(Quasi) uniqueness and restoring dynamics of price-dispersion market equilibria under search cost

Accepted Manuscript (Quasi) uniqueness and restoring dynamics of price-dispersion market equilibria under search cost Francisco Álvarez, José-Manuel Rey

PII: DOI: Reference:

S0304-4068(18)30136-8 https://doi.org/10.1016/j.jmateco.2018.12.002 MATECO 2286

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Journal of Mathematical Economics

Received date : 28 November 2017 Revised date : 9 December 2018 Accepted date : 11 December 2018

Please cite this article as: F. Álvarez and J.-M. Rey, (Quasi) uniqueness and restoring dynamics of price-dispersion market equilibria under search cost. Journal of Mathematical Economics (2018), https://doi.org/10.1016/j.jmateco.2018.12.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

(Quasi)uniqueness and restoring dynamics of price-dispersion market equilibria under search cost∗ † ´ Francisco Alvarez

Jos´e-Manuel Rey‡

Abstract We study existence, uniqueness and restoring dynamics of price-dispersion equilibria in a market for a homogeneous good. We assume that costs are heterogeneous across participating firms. Specifically, we rely on classical extreme value theory and some recent developments on fee-setting mechanisms to consider cost distributions that are Generalized Pareto. Our analysis provides results on the existence and uniqueness of a price-dispersion equilibrium that link the cost dispersion across firms with the search cost of consumers. If the former is large enough compared to the latter, existence and a form of uniqueness of pricedispersion equilibrium arise. In addition, we propose a natural best-response market dynamics that delivers convergence to the price-dispersion equilibrium, even if the market departs slightly from the Diamond paradox equilibrium. Keywords: price dispersion, search cost, equilibrium restoring dynamics, Diamond paradox. JEL Code: C61, D43.

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Introduction

The empirical literature in economics has reported over decades an intriguing finding: many –if not most– markets for a homogeneous good exhibit a significant degree of price-dispersion. Different firms charge different prices, even though all of them sell identical items. This is a pervasive phenomenon that has been documented for a large number of markets. According to Baye et al. (2006), price-dispersion is “ubiquitous and persistent”. In fact, typical price distributions for a vast number of goods have been recently shown to be symmetric with a standard deviation between 19% and 36% (Kaplan and Menzio (2015)). See Baye et al. (2006) for an exhaustive survey on this empirical literature. ∗

We thank two anonymous referees for very helpful suggestions. Any remaining error is our responsibility. FA and JMR are both supported by Grants PRX17/00057 and PRX16/00395, respectively, of the Spanish Visiting Senior Scientist Program Salvador de Madariaga. FA acknowledges the support from the research project ECO2017-86245-P from Ministerio de Ciencia, Innovaci´ on y Universidades. JMR is also partially supported by a Short-Term Harvard University Faculty Grant from Real Colegio Complutense (RCC) at Harvard. † Department of Economic Analysis, Complutense University of Madrid. Campus de Somosaguas, 28223, Spain. E-mail: [email protected]. ‡ Department of Economic Analysis, Complutense University of Madrid, Campus de Somosaguas, 28223, Spain. E-mail: [email protected]; and Department of Psychology, Harvard University, 33 Kirkland St., Cambridge, MA 02138, USA. E-mail: [email protected].

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A preliminary explanation for the phenomenon relies on the fact that consumers must face search costs –this literature was triggered by a seminal paper by Stigler (1961). Even if a consumer is aware that there are firms selling at low prices he still must find those firms, which is costly. The firms in the market anticipate the existence of consumer search costs and raise price over marginal cost. But why should different firms raise prices differently? The Diamond paradox concerns this question (Diamond (1971)). If consumers face search costs, the conformation in which all firms set the monopoly price and consumers search for one price quote only is an equilibrium of the market with no dispersion. Theorists have considered different scenarios to circumvent the Diamond paradox. Reinganum (1979) provided a first way out of the Diamond’s trap by considering different costs across firms, while the consumer side is characterized by a downward sloping demand function. Other market structures with heterogeneous first have been explored, as in MacMinn (1980), which will be commented below. The case of homogeneous costs has been also considered: Burdett and Judd (1983) show that, when firms have equal costs, the presence of search costs yields (non-monopoly) equilibria that typically exhibit price-dispersion, namely different firms set different prices. This paper aims to contribute to the theoretical literature which links price to cost dispersion. We consider a market for a homogeneous good composed of a continuum of firms with different costs and a representative consumer who faces search costs. Our articulated model provides a closed form characterization of a price-dispersion market equilibrium. Existence, uniqueness of the one-shot interaction between buyer and sellers, as well as the stability of equilibrium under an evolutionary market dynamics are analyzed. The stability analysis is particularly relevant in search theory: if price-dispersion equilibria actually coexist with the Diamond equilibrium, to what extent price-dispersion is more likely to be observed in practice? This question is clearly related with the dynamical stability of equilibria. In this paper we characterize a basin of attraction for price-dispersion equilibria and show that the Diamond equilibrium cannot appear under natural conditions. Next we first highlight the basic features of our model regarding firms’ costs, consumer search and market dynamics. We also explain along the way the relevance of our assumptions and findings with reference to the related literature. Then we describe the different steps and main results of our analysis. For most of our analysis we assume that unit production costs follow a Generalized Pareto (GP ) distribution. It turns out that assuming GP cost distributions has significant implications, both practically and theoretically. While their use has a long tradition in economics, our motivation to consider GP distributions is otherwise twofold. One reason comes from extreme value theory: for a large class of unit cost distributions, the conditional distribution of unit costs given that only firms with a cost below some threshold value (say, the consumer’s willingness to pay) operate at the market, is asymptotically a GP distribution –see Balkema and De Haan (1974) and Pickands III (1975). A second argument is based on recent work on optimality of transaction fee rules: the necessary and sufficient condition to implement optimally affine fee-setting mechanisms that are observed in online markets is that sellers’ costs are drawn from a GP distribution (see e.g. Loertscher and Niedermayer (2012)).

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On the consumer side, we assume that the consumer demands one unit of the product and adopts a fixed sample size strategy when searching for the lowest price. This is the classical approach by Stigler (1961). Other works assume an inelastic demand, e.g. MacMinn (1980), Rob (1985) or Carlson and McAfee (1983), while Reinganum (1979) considers a downward sloping demand curve. Under fixed sample size search, the consumer selects ex-ante the number n of sellers to visit and sticks to it regardless the prices she finds along the way. An alternative search strategy is sequential search, in which the consumer decides whether to stop or continue searching after observing each price. A fixed sample size search strategy is consistent with some recent empirical findings, namely De los Santos et al. (2012) and Honka and Chintagunta (2017). Some of these empirical findings, however, are also consistent with sequential search with belief updating –see Santos et al. (2017) for an empirical analysis and Janssen et al. (2017) for a theoretical ground–. In general, it seems there is not conclusive empirical evidence in favor of one particular search strategy. The basic scheme above is similar to that of MacMinn (1980). However, MacMinn (1980) assumes that unit costs obey a uniform distribution, which is embedded in the class of GP distributions. Furthermore, the analysis in MacMinn (1980) is limited to the existence of equilibrium. The assumption that costs are GP distributed allows us to establish the (quasi)uniqueness and stability of the market equilibrium. Such an exhaustive model analysis seems to be unusual in the theoretical literature of price-dispersion. In fact, the existence and uniqueness of price-dispersion equilibria has been largely studied. A classical reference is the paper by Burdett and Judd (1983), in which there is multiplicity of equilibria. Other references include Benabou and Gertner (1993), Rob (1985) and Stahl (1989). All of them assume homogeneous costs among firms, and sequential search strategy on the consumer side. The latter three papers assume some form of heterogeneity in search costs. Benabou and Gertner (1993) analyze a model in which there is (exogenous) inflation, though the distribution of real prices is assumed invariant over time. For the case of identical consumers and inelastic demand, they find a unique price-dispersion equilibrium. Rob (1985) characterizes necessary conditions for a price-dispersion equilibrium with continuous prices, and he also finds some closed form solutions when the search costs are Pareto distributed. Stahl (1989) assumes that consumers are either informed or uninformed, with zero and positive search costs, respectively, and characterizes a unique price-dispersion equilibrium. In the limiting case of uninformed consumers, the equilibrium that he finds collapses to the Diamond equilibrium. Carlson and McAfee (1983) characterize equilibria for a finite number of heterogenous firms and consumers, assuming specific functional forms for both consumer’s search cost and firms’ production costs. They find a price-dispersion equilibrium for the case in which the search costs are uniformly distributed. In general, a closed form characterization of the price distribution depends on the assumptions on the exogenous variables of the model. In this regard, we show that GP price distributions are closed under best responses if firms’ costs are GP distributed, which is key for our extensive analysis. Regarding the market dynamics considered in this paper, both consumer and sellers base their current decisions on past available information. We assume that, while both firms and consumers update their decisions as new information becomes available, firms have capacity to update more rapidly. So, in every period, first firms observe the previous prices and consumer’s search sample size n and adjust their current prices so that the equilibrium among firms is achieved;

