − ) =
(24)
For a firm, is taken as given, and his choice of affects through (24). Differentiating the firm’s profit function with respect to , we have ¢ ¡ = 1 − − + − ( − )
46
Defining ≡ − − [ (1 − − )] from (24) and using the implicit function theorem: Ã ! 2 2 ¢ ( − ) − (1 − − ) ¡ − (1 − − ) − = so ∗ − =− = 1− 1 − − − − [1 − − − − ] {z } | ≡
Moreover,
2 ( − ) − (1 − − ) {} − = , − where = 2 [1 − − − − ]2
where ≡ −1 + 4− − 3−2 − 2−2 Since = −4− (1 − − − − ) 0 and |=0 = 0 0
for all 0 This then implies that 0 for all 0, so
( − ) 2 = − − 0 for all 0 2 |{z} |{z}|{z} 0
0
0
In other words, the profit function of a firm is strictly concave in . Since there is a one-to-one mapping between and , the profit function of a firm is strictly concave in .¥ Proof of lemma 1 If ≥ , all buyers can pay ∗ = , so the queue length at a firm is But if ∈ [ ), only
buyers are served, so the queue length at a firm in the candidate equilibrium is . So = { },
depending on In a candidate equilibrium where all firms post { ∗ = }, with queue length , a ¡ ¢ firm’s expected profit is ∗ = 1 − − ( − ) and a worker’s expected utility is ∗ = 0 ¡ ¢ Now consider this group deviation { Γ}, where is such that the promised expected
utility to deviating buyers is some arbitrarily small and positive amount = 0, and Γ ∈ { buyers
only, all buyers} Since deviating firms cannot do better than pick an optimal price, that is where solves
= − ( − ) = , and = − [( − ) − ](1 − − ), using the F.O.C.s to ( 1), with
the implied deviation profit is = (1 − − − − ) ( − ) and for an appropriately small enough , or equivalently, large enough , ∗ . (We can always find a large enough because the mass of
deviating firms can be made small enough, and since any ∈ [ ), buyers can always afford to pay
for the good.) If ≥ , set Γ = { buyers only}; if , set Γ = {all buyers}.
Hence, this candidate equilibrium is not stable since a profitable group deviation can be found.¥ Proof of lemma 2 First observe that in a segmented equilibrium, it cannot be that ∗ ( ) ∗ ( ), where where ∗ ( )
is the expected utility of a buyer of income ∈ { } - since , buyers can always afford to
purchase whatever buyers are purchasing, which means a buyer will just deviate to the marketplace where the buyers are. Hence, ∗ ( ) ≥ ∗ ( ).
Second, note that buyers must be in a marketplace with firms posting quality (labeled market-
place 1 without loss of generality). Suppose not, and the candidate equilibrium has that buyers are with firms posting quality , and suppose that ∗ ( ) ≥ ∗ ( ) is true. The best that firms in this
marketplace can do is to charge an optimal price, that is, supposing the queue length is e1 , choose ∗ = − [( − ) e1 −1 ](1 − −1 ) and offer these buyers an expected utility ∗ ( ) = ( − ) −1 ,
47
¡ ¢ and their expected profit is ∗1 = ( − ) 1 − −1 − e1 −1 . (This is just solving ( 1) for firms in the marketplace with buyers, and we are silent on what buyers are doing.) However, this is not robust
to a group deviation, because the same firms can simply sell instead to these buyers, so even by ¡ ¢ maintaining this queue length e1 , their deviation profit = ( − ) 1 − −1 − e1 −1 ∗1 , and a
buyer’s deviation expected utility = ( − ) −1 ∗ ( ). Which leads to a contradiction. Hence,
buyers must be in a marketplace with firms posting quality .
Therefore, in a segmented equilibrium buyers are either in a marketplace with firms selling or . Third, suppose buyers are in marketplace 2 with firms selling with a price not the same as that posted in marketplace 1 From the first paragraph of this proof we know that ∗ ( ) ≥ ∗ ( ). Let us suppose that buyers are in marketplace 2 with firms selling and ∗ ( ) ∗ ( ). For this to be
possible, it must be that the price charged to buyers, ( ) ≥ , for otherwise buyers would just deviate to marketplace 1. In this candidate equilibrium, firms must be earning equal expected profit
so if we denote ( ) as the price charged to buyers and ( ) as the queue length there (with ( ) the queue length in the marketplace), it must be that ³ ³ ´ ´ 1 − −( ) ( ( ) − ) = 1 − −( ) ( ( ) − ) =
and since ( ) ( ), it must be that ( ) ( ). Given the queue length at marketplace 1,
which is ( ), the “optimal price” a firm could have charged is ∗ = −
( − ) ( ) −( ) 1 − −( )
(25)
which means that by charging ( ) ≥ , firms in marketplace 1 are not charging the optimal price,
i.e., given that buyers have expected utility ( ) these firms are not at the “top” of their profit © ª function; in fact, firms are charging “too much.” Hence, I can construct a group deviation such that = + , 0 arbitrarily small, which means ( ) buyers have the same level of £¡ ¤¡ ¢ ¢ expected utility ( ), is such that ( ) = 1 − −( ) ( ) − , and because deviating
firms move closer to the optimal price, these deviating firms are earning strictly higher expected profit. Moreover, since , buyers which are not the target income group cannot deviate. Hence, there
does not exist a stable segmented equilibrium with buyers in a marketplace with firms posting quality . Fourth, suppose buyers are in marketplace 2 with firms selling with a price not the same as that posted in marketplace 1, but that ( ) = ( ). In a segmented equilibrium, all firms must be earning the same expected profits. From the second paragraph of this proof, we know that buyers must be in a marketplace with firms selling good, but not only that, if the price were not optimal, in that if the queue length in this marketplace were ( ) and price charged e1 6= ∗ where ∗ is defined in equation (25) then
there exists a profitable group deviation where a group of firms can offer buyers the same expected utility but choose a price closer to the “top” of their profit function. Hence, firms in marketplace 1 must be posting
∗ as defined in equation (25) and their expected profit is ( ) = (1 − −( ) − ( ) −( ) ) ( − ), and buyers have expected utility ( ) = −( ) ( − ). In marketplace 2, ( ) = ( ). If the
price charged in marketplace 2, e2 = ∗ , then marketplace 2 is just a duplicate of marketplace 1, so this
is not a segmented equilibrium. If e2 6= ∗ the firms in marketplace 2 are taking as given that buyers
are offered a level of utility ( ) = ( ), but are not charging them the “optimal” price associated with
their queue length ( ) We know from lemma A.1 that given a level of utility promised to buyers that the
48
profit function is strictly concave, which means that firms in marketplace 2 cannot do better than charge the “constrained” price, which will be to charge e2 = . However, equilibrium requires that firms earn equal expected profit and buyers equal expected utility, which are, respectively, ³ ³ ´ ´ 1 − −( ) − ( ) −( ) ( − ) = 1 − −( ) ( − ) and −( ) ( − ) =
1 − −( ) ( − ) ( )
where ( ) = ∗ and ( ) = (1 − ) (1 − ∗ ), where there are two equations in one unknown, ∗ .
Hence, it is not possible for a segmented equilibrium to have buyers in their own marketplace with firms selling and ( ) = ( ). The last thing to check is if buyers could be in a marketplace with firms posting quality . The best firms can do in this marketplace, supposing the queue length is , is to post { ∗ } where ∗ = −
( − ) − 1 − −
(26)
for = . (If some price 6= ∗ were posted, then given that buyers obtained a certain level of expected
utility, there exists a profitable group deviation, in the spirit of the analysis of the paragraph above.) But if
an optimal price ∗ were posted, the expected utility buyers obtain is ∗ ( ) = − ( − ) and firms obtain ∗2 = (1 − − − − ) ( − ). For firms in marketplace 1 serving buyers, the best these
firms can do is to post an optimal price ∗ which is equation (26) for = when the queue length there is and obtain an expected profit ∗1 = (1 − − − − ) ( − ). For ∗1 = ∗2 , it must mean that
, which then implies that ∗ , which means that any buyer will deviate to the marketplace
where buyers are at and earn a strictly higher level of utility since ∗ ( ) = − ( − ) ∗ ( ).
Hence, this candidate equilibrium is not robust to individual deviations.
As a result, when ≥ , there does not exist any stable segmented equilibrium.¥ Proof of proposition 1 - Proof of part () of proposition 1 Suppose the candidate equilibrium is a pooling equilibrium where all firms sell . If the budget constraint of buyers does not bind, a firm will post the optimal price of ∗ ( ) = as given by equation (9), and ∗ ( ) is given a firm will make expected profit ∗ ( ) given by (10) and a buyer’s expected utility ∗ ( ), a firm by (11). Could a firm unilaterally do better? Given that all buyers have expected utility
will never post e 6= ∗ ( ) when it is selling , since ∗ ( ) is the profit maximizing price. Nor will
∗ ( ) when they a firm unilaterally deviate to sell and post price e : since buyers must be enjoying
come to this deviant firm, the deviant firm faces an effective queue length of , so the price posted has to
∗ ( ) ] (1 − − ) and a deviant firm’s expected profit is e = satisfy (1), which means that e = − [ ¡ ¡ ¡ ¢ ¢ ¢ − − − − − ∗ 1− (e − ) = 1 − ( − ) − ( − ) 1 − − ( − ) = ( ). No
buyer can unilaterally deviate and be made strictly better off either. In other words, this candidate pooling equilibrium is robust to individual deviations. Moreover, this candidate pooling equilibrium is also robust to group deviations.
49
∗ In this candidate pooling equilibrium, all buyers have the same expected utility ( ), so a group ∗ ( ) will attract all buyers provided they can afford it, and a group deviation deviation that offers ∗ ( ) could be attracting either one type of buyer or all. that offers =
There are two types of group deviations to check. In the first, let some firms create a new marketplace © ª selling quality and post , where ∈ { buyers only, buyers only, all buyers}, and
= −
( − ) − 1 − −
(27)
∗ i.e., is the “optimal” deviation price, given , which in turn satisfies ≥ ( ). Can it be that
¡ ¢ () : ≥ ∗ ( ) ⇔ ( − ) (1 − − − − ) ≥ ( − ) 1 − − − − ; and
∗ ( ) ⇔ ( − ) − ≥ ( − ) − () : ≥
(28) (29)
Since ( − ) ( − ), for (29) to be true it must be that − − , or that . But if , ¡ ¢ then ( − ) (1 − − − − ) ( − ) 1 − − − − , which violates (28). Hence @ profitable ¢ ¡ ∗ ( ) but is not an optimal price, i.e., it is not deviation. Now suppose is such that ≥
the deviation price in equation (27). Since the deviation profit function is strictly concave in given a level
∗ ( ), not charging the optimal price as given in (27) implies lower expected profit for the firm, so of
if, with the highest possible expected profit obtained when charging (27) this type of group deviation is not profitable, it will not be profitable with a lower expected profit from charging a non-optimal price. Let us consider the second type of group deviation where quality is sold in this new marketplace and ¡ ª ¢ © the advertisement is , where ∈ { buyers only, buyers only, all buyers} and is ∗ ( ), is chosen optimally, i.e., that such that given satisfying ≥
( − ) − = − 1 − −
Can it be that
¡ ¢ () : ≥ ∗ ( ) ⇔ ( − ) (1 − − − − ) ≥ ( − ) 1 − − − − ; and
∗ ( ) ⇔ ( − ) − ≥ ( − ) − () : ≥
(30)
(31) (32)
If = , then (31) and (32) hold with equality, so no deviating agent is made strictly better off. If , (31) is violated. If , (31) holds, but (32) is violated. Hence, this second type of deviation is not profitable, no matter ( − ) S ( − ) If the deviation price were not the optimal price, then the
deviation profit is even less than as defined in (31), so this type of group deviation is not profitable either. © ª Which means that group deviations like = + for 0, where deviating firms create a
new marketplace for buyers, pricing out buyers, are not profitable - the intuition is that firms, by
charging is on the “downward sloping” part of their profit function, and hence, would make lower than profit than the “optimal” deviation profit of as laid out in (31). Hence, there does not exist any coalitional deviation that will satisfy conditions () − () as laid out in
Definition 3. Hence, a pooling equilibrium as laid out in part () of proposition 1 is stable.
