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The informational divide
Games and Economic Behavior 78 (2013) 21–30
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Games and Economic Behavior www.elsevier.com/locate/geb
The informational divide ✩ Manfred Nermuth a , Giacomo Pasini b,c , Paolo Pin d,∗ , Simon Weidenholzer e a
Institut für Volkswirtschaftslehre, Universität Wien, Austria Dipartimento di Economia, Università Ca’ Foscari di Venezia, Italy Netspar, Network for Studies on Pensions, Savings and Retirement, Tilburg, The Netherlands d Dipartimento di Economia Politica e Statistica, Università degli Studi di Siena, Italy e Department of Economics, University of Essex, United Kingdom b c
a r t i c l e
i n f o
Article history: Received 28 June 2010 Available online 9 November 2012 JEL classification: D43 D85 L11 Keywords: Price dispersion Welfare effects of search Price competition on networks Informational divide
a b s t r a c t We propose a model of price competition where consumers exogenously differ in the number of prices they compare. Our model can be interpreted either as a non-sequential search model or as a network model of price competition. We show that (i) if consumers who previously just sampled one firm start to compare more prices all types of consumers will expect to pay a lower price and (ii) if consumers who already sampled more than one price sample (even) more prices then there exists a threshold – the informational divide – such that all consumers comparing fewer prices than this threshold will expect to pay a higher price whereas all consumers comparing more prices will expect to pay a lower price than before. Thus increased search can create a negative externality and it is not necessarily beneficial for all consumers. © 2012 Elsevier Inc. All rights reserved.
1. Introduction Is more information about prices always good for consumers? In this paper we analyze this question in a consumer search model à la Burdett and Judd (1983) where consumers exogenously differ in the number of prices they compare. In order to assess the welfare effects of increased information we consider the expected prices paid by the various consumers. It turns out that whether more information is indeed beneficial to all consumers will depend very much on which consumers get more information. In line with the previous literature, we find that more information for previously uninformed consumers leads to lower expected prices for all types of consumers (Theorem 3.1) and increases consumer welfare unambiguously. But, surprisingly, increased search by those who already do some search actually harms the uninformed, while benefiting the well-informed, and merely re-distributes welfare from the former to the latter. More precisely, if some searchers search more, the endogenous equilibrium price distribution changes in such a peculiar way that – for a certain number d > 1 (the “informational divide”) – all consumers who compare fewer than d prices face higher expected prices than before, whereas the others face
✩ We are grateful to two referees and to the advisory editor for their useful suggestions. The authors wish to thank Larry Blume, Giacomo Calzolari, Andrea Galeotti, Maarten Janssen, Saul Lach, Marco van der Leij, José Luis Moraga-González, Marco Ottaviani, Mario Padula, and Fernando Vega-Redondo for useful comments on earlier drafts of this paper. Previous versions of this paper were circulated under the titles “Price Dispersion, Search Externalities, and the Digital Divide” and “A Network Model of Price Dispersion.” Corresponding author. E-mail addresses: [email protected] (M. Nermuth), [email protected] (G. Pasini), [email protected] (P. Pin), [email protected] (S. Weidenholzer).
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0899-8256/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.geb.2012.10.016
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lower expected prices (Theorem 3.2). In other words, more information for the informed creates a negative externality for the uninformed. To gain an intuitive understanding of this “informational divide” we observe the following. Expected profits and hence the average selling price depend only on the share of uninformed consumers in the market. This is so because expected profits are equal to the profit a firm can make by selling only to uninformed consumers (at the monopoly price). Thus, if some searchers start to compare more prices, the average selling price remains the same and the resulting change in the equilibrium price distribution can only have a redistributive effect, benefiting some consumers and harming others. We show that the new price distribution “single-crosses” the old one, in such a way that the mean (= the expected price paid by the uninformed) increases and the lower bound decreases. This harms the uninformed and helps those who search a lot. More intuition is given after Theorem 3.2. In addition to the results just described, we show by means of examples that “increased search” can have truly counterintuitive effects: it can lead to higher expected prices for all consumer types that are actually present in the market (here a “type k” consumer is one who samples k prices), and it can even happen that those consumers who engage in more search harm themselves (see Example 3.3 and Example 3.4). The rest of the paper is organized as follows: Section 2 spells out the model. Section 3 presents the results. In Section 4 we discuss some related issues, viz. (i) a network interpretation of our model, (ii) endogenous search, and (iii) policy implications. Proofs are relegated to Appendix A. 2. The model We consider a market for a homogeneous good with N firms and M households (consumers), and write μ = M / N for the number of households per firm. Each firm can produce the good at constant marginal cost, without fixed cost, and sets the price at which it offers the good (all firms set their prices simultaneously). Each household demands one unit of the good, up to a given willingness to pay (assumed greater than the cost). Without loss of generality we normalize the cost to 0 and the willingness to pay to 1 (the same for all firms, respectively households). Households differ in the information they have about the firms’ prices. A household of type k observes the prices of k firms and buys from the cheapest (randomizing with equal probabilities in case of ties), provided the price does not exceed its willingness to pay. We denote by qk the fraction of households of type k. The information structure N (or consumer search distribution) is represented by the vector q = (q1 , . . . , q N ), where of course qk 0 for all k and k=1 qk = 1. Equivalently, we can let qk stand for the number of type k households. We usually do this in the examples. A