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then the consumer updates the value of n as his best response given the price equilibrium, and the next period starts. Additional details for this dynamical setting is provided in section 3.3. Hopkins and Seymour (2002) also study stability of price-dispersion, following a different scheme, namely they consider pricing games with a Rock-Scissors-Paper structure and analyze their stability in mixed strategies. In addition, they consider that all firms have the same unit cost, which constitutes another major difference with respect to our story. Lahkar (2011) also studies stability of price-dispersion equilibria using evolutionary game theory. The departure point for both papers is the scheme by Burdett and Judd (1983), namely equal costs across firms and use of mixed strategy equilibria. Our analysis proceeds as follows. As a first step, we take the consumer’s search behavior as exogenous. We call the price distribution equilibrium among firms with n exogenous an industry equilibrium. We show that, if costs are GP distributed, a price-dispersion equilibrium does exist and it is also GP distributed; moreover, this equilibrium is unique. Then we consider a price dynamics within the industry as follows. Unit costs obey a GP distribution which remains constant over time. In every period, each firm selects its price as the best response given the price distribution (and the consumer’s search size n) in the previous period. The whole set of best responses conforms the industry price distribution in the current period, which in turn is the input distribution to determine the firms’ prices in the next period, and so on. We provide sufficient conditions for convergence of the sequence of price distributions to the industry equilibrium. Roughly, the industry equilibrium is restored as long as the mean of the initial price distribution is not too far from the mean of the unit cost distribution and, additionally, the initial price distribution is sufficiently less dispersed relative to the unit cost distribution. Since the initial price distribution must have been rationalized a priori from the unit cost distribution, the conditions above seem plausible. The next stage of our analysis consists of determining the consumer’s n endogenously in a one-shot interaction: as mentioned above, we assume that the consumer decides the number n of sellers to visit ` a la Stigler (1961), that is, by minimizing total cost defined as the search cost plus the expected minimum price. An equilibrium is now a price distribution together with a value of n, which we call a market equilibrium. The issues of existence and uniqueness of equilibrium for the market are more involved than for the industry. A GP distribution is characterized by three parameters: shape, location, and scale. Our analysis shows that, when costs are GP distributed, only the shape and the ratio of scale to unitary search cost matter for the existence and uniqueness of equilibrium. Roughly, the existence and quasi-uniqueness of equilibrium occur provided that the ratio lies beyond suitable threshold values, which depend on the shape of the cost distribution. The critical threshold values for both existence and quasi-uniqueness are characterized in the paper. Finally, we consider the issue of the restoring dynamics of the market equilibrium. Under the conditions that guarantee existence and quasi-uniqueness of the market equilibrium, we show that the off-equilibrium dynamics considered in this paper restores the price-dispersion market equilibrium for a large range of initial values of n, with n ≥ 2. Our results imply that the (quasi-unique) price-dispersion market equilibrium is asymptotically stable and, furthermore, the Diamond equilibrium cannot emerge provided that sellers’ costs are suffi-

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ciently dispersed and the consumer initially considers a sample size large enough (which is just two for uniform cost distributions). The rest of the paper is organized as follows. In sections 2 and 3 we present the analyses of the industry equilibrium and the market equilibrium, respectively. Within each section, we first characterize the best responses of all agents, then we discuss the existence and uniqueness of equilibrium, and finally we address the issue of the stability of the equilibrium restoring dynamics. In section 4 we conclude with some final remarks. We present our main results in the main body of the document and all proofs in appendix A.

2 2.1

Price-dispersion industry equilibrium Basics

In this section we gather basic assumptions in our setting that will be required for the subsequent analyses. We consider a market for a homogeneous good composed of a continuum of producers –firms or sellers– and a representative consumer who goes shopping to buy one unit of the good. The consumer faces search costs and adopts a fixed sample size search strategy so that, after having obtained n price quotes, he buys from the firm with the lowest price among the observed prices. In section 3 n will be a decision variable for the consumer, whereas in section 2 n will be given. If the consumer buys from a firm with production cost c > 0, the firm obtains a profit p − c, whereas it makes zero profit otherwise. We may assume here that c is an opportunity cost so the seller only incurs cost c if he makes the sale. Each firm decides its own price independently by maximizing expected profits given their production cost. Production costs. Each firm is endowed with a linear production technology. We assume that unit costs across firms are heterogeneous and described by a probability distribution Φ. This will be written as c Φ for the sequel, with c denoting the unit cost of a generic firm. We assume that Φ has no atoms and has compact support [c, c] ⊂ (0, ∞). For the most part of our analysis we will assume that costs obey a Generalized Pareto distribution. Next we present its definition and some properties and then some justification for our assumption. Definition 1 Let k < 0. A random variable x has a Generalized Pareto (GP ) distribution with shape k, location µ > 0, and scale σ > 0, if   1 z − µ −k Pr (x ≤ z) = 1 − 1 + k . σ We will write x GP (k, µ, σ) in that case. Also, we will denote the family of GP distributions with shape k by GP (k, −).   Notice that the support of a distribution GP (k, µ, σ) is the interval µ, µ − σk . The scale parameter σ is directly related with the variance of the distribution, 2 which is given by (1−k)σ2 (1−2k) . In particular, σ serves as a measure of dispersion within a GP family with common shape k. The role of the shape parameter is illustrated in figure 1. Uniform distributions are obtained as a particular case when the shape is k = −1, whereas cumulative distribution functions of GP (k, −)

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are convex for k < −1 and concave for −1 < k < 0. For a more detailed description of GP distributions, see Kleiber and Kotz (2003). When referred to a price distribution, the parameter k can be linked to the elasticity of supply as follows. Assume that the price distribution is given by some GP (k, µ, σ) with probability density function f . Define the elasticity of supply as   f 0 (p) σ ES := µ − − p . k f (p)

In words, ES is the relative change in the density f of the price distribution due to a relative change in price. Notice that the variation at price p is relative to the distance to the upper extreme of the support of the distribution, namely µ − σk − p. It is straightforward to show that ES = 1 + k −1 . Thus the elasticity of supply for a GP price distribution is constant over the support of prices, negative for k ∈ (−1, 0), positive for k < −1, and zero for uniform distributions (k = −1) .

GP (k0 , −) GP (−1, −−)

1

0

µ

µ−

σ k0

µ+σ

GP (k1 , −)

µ−

σ k1

Figure 1: Given location (µ) and scale (σ) parameters, the figure shows the cumulative distribution functions (cdfs) of three distributions from the families GP (k0 , −), GP (1, −) (in red) and GP (k1 , −), with k0 < −1 < k1 < 0. Notice that cdfs of GP (k0 , −) distributions are convex whereas cdfs of GP (k1 , −) distributions are concave. The class GP (1, −) is the family of uniform distributions.

As mentioned in the introduction, the interest in considering GP distributions lies a priori in two facts. A first reason comes from basic results in extreme value theory. Assuming that unit costs are initially drawn from some arbitrary (continuous) distribution, only firms with unit costs below the consumer’s willingness to pay have a chance to enter the market. The unit costs of the firms operating in the market can thus be understood as the tail distribution of some initial unit cost distribution. The Pickands-Balkema-de Haan theorem implies that, for a large family of initial distributions, the tail is asymptotically GP distributed –see Balkema and De Haan (1974) and Pickands III (1975). A second reason to consider GP unit cost distributions is more specifically connected to our framework. Although different in nature, both search costs and transaction fees constitute frictions that market participants take as given and incorporate into their individual decision processes. Recent literature on transaction fees suggests that unit costs must be GP distributed. In particular, some transaction fee schemes that are used in online markets (such as eBay) have been rationalized: such fee-setting mechanisms implement an optimal allocation rule for the intermediary in a Bayesian Nash equilibrium whenever sellers’ unit costs are GP distributed –see Loertscher and Niedermayer (2007) and Loertscher and