We are left to show that there does not exist any other stable pooling equilibria. There are three other possible pooling equilibria. The first of these candidate equilibria is a symmetric equilibrium where all firms
50
post { ∗ ( )}, where ∗ ( ) is the solution to a firm’s profit maximization problem as laid out in − ∗ ( 1), where in a symmetric equilibrium, ∗ ( ) = = − , = = = , so
( − ) − ∗ ( ) = − = ; 1 − − ¡ ¢ ∗ ( ) = 1 − − − − ( − ) ; and
∗ ( ) = − ( − )
(33) (34) (35)
This candidate pooling equilibrium is not even robust to individual deviations. A firm will like to deviate to offer , and since it is offering buyers the same expected utility ∗ ( ), its expected queue length is ¡ ¢ simply , and its deviation price is the e = − [∗ ( ) ] 1 − − , and and its expected profit is ¡ ¡ ¢ ¢ e = 1 − − (e − ) = 1 − − ( − ) − − ( − ) ¡ ¢ 1 − − − − ( − ) = ∗ ( ) .
This candidate equilibrium is clearly also not robust to group deviations as a simple deviation of some firms offering { = all buyers} will make deviating firms and deviating buyers all strictly better off, and
all buyers can afford to pay for the good.
Two other candidate pooling equilibria are: pooling in { e 6= ∗ ( )} ; () pooling in { e 6= ∗ ( )}.
In both of these, since the price posted is not the “optimal price,” there are many profitable group devia-
tions. One example is to take as given that buyers are earning a certain level of expected utility, and offer a
competing advertisement with the current (same) quality offered and which will attract all buyers keeping ¢ ¡ that level of expected utility constant at the optimal , that is, solves − ( − ) = , where is the relevant expected utility level for the quality under consideration, and solves either (27) or
(30), depending on the relevant quality . If it turns out that the implied , then the competing advertisement would target buyers only, and the way the profitable group deviation is constructed remains the same as in this paragraph. Hence, the only stable equilibrium when ≥ is that laid out in part () of proposition 1 Proof of part () of proposition 1 © ª If , a pooling equilibrium in ∗ = is clearly infeasible for buyers since no buyer
can afford good - we know from lemma A.0 that it cannot be that buyers are not served in equilibrium.
Let us consider a candidate pooling equilibrium where all firms post { ∗ ( )}, where ∗ ( ) is
the solution to a firm’s profit maximization problem as laid out in ( 1) for = , where in a symmetric
equilibrium, ∗ ( ), ∗ ( ), and ∗ ( ) are given by (33), (34), and (35) respectively. Applying the reasoning used in the proof of part () of this proposition, this candidate equilibrium is not robust to individual deviations since a deviant firm can be made strictly better off selling instead; and this candidate equilibrium is also not robust to group deviations since a group of firms can deviate to post ª © = which makes all deviating firms and buyers strictly better off. In any other candidate
pooling equilibria where all firms post { e 6= ∗ ( )}, given that when the optimal pice ∗ ( ) were
posted there existed profitable group deviations, any e 6= ∗ ( ) would imply lower expected profit, and
is also not robust to deviations.
© ª Another candidate pooling equilibrium is where all firms post e 6= where e ∈ ( ). Since
e , the buyers are not served since they cannot afford to buy the good, and we know from lemma
51
A.0 that this is not stable. Now suppose all firms posted { e = }. Then the expected utility of a buyer £¡ ¡ ¢ ¤ ¢ e = 1 − − ( − ), and the expected profit of a firm is e = 1 − − ( − ). This is
candidate pooling equilibrium is not stable either, and the intuition why this is not stable is because the price charged by firms when they are selling is not generically “optimal.” Suppose a mass of firms were ¡ ¢ to deviate, and they choose such that deviating buyers have the same level of expected utility,
1 − − e ( − ) =
(36)
Consider any deviant firm. Its profit maximization problem is strictly concave, taking into account that e , from lemma A.1. So using (36) to express its profit maximization problem buyers have to be promised
, we have in terms of
max(1 −
−
Ã
e ) − − 1 − −
!
.
It is easy to show that the above is strictly concave in . Letting be the solution to this problem,
or that
e = 0 ( − ) − − µ ¶ − = ln . e :
(37)
Generically the solution 6= , that is, if the deviant firm could choose, it would pick the “optimal” price
solving the above maximization problem, but generically 6= . If it turns out that , this candidate
pooling equilibrium where all firms post { e = } is not stable since I can construct a group deviation © ª where firms posted , and since , no buyer will go to this new marketplace; deviating e ; deviating firms make higher expected profit because generically the optimal buyers have the same price 6= If, however, ≤ , this candidate pooling equilibrium where all firms post { e = } ª © is not stable either, since the group deviation where all buyers is posted, where solves (37) is profitable since all deviating firms are made strictly better off, and all buyers are indifferent.¥ Proof of lemma 3 The proof is similar to the proof of lemma 1, except that buyers may not be able to afford to pay the deviation price in that proof since , so that deviation may not work, so we prove this lemma here.
In this candidate equilibrium, expected utilities of both and buyers, ∗ ( ) =
∗ ( ) = 0, let and denote the queue lengths in their respective marketplaces, and firms have expected profit ∗ = (1 − − ) ( − ) = (1 − − ) ( − ). Now consider this group deviation ¡ ¢ { = − ( − ) e− 1 − − = e }, where e is the value of such that = (1 − − −
− ) ( − ) = ∗ = (1 − − ) ( − ). The resulting = − ( − ) 0. To ensure that
buyers are not joining this deviation, check that . If, when = e ≤ , then choose a larger
value of (which can be done since it means the mass of firms deviating gets smaller), which means a higher level of , and which will correspond to a higher price , and even though falls, it will still be strictly
positive, so buyers are strictly better off taking part in this group deviation. ¥ Proof of proposition 2
52
First, note that when , any stable equilibrium cannot be pooling. So the question is what form a segmented equilibrium will take, if one exists. Second, note that buyers have to offered the good in marketplace 1 at an optimal price - like in the proof of lemma 2, if buyers were offered anything else there would exist a profitable deviation. Third, note that buyers cannot be offered good since ≤ , so firms would make negative or zero profit. Hence, buyers must be offered the good in marketplace 2, so
any segmented equilibrium, if it exists, is not a PDE but a PSE. Fourth, note that if buyers were offered { } where, given , 6= − ( − ) − (1 − − ), that is, the price is not optimal, then there
would exist a profitable group deviation catering to buyers since deviating firms can offer these buyers the same level of expected utility and move to the “top” of their profit function. Hence, a PSE, if it exists, has both types of firms charging an optimal price. That is, within each marketplace, a firm is solving its profit maximization problem as given by ( 1), and the method to solve it is similar to that laid down when solving a firm’s problem in a pooling equilibrium earlier, except that a firm takes as given the measure of firms in its marketplace, so the set of F.O.C.s to this firm’s problem is given by equations (5) − (7).
To prove that a PSE exists, we have to show that there exists ∗ the fraction of firms in marketplace 1
such that firms make equal expected profit; or that ¡ ¡ ¢ ¢ 1 − − − − ( − ) = 1 − − − − ( − )
(38)
= () , = [(1 − ) ] (1 − ). Differentiating (38) and (38) with respect to yield (38) (38)
(−) ( − ) 0 for 0; 2 (1 − ) ( − ) 0 for 0 = ( − ) (1 − )2
= ( − )
Note that as → 0, = → ∞, and (38) → ( − ) ; and as → 1, → , and (38) → ( − ) (1 − − − − ) | = ≡ 0 where 0 0 for 0. Also, as → 0,
= (1 − ) (1 − ) → (1 − ) and (38) → ( − ) (1 − − − − ) |=(1−) ≡ 1 where
1 0 for 0; and as → 1, → ∞ and (38) → ( − ).
When ( − ) ( − ), for there to be a solution to (38), need to ensure that ( − ) 0 , or ¢ ¡ ( − ) 1 − − − − | = . ( − )
(39)
(39) ∈ (0 1), and (39) also takes a value between (0 1) for 0, where the higher or is, the
higher the value of (39), which makes it harder for (39) to be satisfied. Equation (39) is just condition (0 ) so if condition (0 ) holds, there exists ∗ solving (38), which is just equation (12). Now, having proven that there exists a ∗ such that firms are indifferent between selling or good, we ∗ ∗ ∗ ∗ = ( ) = ∗ and = ( ) = can pin down the equilibrium queue lengths, which are just
[(1 − ) ](1 − ∗ ). Hence, the equilibrium prices for each quality of good as a function of equilibrium queue
lengths are that in equation (13) for = ; all firms earn equal expected profit as given in (14); and buyers’ expected utilities, depending on which marketplace they are in, are given by (15). Notice from equation (12) that since ( − ) ( − ), if we define () ≡ (1 − − − − ), it
∗ ∗ ( ) ( ). But more is clear that 0 () 0 for 0, so from equation (12) it must be that ∗ ∗ ( ) ( ). precisely, since the measure of buyers per firm is = , it must be that
53
∗
∗
∗ Hence, we have that ( ) = − ( − ) ∗ ( ) = − ( − ). On comparing (13) to
∗ ( ) , ∗ ( ) (3) for = , it is clear that since
Notice that in general, we do not know the value of ∗ ( ) relative to , so it can be that ∗ ( ) ≤
∗ ( ) ∗ ( ). But implying that a buyer can unilaterally deviate to marketplace 1 since
if ∗ ( ) then even if a buyer would like to deviate to go to marketplace 1, they will not do so since they cannot afford to purchase a good. Rewriting ∗ ( ) , we have ∗
∗ − − −∗ − 1−
(40)
It is clear that for a function () = − (1 − − ) that 0 () 0, (0) = 1, and (∞) = 0. Hence,
∗ 0. If (40) 1, then for sure the inequality is satisfied. A sufficient condition (40) 0 for all
to ensure that buyers will not deviate to marketplace 1 is ≥ , which was assumed in the statement of this proposition. We maintain this assumption for the rest of this proof.