household of type k = 1 is called uninformed, households of types k 2 are called informed (also searchers or shoppers). We thus obtain a strategic market game among the N firms, where we can take without loss of generality the strategy set of each firm to be the unit interval [0, 1] (a price below 0 would generate losses, and at a price above 1 nobody would buy). Trivial cases apart, this game has equilibria only in mixed strategies, generating price dispersion. From now on, we assume always 0 < q1 < 1. The following known result is stated here for easy reference. A more precise formulation is in Appendix A. Proposition 2.1. Consider a market game with information structure q = (q1 , . . . , q N ) and assume 0 < q1 < 1. Then there exists a unique symmetric equilibrium in mixed strategies: each firm chooses its price at random according to a continuous distribution F ( p ) with support [ p min , p max ], where 0 < p min < p max = 1. Moreover, the equilibrium profit per firm is π = μq1 , and the average selling price (average household expenditure) is pav = q1 . The expected price pk paid by a household of type k is given by
1 pk =
(k = 1, . . . , N ),
p dF k ( p ) p min
where F k ( p ) = 1 − [1 − F ( p )]k is the distribution of the minimum of a sample of size k from the distribution F . The distribution F k shifts more and more mass near p min , as k increases, so that pk → p min for k → ∞. It is easy to see1 that the expected price pk can also be written as
1 pk =
k
1 − F ( p)
dp .
(1)
p min
This implies the well-known fact that pk is a strictly decreasing, convex function of k (cf. Burdett and Judd, 1983, p. 961). We write p = ( p 1 , . . . , p N ) for the list of expected prices paid by the various types. Note that if N > K , we have expected prices pk even for some types k > K , although there are actually no such consumers. Such a pk is simply what a (hypothetical) consumer would expect to pay if she sampled k prices from the given distribution F .
1
pk =
p . F k ( p ) dp =
pk[1 − F ( p )]k−1 F ( p ) dp =
pk[1 − F ( p )]k−1 dF ( p ) = [1 − F ( p )]k dp, where the last equality follows by partial integration.
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3. The informational divide The boundary between households who have access to the internet and those who do not is sometimes referred to as the “digital divide”, and there is evidence that households above the divide pay lower prices on average than those below (see e.g. Baye et al., 2003). This is trivially true in our model, too, wherever we put the “divide”, simply because the expected prices pk decrease with k. In what follows, we will demonstrate the existence of a much less obvious, but perhaps even more deplorable kind of divide which we call the “informational divide”: if some of those consumers who already search begin to search even more (in a certain well-defined sense, see Definition 1 below), then the equilibrium price distribution changes in such a way that all types below a certain threshold (our “informational divide”) face higher expected prices than before (in particular, the uninformed households always suffer), while those above the divide face lower prices than before (Theorem 3.2). That is, (a certain way of) increasing the information in the market favors only the high types, and harms the low types. Nobody here searches less than before, but the increased activity of some produces a negative (pecuniary) externality for the others (the low types who do not change their behavior). This is made precise in the following. Consider two different search distributions q = (q1 , . . . , q N ) and q˜ = (˜q1 , . . . , q˜ N ), and denote the associated equilibrium quantities by F , F˜ , pk , p˜ k , etc. Definition 1. q˜ has fewer low types and more high types than q if there exists a threshold type , 1 < N, such that
q˜ k qk
for k < ,
q˜ < q ,
q˜ k qk
for k > .
This definition is the discrete analogue of the rotation ordering of Johnson and Myatt (2006) for continuous functions.2 We say simply that households search more under q˜ than under q, or that q˜ has more search than q. We want to see what happens to the expected prices pk if households search more in the sense of Definition 1, and consider the following two cases in turn. First, we consider the case when uninformed consumers start to compare more prices, and second, we consider the case when already informed consumers compare more prices. Uninformed Consumers begin to search. Consider first the case that only the uninformed households (type 1) search more. This means that q˜ 1 < q1 and q˜ k qk for all k 2. Here we find, in line with the previous literature: Theorem 3.1. Assume q˜ 1 < q1 and q˜ k qk for all k 2. Then p˜ k < pk for all k. That is, if some uninformed households begin to search, the expected prices for all types go down. The average selling price, pav = q1 also goes down. Intuitively, there is more search and the market becomes more competitive. Informed Consumers search more. Consider next the case that only the informed households (types k 2) search more. This means that q˜ 1 = q1 and q˜ has more search than q. Here the situation is not so transparent: there is also “more search”, but the average selling price pav = q1 = q˜ 1 remains the same. This suggests (but does not prove) that not all prices pk can go down. Our main result is the following. Theorem 3.2. Assume that households search more under q˜ than under q, and q˜ 1 = q1 . Then there exists a number d, d > 1 such that p˜ k > pk for k < d and p˜ k < pk for k > d. Sketch of proof. The proof depends on a mathematical result about the order statistics of distributions with the singlecrossing property (cf. Chateauneuf et al., 2002).3 First we show that expected prices (the order statistics) can be written as integrals of the quantile function with respect to a certain log-supermodular density (Lemma 4.2). Then we show that the two distributions F and F˜ have the single-crossing property (Fig. 1). This follows from Eq. (2) and Lemma 4.3. Finally we show that this property implies the existence of a divide d. This follows from Lemma 4.4 and Lemma 4.5 (essentially, the single-crossing property is preserved under integration with respect to a log-supermodular density, cf. Athey, 2002). Details are in Appendix A. 2 We call d the informational divide. All types below the informational divide pay higher expected prices, and all types above it pay lower expected prices. In particular, the uninformed households (k = 1) always suffer because d > 1. Loosely speaking, the distribution F˜ has less weight “in the middle” (near the crossing point), and is in some sense more “spread out” than F . It has a higher mean and a lower minimum price. The higher mean hurts non-searchers, the lower minimum price helps consumers who search a lot.