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Niedermayer (2012). Rephrasing, according to this research, some transaction fees observed in real markets are rational (by game theoretic standards) on the basis that sellers’ costs are GP distributed. Consumer’s search strategy. The consumer decides ex-ante the number n of prices to consider, and then buys at the lowest price after having checked them all. This is a fixed sample size strategy. We assume throughout the paper that the consumer checks at least one price, n ≥ 1. In section 2, n is exogenously given, whereas in section 3 it will be decided by the consumer. In particular, we will assume that he adopts the Stigler’s criterion to select n, that is, by minimizing the expected purchase cost plus the search costs (Stigler (1961)). To obtain a price quote from a seller has a cost τ > 0, so τ is the search cost per price quote. The details are postponed until section 3, where n will play an endogenous role (see section 3.1.) There are two basic models in the search-theoretic literature regarding search strategy: fixed sample size and sequential. Under a sequential search, the consumer decides whether to buy at the last observed price or to obtain a further price quote. The theoretical analysis is different depending on the search strategy under consideration. Roughly, a sequential search strategy implies that the consumer sets a reservation price and searches until finding a price not larger than the reservation price. Generally, this entails the use of mixed strategies (Stahl (1989)). With some exception, the models undertaking sequential search strategy assume identical firms, which constitutes a fundamental difference with respect to our setting. While sequential search has been a more popular assumption in the theoretical literature, there is not conclusive empirical evidence so far in favor of one search strategy over the other. Recently two empirical studies point to fixed sample size as a plausible consumer search strategy. Indeed, De los Santos et al. (2012) and Honka and Chintagunta (2017) consider different markets –online books and car insurance, respectively–. Beyond analyzing different markets, both papers use different methods to discriminate empirically in favor of fixed sample size search. Essentially, De los Santos et al. (2012) focuses on the observed sequence of searches of the consumers. If consumers search sequentially, they should buy at their last search, which is not necessarily the case under fixed sample search. More generally, under sequential search there should be some dependence between the number of searches and the history of searches. Honka and Chintagunta (2017) rather concentrates on the structure of the consideration set –the prices that are actually observed–, abstracting from the order in which they were observed. Roughly, the proportion of prices in a consideration set that lie below the population average price is independent of the cardinality of the consideration set under a fixed sample size strategy, but not under sequential search. Diamond equilibrium. Suppose that the consumer has a finite willingness to pay v, which is exogenously given. The term willingness to pay is used here as valuation in auction theory, so it represents the consumer’s (monetary) utility. Logically, there must be no firm in the market with production cost strictly higher than v. Moreover, since there is a continuum of firms with costs supported in the interval [c, c], we may assume that c = v, that is, there are firms with costs arbitrarily close to v but not beyond that value. Assuming expected profit-maximization on the firms’ side and search costs on the consumer’s side, the Diamond equilibrium in our setting is defined by n∗ = 1 and the atomic

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distribution p∗ = c, i.e. the consumer searches for one price only and all firms set the monopoly price. We proceed gradually in the rest of this section. First we characterize the best response of a generic firm (subsection 2.2). Then we introduce the equilibrium concept and show the existence and uniqueness of equilibrium for the class of GP cost distributions (subsection 2.3). Finally we determine its stability character under a best-response dynamics (subsection 2.4).

2.2

Firm’s problem

We assume that n ≥ 2 for the sequel, except otherwise stated. Since n is exogenous here, all decisions are made by firms. The continuum of firms’ prices will be represented by a probability distribution with support [p, p], termed hereafter the price distribution F . Thus F (p) gives the fraction of firms that set a price not larger than p. From a firm’s perspective, its own unit cost c, the price distribution F , and the number n of price quotes that the consumer obtains, are known. Now, denoting the firm’s own price by p, its expected profit is given by πF (p, n) := (p − c) (1 − F (p))n−1 .

(1)

Expression (1) actually gives the firm’s profit (i.e. the markup p − c) given that p is the lowest price among n prices randomly drawn from F . Notice that profit is zero otherwise. The subscript F emphasizes the dependence of the expected profit with respect to the price distribution F and it may be omitted where it leads to no ambiguity. It is understood from (1) that, if p ≥ p, then πF (p, n) = 0. In words, if a firm’s price is higher than the rivals’ maximum price, it has zero probability of selling (recall that n ≥ 2) and it thus makes zero profit. Also, if p ≤ p, then πF (p, n) = p − c, that is, if the firm prices its product below all its rivals, it makes the sale and earns the markup. Next, we define the firm’s best response p∗ given the price distribution F . Definition 2 Let n ≥ 2. The best response of a firm with unit cost c ∈ [c, c] given the price distribution F is p∗ := argmaxp≥c πF (p, n),

(2)

where πF (p, n) is given by (1). Notice that each firm’s best response makes use only of its own cost rather than the whole cost distribution Φ. This fact simplifies greatly the informational structure of the model since it does not require Φ to be common knowledge across firms, as it is standard under incomplete information. However, Φ plays a key role in the equilibrium analysis, namely, we will show below that, if Φ belongs to the class of GP distributions, there are fairly simple conditions for the existence, uniqueness and stability of an industry equilibrium (to be defined below). Some additional notation will be helpful. Given a cumulative distribution distribution (cdf) F with probability density function (pdf) f , its hazard function h is defined by f (p) h(p) := , p ∈ [p, p). 1 − F (p) Notice that in the case c ≥ p, p∗ = p is a best response for the firm. The next proposition characterizes the best response of a generic firm in the more significant case c < p.

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Proposition 3 Let n ≥ 2, c < p, and F differentiable in [p, p] with h nondecreasing in [p, p). Let p∗ be the best response defined in (2). Then (i) If (n − 1)f (p)(p − c) ≥ 1, then p∗ = p.

(ii) If (n − 1)f (p)(p − c) < 1, then p∗ = p0 ∈ (p, p) where p0 is obtained as a solution of the equation (n − 1)h(p)(p − c) = 1.

(3)

Furthermore, p0 is unique. Proposition 3 implies that p∗ is well-defined as a function of the random variable c. In addition, it can be interpreted in marginal terms. To ease that interpretation, differentiate (1) with respect to p to obtain dπ = (1 − (n − 1)h(p)(p − c)) dp. (1 − F (p))n−1

(4)

Notice that, at the margin, what is relevant for the firm’s decision is the effect of a small variation in price conditional on the event that the new price is minimal in a random sample of size n, which is the expression on the left-hand side of (4). The right-hand side of (4) decouples the effects of such variation on the markup and on the probability of being the minimum price, which correspond with the positive and negative terms, respectively. At an interior solution both effects cancel out, leading to condition (3). At the corner solution p∗ = p the negative effect on the probability of being the minimum price is larger than the positive effect on the markup, which is the condition in part (i) of proposition 3. It is straightforward to show from proposition 3 that the optimal price p∗ increases as c increases, so that firms with higher costs set higher prices. Additionally, both the optimal price and markup decrease as the size of the price sample n increases. Furthermore, the markup also decreases with c, that is, firms with higher costs make lower profits. These all seem natural features of the firms’ behavior in our setting. An additional interpretation for the solution of the firm’s problem comes from auction theory. In fact, the firm’s problem is quite similar to the problem that characterizes the optimal bidding in a first-price single-unit auction in which rivals bids (not valuations) are sampled from F . Let us define Myerson’s virtual valuation as 1 − F (p) Jn (p) := p − . (n − 1)f (p) Using Jn (p), it follows from (4) that:   dπ sign = sign{c − Jn (p)}. dp

Since c is the firm’s marginal cost, the latter expression allows us to interpret Jn (p) as the expected marginal revenue of setting the price p. Notice that proposition (3) follows assuming that Jn is non-decreasing, which in particular implies that h is non-decreasing. Thus, an interior solution satisfies c = Jn (p), which is part (ii) of proposition (3). The condition for a corner solution, which is part (i) of the proposition, follows analogously.1 1 We thank an anonymous referee for pointing out this alternative interpretation of the firm’s pricing decision. A more detailed discussion on Myerson’s virtual valuation can be found in Bulow and Roberts (1989).

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2.3

Industry equilibrium

The concept of equilibrium of the industry follows naturally from the firms’ individual best responses. A price distribution F constitutes an industry equilibrium if it is formed by the best responses to F , one for each firm as defined in (2). Formally, the equilibrium concept can be defined as a fixed point of a best response operator defined on the space of probability distributions, as follows. Definition 4 Let a pair (Φ, n) be given, with c Φ and n ≥ 2. For any price distribution F with support [p, p], let γ(F ) denote the best response operator defined by argmaxp∈[p,p] πF (p, n) γ(F ), (5) where πF (p, n) is the profit function defined in (1). Then, a price distribution F ∗ is a (price-dispersion) industry equilibrium if F ∗ = γ(F ∗ ). More precisely, F (p) = γ(F )(p) for all p ∈ [p, p]. Given a consumer’s n, the equilibrium among firms defined above is called an industry equilibrium. In a later section, n will be endogenous and thus a part of the equilibrium profile, denoted as market equilibrium, which includes both firms’ and consumer’s best responses. Either referred to industry or market equilibrium, the qualifier price-dispersion, which is standard in the literature of search cost, applies whenever F ∗ is a non-degenerate distribution. From definition 4 the existence of equilibria amounts to finding fixed points of the best response operator γ. Under the assumptions of proposition 3, given n ≥ 2 and the unit cost distribution Φ, γ maps a price distribution, say F0 , into a price distribution, say F1 . Of course, the set of fixed points of γ depends on Φ. Theorem 5 below is a key preliminary result of the paper. It states that the set of fixed points of the best response operator γ can be usefully characterized when the unit cost distribution Φ is GP . Theorem 5 Let n ≥ 2, k < 0, and assume that the distribution Φ of unit costs in the industry is GP with shape k, specifically c GP (k, µc , σc ). Then (i) There is a unique industry equilibrium F ∗ within the class GP (k, −), specifically F ∗ is GP (k, µp , σp ), with 1 n−1 σc and σp = σc . n−1−k n−1−k  (ii) In equilibrium, a firm with unit cost c ∈ suppΦ = µc , µc − price  σp  n−1 −k p∗ = c+ µp − . n−1−k n−1−k k µp = µc +