This PSE is robust to individual deviations. A firm in marketplace where quality is sold cannot do better by posting some 6= ∗ ( ) since ∗ ( ) is the optimal price. A firm within a marketplace
cannot do better changing the quality sold: if a firm in marketplace 1 tried to sell instead, the queue
∗ ∗ = ( ), which means if it priced it at an optimal price for this queue length it faces is still ¡ ∗ ∗ ∗ ∗ ¢ ∗ − ∗ − (1−− )], its deviation profit = 1 − − − length, = −[( − ) ( − ) ¡ ∗ ∗ ¢ − ∗ − ∗ − 1− ( − ) = ( ). A firm in marketplace 2 will not try to sell since ≤
A firm also cannot do better by changing marketplaces either, since firms in all marketplaces earn the same
∗ ( ) ∗ ( ) no buyer has any incentive to change expected profit. As for buyers, since
marketplaces; and even though buyers would prefer to be in marketplace 1, they cannot afford to pay for this since ∗ ( ), which means they obtain zero utility from being in the marketplace 1, so they no buyer would deviate unilaterally either. What remains is to show that this PSE is robust to group deviations. Consider first group deviations catering to buyers. Suppose a mass of firms decide to deviate to post ∗ ∗ ( ) }, since ∗ ( ) is an optimal price, given ( ), these firms { e 6= ∗ ( ) e 6=
cannot do any better by charging e 6= ∗ ( ). There is clearly no way to offer buyers a deviation © ª ¡ ¢ advertisement with quality sold: suppose the advertisement were , where were
∗ ( ) and is defined in equation (27), then it must optimal in that solves = − ( − ) =
∗ ( ), but this implies a deviating firm’s profit is = (1 − − − − ) ( − ) be that
∗ ( ), which is not profitable. Now if = but were not optimal, then the implied profit to a firm is even lower than if had been optimal, so this does not constitute a profitable deviation either. Now consider group deviations catering to buyers. Suppose
∗ ( ) } were posted by a group of firms. Since ∗ ( ) is an optimal price, { e 6= ∗ ( ) e 6=
given ∗ ( ), these firms cannot do any better by charging e 6= ∗ ( ), so this deviation is not
profitable. Now consider deviations like this { e ≤ e }, where were offered to buyers - since
≤ , these deviating firms posting e ≤ will earn zero or negative expected profits, so these group deviation are not profitable.
Now consider group deviations targeting all buyers. Suppose { e eall buyers} - since ≤
deviating firms are making zero or negative expected profits, so this is not a profitable group deviation.
Now suppose deviating firms offer { e eall buyers}. To be able to attract all buyers, the deviating utility
54
∗
∗ ∗ promised ≥ ( ) ∗ ( ). Suppose = ( ) = − ( ) ( − ). If the deviating
firms could choose an optimal price associated with this quality, with associated queue length which
∗ ( ). means they are doing the best they can, then it must be that = − ( − ), or that
But this would then imply that deviation profit = (1 − − − − ) ( − ) ∗ ( ) ¥ Proof of proposition 3
Consider equation (12) but for ease of notation in this proof, let us drop the “∗ ” Recall that = , = (1 − ) (1 − ), (12) 0 and (12) 0. It is straightforward to verify that (12) (12)
= ( − ) − 0; µ ¶ − − = ( − ) 0. 1−
Hence, 0 unambiguously. However, this is not enough to help us think about what happens to equilibrium queue lengths. Suppose the economy initially had 0 fraction of buyers, and the equilibrium fraction of firms is 0 , and the equilibrium queue length at a firm is 0 = 0 0 . Now suppose this economy had 1 fraction of buyers instead, 1 0 , and the new equilibrium fraction of firms is now 1 0 , and the associated new queue length at a firm is 1 = 1 1 . For 1 0 , it is equivalent to 1 1 0 0 . But it can be shown, writing = 1 − 0 0 and = 1 − 0 0 that 0 + 0 0 1 = ⇔ 1. 1 0 + 0 0 This is very intuitive. It simply says that if the percentage change of exceeds the percentage change of , then will rise. Rewriting equation (12) as ¡ ¡ ¢ ¢ ≡ ( − ) 1 − − − − − ( − ) 1 − − − −
and applying the Implicit Function Theorem,
(1−) 0 2 + 1 (1−) 2 0 = 1 1 0 + 1 1−
where 0 ≡ ( − ) − 0 for 0, and 1 ≡ ( − ) − 0 for 0. Also, h i (1−) 1 1 ∙ ¸ + 0 1 1− (1−) (1 − ) 1 since = 1 1 (1 − ) 0 1 + 1 1− where the latter inequality holds since . This implies that as goes up, goes up, but not by as ∗ ∗ ( ) goes up. From equation (12), this implies that ( ) much in percentage terms, so overall
goes up. From equation (13), since for a function () = − (1 − − ), (0) = 1 (∞) = 0, 0 () 0,
this implies that ∗ ( ) goes up, and ∗ ( ) goes up. Clearly, each firm’s expected profit has gone up
∗ ( ) and ∗ ( ) have decreased.¥ since queue lengths have gone up. This thus implies that both
Proof of proposition 4 We know from the proof of proposition 2 that a PSE exists under condition (0 ). From part () of corollary 4, ∗ ( ) . Suppose = − , 0 arbitrarily small. Then a buyers will
55
∗ unilaterally deviate to marketplace 1 where the buyers are since ( ) ∗ ( ). In fact, for
∈ [∗ ( ) ) the PSE is not robust to individual deviations, and hence cannot be stable, since a
buyer will want to change marketplace. The question is whether PSE can be stable for ∈ ( ∗ ( )). Let ∗ = ∗ ( ), = . Let us consider group deviations first before discussing individual deviations.
In a PSE, ∗ solves ∗ ( ) = ∗ ( ), where ∗ = ∗ ( ) is defined in (14), and ∗ = ∗
∗
∗ ( ) = − ( − ) = [(1−− )∗ ] ( − ∗ ), for = , ∗ = ∗ ( ). Just like in the proof of
∗ ∗ } or { e 6= ∗ e 6= } proposition 2, any group deviations to non-optimal prices { e 6= ∗ e 6=
are not profitable, so I will not repeat the analysis here. Another type of group deviation of this form
{ either or all buyers} is also not profitable - suppose the deviation price posted is an optimal
∗ and is defined in equation (27); clearly this deviation is not price, i.e., solves − ( − ) =
profitable. Which implies that any other deviations of this type = where the price posted is not optimal
yields even lower deviation profit and are also not profitable. The are two group deviations left to consider, whose analysis is different when . The first is when the group deviation is of this form: { = s.t. = ∗ }, where
¢ 1 − − ¡ − = ∗ , and ¢ ¡ = (1 − − ) − =
(41) (42)
It is sufficient to consider a group deviation where = , because we know that in a group deviation where the constrained price is charged that firms would like to increase price even more, but they cannot because of the income constraint that binds. Hence, if at the constrained optimal price of = deviating firms cannot be made better off, they are strictly worse off for all deviating prices strictly less than . (There is no way for deviating firms to actually charge the “unconstrained” optimal price, i.e., the such that = ∗ because .) For now, note that for (41) to hold, if were to increase, has to decrease. We have to check whether for ∈ ( ∗ ) there exists a profitable group deviation of this form { = s.t. ∗
∗ is also a number in a PSE. = ∗ } Note that ∗ = − ( − ) is a number since
When = + , for 0 arbitrarily small, where this is the smallest -value we have to consider,
if deviating firms charged = , then from equation (42), = 0, and since ∗ 0, this group deviation is not profitable, so by continuity when = + the deviation profit is very small and close to 0. Now let us consider the largest -value possible, which is = ∗ − for 0 arbitrarily small, but
for ease of exposition suppose the largest -value possible is = ∗ , so we can write − = − ∗ = ¡ £ ¡ ∗ ¤ ∗ ¢ ∗ ∗¢ ∗ − ∗ − 1 − − ( − ) 1 − − (from equation (13) for = ), and −∗ = ( − ) ∗ , and we also know from (from equation (13) for = ). For (41) to hold, it must be that ¢ ¡ ∗ ∗ ∗ ∗ corollary 4 that , so . Moreover, since = (1 − − ) − , ∗ ) ( − ), where () ≡ [(1 − − − − ) (1 − − )], since when = = ∗ , ∗ − = ( ¡ ∗ ¢ ∗ ∗ ) ( − ) ∗ = 1 − − ( ) ( − ). In other words, at the largest = (1 − − ) (
possible -value, the group deviation { = s.t. = ∗ } is profitable.
Since at the lowest possible -value the group deviation is not profitable, i.e., ≡ − ∗ 0 but it is
profitable at the largest possible -value, i.e., 0, what remains is to show that this difference function
is strictly increasing in for ∈ ( ∗ ). From (42), writing = , noting that if changes
and the economy still remains in a PSE that the equilibrium outcome in the PSE remains unchanged (in particular ∗ remains unchanged). Which means = . Differentiating (42) with respect to
56
, and applying the Implicit Function Theorem on (41) by setting = to obtain ,
=
1 − − ∗ Ψ, where ( − ) (1 − − − − )
Ψ ≡ ( − ) (1 − − − − ) − ( − ) − .
(43)
If it can be shown that Ψ 0 for 0, then = 0. It is clear that Ψ 0 if and only if
( − ) − − ( − ) 1 − − −
(44)
but the (44) is strictly decreasing in , so if at the largest value of = ∗ equation (44) holds, then Ψ 0. Therefore, letting = ∗ , equation (44) becomes ∗
∗ − − ∗ ∗ ∗ − 1 − − − 1 − − − −
(45)
Defining () = − (1 − − − − ), it can easily be shown that () is a strictly decreasing function
in , with () 0 for all 0. Hence, since from earlier in this proof (for this particular group deviation) ¡ ¢ ∗ ∗ , it must be that ( ) , or that equation (45) holds. Which thus implies that we know that Ψ 0 0, and 0. Since 0 at = and 0 at = ∗ , with 0,
there exists one value of such that = 0, or that there exists some critical value of , which is called ∈ ( ∗ ) such that for ≤ the PSE as solved for earlier is robust to this type of group deviation.
In lemma A.3 (at the end of the Appendix) I show that if we had instead considered group deviations
where buyers are offered quality at a price they can afford, but deviating firms’ expected profit = ∗ and asked if deviating buyers have a utility ∗ , the resulting cutoff value of where the PSE is robust as solved for, because basically we are trying to find the value of such to group deviations is exactly
that = ∗ and = ∗ and there is exactly one value of that satisfies both equations. The idea is ∗ ), if we gave the same expected payoff to one deviating group (say, the deviating that for all ∈ (
firms), and the other deviating group (say, the buyers) can be made better off, there are many ways to
actually split this “surplus” from deviating between two groups, and there are many group deviations which ], there is no way to keep as can make both deviating parties strictly better off. However, for ∈ (
given the expected payoff to one deviating group and make the other deviating party strictly better off.