2
We thank the advisory editor for making us aware of this relationship. Loosely speaking, if F˜ is more “spread out” than F in the sense of single-crossing, the expectations p˜ k of its order statistics are also more spread out in the sense of the “divide”. One can check that, in general, the two equilibrium distributions are not comparable in terms of other usual dispersion measures like variance, second-order stochastic dominance, mean-preserving spread, monotone single-crossing, right or left monotone single-crossing. 3
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Fig. 1. The solid line is the graph of the function x = F ( p ) which is the same as the graph of p = F −1 (x). The dashed line is the same for F˜ . A resp. B is the area to the left resp. right of the solid curve. A 1 resp. B 1 is the area between the two lines. The thick bars below [resp. above] the horizontal axis indicate the bottom and top decile of prices under the distribution F [resp. F˜ ]. Clearly the latter are more extreme. The picture is drawn for q = (20, 20, 10, 0), q˜ = (20, 0, 10, 20).
A more precise intuition is as follows. For a given search structure q, a market equilibrium is given by a mixed pricing strategy F ( p ): each firm chooses its price at random from the distribution F , and all prices in the support of F give the same expected profit. This profit is ‘price times market share’. The market share of a firm charging price p is an increasing function of 1 − F ( p ) (the number of more expensive firms). If we now introduce “more search” (by already informed consumers), this market share function changes: it becomes smaller for expensive firms (small values of 1 − F ( p )) and larger for cheap firms (large values of 1 − F ( p )), so that no longer all prices give the same profit. To restore equilibrium, we must increase 1 − F ( p ) where it is small (i.e. for p above a certain p¯ ) and decrease 1 − F ( p ) where it is large (for p < p¯ ). Therefore the new equilibrium distribution function F˜ , corresponding to “more search”, crosses the old F in one point p¯ < 1, such that F˜ ( p ) lies above [resp. below] F ( p ) to left [resp. right] of p¯ (see Fig. 1). Moreover, F˜ has a higher mean than F , because now more consumers buy from (relatively) cheap stores, but the total (hence also average) profit remains the same (it is anchored by the profit μq1 a firm can make by selling only to the uninformed, charging p max = 1). Finally, since F˜ has a higher mean than F , consumers who do not search expect to pay a higher price under F˜ ; the same may also be true (depending on parameters) for consumers who search only little so that they have a high probability of buying from a firm in the upper deciles. By contrast, consumers who search a lot have a high chance of finding a firm in a lower decile, and will expect to pay lower prices under F˜ than under F . It may be that d > K , i.e. there are actually no consumers above d. In this case, all types k of consumers with qk > 0 will expect to pay a higher price, see Example 3.3. It is important to realize that firms’ profits and hence also total consumer welfare depend only on the fraction of uninformed consumers. More search by already informed consumers does not affect total consumer welfare. However, it amounts to a redistribution of consumer welfare from relatively uninformed consumers to relatively informed consumers. In this sense, more search by informed consumers imposes a negative externality on relatively uninformed consumers and may impose a positive externality on highly informed consumers. We will now present an example where some consumers who previously compared two prices start to compare three prices. Example 3.3. Consider the case where consumers only sample at most three firms, i.e. K = 3 and consider the consumer search distribution q = (5, 5, 10). Using (1) we can numerically compute the expected prices paid by the different types of consumers, obtaining p 1 0.3761, p 2 0.2408, and p 3 0.1915. Suppose now that one consumer who previously sampled two firms samples now three firms, obtaining a new consumer search distribution q˜ = (5, 4, 11). Computing expected prices under this new consumer search distribution we now obtain that p˜ 1 0.3832 > p˜ 1 , p˜ 2 0.2434 > p 2 , and
M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
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p˜ 3 0.1919 > p 3 . In addition, we find that a consumer of type 4 would actually pay a lower price (i.e. p˜ 4 < p 4 ), but there are no such consumers represented in the population. At a first glance, the previous example seems to be in contradiction with the observation that as the number of uninformed consumers is the same under both consumer search distributions also the expected selling price has to be constant. This contradiction is however easily resolved if one takes into account that the consumers who switched from sampling two firms to three firms now pay a lower price, i.e. p˜ 3 < p 2 .4 It is clear from the above that if some shoppers search more they can exert a negative externality on other households. Perhaps surprisingly, it is even possible that all consumers who search more hurt themselves: Example 3.4. Starting from the initial search distribution q = (1, 20 000, 0, 0, 0, 10 000), suppose that all type 2 households switch to become type 3 (begin to compare three prices instead of two), resulting in the new search distribution q˜ = (1, 0, 20 000, 0, 0, 10 000). Clearly q˜ has more search than q and q˜ 1 = q1 , i.e. the average selling price pav = p˜ av remains the same, and one can check that p˜ k > pk for k 5 and p˜ 6 < p 6 (the informational divide is d = 5). But even more is true: the expected price p˜ 3 paid by the new type 3 households under distribution q˜ is not only higher than p 3 , but even higher than p 2 , the expected price paid by the same households before the change. Thus the increased search by these former type 2 households benefits only the highest type 6 in the population, and actually makes those who search more worse off than before. 