(6) σc k



sets the (7)

Part (i) of theorem 5 shows that, if unit costs are GP distributed, the shape of the distribution of unit costs is transferred to the equilibrium distribution of prices. Notice that GP (k, −) distributions with k < 0 have non-decreasing hazard functions, so that proposition 3 applies. Furthermore, all equilibrium prices σ are characterized by equation (3). Since µp > µc and µp − kp = µc − σkc , the domain of equilibrium prices is obtained by shrinking the support of industry costs from the left endpoint. Notice that, since σp < σc , the distribution of prices at equilibrium is less dispersed than the distribution of costs. Notice that, under the assumption that the cost distribution is GP , the virtual valuation function, Jn is linear in p, which simplifies our analysis to a large

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extent. Clearly, linearity of Jn is equivalent to linearity of the inverse hazard function, which characterizes the family of GP probability distributions. Part (ii) of theorem 5 gives the specific structure of each firm’s solution in terms of parameters of the industry. In general, a firm sets its price p∗ as a convex linear combination of its own cost and the maximum price under F ∗ , i.e. σ p = µp − kp . Thus, any firm with unit cost strictly less than p has a strictly positive markup at equilibrium, but firms with larger costs have smaller markups than firms with smaller costs. As n increases, c is given more weight in that combination and, in the limit n → ∞, the markup disappears, i.e. p∗ → c or, equivalently, F ∗ converges to Φ, which seems very natural. The model outcome is also the price-equal-marginal cost rule if unit costs are identical across firms with probability 1. To check that, it suffices to take k → −∞ in the expressions in theorem 5. It is remarkable that theorem 5 renders the Diamond paradox equilibrium –with no price-dispersion– as the limiting case n → 1. Indeed, if we set n = 1 in (6) and (7), then F ∗ is the degenerate distribution with unit mass at p∗ = µc − σkc , which corresponds with the consumer’s willingness to pay v = c, as mentioned in section 2.1. Thus, all firms set the monopoly price and the consumer searches for one price only, which corresponds to the Diamond equilibrium of the industry in our setting. Theorem 5 implies that, as long as n > 1, the same industry will exhibit price-dispersion and, moreover, it gives account of the industry price structure in terms of parameters of its cost distribution and consumer’s seach.

2.4

(Industry) equilibrium restoring dynamics

In this subsection we analyze the stability of the industry equilibrium under a best response dynamics for firms. Suppose that, at the initial period t = 0, the industry price distribution is given by some F0 which does not constitute an equilibrium. Then, each firm plays a best response given F0 , so that the industry price distribution at period t = 1 is given by F1 = γ(F0 ), where γ(F ) is the best response operator defined in (5), and so forth. Thus, the dynamics of the price distribution is defined by Ft+1 = γ(Ft ), t = 0, 1, 2, . . . ,

(8)

plus some initial distribution F0 , that can be understood as a perturbation of the industry equilibrium F ∗ . The main issue here consists of providing sufficient conditions on F0 so that the sequence of distributions defined by (8) converges towards F ∗ asymptotically, which means that the sequence of industry prices pt with distribution Ft converges towards the equilibrium prices p∗ in distribution as t → +∞. These conditions may also provide a basin of attraction of the industry equilibrium. Lemma 6 Let n ≥ 2 and k < 0. Assume that c GP (k, µ0 , σ0 ), with σ0 , and n−1 n−1 σ0 ≤ σc . n−1−k

µ0 − µc ≤

Let F1 = γ(F0 ), with γ defined by (5). Then,

11

GP (k, µc , σc ) and let F0 be

(9a) (9b)

(i) F1 is GP (k, µ1 , σ1 ), with σ0 − kµ0 n−1 + µc , and n−k−1 n−k−1 n−1 σ1 = σc . n−1−k

µ1 =

(ii) In addition, µ1 − µc ≤

σ1 . n−1

(10a) (10b)

(11)

Part (i) of lemma 6 guarantees that, if Φ is GP with shape k, the operator γ is well defined within the class GP (k, −) as long as conditions (9a)-(9b) on location and scale hold for the input distribution. Moreover, (9a)-(9b) constitute sufficient conditions on F0 for the sequence {Ft }t∈N defined by (8) to converge to the industry equilibrium. This is the content of the following proposition. Proposition 7 Let n ≥ 2 and k < 0. Assume that c GP (k, µc , σc ) and let F0 be GP (k, µ0 , σ0 ), such that conditions (9a)-(9b) are satisfied. Let {Ft }t∈N be the sequence of distributions obeying Ft+1 = γ(Ft ), with γ defined by (5). Then (i) The dynamics γ : GP (k, −) 7→ GP (k, −) is well defined. Furthermore, for any t ≥ 0, the distribution Ft+1 is GP (k, µt+1 , σt+1 ), with location and scale (µt+1 , σt+1 ) obeying the dynamical system µt+1 =

−k 1 n−1 µt + σt + µc , n−k−1 n−1−k n−1−k n−1 σt+1 = σc . n−1−k

(12a) (12b)

(ii) The sequence {Ft }t∈N converges pointwise to the industry equilibrium F ∗ characterized by theorem 5. Moreover, the pair (µp , σp ) corresponding to F ∗ is a semi-stable equilibrium of the parameter dynamics. Conditions (9a)-(9b) define a region of initial conditions, Ω0 , in the parameter space (µ, σ) of the class GP (k, −) so that the distribution dynamics (8) defined by γ is equivalent to the associated parameter dynamics defined in proposition 7(i). It follows from theorem 5(i) that h(µp , σp ) lies ion the boundary of Ω0 . Moreover, σ

p the set Ω0 lies inside the region 0, µp + n−1 × [0, σp ]. As a consequence, the equilibrium pair (µp , σp ) of the parameter dynamics is asymptotically stable with respect to µ and semi-stable with respect to σ. Thus, the equilibrium distribution F ∗ is restored under the industry dynamics (8) after a small perturbation of the location µp , and/or a leftward perturbation of the scale σp . Actually, notice that the scale dynamics adjusts the equilibrium value σp forever after one iteration, and then only the location dynamics is relevant. This is a linear dynamics which is asymptotically stable, so once the scale is adjusted, the location µt increases monotonically towards its equilibrium value µp .

3

Price-dispersion market equilibrium

In this section we consider that the number n ≥ 2 of price quotes obtained by the consumer is not given but rather is to be determined endogenously. We thus assume that all firms set their prices independently, according to the decision

12

scheme of the previous section, and –also independently– the consumer selects the size n of the price sample optimally. In equilibrium, the firms’ and consumer’s optimal decisions determine both a price distribution F ∗ and a sample size n∗ . Below we first characterize the best response of a consumer facing search costs (subsection 3.1). Then we introduce the concept of market equilibrium and prove its existence and quasi-uniqueness given that the cost distribution of the industry is GP (subsection 3.2). Finally, we consider a best-response type of dynamics within the market and determine the stability of the equilibria found in the previous section (subsection 3.3).

3.1

Consumer’s problem

We must provide the consumer with a criterion to select n. The usual reason why the consumer might not check (or know from the onset) all available market prices –the whole price distribution in the setting of this paper– is because to acquire price information is costly. This is referred as consumer’s search cost in the literature (see e.g. Baye et al. (2006)). In the simplest –and most common– version of a search cost model, the consumer must face a cost c(n) = τ n in order to obtain n price quotes, where τ > 0 is the unit search cost –see Stigler (1961), which is the seminal work considering linear search costs. Given τ , the consumer chooses n by balancing out the total search cost and the expected saving in price, taking into account that he selects the lowest price in the considered price sample. As mentioned, we assume that the consumer checks at least one price.2 For any n ≥ 1, let pF (n) = {p1 , . . . , pn } denote a random sample of size n from a price distribution F . Thus the price he expects to pay after checking the sample pF (n) is given by GF (n) := IE [min pF (n)] ,

(13)

where IE stands for expected value. Given τ and F , the consumer reacts by choosing n∗ optimally, as defined next. Definition 8 Given a price distribution F and τ > 0, the best response of a consumer with unit search cost τ is defined by n∗ := argminn∈N+ {GF (n) + τ n},

(14)

with GF (n) as in (13). Thus, n∗ is the sample size that minimizes total cost, defined as the expected purchase cost plus the cost of search. As in the case of the industry equilibrium, we first characterize the consumer’s best response and then analyze the equilibrium. In line with the previous section, we assume that the price distribution F belongs to the family GP (k, −). Since n can only take integer values, it will be convenient to make use of the floor function that maps a real number x to the largest positive integer not greater than x, i.e. bxc := max {m ∈ Z+ : m ≤ x}. 2

This is a standard assumption in the search cost literature. Sometimes it is stated slightly differently, assuming that the first price quote is obtained (or the first shop is visited) for free, by arguing that the costly activity is not the visit itself but the comparisons among different price quotes. So the consumer has no search cost for visiting the first shop, as there is no comparison yet to be made. More generally, we might simply assume that there is an a priori consumer’s willingness to search such that she visits at least one shop.