∗ all buyers}, which is the last left Now let us consider this group deviation { = s.t. =
∗ ∗ , in this group deviation, at all levels of , the associated profit is lower to consider. Since in a PSE, ∗ . Hence, the associated profit function since deviating buyers have to be promised more through =
is lower than the profit function associated with a group deviation { = s.t. = ∗ } which
we had considered in the preceding paragraphs, and it is also monotonically increasing in for the relevant range of . As a consequence, the more “binding” of the two group deviations is the one we had considered
earlier targeted at buyers, so we do not have to consider this particular group deviation. Hence, the ¡ ¢ ∈ is such that for ≤ the PSE solved for earlier is robust to group deviations. critical value The last step is to consider individual deviations. Like in the proof of proposition 2, a firm in marketplace
where quality is sold cannot do better by posting some 6= ∗ since ∗ is the optimal price. A firm in marketplace 1 cannot do better changing the quality sold by the same reasoning in the proof of proposition
2. Now consider a firm in marketplace 2 deviating by selling quality instead (this part is different from the
57
∗ proof of proposition 2). This firm takes as given the queue length , and it could try to post instead at the
“constrained” optimal price, which is . When considering group deviations earlier in this proof, a group of deviating firms in posting { = s.t. = ∗ } is basically choosing s.t. = ∗ , but in an ∗ . Hence, values of such that a individual deviation, this individual firm takes as given the queue length
group deviation offering to at a constrained price is not profitable (i.e., when you can choose ) tells ∗ is taken as given is also not profitable. Hence, we do us that an individual deviation where queue length
not have to separately check for an individual deviation where a firm offers quality in marketplace 2. As for the rest of individual deviations, note that a firm also cannot do better by changing marketplaces either ∗ ∗ no buyer has any incentive since all firms earn the same expected profit. As for buyers, since
to change marketplaces; and even though buyers would prefer to be in marketplace 1, they cannot afford to pay for this since ∗ ( ), so no buyer would deviate unilaterally either. ¡ ¢ ∈ , is truly such that for ≤ the PSE as solved Hence, the critical value of solved for, for earlier is robust to both group and individual deviations, and is thus stable.¥ Proof of lemma 4 In an equilibrium all firms have to make the same expected profit. Let and buyers be in marketplaces 1 and 2 respectively. If the quality offered in both marketplaces is , it must be that ¡ ¡ ¢ ¢ 1 2 ∗ = ∗ ( ) = (1 − − ) 1 − = (1 − − ) 2 −
(46)
and are the queue length and posted price respectively in marketplace = 1 2 and a buyer’s where
] ( − ) Notice that if 1 = 2 , then for expected utility in marketplace is = [(1 − − ) 1 2 = , and 1 = 2 . But if the advertisements in the two marketplaces are equation (46) to hold,
identical, one marketplace is simply a duplicate of the other, thus violating what is means to be a segmented 1 2 6= . equilibrium (this is in fact a pooling equilibrium). Hence, 1 6= 2 and hence, from equation (46)
, is Suppose either, or both 1 and 2 are not optimal, whereby an optimal price, given queue length
− (1 − − ). If either or both 1 and 2 are not optimal, then it is possible to = − ( − )
create a profitable group deviation whereby the agents in the marketplace with the non-optimal price who are currently getting expected utility are attracted, with the same expected utility promised, and deviating
firms choose such that − ( − ) = , and since the price can always be afforded by all agents,
deviating firms are able to move to the “top” of their profit function, taking as given that is offered, and
can hence do strictly better. Now suppose that both prices are optimal, that is, expected utilities of both 1
2
1 2 = , which is types of buyers must satisfy − ( − ) = − ( − ), but this then implies that 1 2 6= ).¥ a contradiction (since we know that in an equilibrium 1 6= 2 and
Proof of lemma 5 Denote the price posted in marketplace as , = 1 2. First note that in any PDE, 1 ≮ 2 . Suppose not. Suppose 1 2 . This implies, from firms’ equal expected profit condition, 1 = 2 ≡ ∗ ,
1 2 1 2 . If the implied , where = where = (1 − − ) ( − ) for = 1 2, that
] ( − ) for = 1 2 a buyer will deviate to marketplace 1 since he can afford to [(1 − − )
1 2 , a buyer will deviate to marketplace 2 since he can afford the purchase the good; if the implied ∗
1 2 = ≡ . However, in this candidate PDE with 1 2 , if we take two good. So it must be that
58
prices, 1 and 2 , and let ∗ be the fraction of firms in marketplace 1, that is, ∗ solves 1 = 2 ≡ , then ∗
1 = 2 ≡ (there are two equations in one unknown ). Hence, it cannot generically it is not true that
be that 1 ≮ 2 , or equivalently, that 1 ≥ 2 . However, if 1 = 2 , from the firms’ equal expected profit
1 2 1 2 = , and = , so this is not a PDE since one marketplace is a duplicate of another. condition,
Hence, in a candidate PDE it must be that 1 2 . Next, note that if the queue length in marketplace 1∗ and 1 6= 1∗ 1 were , where
1∗
1∗ = −
1∗ − ( − ) 1∗ − 1−
(47)
that is, the price posted 1 is not the “optimal price” 1∗ , then there is a profitable group deviation: a group ¡ ¢ of firms can offer a combination giving these buyers the same level of expected utility, which increases deviating firms’ expected profit, since they are moving to the “top” part of their profit function, and buyers can afford to pay by assumption (1). Hence, in a PDE, 1∗ as given by equation (47) must 1∗ . be the price posted in marketplace 1 when the queue length is 1∗ , it must be that that 1∗ Anticipating the result from part () of this lemma that . 2∗ Next, want to show that 2∗ = . For sure we know that ≤ , but the reason we know this has ∗
1∗ 2∗ ∗ ∗ ≥ , where = [(1 − − ) ]( − ∗ to hold with equality is as follows. In an equilibrium, )
∗ , for otherwise is the expected utility of a buyer in marketplace = 1 2, when the queue length is 1∗ 2∗ = , using the same argument as in this proof a buyer will deviate to marketplace 2. But if ∗ 2∗ 1∗ 2∗ earlier, we have two equations ( 1∗ = and = ) in one unknown, , so it is not generically
true that both equations can be satisfied. Hence, in a PDE, it must be that this inequality is satisfied: 1∗ 2∗ . (This proves part () of this proposition.) But for a buyer not to deviate to marketplace ¡ ∗ 2∗ ∗¢ 2∗ 2∗ = [(1 − − ) ] − 2 which is just a 1, it must be that 1 . Now note that since
2∗ 1∗ 2∗ 1∗ = − ( − ). For , since = number, denote a hypothetical which is such that 1∗ −
∗
∗
1 ( − ) (this is true because 1∗ satisfies equation (47)), it must be that . But then,
∗
1 ≡ − [( − ) − ](1 − − ), and we know that 1 . In other words, if firms
could charge the optimal price in marketplace 2, they would charge a price , but clearly they cannot since ∗
1 . Hence, firms in marketplace 2 face a “binding” income constraint of buyers. We know from the proof of a lemma A.1 that taking as given the expected utility buyers must receive, a firm’s deviation profit when it is allowed to choose a price-queue length pair is strictly concave. Hence, if the optimal price firms in marketplace 2 would like to charge exceeds buyers’ income, then the price posted by these firms would be as close as possible to the optimal price, and since any price charged can be up to ∗
, these firms will indeed post 2 = . (This completes the proof of part () of this proposition.) 1∗ In other words, a PDE, if it exists, will have 2∗ = and as defined in equation (47), and firms
make equal expected profit operating in either marketplace, or that ∗ is the solution to ¢ 1∗ ¡ 2∗ − (1 − − ) 1∗ ) ( − ) − = (1 − {z } | {z } |
(48)
= 2∗
= 1∗
1∗ Since 2∗ = , for the equal profit condition, equation (48) (which is just equation (16) in the main 1∗ 2∗ 1∗ 2∗ . And hence, . text) to hold,
(This proves part () of this proposition, and hence completes the proof of this proposition.)¥ Proof of lemma 6
59
First, we show that there exists a PDE at arbitrarily close to but a bit lower than . In a PDE, ∗2 1∗ 2∗ ∗ ∗1 = , where, substituting in as defined in equation (47), letting = and dropping the “ ” for
ease of notation, we have 2 1
2
= (1 − − ) ( − ) , ¡ ¢ 1 1 1 1 − = (1 − − ) 1 − = (1 − − − ) ( − )
(49) (50)
1 2 = ; = (1 − ) (1 − ). To show there exists a PDE we have to show there exists with
a ∗ solving 1 = 2 . Differentiating (49) with respect to , it is easy to show that 2 0 ¡ ¢ 2 2 2 0; as → 0 → (1 − ) so 2 → 1 − −(1−) ( − ); as → 1 → ∞ so for all 1 0; 2 → ( − ). Differentiating (50) with respect to , it is easy to show that 1 0 for all ¡ ¢ 1 1 1 1 − −() − () as → 0 → ∞ so → ( − ); as → 1 → so → 1 − ( − ).
For a PDE to exist, there has to exist a ∗ such that 1 = 2 , and in order for that to happen, the condition
required is, since 1 |=0 = ( − ) ( − ) = 2 |=1 , that 2 |=1 1 |=1 or that ³ ´ − (51) 1 − − − () −() − ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ − When → , ( − ) → ( − ) 1 − − − − 1 − − , so (51) → 1 − − − − 1 − − . ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ Since 1, 1 − − − () −() 1 − − − − 1 − − − − 1 − − , and hence (51) −
holds when → . In other words, there exists a PDE for values very close to . (It is easy to see
from (51) that as falls, then (51) falls, which is why (51) need not be satisfied, or that there need not exist a ∗ such that = , i.e., a PDE need not exist for low enough values of . ) Second, we show that this PDE is robust to group deviations for arbitrarily close to but just under . −
This is actually done in the proof of proposition 5 below where I show that when → , ∗ = ∗ ( )
, where refers to any deviation profit to a firm from a group deviation. In particular, group deviations © ª where are also not profitable. In other words, there does not exist a profitable group deviation
for values of close, and just under, .
Third, this PDE is robust to individual deviations. No buyer can go to the wrong marketplace and be made strictly better off. No firm can go to the other marketplace and be made strictly better off either. A firm in marketplace 1 cannot do better than post this optimal price 1∗ : if it posted a non-optimal price it would 1∗ , but the make strictly less profit; if it posted quality instead, has to take as given that the queue length is
best it can do when posting a different quality is to charge the optimal price associated with this queue length, 1∗
1∗
1∗
1∗
1∗ − 1∗ − ) ( − ) (1 − − − ) ( − ) = 1∗ but then deviation profit = (1 − − − .