4. Concluding remarks We conclude by discussing an alternative interpretation of our model in the context of networks, providing a short discussion on endogenous search, and by drawing some policy implications. As regards the first point, our model specification can also be interpreted as a model of price competition on a network in the following sense5 : Firms and consumers represent nodes in a bipartite network, i.e. a network where there are only links between firms and consumers. A link between a consumer and a firm in this network indicates that a consumer observes the price of this firm and may buy from this firm and that the firm may sell to this consumer. If we assume that firms only know the degree distribution of consumers, i.e. the probabilities that a given consumer has one link, two links, and so forth, we obtain a network game of incomplete information.6 Alternatively, we could also assume that the network is symmetric in the sense that every firm faces the same consumer degree distribution and analyze this game of complete information. Within this network interpretation, the consumer degree distribution essentially plays the role of the consumer search distribution in the consumer search model. In this sense, the equilibrium we find in the search context and the comparative static exercise of varying the consumer search distribution translate into our network context. In particular, the comparative static exercise of introducing more search can be interpreted as increasing the density of the network. We then have that adding links to consumers who are only linked to one firm decreases expected prices for all consumers, whereas adding links to consumers with already more than one link amounts to a redistribution of social welfare from consumers who just have a few links to well-connected consumers. In the present model, the search distribution (the network) is exogenous. A natural question is what happens if consumer types are endogenous, derived from some form of optimal search?7 Assume, for example, that the search costs of some consumers who already have low search costs (and compare prices) become even lower. Then these consumers will search more, and this will lead firms to adjust their pricing policy so that we observe an informational divide, by Theorem 3.2. Let’s call this the first-round effect. But with endogenous search there is a second-round effect: the new price distribution may be such that it now becomes worthwhile for other consumers (whose search cost has not changed) to search more (or less). This will in turn change the equilibrium price distribution again, and it seems very hard to say in general where this process will end. Clearly, a sufficient condition for our results to hold is that the second-round effects are negligible. It is easy to give examples where this is the case, so that a lowering of the search cost for some consumers who are already searching leads to a new equilibrium (after optimal adjustment of all search behavior) which exhibits an informational divide. Second-round effects will be negligible if consumer types are robust, in the sense that a consumer does not change her search behavior in response to very small price changes. This may happen with linear search costs that take only a few, discrete values; it may also happen with nonlinear search costs: for example, when a consumer goes to a certain shopping mall every weekend, she can compare the prices of all the shops there, at a very low marginal search cost, but she might not
4
Similar observations are known as Simpson’s paradox in statistics. This second interpretation of our model is related to the literature of competition on networks. See e.g. Kranton and Minehart (2001), Corominas-Bosch (2004), Blume et al. (2009), Piccione and Spiegler (2012), or Galeotti (2010). 6 This incomplete information setup stipulates the use of the Bayesian-Nash equilibrium concept for networks, as e.g. in Jackson and Yariv (2007) or Galeotti et al. (2010). 7 Such models have been studied extensively in the literature, see e.g. Burdett and Judd (1983), Janssen and Moraga-González (2004), Burdett and Smith (2010) and the survey of Baye et al. (2006). 5
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M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
drive to the other end of town to check the prices at another shopping mall.8 Such a structure of search costs implies that, at least for a certain price range, a consumer’s type is essentially independent of prices, i.e. exogenous. A similar argument can be made for consumers who do, or do not, have access to the internet, etc. These examples clarify that allowing for endogenous search would entail assumptions on how individual search distributions change with the number and location of firms. Lach and Moraga-González (2012) state that since all the existing theoretical studies in this field provide results under very specific assumptions, they prefer to follow a reduced form approach to the problem. Our conclusion is that assuming the search distribution to be exogenous is not worse than the existing viable alternatives. Finally, we note that in our model, search is not socially productive. Thus, if search is costly, the first best outcome is that consumers do not search and trade at the monopoly price. This need not be the case in a more general model: for example, when consumers or products are heterogeneous, search may be socially productive. In such a model, an increase in search intensity might not only lead to redistribution, but also increase the social surplus.9 Our final considerations start from the fact that it has become widely accepted in economics that consumer welfare can be increased by making the comparison of prices easier. If previously uninformed consumers start to compare prices this is definitely the case. However, if only already informed consumers compare more prices total consumer welfare stays unaffected. Moreover, it will result in a redistribution of consumer welfare from relatively uniformed to informed consumers. The internet – and with it the onset of price comparison sites – has been largely praised by economists and policy makers as a way to ease consumer search and thereby increase competition. Consequently, policies to improve internet accessibility, and to promote the use of price comparison sites, are also seen as tools to increase consumer welfare. Our findings suggest that it may be more important to make the internet accessible to a broader audience than to make existing connections faster. Appendix A First we introduce some more terminology and notation. The average number of searches (or links) per household is
κ = E q [k] = k kqk ; it is a measure for the intensity of search in the market (density of the network); the number of links per firm is μκ . For q = (q1 , . . . , q N ), q1 > 0, define the auxiliary function ϕ (x) := kqk xk−1 for x ∈ [0, 1] k
and if q1 > 0
ψ(x) :=
1 q1
kqk xk−1
for x ∈ [0, 1].