13

Proposition 9 Let τ > 0 and k < 0, and let F be GP (k, µ, σ). Then there exists a unique solution for the consumer’s problem (14), which satisfies σ  σ n∗ = bk + g ≥ (k − 1)(k − 2), c ≥ 2 ⇐⇒ τ τ where g is strictly increasing and concave.

As it may be expected, the optimal search n∗ decreases as the unit search cost τ increases. Notice that the optimal choice n∗ of a consumer facing a GP price distribution does not depend on the location µ. Naturally, only the parameters of the price distribution that account for price variability matter in order to determine how many prices to check; in particular, n∗ can only increase as either the shape k or the scale σ of prices increase. Interestingly, the effect of the shape on n∗ is linear (via the floor function) and independent of the search cost. In contrast, only the scale matters in relation to the search cost, that is, n∗ is non-decreasing with respect to the ratio σ/τ and, in addition, the marginal effect of that ratio on the consumer’s choice decreases as the ratio increases. Finally, proposition 9 provides necessary and sufficient conditions for the consumer to search for at least two price quotes. As an example, the uniform distribution requires σ/τ ≥ 6.

3.2

Market equilibrium

We next turn attention to the equilibrium solution involving both sides of the market. The concept of market equilibrium will be the natural extension of the industry equilibrium studied in section 2. Notice that, given the cost distribution Φ and the unit search cost τ , both firms’ pricing decisions, given by γ(F ), and consumer’s search strategy, given by n∗ = n∗ (F ), depend on the (observed) price distribution F only, which thus constitutes the mechanism transmitting the optimal decisions of agents in the market. The market equilibrium is thus constituted by a pair (F ∗ , n∗ ) such that no single agent, either a firm or consumer, has incentive to deviate from his decision. This equilibrium concept is formally presented next. Definition 10 Let c Φ be the unit cost distribution of the industry, and let τ > 0 be the unit search cost of the consumer. A (price-dispersion) market equilibrium is a pair (F ∗ , n∗ ) formed by a (non-degenerate) price distribution F ∗ and a price sample size n∗ such that 1. argmaxp≥c πF ∗ (p, n∗ ) (5) with n = n∗ , and

F ∗ , that is F ∗ = γ(F ∗ ), where γ is as defined in

2. n∗ = argminn∈N+ {GF ∗ (n) + τ n}, where GF ∗ is given by (13) with F = F ∗ . The main issue now is to establish the existence and uniqueness of market equilibrium, and then determine whether an equilibrium is restored under some perturbation of the price distribution F ∗ or the consumer’s search n∗ . When unit costs are GP distributed, precise answers to these questions can be provided in terms of the ultimate parameters that control the interaction in the market, namely τ on the consumer side, and k, µ and σ on the industry side. In particular, equilibrium solutions (F ∗ , n∗ ) can be computed from the values of the parameter bundle (τ, k, µ, σ). Notably, for a given shape k of the cost distribution, a pricedispersion equilibrium exists as long as the dispersion of firms’ costs is sufficiently large relative to the search cost, i.e. the ratio σc /τ is large enough. Furthermore,

14

if this ratio is beyond certain threshold value, only two equilibria can coexist and they are naturally very close to each other. The specific statements regarding these questions are gathered in the following theorem, which is the central result in this section. Theorem 11 Let k < 0, c GP (k, µc , σc ) and τ > 0. Then there exist two threshold values, θ0 (k) and θ1 (k), satisfying • θ0 (k) ≤ θ1 (k) for k < 0,

• θ0 (k) = θ1 (k) ≥ 2 for −1 ≤ k < 0,

• θ0 (k) → +∞ (linearly) as k → −∞,

such that (i) If

σc τ σc τ

< θ0 (k), then a (price-dispersion) market equilibrium does not exist.

≥ θ0 (k), then there exists at least one (price-dispersion) market (ii) If equilibrium. Furthermore, any equilibrium (F ∗ , n∗ ) satisfies that F ∗ is GP (k, µ∗p , σp∗ ), with n∗ − 1 1 σc + µc , σp∗ = ∗ σc and −k−1 n −1−k  ∗ σp n∗ = bk + g c, τ

µ∗p =

n∗

where g is as in proposition 9. In addition, assume that there are two equilibria, (F ∗ , n∗ ) and (F ∗0 , n∗0 ), ∗0 ∗0 ∗ where F ∗ is GP (k, µ∗p , σp∗ ) and F ∗0 is GP (k, µ∗0 p , σp ). Then, if n > n , it ∗0 ∗ ∗0 ∗ holds that µp < µp and σp > σp . (iii) If στc ≥ θ1 (k), then there exist at most two (price-dispersion) market equilibria. In addition, in the multiple case, the two equilibria are of the form (F ∗ , n∗ ), (F ∗0 , n∗ + 1), that is their corresponding consumer’s n are consecutive integers, n∗ and n∗ + 1. It is an interesting exercise to consider the limiting case of the price-dispersion equilibrium described in theorem 11 as the search cost τ vanishes. As τ → 0, it follows from the proof of theorem 11 that n∗ → +∞ and, in turn, the cost distribution GP (k, µc , σc ) emerges as the limiting price distribution. This corresponds to the Walrasian equilibrium in which each firm sets its price equal to marginal cost and all firms make zero profit with probability one. Indeed, since GF (n) → µc as τ → 0 almost surely –see (13)–, the consumer will find the firm with the lowest price, which will be the only one making the sale (with zero profit). A standard game-theoretic argument ` a la Bertrand for the interaction among firms shows that “price equal to marginal cost” describes an equilibrium. This is the cost distribution GP (k, µc , σp ) which thus constitutes the Walrasian equilibrium. Finally, it is worth mentioning that the price-dispersion equilibria characterized in theorem 11 stand between two polar equilibria: Walrasian equilibrium and Diamond equilibrium, whose market-clearing prices are positioned at the lower and upper extreme of the cost distribution, respectively, with probability one. When n is arbitrarily large, due to a small search cost, the Walrasian equilibrium emerges, as just explained. In contrast, the Diamond case is obtained in the limiting case n = 1, which leads all firms to set the same price: the upper extreme of the cost distribution, which corresponds to the consumer willingness to pay.

15

3.3

(Market) equilibrium restoring dynamics

In this subsection we analyze the market dynamics when it is initially out of equilibrium, in particular whether the price-dispersion equilibrium is recovered after perturbation. We adopt an evolutionary myopic approach, namely firms and consumer play alternatively over time using their best responses given available infomation of the previous period only. Within each period, say t, firms decide prices Ft , then consumer selects the price sample size nt and then a new period starts. In each period t, firms move simultaneously over a number of successive rounds as described in section 2.4, until they reach the industry equilibrium Ft∗ for the value nt prevailing at the beginning of the period. Once the price equilibrium of the industry of that period is reached, the consumer moves by playing his best response to Ft∗ . It can be argued that no concrete model of market dynamics will be fully convincing. We next provide some complementary reasons to consider the market dynamics described above. In regard to sequential moves, a first intuition from everyday market interaction suggests that firms operate –update their decisions– at a much higher frequency than consumers do. Our dynamics gives account of that asymmetry by assuming a first stage in each period in which firms interact repeatedly to adjust prices, given the consumer’s search behavior. Also, we simply rely on game-theoretic grounds. Roughly, we mimic settings in which two players take turns alternatively over discrete time. A leading example of such dynamics in a pricing game is Maskin and Tirole (1988), though they use Markov perfect equilibrium rather than off-equilibrium best strategies. In our setting, we replace two players with two kinds of players: firms and consumers. As said in section 2.1, the consumer is representative: many identical consumers, each one behaving individually as described in section 3.1, would do the same. Representativeness implies that consumers do not compete among themselves. However, firms do. We assume that when firms are to interact among them, they do until they resolve their within-type price competition. We must notice that our evolutionary approach does not necessarily conform a Nash equilibrium of a dynamic market game. In regard to the local rationality of agents, we assume that agents are not expected to implement a complex intertemporal strategy bur rather to play an adaptive strategy given the information currently available. This is in line with the literature of learning –see e.g. the search cost model in Hopkins and Seymour (2002) and references therein. More generally, the use of best-reply dynamics to analyze off-equilibrium paths in games in which players take turns alternatively over time has a long tradition in economics. Just as an example, Johnson et al. (2002) show experimental evidence of the so called bounded rationality when players are engaged in sequential bargaining. The dynamics is depicted schematically in figure 2. Assume that the value nt is observed by firms at the start of period t, and then the industry reacts by forming the corresponding industry equilibrium, as in theorem 5. This stage corresponds to the node Industry in the figure, at which the ad hoc equilibrium Ft∗ emerges from the inputs nt and Φ. Then the consumer reacts to Ft∗ by selecting the ad hoc (optimal) price sample size, denoted by nt+1 , obtained as in proposition 9 with Ft∗ and τ as inputs. This is the stage represented by the node Consumer in the figure. The new period starts with nt+1 given. Assuming that Φ is GP (k, µc , σc ), it follows from theorem 5 and proposition