A firm in marketplace 2 is posting the “best” constrained price, so any price higher is not feasible, in that
buyers cannot afford it, and any price lower yields strictly lower expected profit. A firm in marketplace 2 could try to post quality instead when queue length is taken as given (since it is the only firm deviating © ª it cannot “choose” the queue length), but in a group deviation where the queue length can be “chosen” by a group of deviating firms, if this group deviation is not profitable (which is proven in the
proof of proposition 5 below when is just arbitrarily less than ), then when queue length is restricted 2∗ , the individual deviation is not profitable either. to be
Fourth, we note that there is no other stable equilibrium. This is because we know from part () of proposition 1 that there does not exist any stable pooling equilibrium for . Hence, any potential stable equilibrium is a segmentation equilibrium. Since buyers have to consume , and at what corresponds to an “optimal price” in any equilibrium, for otherwise there is a profitable group deviation , the
60
question is what buyers are consuming. Since we know from proposition 4 that there does not exist a PSE for arbitrarily close to but just under , the only possibility is that buyers are consuming . Hence, this PDE is the unique stable equilibrium. Proof of lemma 7 1∗
1∗
2∗
2∗ 1∗ 2∗ − 1∗ − Since ∗ ( ) = 1∗ − ) ( − )−(1−− ) ( − ) = , let ≡ − = (1−
1∗ 2∗ = ∗ and = (1 − ) (1 − ∗ ). Note that where ∗ is the solution to the above, with 2∗ 2∗ 1∗ 2∗ 1∗ 1∗ − = −(1 − − ) 0; ) ( − ) ∗ − ( − ) − ∗ 0 = ( | {z } | {z } 0
0
Therefore, ∗ = −( )( ∗ ) 0. Hence, when falls, ∗ goes up so: this results in
1∗ 2∗ decreasing and increasing, with ∗ = ∗ ( ) decreasing monotonically (as is apparent from +
1∗ considering the expression for 1∗ ), and increasing monotonically. In fact, notice that as → , +
∗ → should a PDE exist at this low value of . As falls, the price paid by buyers is falling, but
2∗ , which decreases their probability of obtaining the good, and these buyers face a higher queue length
2∗ could rise or fall.¥ their overall expected utility
Proof of proposition 5 Now let us check when a PDE is stable. Define ∗ = ∗ = 1∗ = 2∗ . We check group deviations first, before checking individual deviations. There are three types of group deviations to check: () deviations catering to buyers; () deviations catering to all buyers; and () deviations catering to buyers. First, consider () deviations catering to buyers only. If deviating firms offered , there is no way ¡ ¢ 1∗ since in marketplace 1 (with buyers only) the firms they can come up with such that ≥
there are already posting an optimal price. Now suppose a group of firms posted a deviation advertisement © ª of . Given that these firms cannot do better than to charge an “optimal” price, that is, for a
given , is such that (27) is satisfied, we thereby obtain that = (1 − − − − ) ( − ) and
1∗
1∗ 1∗ = − ( − ), it must be that . However, this implies = − ( − ), and for ≥ 1∗ −
1∗ = (1 −
1∗ −
1∗ −
) ( − ). Hence, this is not a profitable group deviation. Now suppose © ª a group of firms posted a deviation advertisement of but were not “optimal;” since when
were optimal it is not possible to find a profitable group deviation, in this deviation where were not optimal deviating firms make even less profit, and hence, this sort of group deviation is also not profitable.
Hence, there does not exist any profitable group deviation for type () deviations catering to buyers only, for all values of , including when is just arbitrarily close to, but smaller, than . () Now consider group deviations catering to all buyers, or “all” for short. Let us start with a group deviation { all} where is optimally chosen, i.e., equation (27) is satisfied for a given . In a PDE, 1∗ 2∗ 1∗ , so in this group deviation, the promised deviation utility has to be such that ≥ . But
this means that the analysis used in considering type () deviations earlier in this proof when the group © ª deviation was applies directly, since we also know that all buyers can afford to purchase good ª © from assumption (1). What about other deviations attracting all buyers? What about ?
61
¡ ¢ 1∗ Can it be that ≥ and ≥ ∗ ? If were chosen optimally, then for = − ( − ) ≥ 1∗
1∗ 1∗ = − ( − ) to be true, it must be that ≤ . But = (1 − − − − ) ( − ) ≥ ∗ = 1∗
1∗
1∗ − 1∗ 1∗ ) ( − ) requires that ≥ . Which implies that = has to hold, but then (1 − − −
1∗ , and = ∗ which means this is not a feasible deviation since no deviating agent is made strictly = ª ¡ © ¢ better off. What about a group deviation where is not chosen optimally? Since in
the best deviation firms can only make equal profit as in the PDE, any other non-optimal price must imply lower expected profit, which means it is not a profitable deviation.
Hence, there does not exist any profitable group deviation for type () deviations catering to all buyers only, for all values of , including when is just arbitrarily close to, but smaller, than . Lastly, let us consider () type deviations catering to buyers only. Since in the candidate equilibrium © ª buyers are charged the constrained price 2∗ = , can deviating firms do better by posting ¢ ¡ 2∗ ? We know from the proof of lemma such that satisfy equation (30) and = − ( − ) ≥ 2∗ , firms would like to charge a hypothetical price exceeding 5 that taking as given that buyers enjoy
2∗ but cannot, and since the firm’s profit function is strictly concave taking as given (from lemma A.1), the
firm is already choosing the constrained maximal price . Hence, it cannot charge a lower, but feasible price to buyers without lowering its expected profit. Hence, this group deviation is not profitable. Now let us © ª consider group deviations catering to buyers where the good is offered instead. Suppose ¡ ¢ is such that is an optimal price, i.e., satisfy equation (27) and
=
2∗
¢ 1 − − ¡ 1 − − − 2∗ − ( − ) ≥ = ( − ) = 2∗
(52)
( ) = (1 − − − − ) ( − ) = (1 − − )( − ) whereas (53) ¢ 1∗ 1∗ 1∗ ¡ 2∗ 1∗ − − ∗ = (1 − − − ) ( − ) = (1 − − ) 1∗ ) ( − ) (54) − = (1 − 1∗
1∗ 1∗ Comparing (53) to (54), for ( ) ≥ ∗ , , which implies that = − ( − ). In
other words, no matter what value is, this deviation will optimally not attract buyers.
Hence, there does not exist any profitable group deviation for type () deviations catering to buyers only, for all values of , including when is just arbitrarily close to, but smaller, than . © ª − Claim 1 Let → . This group deviation is not profitable.
Proof. First suppose that is an optimal price, or that satisfies (27). We know from the proof −
−
1∗ 1∗ of lemma 5 that = 2∗ , which means that when → , → in such a way that −
+
1∗
1∗ 2∗ 2∗ 1∗ 1∗ − ( − ) → 1∗ We also know that → and → and → , where =
− ( − ). Hence, for values very close to , for (52) to hold, since = − ( − ) and 1∗
2∗ 1∗ 1∗ → = − ( − ), it must be that , but from comparing (53) and (54), we know that © ª − ∗ . In other words, as → , there does not exist a profitable group deviation such 2∗ and ≥ ∗ . And this is when the considered is an optimal price; which means that for that ≥ ¡ ¢ all other possible non-optimal deviation prices, i.e., do not satisfy equation (27), there is no way to ª © 2∗ and ≥ ∗ . construct a profitable group deviation such that ≥
Having considered the largest value of , let us now consider the smallest possible value of . From
the proof of lemma 6, it is clear from equation (51) that for low enough values a PDE need not exist. So let
62
min us denote the value of such that (51) holds with equality, i.e., let be such that
´ min − ³ 1 − − − () −() = −
(55)
© ª + min This group deviation is profitable. Claim 2 Let →
+
min , this is the Proof. Let us suppose that is an optimal price, or that satisfies (27). When →
minimum value of at which a PDE exists, or equivalently, using the result of lemma 7 that ∗ 0, ∗
∗
∗
i.e., that is the highest possible value such that a PDE exists, or that → 1 This implies, as → 1, ¡ min ¢ ¡ ¢ 2∗ 2∗ = (1 − ) (1 − ) → ∞, ∗ → − = 1 − − − () −() ( − ), and → 0 that ∗
2∗ 2∗ , is such that, since → 0, is some large Hence, as → 1, the such that = − ( − ) = ¡ ¢ min number, implying that (1 − − − − ) → 1, so = → ( − ) (using equation (53)). For ¡ ¡ ¢ ¢ ∗ min min − − () −() ( − ) ( − ), or = at this value, it must be that 1 −
³
´ − 1 − − − () −() −
+
min , this group deviation but this is just condition (0 ) which is a maintained assumption. Hence, as → © ª is profitable.
© ª The above two claims show that at the highest possible value this group deviation is
not profitable since ∗ , but that at the lowest possible value this group deviation is profitable, or
that ∗ . We also know from lemma 7 that ∗ falls monotonically as falls, so the question is whether changes monotonically in . For sure, there exists at least one critical value of such that ∗ = . 2∗ as given in equation (52), there are two opposing forces When falls, considering the expression for 2∗
2∗ 2∗ 2∗ : ( − ) goes up as falls, but [(1 − − ) ] decreases as falls (since increases as on
2∗ 2∗ falls from lemma 7). Hence, it is not clear how changes. Suppose as decreases that in a PDE
goes up monotonically, which then implies that decreases monotonically. If it does, then there exists a unique level such that for all higher than that critical value, ∗ and for all lower than that 2∗ is not monotonically decreasing in in a . We know critical value, ∗ . It turns out that © ª ∗ 2∗ that at values close to that (where the deviation is ). As starts to fall,
2∗ rises, which implies that falls, and falls. However, when is low enough, starts to fall! Which
implies that and rise. Since is not changing monotonically when changes, is positive-valued −
+
min , but ∗ is monotonically increasing in , it is not clear if there is a when → and when →
unique value where ∗ = . Numerically, for the values I’ve considered so far, changes smoothly, and
s.t. ∗ = . Even if there were multiple turning there is one turning point, so there is still a unique
points of the ( ) function implying multiple cutoffs of such that ∗ = , if we number the cutoffs as the highest cutoff, then for ∈ [ ) ∗ ≥ , so this from highest to lowest and denote
region of PDE is robust to group deviations. If however, there is a unique cutoff even if the function is
, then for ∈ [ ) not monotonic in such that ∗ = , then again, if we label this unique cutoff
∗ ≥ , and again this is the region where there does not exist profitable group deviations.17
Note that if there exists a unique such that ∗ = , then the above proof where we solved for 2∗ allocations such that = and asked if ∗ would lead to the same cutoff as if we had solved 17
63
Now let us consider individual deviations. On the firm’s side, no firm in either marketplace can be made strictly better off going to the other marketplace since all firms earn the same expected profit. A firm in marketplace 1 with the optimal 1∗ posted cannot offer any other price which will allow it to offer buyers 1∗ and make higher expected profit since 1∗ the same is an optimal price; neither can this firm post a 1∗ , and different quality, because being one deviating firm it faces the same queue length as everyone else of 1∗ , but since quality has a lower surplus, posting the “optimal” price associated with it takes as given 1∗ means this deviating firm will be strictly worse off. As for a firm in marketplace 2, given that it is this
already charging the constrained optimal price 2∗ = for quality , if it were to charge any price lower 2∗ than 2∗ will make strictly lower expected profit, and cannot charge a higher price than since buyers
cannot afford it. If a firm in marketplace 2 were to offer quality instead, then since it is the only firm 2∗ , and the best it could do were to deviating, it takes as given the queue length in marketplace 2, which is 2∗ . However, when we considered the offer the optimal price satisfying equation (27) with queue length © ª group deviation earlier in this proof, deviating firms are free to choose the in their group
deviation, so if by being freely able to choose the queue length the group deviation is not profitable, then an 2∗ is also not profitable. If there were one individual deviation where the queue length is restricted to be such that ∗ = , then for all , not only is this the range of that are robust to this group © ª deviation , it is also robust to individual deviations where a firm in marketplace 2 posted
quality good instead. Of course, if an individual firm in marketplace 2 were to post at a non-optimal
2∗ , it will be even worse off. Now consider the buyer’s problem: a buyer would price with queue length 1∗ 2∗ , and a buyer in marketplace 2 would like to go to choose not to go to marketplace 2 since
marketplace 1 but cannot afford to since 1∗ . Because it is not clear how many solutions there are such that ∗ = , it is hard to relate to the other cutoff value of where a PSE is stable. Even if there is a unique , this/these cutoff(s) where a PSE is stable, but for it is not clear analytically how it compares to the other cutoff value of , then a PDE is stable, in that it is robust to both group and individual deviations. sure, for all ≥
value, and that , which means there is a range Numerically, it does appear that there is a unique
of values for which both PDE and PSE are stable.¥ Proof of proposition 6 2∗ This proof is similar to the proof of proposition 3. In a PDE, it must be that ∗ ( ) = 1∗ = 1
1
2
1 − ) ( − ) − (1 − − ) ( − ), For ease of notation, drop the “∗ ” and define ≡ (1 − − −
1 2 = and = (1 − )(1 − ). We obtain, using the Implicit Function Theorem, that where (1−) 0 2 + 1 (1−) 2 0 = 1 0 1 + 1 1−
2∗ for allocations such that ∗ = and asked if , just like in the proof of PSE. The idea is that if 2∗ we can find group deviations such that ≥ and ∗ ≥ with at least one strict inequality, since no group can be made worse off, the question is how the “gain” from deviating in a group are split between the buyers or sellers. In what was done above, sellers appropriated all the surplus, but the other extreme case where buyers appropriated all the surplus, or anything in between where the gains from deviating are shared would imply a profitable group deviation.