k
We have ψ(0) = 1, ψ(1) = κ /q1 , and ψ is continuously differentiable and strictly increasing in x. Clearly
ϕ (x) = q1 ψ(x).
Proposition 4.1. Consider a market game with information structure q = (q1 , . . . , q N ) and assume 0 < q1 < 1. Then there exists a unique symmetric equilibrium in mixed strategies: each firm chooses its price at random according to a continuous distribution F ( p ) with support [ p min , p max ], where
0 < p min =
1
ψ(1)
=
q1
κ
< 1,
p max =
1
ψ(0)
=1
and the distribution is given implicitly by the equilibrium condition
p ψ 1 − F ( p ) ≡ 1,
p min p p max
(and of course F ( p ) = 0 for p p min and F ( p ) = 1 for p p max ). Moreover, the equilibrium profit per firm is average selling price (average household expenditure) is pav = q1 .
π = μq1 , and the
The equilibrium condition gives us an explicit formula for the inverse p = F −1 (x) (also known as the quantile function) of the distribution function x = F ( p ) (restricted to its support, of course), viz.
p = F −1 (x) =
1
ψ(1 − x)
.
(2)
Proposition 4.1 is known at least since Burdett and Judd (1983) (see especially the proof of Lemma 2 there); for the reader’s convenience, we give a brief sketch of the proof. Sketch of proof of Proposition 4.1. First one establishes that the equilibrium price distribution F must be continuous and strictly increasing on a support of the form [ p min , p max ] with 0 < p min < p max = 1 (otherwise, one can easily find profitable 8 9
Such a strongly convex cost structure is perhaps no less realistic than the assumption of linear costs frequently found in the literature. This was pointed out by a referee.
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27
deviations). Given this, we observe that every price in the support must give the same (expected) payoff to a firm. If the firm charges p max = 1 it gets only its share of uninformed consumers, so π = p max .μq1 = μq1 is the equilibrium profit. If it charges p min it gets all households to which it is linked, so π = p min .μκ , hence p min = q1 /κ . Total profits are N π and must be equal to total household expenditure Mpav , so pav = ( N / M )μq1 = q1 . A firm charging any price p ∈ [ p min , 1] is observed by μqk k households of type k and makes a sale to such a household iff the other (k − 1) prices observed by the household are higher than p, which occurs with probability [1 − F ( p )]k−1 . Thus the firm’s profit is
p.
N
k−1
μqk k 1 − F ( p )
= p .μq1 .ψ 1 − F ( p ) = π = μq1 .
k =1
This implies
p .ψ 1 − F ( p ) ≡ 1 or F ( p ) = 1 − ψ −1 (1/ p ).
(3)
2
The following lemma gives an explicit formula for the expected prices pk . It enables us to compute the expectations pk of the many distributions F k , k = 1, 2, . . . in terms of the single function ψ . Lemma 4.2. Let F be a price distribution with associated prices pk . Then, for k = 1, 2, . . . :
1 pk =
F −1 (1 − x) dxk =
0
1
1
1
ψ(x)
dxk =
0
kxk−1
ψ(x)
dx.
(4)
0
Proof of Lemma 4.2. 10 The expected price pk is the mean of the distribution F k ( p ) = 1 − [1 − F ( p )]k . Since F 1 = F , the expected price p 1 is given by (integrating by parts)
1
1 p dF ( p ) = p F ( p )
p1 =
p min
p min
1 −
F ( p ) dp .
(5)
p min
In Fig. 1, we have A + B = 1, where A is the area in the unit square to the right (under) the curve x = F ( p ) and B is the area to the left of this curve. In Eq. (5) the first term on the right is equal to 1 and the second is the area A. Therefore p 1 is equal to the area B:
1 p1 = B =
F −1 (x) dx,
0
where we compute the area B by integrating along the x-axis. Clearly, this can also be written as
1 p1 =
F −1 (1 − x) dx,
0
which proves the assertion for k = 1. By the same logic, the expected price pk is given by the area to the left of the curve x = F k ( p ):
1 pk =
F k−1 (1 − x) dx.