16

Φ

nt

Industry

τ Ft∗

Consumer

nt+1

Figure 2: The nodes Industry and Consumer represent the industry equilibrium (as in theorem 5) and the consumer’s optimal choice (as in proposition 9), respectively. Notice that nt+1 = n?t+1 is optimal according to (14) given the inputs at period t. We drop the superscript here for convenience. 9 that the market dynamics described above is steered by   σc nt − 1 c, t = 0, 1, 2, . . . nt+1 = bk + g τ nt − 1 − k

(16)

Thus, equation (16) defines a two-period nonlinear dynamics for the (optimal) consumer’s n that implicity incorporates the industry’s (optimal) reaction interperiod. Clearly, a market equilibrium occurs whenever nt = nt+1 := n∗ holds in (16). The equilibrium value n∗ then determines µ∗p and σp∗ from the formulae in theorem 11(ii). As mentioned in proposition 9, the consumer reacts to the scale parameter of the price distribution, σt , but not to the location, µt , while lemma 6 shows that, when firms move, in one period they attain the industry equilibrium value of the scale parameter which corresponds to the current value of n. Thus, the dynamics in (16) entails that firms and consumer play at alternate periods except for the adjustment on the location parameter of the price distribution, which is irrelevant for the consumer. We focus on the case of quasi-unique equilibrium established in theorem 11, i.e there are two equilibria only, whose consumer’s n correspond to consecutive values n∗ and n∗ + 1, with n∗ ≥ 2. In turn this implies that n∗ and n∗ + 1 are the only two equilibria of the dynamical system (16). Theorem 11(iii) gives a sufficient condition for this equilibria configuration. Proposition 12 Let k < 0, c GP (k, µc , σc ) and τ > 0. Assume that στc ≥ θ1 (k), with θ1 (k) as in theorem 11, so that there exist at most two equilibria of (16), n∗ and n∗ + 1, with n∗ ≥ 2. Then (i) Any orbit {nt }t∈N of (16) starting at n0 ≥ n∗ + 1 converges to n∗ + 1 as t → +∞.

(ii) There exists n(k) < n∗ such that any orbit {nt }t∈N of (16) starting at an integer n0 ∈ [n(k), n∗ ] converges to n∗ as t → +∞. √ (iii) If |k| ≤ 3, then it is n(k) = 2 in (ii).

Throughout the paper we have considered that n ≥ 2 in order to determine the existence of price-dispersion market equilibria under GP cost distributions. The Diamond equilibrium, in which all firms set the monopoly price and the consumer selects n∗ = 1, is also an equilibrium of the market. The fact that this equilibrium always exists in the presence of search frictions –even when all firms have identical costs– is the content of the Diamond paradox (Diamond (1971)). A key question then is whether, starting off equilibrium, the Diamond equilibrium will be the observable equilibrium after the market adjustment. In other

17

words, under a plausible market dynamics, will a given initial arbitrary market configuration (F0 , n0 ) converge towards the Diamond equilibrium or rather towards a price-dispersion equilibria? Proposition 12 provides an answer to that question for GP distributed costs in terms of the starting search value n0 , under the assumptions in theorem 11(iii). Notice that n0 in turn might have been obtained rationally from some initial price distribution F0 by solving problem (14). Proposition 12 establishes two different convergence regions for n0 under the market dynamics defined in (16). The first region includes all values of n0 larger than the equilibrium value n∗ + 1: the dynamics (16) leads the market to the closest price-dispersion equilibrium, corresponding to n = n∗ + 1. This is part (i) of the proposition. The second region comprises values n0 smaller than the equilibrium values. This is the delicate area, as the initial value of n0 might be close to the Diamond equilibrium. Proposition 12(ii) defines a critical value n(k), such that convergence to a price-dispersion equilibrium is guaranteed for sequences {nt }t that depart from some value n0 greater than or equal to n(k). The notation emphasizes that n(k) depends on k, which is the shape parameter of the distribution of unit cost. As a main conclusion, given a GP unit cost distribution, the Diamond equilibrium will not emerge under the market dynamics considered in this section, as long as the consumer initially obtains a number of price quotes larger than the threshold value n(k). In fact, for a class of GP (k, −) distributions that includes those with concave or linear shapes the common threshold value is n(k) = 2. To illustrate the significance of this result, consider the case of uniform distributions GP (−1, −). Assume that unit costs are GP (−1, µc , σc ), and they are sufficiently dispersed relative to the consumer’s search cost τ , so that the condition in theorem 11(iii) is satisfied. In fact, it must be σc /τ > 12 in this case. Assume that the consumer starts shopping by comparing prices from at least two sellers (i.e. n0 = 2), which is in line with some recent experiments on single-option aversion –see Mochon (2013). Then the adjustment dynamics defined in figure 2 leads the market to a price-dispersion equilibrium (F ∗ , n∗ ) in which n∗ ≥ 2 and prices p∗ are uniformly distributed, moreover they are defined by formulae in theorem 11(ii).3 A final remark on the convergence to the Walrasian equilibrium is in order. Assume that the market is at some equilibrium for a given strictly positive search cost and, suddenly, the search cost vanishes. In the next period the consumer plays n∗ → ∞. As shown in the previous section, the firms’ best reaction is then to set price equal to marginal cost. Thus, there is convergence to the Walrasian equilibrium in just one period. In other words, our model requires of positive search cost to sustain positive markups at the equilibrium. Whenever the search cost exists, price-dispersion emerges and typically persists under perturbation.

4

Conclusions

It has long been repeated –and reported– that the law of one price in markets for homogenous goods seems to be an exception rather than the rule. There is a vast theoretical literature on the rationalization of that fact. Two basic ingredients for the phenomenon seem to be the existence of consumer’s search cost and 3

A graphical illustration of the dynamics is provided in appendix B.

18

some fundamental heterogeneity. This heterogeneity can be in the pricing strategy (mixed strategies in Burdett and Judd (1983)), or in the search cost across consumers (Rob (1985)) or in the firms’ costs (MacMinn (1980)). A number of papers consider more than one kind of heterogeneity. Regarding search costs, two different consumer search strategies have been suggested in the literature: sequential and fixed size sample, with a growing body of empirical literature trying to discriminate among them from observed consumer’s behavior. After this long journey, some questions remain unsettled. First, in many models price-dispersion equilibria co-exist with the Diamond paradox, in which all firms charge the same (monopoly) price and consumers visit only one shop, with no price-dispersion. Second, the analysis of out-of–equilibrium dynamics, which is crucial when there is multiplicity of equilibria, are relatively scarce. A crucial point for this scarcity of results may partly be the difficulty in finding best-response dynamics in closed form. This paper contributes to the theoretical analysis by adopting a practical approach as a starting point. How are costs expected to be distributed across firms (sellers) in real markets and what search strategy do consumers follow? Regarding the first question, there is theoretical support for Generalized Pareto (GP ) cost distributions, that comes from two different angles: extreme value theory and fee-setting mechanisms in online markets. On the second issue, recent empirical studies show that the consumer’s search behavior is consistent with a fixed sample size search strategy in some markets (De los Santos et al. (2012) and Honka and Chintagunta (2017)) . We show that considering GP distributions and fixed sample size search has strong implications. First, it simplifies the analysis as it delivers closed-form best responses. Second, it allows us to find simple sufficient conditions for existence and uniqueness, namely the cost dispersion across firms must be large enough as compared to the consumer’s search cost. Third, we show that a evolutionary type of best-response dynamics may restore the price-dispersion equilibrium after perturbation, driving the market away from the Diamond equilibrium, even if the perturbation takes the market near the Diamond paradox.