64
1
2
1 − 1 2 where 0 ≡ ( − ) 0 for 0, and 1 ≡ ( − ) − 0 for 0. But not only that, h i (1−) 1 ∙ ¸ 0 1 + 1 1− (1−) (1 − ) 1 since = 1 1 (1 − ) 0 1 + 1 1− 1 . In other words, as increases, increases, but by proportionwhere the latter inequality holds since 1∗ increases; ally less. Therefore, as increases, the equilibrium objects change in the corresponding way: 2∗ increases (since the equal expected profit condition has to hold); 1∗ increases (this is immediate from 2∗ 2∗ 1∗ the expression of 1∗ ); remains the same (since = , and has not changed); decreases (since 1∗ 2∗ 2∗ rises); also falls (since rises); and ∗ ( ) goes up.¥
Proof of proposition 7 Consider a candidate equilibrium where buyers of the same type do not have the same expected utility. If two buyers of the same type are in different marketplaces, the buyer with a lower expected utility can simply deviate to the marketplace where the buyer with a higher expected utility is. Hence, this candidate equilibrium is not robust to individual deviations. If these two buyers are in the same marketplace, there is no way a firm can distinguish between two buyers to offer them, from an ex ante point of view, two different expected utility bundles, by the assumption of anonymity. Hence: Claim 3 All buyers of the same type must have the same level of expected utility. Now consider a candidate equilibrium where firms, which are ex ante identical, are earning different expected profit levels in different marketplaces. Then a firm earning lower expected profits will simply deviate to the marketplace where firms are earning higher expected profit. If two firms were earning different expected profits within the same marketplace, then nothing prevents a firm from deviating to post the advertisement which will enable him to earn higher expected profits. Hence, this not robust to individual deviations either, so this type of candidate equilibrium cannot be an equilibrium. Therefore: Claim 4 When firms are not posting the same advertisements within a marketplace, firms must all be making the same level of expected profit. Claims 3 and 4 imply that in an equilibrium agents who are ex ante identical must have the same expected payoff. This does not mean that firms in a marketplace are posting the same advertisement. Consider a candidate equilibrium where firms in a marketplace post different schedules but have the same expected profit. Let us suppose that firms in this marketplace are selling a good of quality ∈ { }, but with different prices. So there is a distribution (). Buyers in this marketplace (which may or may
not be only of one type) are all earning the same expected utility (by Claim 3), that is, buyers are applying to firms with mixed strategies in such a way that they trade off the rate at which they go to a firm posting a particular price with the waiting time summarized by the implied queue length. Consider two prices 1 , 2 , 1 6= 2 , drawn from (), and posted by firms in a marketplace with associated queue lengths 1 and
2 , and associated expected profits (1 ) and (2 ) respectively. It must be that (1 ) = (2 ) ≡ , or ¡ ¡ ¢ ¢ 1 − −1 (1 − ) = 1 − −2 (2 − ) =
65
(56)
and on the buyers’ side, (1 ) = (2 ) ≡ , or (1 − −2 ) (1 − −1 ) ( − 1 ) = ( − 2 ) = 1 2
(57)
There are three cases to consider. Case (). Suppose both 1 and 2 are optimal, in that ( − ) 1 −1 ; and 1 − −1 ( − ) 2 −2 = − 1 − −2
1
= −
2
(58) (59)
but 1 6= 2 . Equations (56) and (57) can be rewritten as
=
¡ ¡ ¢ ¢ 1 − −1 − 1 −1 ( − ) = 1 − −2 − 2 −2 ( − ) ; −1
=
−2
( − ) =
( − )
(60) (61)
Since 1 6= 2 , we know from (58) and (59) that 1 6= 2 . But (60) and (61) imply that 1 = 2 , which is a contradiction. Hence it cannot be that both prices 1 and 2 are optimal. Case (). When considering two prices 1 and 2 suppose only one price is optimal: without loss of generality, suppose 1 is optimal and defined in equation (58), and 2 is not optimal, in that given queue length 2 2 6= −
( − ) 2 −2 1 − −2
Equations (56) and (57) can be rewritten as
=
¡ ¡ ¡ ¢ ¢ ¢ 1 − −1 − 1 −1 ( − ) = 1 − −1 (1 − ) = 1 − −2 (2 − ) ;
= −1 ( − ) =
−1
1− 1
( − 1 ) =
−2
1− 2
(62)
( − 2 ) = (2 ) 6= −2 ( − )
If 1 = 2 , then 1 6= 2 since 1 is optimal and 2 is not, but this then implies that (1 ) 6= (2 ), which is a contradiction. Therefore, 1 6= 2 . But if 1 6= 2 , equation (62) implies that 1 6= 2 . Now consider a
group deviation where a group of firms propose an advertisement to attract buyers in this marketplace away from the { 2 } advertisement by offering such that it is an optimal price, i.e., solves and
= − ( − ) =
(63)
= −
(64)
−
( − ) 1 − −
where is the deviation utility offered. The idea is that since 2 is not optimal, and by definition not at the top of the expected profit function given that buyers enjoy , these deviating firms can offer the same but move to the “top” of their expected profit function by picking solving (63), and earn strictly higher
expected profit = (1 − − − − ) ( − ) . However, from comparing (63) to (57), it is clear
that = 1 , and hence, = (moreover, = 1 ), which contradicts the fact that 1 is optimal whereas 2 is not. In other words, it cannot be that the 2 posted is not optimal and 2 6= 1 .
66
Cases () and () imply that if () is non-degenerate, it can only consist of non-optimal prices. Case (). Lastly, consider two prices 1 , 2 , 1 6= 2 , where neither price is optimal, but equations (56) and (57)
are satisfied. Without loss of generality suppose 1 2 . Let us consider price 1 . Since 1 is not optimal, a group of firms can get together and offer a competing advertisement to the same buyers in that marketplace: ª © same buyers , where deviating utility satisfies (63) and satisfies (64) If is weakly less than the income of these buyers, this deviation is feasible, and because the firm is now choosing an optimal
price given that deviating buyers are no worse off, it must be that deviating firms’ expected profit . If, however, exceeds the income of these buyers, since 1 (because buyers could afford to pay 1 but not ), then it must mean, because of the strict concavity of the profit function, that deviating firms can offer any e 1 and because e enables the firms to “move up” their profit function that they will make strictly higher expected profits anyway, or that . And since buyers in that marketplace could afford both 1
and 2 where 1 2 , they can also afford to pay some e 1 In either case, this candidate equilibrium is not robust to a group deviation. Hence case () cannot be supported in equilibrium.
After considering cases () − (), it is clear there cannot exist a distribution () in prices when firms
are posting a particular quality in a marketplace. But nothing in the above depended on comparing the
surplus of goods of different qualities, which means that in all Cases () − (), there does not exist a distribution () in prices when firms are posting a particular quality in a marketplace.¥ Proof of proposition 9 Note that buyers must be in a marketplace, say, marketplace 1, where is sold at an optimal price, −1 (1−−1 )] (it does not matter that is, if the queue length there is 1 , then 1 = = −[( − ) 1
if they are by themselves in marketplace 1, or with other types of buyers also). Suppose not. Suppose © ª buyers are in marketplace 1 where 1 6= is posted, with expected utility 1 . Since the price posted is ¡ ¡ ¢ ¢ not an optimal price, a group of firms can deviate and choose such that [(1−− ) ] − = 1
and make strictly higher expected profits. For instance, choose such that − ( − ) = 1 , with ¡ ¢ defined in equation (27), that is, given that buyers have to receive 1 deviating firms choose to move
to the “top” of their profit function, which can always be afforded by buyers. Hence this advertisement © ª 1 6= is not robust to group deviations. Now instead suppose buyers are in marketplace 1 with © ª −1 (1 − −1 )] and these queue length 1 where 6= is posted, where = − [( − ) 1
buyers have expected utility 1 . Like in the argument above, there exists a profitable group deviation where ¡ ¢ firms choose such that − ( − ) = 1 , with defined in equation (30) and make strictly © ª higher expected profits. Now suppose buyers are in a marketplace where = is posted, so
1 = −1 ( − ), and a firm’s expected profit is 1 = (1 − −1 − 1 −1 ) ( − ) . But then there © ª −1 ( − ) 1 , and exists a profitable group deviation = 1 = which yields =
= (1 − −1 − 1 −1 ) ( − ) 1 . Hence, in an equilibrium, if it exists, buyers must be in a marketplace where is sold at what is an optimal price, that is, if the queue length at that marketplace is ∗ ∗ 1 , then 1 = , and let firms’ expected profit be denoted by 1 , and buyers’ expected utility be 1 .