0
−1
we re-write x = F k ( p ) = 1 − [1 − F ( p )]k as F ( p ) = 1 − (1 − x)1/k or p = F −1 [1 − (1 − x)1/k ] = F k−1 (x). This
To find F k implies
1 pk =
F −1 1 − x1/k dx.
0
10 It may well be that this formula is known. But as we do not know a reference, we give a proof. It is also easy to see that it generalizes to any support of the form [a, b].
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M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
The substitution g (t ) = t k transforms this to
1 pk =
F
−1
1
(1 − t ) g (t ) dt =
0
F −1 (1 − t )kt k−1 dt .
0
The result follows by formula 2.
2
Using this lemma, we can check that the average of the pk ’s is indeed equal to the average selling price pav = q1 :
q·p=
1
k qk ku
qk pk =
k
k −1
ψ(u )
du = q1 = pav .
0
Proof of Theorem 3.1. Obviously the assumption implies that q˜ k /˜q1 qk /q1 for all k, with strict inequality for some k, so ˜ x) > ψ(x) for all x ∈ (0, 1). This in turn implies p˜ k < pk for all k by Eq. (4). 2 that ψ( Lemma 4.3. (“Single Crossing”) Assume households search more under q˜ than under q, and q˜ 1 = q1 . Then there is a number b ∈ (0, 1) ˜ x) < ψ(x) for x ∈ (0, b) and ψ( ˜ x) > ψ(x) for x ∈ (b, 1]. such that ψ(
˜ b) = ψ(b), i.e. ψ˜ and ψ cross only once in (0, 1], at the point b. Of course, by continuity, ψ( Proof of Lemma 4.3. Since q˜ 1 = q1 > 0, it suffices to prove the assertion for ϕ = q1 ψ . By assumption, qk − q˜ k 0 for k < , q − q˜ > 0, and qk − q˜ k 0 for k > . We know that ϕ (0) − ϕ˜ (0) = q1 − q˜ 1 = 0 and ϕ (1) − ϕ˜ (1) = κ − κ˜ < 0. For 0 < x 1 we have
ϕ (x) − ϕ˜ (x) =
k(qk − q˜ k )xk−1 +
k =1
N
k(qk − q˜ k )xk−1 = x−1 A (x) − B (x) ,
k=+1
where
A (x) =
−1
k(qk − q˜ k )
k =1
1 xl−k
+ (q − q˜ )
is nonincreasing in x and (qk − q˜ k ) > 0 ∀x, and N
B (x) =
k(˜qk − qk )xk−l
k=+1
is strictly increasing in x (because at least one coefficient k(˜qk − qk ) must be positive), and tends to zero for x → 0. Therefore the function f (x) := A (x) − B (x) is strictly decreasing on (0, 1], positive for x near zero, and negative for x = 1 (because ϕ (1) − ϕ˜ (1) < 0). This implies the assertion. 2 The following lemma is the mathematical core of the proof. Lemma 4.4. Let g (x) be a nonnegative continuous function on the interval [0, 1] which is strictly positive except in at most finitely many points, and choose b with 0 < b < 1. For i = 1, 2, . . . define
b
b i
A i :=
g (x) dx = 0
g (x)ix 0
Then
Bi Ai
<
B i +1 A i +1
for i = 1, 2, . . .
(obviously A i , B i are always positive).
i −1
1 dx,
1 i
B i :=
g (x) dx = b
b
g (x)ixi −1 dx.
M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
29
Intuitively, the distribution H i (x) = xi on the interval [0, 1] has more weight on the right if i increases, hence B i should increase relative to A i (one might also think that B i − A i increases in i, but one can show by examples that this is not true in general).11 Proof of Lemma 4.4. Fix b ∈ (0, 1). We note that for positive x the function xα is defined for all
b Aα =
1
g (x)α xα −1 dx,
Bα =
0
Bα
g (x)α xα −1 dx,
b
where α is now a continuous variable. For we obtain
Aα
α ∈ R and write
b
= 01 b
g (x)( bx )α −1 dx g (x)( bx )α −1 dx
α 1 these integrals are certainly finite and positive. Dividing A α , B α by αbα −1
.
If α increases, the numerator decreases because bx < 1 for 0 < x < b; and the denominator increases because b < x < 1. Therefore A α / B α is strictly decreasing in α . 2
x b
> 1 for
In Lemma 4.5 below we prove a result about the order statistics of single-crossing distributions which is valid not only for distributions F of the form given in Proposition 4.1, but more generally. For purposes of exposition only, let us now use the term “price distribution” for any continuous distribution F which has a support of the form [ p min , 1] with 0 < p min < 1, and is strictly increasing and differentiable on its support (the equilibrium distributions of Proposition 4.1 have these properties). We denote by F −1 (x) the inverse of the function x = F ( p ) (restricted to its support, of course). Clearly, this inverse exists and is strictly increasing and differentiable on [0, 1]. Finally, we denote by pk the expected value of the minimum of a sample of size k from the distribution F (the kth-order statistic), and call the numbers pk (k = 1, 2, . . .) the associated (expected) prices for F . Lemma 4.5. (The “Divide”) Consider two price distributions F and F˜ , with p˜ min < p min and assume that there exists a p¯ ∈ ( p min , 1) such that F˜ ( p ) > F ( p ) for p ∈ ( p˜ min , p¯ ), and F˜ ( p ) < F ( p ) for p ∈ (, p¯ , 1). Then there exists a “divide” d 1 such that p˜ k > pk for k < d and p˜ k < pk for k > d. Proof of Lemma 4.5. By continuity, F˜ ( p¯ ) = F ( p¯ ) and F˜ (1) = F (1) = 1. Write b := 1 − F ( p¯ ) = 1 − F˜ ( p¯ ) and define the function h(x) := F −1 (1 − x) − F˜ −1 (1 − x). By Lemma 4.2
1 pk − p˜ k =
b k
h(x) dx = 0
1 k
h(x) dxk .