19

A

Proofs

Proof of proposition 3. Any price lower than p will do no better than p while any price above p will do no better than p, so we can restrict the search of solutions of problem (2) to the interval [p, p]. In that interval, πF (p, n) is continuous and differentiable with respect to p –laterally at the endpoints. Notice that, for p ∈ [p, p], sign πF0 (p, n) = sign H(p), where the prime denotes derivative with respect to the first argument, and H(p) := 1 − F (p) − (n − 1)f (p)(p − c). It is understood above that derivatives at p and p are one-sided. Notice that p = p is always a critical point of πF (p, n) which corresponds to a minimum, given that c < p. The assumption in part (i) implies that H(p) ≤ 0. Using the monotonicity of h, we have, for p ∈ (p, p), p−c>

1 1 ≥ , (n − 1)h(p) (n − 1)h(p)

so that H(p) < 0. It follows that p∗ = p. On the other hand, the inequality in part (ii) gives H(p) > 0. Weierstrass theorem guarantees that πF (p, n) reaches its maximum at some interior point p0 , given that both p and p are minima. It follows that H(p0 ) = 0 for some p0 ∈ (p, p), that is 1 (p0 − c)h(p0 ) = , (17) n−1

which is equation (3). Furthermore, since the left-hand side of expression (17) is strictly increasing in (p, p), p∗ = p0 is unique.

QED The next lemma is used in the proof of theorem 5. Lemma 13 Let x be a random variable with distribution GP (k, µ, σ), and let y = a + bx, where a ∈ R and b > 0. Then y has distribution GP (k, a + bµ, bσ). Proof of lemma 13. Using definition 1, we have 

   − 1 k z−a 1 z−a Pr(y ≤ z) = Pr x ≤ =1− 1+k −µ b σ b  − 1 k 1 = 1 − 1 + k (z − (a + bµ)) . bσ The last expression above is the cumulative distribution function (cdf) of a variable with distribution GP (k, a + bµ, bσ). QED Proof of theorem 5. In the proof we will use the fact that, if x has distribution GP (k, µ, σ), its hazard function is given by 1 h(x) = σ



x−µ 1+k σ

20

−1

.

(18)

In particular, h is strictly increasing in [µ, µ − σk ). Notice that h(p) = σp−1 , for F ∗ is GP (k, µ  p , σp ).  Take c ∈ µc , µc − σkc . Using (6), we have p−c≤

σp σc 1 = = . n−1−k n−1 (n − 1)h(p)

It thus follows from proposition 3 that the best response of a firm with cost c is given by 1 p∗ − c = , (n − 1)h(p∗ ) that is,

p∗ =

σp − kµp n−1 + c, n−1−k n−1−k

(19)

which is equation (7). Since c GP (k, µc , σc ), it follows from (19) and lemma 13 that p∗ has distribution GP (k, µp , σp ), where µp , and σp are as in (6). Thus F ∗ is the distribution of p∗ and in turn an industry equilibrium. Clearly, F ∗ is unique within the class GP (k, −). QED Proof of lemma 6.  (i) Consider c ∈ µc , µc −

σc k

 . Using (9a) and (18), we have

µ0 − c ≤

σ0 1 = . n−1 (n − 1)h(µ0 )

According to proposition 3(i), the best response of a firm with unit cost c satisfies (3), which gives p∗ =

σ0 − kµ0 n−1 + c. n−k−1 n−k−1

Since c GP (k, µc , σc ), it follows from lemma 13 that p∗ with parameters as given in (10a)-(10b).

GP (k, µ1 , σ1 ),

(ii) Combining identities (10a)-(10b) with (9a)-(9b) we have µ1 − µc =

σ0 σ0 σ1 −k(µ0 − µc ) + ≤ ≤ . n−k−1 n−k−1 n−1 n−1

QED Proof of proposition 7. (i) It follows from lemma 6 that F1 is GP (k, µ1 , σ1 ) with µ1 − µc ≤

n−1 σ1 and σ1 = σc . n−1 n−1−k

(20)

Applying lemma 6 repeatedly we have that Ft is GP (k, µt , σt ), with µt and σt satisfying conditions (20) just replacing subindex 1 with t. This implies that γ is well defined within the class GP (k, −). Furthermore, applying lemma 6(i) iteratively, we obtain the best response dynamics (12) for the parameters (µt , σt ) of GP (k, −) distributions.

21

(ii) Notice that σt = σp for t ≥ 1, where σp is the equilibrium value defined in theorem 5(i). It thus suffices to study the dynamics of µt , which is of the form −k µt+1 = µt + ψ, n−k−1 where ψ is some appropriate constant. Since n ≥ 2, the µ-dynamics is monotonically asymptotically stable. It follows from (9a) that µ0 ≤ µp , where µp is the equilibrium value defined in theorem 5(i). Thus µt increases monotonically to its limiting value. It can be easily checked that this limiting value is µp . QED The next lemma is used in the proof of Proposition 9. Lemma 14 Let n ≥ 1. Assume that each variable in p(n) = {p1 , . . . , pn } is independent and identically distributed with distribution GP (k, µ, σ). Then G(n) := IE [min p(n)] = µ +

σ . n−k

Proof of lemma 14. Let F denote the cdf of a distribution GP (k, µ, σ). Given n ≥ 1, let  Fmin denote the  cdf of the variable min p(n) whose support is clearly µ, µ − σk . For any z ∈ µ, µ − σk , we have 

1 Fmin (z) = 1 − (1 − F (z)) = 1 − 1 + k (z − µ) σ n

− n k

.

A change of variable thus gives G(n) =

Z

µ− σ k

µ

n σ

 − n −1 k σ 1 1 + k (z − µ) zdz = µ + . σ n−k

QED Proof of proposition 9. For any n ≥ 2, let us define ∆G(n) := |G(n)−G(n−1)|. From lemma 14, G is decreasing and convex. Therefore, the solution satisfies n∗ ≥ 2 ⇐⇒ ∆G(2) ≥ τ . Direct substitution allows to write this latter inequality as στ ≥ (k − 1)(k − 2). Whenever the solution satisfies n∗ ≥ 2, the solution to problem (14) is given by the largest integer n satisfying ∆G(n) ≥ τ or, equivalently, the floor of the solution of ∆G(n) = τ . Using lemma 14 this equation can be written as σ = (n − k)(n − k − 1). τ This equation has two real roots in n − k, one positive and the other negative, which guarantees uniqueness of solution to problem (14). The positive root gives the possible solution to the consumer’s problem, namely the floor of σ  n=k+g , (21) τ
√ with g(x) = 12 1 + 1 + 4x which is increasing and concave. QED

The next lemma is used the proof of theorem 11.

22

Lemma 15 Let k < 0 and define, for n ≥ 2, θL (n) := Then


n−k n−k (n − k)2 − 1 . (n − 1 − k)2 and θH (n) = n−1 n−1

(i) θL has a strict global minimum n0 (k) ≥ 2.

(ii) θL (n) < θH (n) for all n ≥ 2.

(iii) θL (n + 1) < θH (n) for all n ≥ 2.

(iv) There exists n1 (k) ≥ 2 such that θH (n) < θL (n + 2) for all n > n1 (k).

Proof of lemma 15 (i) Let k < 0 be given. Formally, the derivative of θL with respect to n can be written as n−k−1 0 θL (n) = λ(n), (n − 1)2

where λ(n) = 2n2 −(3+k)n+1+2k−k 2 . Since λ(1) < 0, λ has a unique real + + 0 root in (1, ∞), say n+ 0 = n0 (k). Since sign θL (n) = sign λ(n), n0 constitutes a global minimum of θL . It suffices to take n0 (k) = max{2, bn+ 0 (k)c}.

(ii) Notice that, for n ≥ 2,

θH (n) − θL (n) = 2

n−k (n − k − 1) > 0. n−1

(iii) Clearly, θL (n + 1) < θH (n) ⇐⇒

n+1−k (n − k)2 − 1 (n − k) < , n n−1

and the latter inequality holds for n ≥ 2.

(iv) We have

2(n + 1 − k) 2 (n − k 2 + k − 1) > 0, n2 − 1 √ provided that n > n+ k 2 − k + 1. Notice that n+ 1 (k) := 1 (k) ≥ 1. Now + take n1 (k) = max{2, bn1 (k)c}. θL (n + 2) − θH (n) =

QED Proof of theorem 11. It follows from the proofs of theorem 5 and proposition 9 that the equations that conform any market equilibrium are 1 σ c + µc , (22a) n−1−k n−1 σp = σc , (22b) n−1− k  σ p n = bk + g c. (22c) τ
√ with g(x) = 12 1 + 1 + 4x . Notice that (22b) and (22c) conform two equations in the unknowns σp and n and that for any pair (σp , n) satisfying those equations, (22a) determines a unique solution in µp . It thus suffices to study the system (22b)–(22c). µp =

23

σp =

n 4

n−1 σ n−1−k c

n=k+g 3

σp
τ

2 1 0

σp

Figure 3: The blue curve –defined by equation (22b)– passes through (0, 1), whereas the orange curve passes through at (0, 1 + k). Recall that k < 0, so in the graph we have assumed 1 + k > 0, which is qualitatively irrelevant. The red curve is obtained as the integer floor of the vertical values of the orange curve, so it corresponds to equation (22b). The solutions to the system (22b)–(22c) are the intersections of the blue curve and the red curve. In the figure, the two green points correspond to price dispersion equilibria with n∗ = 2 and n∗ = 3, respectively. The red point corresponds to the Diamond equilibrium, at (σp , n) = (0, 1). Define the auxiliar functions sa and sb as follows:  n−1 τ (2n − 1 − 2k)2 − 1 and sb (n) = σc . sa (n) = 4 n−1−k Notice that sa (n) is the solution for σp of n=k+g