Since in an equilibrium buyers are in a marketplace with firms selling at an “optimal price,” the question is what buyers are consuming, and if they are in the same marketplace as the buyers. Let us suppose a candidate equilibrium is a segmented equilibrium. There are 4 possibilities to
67
consider. () Suppose buyers are in their own marketplace, marketplace 2, also with firms selling , © ª = − with an implied queue length of 2 , and firms are posting 2 6= e , 2 ≤ , and e [( − ) 2 −2 (1 − −2 )]. This is clearly not stable because price 2 is not optimal, so if in this can¡ ¢ didate equilibrium buyers have expected utility 2 then a group of deviating firms can choose
such that − ( − ) = 2 and make strictly higher expected profit, and since any price charged can
be up to , we know from assumption (1) that buyers can afford this price. () Now suppose in a © ª candidate equilibrium buyers were offered good at an optimal price, i.e., firms post 2 = e ,
−2 (1 − −2 )], with 2 ≤ , expected utility 2 = −2 ( − ), and expected e = −[( − ) 2
profit 2 = (1 − −2 − 2 −2 ) ( − ). This is not stable either, because there exists a group deviation
where this is posted { = 2 = −[( − ) 2 −2 (1 − −2 )] }, and = −2 ( − ) 2 ,
and = (1 − −2 − 2 −2 ) ( − ) 2 , and we know from assumption (1) that buyers can afford
to purchase good at the deviation price. () Suppose in a candidate equilibrium buyers were sold © ª good at a non-optimal price, i.e., that firms were posting 2 6= e . If e ≤ , then there exists a profitable group deviation where a group of firms can simply charge e and move to the “top” of their
profit function while giving buyers the same level of expected utility, and firms are made strictly better
e off. If, however, e , say, in particular, 2 = where 2 6= , that is, firms are already charging
the constrained profit maximizing price, then we cannot appeal to this argument about charging e since © ª that is not affordable. So suppose firms posting 2 6= e is a candidate equilibrium. It must be that the fraction of firms ( ∗ 1 − ∗ ) sorting themselves into the two marketplaces is such that ∗1 = ∗2 ( ), where ∗2 ( ) is a firm’s expected profit in marketplace 2. On top of that, it must be that 1∗ ≥ 2∗ ( )
(where 2∗ ( ) refers to buyers’ expected utility in marketplace 2), for otherwise buyers can deviate to marketplace 2; but in this case, buyers can afford to purchase good in marketplace 1, so for this to be an equilibrium it must be that 1∗ = 2∗ ( ) But there is one unknown, ∗ , that has to satisfy two equations (∗1 = ∗2 ( ) and 1∗ = 2∗ ( )) simultaneously, which is not possible. Hence, this cannot be an equilibrium. © ª () Now suppose buyers are in a marketplace where 2 = e is posted when the queue length there
is 2 , again 2 ≤ . In this candidate segmented equilibrium, 2 = (1 − −2 − 2 −2 ) ( − ) and since © ª buyers are in marketplace 1 with when queue length is 1 , ∗1 = (1 − −1 − 1 −1 ) ( − ). In an equilibrium, ∗1 = 2 , but this then implies that 1 = 2 , and 1 = 2 , so one market is just a duplicate
of the other, which means this is not a segmented equilibrium, which is a contradiction. Hence, given that the equilibrium market structure must have buyers in a marketplace where is sold at an optimal price, there does not exist any stable segmented equilibrium in Case (). ¥ Proof of proposition 10 From proposition 9 we know there does not exist any stable segmented equilibrium. So the question is if we can find stable pooling equilibria. The proof is in two steps. First, I show that a pooling equilibrium in { ∗ = } is stable. Second, I show that there does not exist any other stable pooling equilibrium.
Consider the candidate pooling equilibrium where all firms post { ∗ ( )}, where ∗ ( ) is defined
in (21), a firm’s expected profit ∗ ( ) is given in (22), and a buyer’s expected utility ∗ ( ) in (23).
Note that the budget constraints of buyers are not binding here because of assumption (1) - all buyers can afford to purchase quality at ∗ ( ) ∈ ( ). Could a firm unilaterally do better? Given that
all buyers have expected utility ∗ ( ), a firm will never post e 6= ∗ ( ) when it is selling , since
68
∗ ( ) is the profit maximizing price. Nor will a firm unilaterally deviate to sell and post price e :
if it were to do so, it has to offer deviating buyers ∗ ( ), and the deviant firm faces the same queue ¡ ¢ length , so the price posted has to satisfy (1), that is, e = − [∗ ( ) 1 − − ], and its expected ¡ ¡ ¡ ¢ ¢ ¢ − ) = 1 − − ( − ) − − ( − ) 1 − − − − ( − ) = profit is e = 1 − − (e ∗ ( ). In other words, this candidate pooling equilibrium is robust to individual deviations.
Moreover, this candidate pooling equilibrium is also robust to group deviations. There are two group
deviations to consider: () group deviations catering to buyers only, and () group deviations catering to either buyers only or deviations catering to all buyers. First consider () group deviations catering to buyers only. Suppose the advertisement were © ª ¡ ¢ 6= ∗ ( ) , where is such that = ∗ ( ). Since is not an optimal price, this ¡ ¢ results in a deviation profit ∗ ( ). In fact, any is such that ∗ ( ) will imply a lower deviation profit . Hence, this group deviation is not profitable. Suppose the advertisement were © ª ¡ ¢ = , where is such that = − ( − ), i.e., is an optimal price, with
given in equation (30), which implies that = (1 − − − − ) ( − ); buyers can always
afford to purchase quality good by assumption (1). For ≥ ∗ ( ), since ( − ) ( − ), it
must be that , but this then implies that ∗ ( ), so this group deviation is not profitable. Now © ª consider a group deviation where the advertisement were 6= - if by charging a = in the previous paragraph the deviant firms cannot be made strictly better off, charging 6= must
imply even lower expected profit, so this type of group deviation is not profitable either. Hence, none of the possible group deviations catering to buyers only are profitable.
Consider () group deviations catering to buyers only or to all buyers. Let Γ ∈ { buyers only, all ª © buyers}. The first type of advertisement is 6= ∗ ( ) Γ . Reasoning as in the above paragraph,
and noting that buyers can afford to purchase good by assumption (1), it is easy to show that this ª © group deviation is not profitable. Now suppose the advertisement is = Γ , where ≤ Since buyers can afford to purchase the quality good at price , we can use the same reasoning as in the above paragraph to show that this type of group deviation is not profitable either. Lastly, suppose a group ª © deviation where the advertisement were 6= Γ were posted instead. If ≤ , then clearly
this group deviation is not profitable. However, if , then we cannot appeal to the reasoning used so far. For example, could be = and . But this group deviation is not profitable either as I will show now. To see this, suppose in this group deviation, the promised = ∗ ( ) = − ( − ).
Construct a hypothetical queue length which is such that − ( − ) = ∗ ( ); that is, if the price
affordability was not an issue, this would be the hypothetical queue length such that the a buyer’s expected
utility exactly equals ∗ ( ) - the associated hypothetical price . Since we are in Case (), it implies that , and the associated hypothetical profit function if firms could select this would be
= (1 − − − − ) ( − ) ∗ ( ) where the last inequality is immediate from observing that
. By construction, this is the highest profit a deviant firm could obtain, if it could charge the price
such that the queue length is , which means that any other 6= would imply a strictly lower level of
expected profit than , but ∗ ( ) anyway. Hence, this type of group deviation is not profitable. © ª Since the candidate pooling equilibrium where all firms post ∗ ( ) = does not have a profitable individual firm or buyer deviation, and is robust to group deviations, this pooling equilibrium is stable.
}, Next we check if there are other stable pooling equilibria. There are three possibilities: () { =
69
¡ © ¢ ª where given e, = −[( − ) e− 1 − − ]; () { 6= }; and () 6= ∗ ( ) = .
We know from the proof of proposition 9 that buyers must be in a marketplace where is sold at an optimal price, for otherwise there is a profitable group deviation. Hence, all these 3 pooling equilibria, which entails buyers being in a marketplace where is not sold at an optimal price are not stable.¥ Proof of proposition 11 Consider first symmetric equilibria where firms within a marketplace are posting the same advertisement. We know from lemma A.0 that it cannot be that an equilibrium has a group of buyers who are inactive, which means that in equilibrium, all buyers must be active. Claim 5 In any equilibrium, buyers must be offered a quality with an associated optimal price. Proof. Suppose not. Then there exists a profitable deviation where a group of firms post the same ¡ ¢ quality and offer these buyers the same level of expected utility, but then choose a such that the
deviating firms move to the “top” of their expected profit function, and are strictly better off. By assumption (1) these buyers can afford to purchase both and quality goods. Claim 6 In any equilibrium where buyers are offered quality , the price posted must be an optimal price. Proof. If the price offered to buyers for quality were not optimal, there would exist a profitable group deviation, and by assumption (1) buyers can afford to pay for the quality good. Hence, the only time the budget constraint may bind is for buyers when they are offered quality . Claim 7 Suppose in a equilibrium buyers are offered quality good at a non-optimal price e ≤ . This cannot be part of a stable equilibrium.
Proof. First note that this candidate equilibrium is a segmented equilibrium because buyers have to be in a marketplace with an associated optimal price for a quality. Suppose e where e is not an optimal price. Then there exists a profitable group deviation where buyers are offered the same quality e , and since this price is not optimal, deviating firms can pick a price good, same expected utility, denoted
closer to the “top” of their expected profit function and be made strictly better off. That is, offer ≤ e = [(1 − − ) ]( − ). Hence, any non-optimal e cannot where e such that =
be part of a stable equilibrium. Now suppose e = . Let and buyers be in marketplaces 1 and 2
respectively, with the expected utility of buyers in marketplace = 1 2 If firms in marketplace 1 offer
quality , and from an earlier claim we know that the price offered is an optimal price, then it must be
that 1 = 2 , because if 1 2 , buyers can afford to purchase in marketplace 1 and will unilaterally deviate, and if 1 2 , buyers can afford to purchase in marketplace 2 and will unilaterally deviate. So 1 = 2 . However, in this candidate equilibrium, the equilibrium ∗ , the ratio of firms in marketplace 1, has to satisfy two equations, firms’ equal expected profit condition, and 1 = 2 , which is not possible. Now suppose firms in marketplace 1 offer quality at an optimal price to buyers, and firms in marketplace 2 offer quality at e ≡ 2 = to buyers. In an equilibrium it must be that 1 ≥ 2 ,
and 1 2 so that no individual buyer has any incentive to deviate. Can 1 = 2 , and 1 2 , where 1
1 =
2
¢ ¢ 1 − − ¡ 1 − − ¡ 1 − 1 2 − ( − ) ; = = − 2 ; and 1 2
70
(65)
¢ ¢ 1 1 1 ¡ 2 ¡ 1 = (1 − − − 1 − ) ( − ) = (1 − − ) 1 − = 2 = (1 − − ) 2 − ? If there exists
∗ , the ratio of firms in marketplace 1 in equilibrium, such that 1 = 2 , then it is not true that 1 = 2
generically. Hence, a candidate equilibrium must have 1 2 , and 1 2 . Now consider this group
deviation { } where : = − ( − ) = 2 . This implies, from equation (65), since
( − ) = ( − ), that 1 . However, = (1 − − − − ) ( − ) 1 = 2 , implying
that this group deviation is profitable, and hence this candidate equilibrium cannot be possible.