h(x) dx + 0
b
The single-crossing assumption for F and F˜ implies a similar property for the inverse functions, viz. h(x) > 0 for 1 > x > b and h(x) < 0 for 0 < x < b. Therefore,
pk − p˜ k = B k − A k , b 1 where A k = 0 |h(x)| dxk , B k = b |h(x)| dxk . The numbers A k , B k correspond to the size of the areas A 1 , B 1 in Fig. 1, computed w.r.t. the measure dxk (after replacing x with (1 − x)) on the vertical axis. The function g (x) := |h(x)| satisfies the assumptions of Lemma 4.4, hence B k / A k increases strictly with k. Thus, if pm − p˜ m = B m − A m > 0 for some m, so that B m / A m > 1, we have also B k / A k > 1, hence pk − p˜ k = B k − A k > 0 for all k m. In other words, if the expected price decreases for some type m, then also for all higher types. We know that p˜ min < p min , therefore p˜ N < p N for N sufficiently large (because a household who searches long enough must find a price arbitrarily close to the minimum). The divide d is the first index such that p˜ d pd . 2 Proof of Theorem 3.2. Let households search more under q˜ than under q and q˜ 1 = q1 Denote by F , F˜ the corresponding equilibrium price distributions with associated expected prices pk , p˜ k . By Eq. (2),
h(x) := F −1 (1 − x) − F˜ −1 (1 − x) =
1
ψ(x)
−
1
˜ x) ψ(
.
11 Lemma 4.4 can be seen as a ‘strict’ version of Lemma 5 in Athey (2002), using special properties of the log-supermodular density because direct appeal to Athey’s lemma would not give us the strict monotonicity needed for the proof of Theorem 3.2.
α xα −1 . We state it
30
M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
By Lemma 4.3, there is a point b, 0 < b < 1, such that h(x) < 0 for 0 < x < b and h(x) > 0 for b < x < 1. These are exactly the properties used in the proof of Lemma 4.5, so that there is a divide d 1. It remains to show that d > 1, i.e. not all prices go down. Using vector notation, write q · p = k qk pk . Then the expected average prices satisfy
q˜ 1 = p˜ av = p˜ · q˜ = q1 = pav = p · q.
(6)
Moreover, the components of p = ( p 1 , . . . , p N ) are strictly decreasing and q˜ FOSD q. Therefore q˜ · p < q · p. If all prices p˜ k were less than pk , we should have p˜ p, hence q˜ · p˜ q˜ · p < q · p, contradicting (6). 2 References Athey, S., 2002. Monotone comparative statics under uncertainty. Quart. J. Econ. 67 (1), 187–223. Baye, M.R., Morgan, J., Scholten, P., 2003. The value of information in an online consumer electronics market. J. Public Policy Mark. 22 (1), 17–25. Baye, M.R., Morgan, J., Scholten, P., 2006. Information, search, and price dispersion. In: Hendershott, T. (Ed.), Handbook on Economics and Information Systems. Elsevier, pp. 323–376. Blume, L.E., Easley, D., Kleinberg, J., Tardos, V., 2009. Trading networks with price-setting agents. Games Econ. Behav. 67 (1), 36–50. Burdett, K., Judd, K.L., 1983. Equilibrium price dispersion. Econometrica 51 (4), 955–969. Burdett, K., Smith, E., 2010. Price distributions and competition. Econ. Letters 106, 180–183. Chateauneuf, A., Cohen, M., Meijlison, I., 2002. Comonotonicity-based stochastic orders generated by single crossings of distributions, with applications to attitudes to risk in the Rank-dependent Expected Utility model. Universitè de Paris I, CERMSEM. Corominas-Bosch, M., 2004. Bargaining in a network of buyers and sellers. J. Econ. Theory 115 (1), 35–77. Galeotti, A., 2010. Talking, searching and pricing. Int. Econ. Rev. 51 (4), 1155–1174. Galeotti, A., Goyal, S., Jackson, M.O., Vega-Redondo, F., Yariv, L., 2010. Network games. Rev. Econ. Stud. 77 (1), 218–244. Jackson, M.O., Yariv, L., 2007. Diffusion of behavior and equilibrium properties in network games. Amer. Econ. Rev. (Pap. Proc.) 97 (2), 92–98. Janssen, M.C.W., Moraga-González, J.L., 2004. Strategic pricing, consumer search and the number of firms. Rev. Econ. Stud. 71 (4), 1089–1118. Johnson, J.P., Myatt, D.P., 2006. On the simple economics of advertising, marketing, and product design. Amer. Econ. Rev. 96 (3), 756–784. Kranton, R.E., Minehart, D.F., 2001. A theory of buyer–seller networks. Amer. Econ. Rev. 91 (3), 485–508. Lach, S., Moraga-González, J.L., 2012. Heterogeneous price information and the effect of competition. Mimeo. Piccione, M., Spiegler, R., 2012. Price competition under limited comparability. Quart. J. Econ. 127, 97–135.