σ  p

τ

,

whereas sn (b) is simply the right hand side of (22b). Figure 3 depicts the geometry of the system (22b)–(22c). The orange and blue functions in figure 3 are σp = sa (n) and σp = sb (n), respectively. Notice that a given n ∈ N is part of a solution to (22b)–(22c) if and only if sa (n) ≤ sb (n) ≤ sa (n + 1), or, equivalently, 0 ≤ sb (n) − sa (n) ≤ 2τ (n − k). This condition can be rewritten in terms of the functions θL and θH defined in lemma 15, namely σc θL (n) ≤ ≤ θH (n). (23) τ The geometry of the intervals [θL (n∗ ), θH (n∗ )] was obtained in lemma 15. Define θ0 (k) = θL (n0 (k)) and θ1 (k) = θL (n1 (k)), where n0 (k) and n1 (k) are given in parts (i) and (iv) of lemma 15, respectively. By definition, θ0 (k) ≤ θ1 (k). Because of the overlapping property of the intervals [θL (n∗ ), θH (n∗ )] implied by (ii)-(iii) in lemma 15, it holds that [ [θ0 (k), ∞) ⊂ [θL (n), θH (n)]. n≥n0 (k)

This means that if στc ≥ θ0 (k), there there exists at least an interval, say [θL (n∗ ), θH (n∗ )] such that στc ∈ [θL (n∗ ), θH (n∗ )], which in turn implies that n∗ is part of an equilibrium because of (23).

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Also, since θ0 (k) is the global minimum of θL , σc ∈ / τ

[

σc τ

< θ0 (k) implies that

[θL (n), θH (n)],

n≥n0 (k)

and thus there is no solution for (23), in turn an equilibrium does not exist. Notice that assertion (iii) of the proposition follows from the geometry of equilibria depicted in figure 3. Finally, while for n ≥ n0 (k), two consecutive intervals [θL (n), θH (n)] and [θL (n + 1), θH (n + 1)] always overlap, part (iv) in lemma 15 implies that the intersection [θL (n), θH (n)] ∩ [θL (n + 2), θH (n + 2)] is void for all n > n1 (k). It follows that, if στc ≥ θ1 (k), then στc belongs at most to two consecutive intervals, which gives the form of uniqueness claimed in part (iii) of the proposition. QED Proof of proposition 12 (i) Consider some initial n0 > n∗ + 1 and let {nt }t be the sequence obeying (16). To simplify the exposition, write equation (16) as a composition, nt+1 = bρ(nt )c. Since n∗ + 1 conforms an equilibrium, we have n∗ + 1 = bρ(n∗ + 1)c or, equivalently n∗ + 2 > ρ(n∗ + 1) ≥ n∗ + 1. Notice that ρ is strictly increasing, concave and bounded from above. Thus, for m ≥ 2, it is ρ(n∗ + m) > ρ(n∗ + 1) ≥ n∗ + 1. Clearly, the floor function preserves the previous inequalities. Thus, n∗ + 1 constitutes a lower bound for any orbit {nt }t that starts at n0 > n∗ + 1. Next we prove that ρ(n∗ + m) < n∗ + m holds for all m ≥ 2. Indeed, since ρ is bounded, it must be ρ(n∗ + M ) < n∗ + M for some finite M . Suppose that M > 2 and that ρ(n∗ + M − 1) ≥ n∗ + M − 1 holds. We thus have n∗ + M > ρ(n∗ + M ) > ρ(n∗ + M − 1) ≥ n∗ + M − 1. The first and third of this chain of inequalities defines n∗ + M − 1 as an equilibrium, which contradicts that there is no equilibria for values of n larger than n∗ + 1. Thus, if ρ(n∗ + m) < n∗ + m holds for some m = M > 2, it also holds for m = M −1. By induction, it holds for all m > 1, as claimed above. Thus, we have nt+1 = bρ(nt )c ≤ ρ(nt ) < nt for t = 0, 1, . . ., with {nt }t bounded from below by n∗ + 1. Hence, {nt }t must converge to n∗ + 1.

(ii) Let {nt }t be an orbit of (16) starting at n0 < n∗ . We will first show that the sequence is bounded from above by n∗ . Consider m < n∗ . Since ρ is strictly increasing and n∗ is an equilibrium, it holds that bρ(m)c ≤ bρ(n∗ )c = n∗ . Thus, nt < n∗ implies nt+1 = bρ(nt )c ≤ n∗ . Now let n0 (k) be as in 15(i) and assume n0 ≥ n0 (k) holds. We will show next that {nt }t is increasing. Suppose that ρ(m) ≤ m for some n0 (k) ≤ m < n∗ . Notice that σc ≤ θL (m) τ

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where θL is the function defined in Lemma 15. Since n∗ constitutes an equilibrium, it follows from (23) that θL (n∗ ) ≤

σc . τ

The two latter inequalities imply that θL (n∗ ) ≤ θL (m), which contradicts the fact that θL strictly increases to the right of n0 (k), as proved in Lemma 15. Therefore, for any integer m such that n0 (k) ≤ m < n∗ , it must be ρ(m) > m , in turn bρ(m)c ≥ m. In fact the inequality is strict, since m cannot be an equilibrium. It follows from above that the sequence {nt }t satisfies nt+1 = bρ(nt )c > nt whenever n0 (k) ≤ nt < n∗ . Being bounded from above by n∗ , {nt }t must converge to n∗ . + (iii) From Lemma 15, we have n0 (k) = max{2, bn+ 0 c}, where n0 is the unique 2 2 positive root of λ(n) = 2n − (3 + k)n + 1 + 2k − k . Since λ(n) is strictly convex with λ(1) < 0, it is √ n+ 3. 0 ≤ 2 ⇐⇒ λ(2) ≥ 0 ⇐⇒ |k| ≤

QED

B Graphical representation of the market dynamics Assuming that Φ is GP (k, µc , σc ), the market dynamics described in the main text is steered by the planar system σ  t c, (24a) nt+1 = bk + g τ nt − 1 σt = σc . (24b) nt − 1 − k Expression (24a) gives the consumer’s reaction to σt , according to proposition 9, whereas (24b) gives the industry response versus nt according to theorem 5. A market equilibrium occurs whenever nt = nt+1 := n∗ holds in (24a)–(24b). The value n∗ determines the equilibrium value σp∗ from (24a) or (24b). The equilibrium value µ∗p is obtained from the formula in theorem 11(ii). We focus on the case of quasi-unique equilibrium established in theorem 11, i.e there are two equilibria only, whose consumer’s n correspond to consecutive values n∗ and n∗ + 1, with n∗ ≥ 2. In turn this implies that n∗ and n∗ + 1 are the only two equilibria of the dynamical system (24a)–(24b). Theorem 11(iii) gives a sufficient condition for this equilibria configuration. Figure 4 illustrates the convergence to a price-dispersion equilibrium for a small initial value n0 < n∗ . To ease the sketch of the dynamics, assume that the argument of the floor function in (24a) returns integer values n1 , n2 ,... so that the orbit of n0 lies on the orange curve in figure 4. The blue curve in the figure corresponds to the graph of (24b),and then the green point (σ ∗ , n∗ ) at which both curves intersect constitutes a price-dispersion equilibrium in the (σ, n)–plane. Notice that the point (0, 1) in red corresponds to the Diamond equilibrium. Starting with the initial search value n0 , firms and consumer respond alternately in each period, giving rise to the orbit n0 → σ1 → n1 → σ2 → n2 → . . . ,

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depicted by the black line inside the region between both reaction curves. Essentially, the figure shows how the market gets away from the Diamond equilibrium and converges to the price-dispersion equilibrium. The dynamics starting from some n0 > n∗ can be represented similarly.

n σt =

nt −1 σ nt −1−k c

nt+1 = bk + g

n∗ n 2

σt τ


c

n1

n0 1 0

σ1

σ2 σ ∗

σp

Figure 4: The blue and orange curves give the reactions of industry and consumer, respectively. To ease the presentation, here we assume that the referenced n-values in the figure are integer, so that the orbit of n0 is embedded in the orange curve. The price-dispersion equilibrium (σ ∗ , n∗ ) is at the green point, where both reaction curves meet. The Diamond equilibrium is at the red point (0, 1). The market dynamics starts initially at some n0 satisfying 1 < n0 < n∗ and then industry and consumer respond alternately in each period. At period t, the configuration of the market is given by (σt , nt ). The figure shows how the market dynamics approaches the price-dispersion equilibrium even for search values n0 initally close to the Diamond equilibrium.

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