Since the only type of equilibria where the budget constraint of can bind cannot be supported in equilibrium, we are left to consider equilibria where no budget constraint binds for any buyer. Claim 8 In any equilibrium, the queue length at a marketplace must be . Proof. In a pooling equilibrium, the queue length in the one marketplace where everyone is in must obviously be . The question is whether the queue length is in all marketplaces in a segmented equilibrium. Denote the price and queue length in marketplace = 1 2 as and respectively. Suppose first that 1 2 Since the budget constraints of all buyers do not bind, then both prices prevailing in the ¡ ¢ marketplace must be optimal, that is, 1 = 1 and 2 = 2 , = − ( − ) − 1 − − , where is the quality offered in marketplace . The reason both prices have to be optimal is that if a price
posted were not optimal, then there exists a profitable deviation where a group of firms can get together can offer deviating buyers the same level of utility and earn strictly higher profit by moving to the “top” of their expected profit function. Hence, we can write the expected profit functions as ³ ³ ´ ´ 1 1 2 2 1 = 1 − − − 1 − (1 − 1 ) = 1 − − − 2 − (2 − 2 ) = 2
(66)
If 1 = 2 , from (66) we have 1 = 2 , which is a contradiction since we started off with 1 2 . Now if 1 6= 2 , it must be that (1 − 1 ) = (2 − 2 ) since we are in Case () where ( − ) = ( − ).
But if (1 − 1 ) = (2 − 2 ), from equation (66) we still have that 1 = 2 , which is a contradiction.
Hence, it cannot be that 1 2 . Now suppose 1 2 instead, and repeating the analysis done in this paragraph, it can be shown that it cannot be that 1 2 . Hence, we have 1 = 2 = .
Summarizing the above, we have that in a equilibrium the queue length in a marketplace must be , the price charged in any marketplace must be optimal, that is, for a quality offered, firms charge £ ¡ ¢¤ ¡ ¢ = − ( − ) − 1 − − and earn expected profit = 1 − − − − , and buyers obtain expected utility = − where = ( − ) for = . But are these equilibria stable?
Lemma 8 Equilibria where the queue lengths in all marketplaces are ; the prices charged in all marketplaces that are open must be optimal, in that given a queue length of , for a quality = offered, firms charge £ ¡ ¢¤ = − ( − ) − 1 − − ; all buyers have expected utility = − ; and all sellers earn expected ¡ ¢ profit = 1 − − − − are stable. Proof. There is clearly no profitable individual deviation since all buyers are earning the same and
all firms are earning the same . So let us check if there exists any profitable group deviation. Suppose a group deviation were proposed such that the price were optimal, but this then implies that for ³ deviation ´ − − − = ≥ , ≤ , and for = 1 − − ≥ , it must be that ≥ , implying
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that = , and hence = , and = , so this group deviation is not profitable since no one is strictly better off. And this is true no matter the quality offered. Now suppose a group of firms propose deviating to a non-optimal price, taking as given that has to be offered to deviating buyers. But given , the optimal price is already offered, any non-optimal price (no matter what quality it is) will result in a strictly lower expected profit level, so this group deviation is not profitable either. There is basically no way to make someone better off without making someone else worse off. Hence, all equilibria where all buyers ¡ ¢ have expected utility = − and all sellers earn expected profit = 1 − − − − are stable. The above lemma states that all equilibria with properties described in lemma 8, so long as the prices
are affordable to the agents in question, are stable. Lemma A.3: In the proof of proposition 4, both group deviations { = = s.t.
= ∗ ( ) } and { = = s.t. = ∗ } yield the same cutoff .
Proof. Let ∗ = ∗ ( ). It has to be shown that when a group deviation { = = s.t.
is reached, where for ∈ ( ] the PSE is = ∗ ( ) } is considered that the same cutoff
∗ ( )) the PSE is not stable. First notice that when = ∗ = ∗ ( ), stable, and for ∈ ( ¡ ∗ ¢ ∗ ∗ = ( ), it must be that for = (1 − ) ( − ) = ∗ = ∗ = 1 − (∗ − ), where
∗ ∗ , and since ( = ∗ ) = [(1 − ) ] ( − ), that ( = ∗ ) = , and ( = ∗ ) = ∗ ∗ , it follows that ( = ∗ ) ∗ . Defining ( ) = ( ) − ∗ , clearly since in a PSE
( = ∗ ) 0. Since the largest possible value we have to consider is = ∗ − for arbitrarily
∗ , so the effect small, the such that = ∗ , has the property that ( = ∗ − ) ( = ∗ ) =
on is not clear if we just consider the expression for . However, upon differentiating with respect
to , one obtains 1 − − =
½
Ψ ( − ) −
¾
,
where Ψ = ( − ) (1 − − − − ) − ( − ) − , exactly as that defined earlier in equation (43)
in the proof of proposition 4, and use can be made of what was shown earlier that Ψ 0 for all 0.
Hence, 0 for all 0. In other words, as falls from ∗ to ∗ − , correspondingly falls,
∗ , and this last equality implies that buyers, which are not i.e., ( = ∗ − ) ( = ∗ ) =
the target income group of this deviation, will optimally choose not to join in this group deviation. Since
∗ ∗ , when falls by a small amount , ( = ∗ − ) ( = ∗ ), and by a ( = ∗ ) =
continuity argument, ( = ∗ − ) ∗ . Hence, ( = ∗ − ) 0.
Now suppose = . This group deviation is not profitable since the price charged to deviating buyers
exactly equals the cost of production, so = 0, and there does not exist a such that = ∗ since ∗ 0. If we let increase a little from it is clear that there still need not exist a such that = ∗ .
Notice that since = (1 − ) ( − ), as → ∞ that → ( − ), and → 0. Since and
are inversely related, → ∞ results from considering the smallest possible value of , which is denoted as , and = ∗ + and since ∗ 0. When increases away from , by a continuity
min denote the smallest value argument, increases from zero, but is very close to zero. Hence, letting ¡ ¢ min such that there exists a solving = ∗ , = 0. ¡ ¢ min 0, and at the largest value to consider, At the smallest value to consider =
( = ∗ − ) 0, and since 0 for all 0, there is exactly one cutoff such that
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= 0. Moreover, this cutoff value must be where is such that ( ) = 0 as in the proof of propo-
sition 4, since the deviation payoffs ( ) are monotonically changing in when we consider this ¡ ¢ group deviation { = = s.t. = ∗ ( ) }, and the deviation payoffs are also © ª monotonically changing in when we considered the group deviation = = = ∗ earlier in the proof of proposition 4, so no matter which of the two group deviations we are considering,
is such that for all ∈ ( ] the PSE is stable, i.e., it is not possible to find a profitable this cutoff
∗ ( )), there are many ways to find a profitable group deviation group deviation, and for ∈ (
which deviating firms can post to attract buyers, but not buyers, which will deliver that ≥ ∗ and
≥ ∗ , with at least one group of agents strictly better off. The two group deviations { = =
s.t. = ∗ ( ) } and { = = s.t. = ∗ } deliver the “extreme” scenarios where one set of agents are only made indifferent while the other set of agents obtain all the “surplus,” but anything in between where the surplus from the deviation are shared between the two sets of agents is possible.¥ Comment #1 () The assumption of firms’ commitment to advertisements Firms in this model are assumed to commit to their advertisements, which is an assumption used in
the canonical Burdett et al. (2001) paper, and widely used elsewhere. The reason for assuming that firms can commit to their advertisements is so that firms can perfectly direct buyers’ search behavior - buyers know exactly what firms are posting, know that no firm will renege, and hence, can choose visit strategies accordingly. If the commitment assumption were weakened, it adds a layer of complication to the analysis, where buyers’ search behavior takes into account what else firms would do conditional on a meeting. As a first step in this paper, without clouding the main message about firms taking into account buyer heterogeneity in their advertisement choice, I assume commitment on firms’ side, like in Burdett et al. (2001). Papers which have tried to relax this commitment assumption include Camera and Selcuk (2009)18 where they propose a directed search model of the labor market where ex post renegotiations are allowed; and more recent papers like Doyle and Wong (2013)19 and Gomis-Porqueras et al. (2017)20 where they do away with the commitment assumption. () Buying a good versus buying multiple goods A buyer in this product market is modeled as only interested in seeking to purchase one good. One can think, because a buyer’s utility conditional on purchasing a good of quality = is valuation minus price, or − , that this encapsulates, in a reduced form way, the idea that a buyer is interested in purchasing
other goods but those decisions are not modeled, which is why a buyer would like, ceteris paribus, to pay a lower price for this good in question. Which is why the price paid for the good enters the utility function. () Are really incomes? 18
Camera, Gabriele, and Cemil Selcuk, 2009, “Price Dispersion with Directed Search,” Journal of the European Economic Association, 7(6), 1193-1224. 19 Doyle, Matthew and Jacob Wong, 2013, “Wage Posting without Full Commitment,” Review of Economic Dynamics, 16(2), 231-252. 20 Gomis-Porqueras, Pedro, Benîot Julien, and Chengsi Wang, 2017, “Strategic Advertising and Directed Search,” International Economic Review, 58(3), 783-805.
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Buyers are modeled as having incomes , = in this paper, and incomes measure the ability to pay. More generally, I think of this income as an amount that a buyer has allocated to the purchase of a good in the product market I have modeled, so it does not have to literally mean an income earned by a buyer (e.g., wage income). In real life, a person with a particular level of income has a “portfolio choice” problem in that amounts are budgeted to various consumption items. I take as a starting point in this paper the fact that some buyers have more income to buy a particular good, say a baby stroller, and others have less income to devote to this good. And I would imagine that a higher income earner could allocate more resources to consumption in each of the categories he would like to consume. Comment #2 I have no idea which case fits “reality” better, though clearly Case () is a knife-edge case where any slight change in any of the four parameters ( ) will bring us back to Case () or (). Faber and Fally (2015)21 use detailed matched home and store scanner consumption microdata and they observe that “On the consumption side, we find that rich and poor households on average strongly agree on their ranking of quality evaluations across products, but that higher income households value higher quality attributes significantly more. On the production side, producing attributes that all households evaluate as higher quality increases both the marginal as well as fixed costs of production. In combination, these two features give rise to the endogenous sorting of larger, more productive firms into products that are valued more by wealthier households” which suggests that empirically, higher quality goods have more of a pie to split via a higher surplus. Even though buyers in my model have the same preferences and firms are ex ante identical so I cannot directly appeal to their observation, it does make one wonder if a higher quality good does have a higher surplus relative to a lower quality good. 21
Faber, Ben and Thibault Fally, 2015, “Firm Heterogeneity in Consumption Baskets: Evidence from Home and Store Scanner Data,” mimeo.
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