Contents lists available at SciVerse ScienceDirect
Games and Economic Behavior www.elsevier.com/locate/geb
The informational divide ✩ Manfred Nermuth a , Giacomo Pasini b,c , Paolo Pin d,∗ , Simon Weidenholzer e a
Institut für Volkswirtschaftslehre, Universität Wien, Austria Dipartimento di Economia, Università Ca’ Foscari di Venezia, Italy Netspar, Network for Studies on Pensions, Savings and Retirement, Tilburg, The Netherlands d Dipartimento di Economia Politica e Statistica, Università degli Studi di Siena, Italy e Department of Economics, University of Essex, United Kingdom b c
a r t i c l e
i n f o
Article history: Received 28 June 2010 Available online 9 November 2012 JEL classification: D43 D85 L11 Keywords: Price dispersion Welfare effects of search Price competition on networks Informational divide
a b s t r a c t We propose a model of price competition where consumers exogenously differ in the number of prices they compare. Our model can be interpreted either as a non-sequential search model or as a network model of price competition. We show that (i) if consumers who previously just sampled one firm start to compare more prices all types of consumers will expect to pay a lower price and (ii) if consumers who already sampled more than one price sample (even) more prices then there exists a threshold – the informational divide – such that all consumers comparing fewer prices than this threshold will expect to pay a higher price whereas all consumers comparing more prices will expect to pay a lower price than before. Thus increased search can create a negative externality and it is not necessarily beneficial for all consumers. © 2012 Elsevier Inc. All rights reserved.
1. Introduction Is more information about prices always good for consumers? In this paper we analyze this question in a consumer search model à la Burdett and Judd (1983) where consumers exogenously differ in the number of prices they compare. In order to assess the welfare effects of increased information we consider the expected prices paid by the various consumers. It turns out that whether more information is indeed beneficial to all consumers will depend very much on which consumers get more information. In line with the previous literature, we find that more information for previously uninformed consumers leads to lower expected prices for all types of consumers (Theorem 3.1) and increases consumer welfare unambiguously. But, surprisingly, increased search by those who already do some search actually harms the uninformed, while benefiting the well-informed, and merely re-distributes welfare from the former to the latter. More precisely, if some searchers search more, the endogenous equilibrium price distribution changes in such a peculiar way that – for a certain number d > 1 (the “informational divide”) – all consumers who compare fewer than d prices face higher expected prices than before, whereas the others face
✩ We are grateful to two referees and to the advisory editor for their useful suggestions. The authors wish to thank Larry Blume, Giacomo Calzolari, Andrea Galeotti, Maarten Janssen, Saul Lach, Marco van der Leij, José Luis Moraga-González, Marco Ottaviani, Mario Padula, and Fernando Vega-Redondo for useful comments on earlier drafts of this paper. Previous versions of this paper were circulated under the titles “Price Dispersion, Search Externalities, and the Digital Divide” and “A Network Model of Price Dispersion.” Corresponding author. E-mail addresses: [email protected] (M. Nermuth), [email protected] (G. Pasini), [email protected] (P. Pin), [email protected] (S. Weidenholzer).
*
0899-8256/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.geb.2012.10.016
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M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
lower expected prices (Theorem 3.2). In other words, more information for the informed creates a negative externality for the uninformed. To gain an intuitive understanding of this “informational divide” we observe the following. Expected profits and hence the average selling price depend only on the share of uninformed consumers in the market. This is so because expected profits are equal to the profit a firm can make by selling only to uninformed consumers (at the monopoly price). Thus, if some searchers start to compare more prices, the average selling price remains the same and the resulting change in the equilibrium price distribution can only have a redistributive effect, benefiting some consumers and harming others. We show that the new price distribution “single-crosses” the old one, in such a way that the mean (= the expected price paid by the uninformed) increases and the lower bound decreases. This harms the uninformed and helps those who search a lot. More intuition is given after Theorem 3.2. In addition to the results just described, we show by means of examples that “increased search” can have truly counterintuitive effects: it can lead to higher expected prices for all consumer types that are actually present in the market (here a “type k” consumer is one who samples k prices), and it can even happen that those consumers who engage in more search harm themselves (see Example 3.3 and Example 3.4). The rest of the paper is organized as follows: Section 2 spells out the model. Section 3 presents the results. In Section 4 we discuss some related issues, viz. (i) a network interpretation of our model, (ii) endogenous search, and (iii) policy implications. Proofs are relegated to Appendix A. 2. The model We consider a market for a homogeneous good with N firms and M households (consumers), and write μ = M / N for the number of households per firm. Each firm can produce the good at constant marginal cost, without fixed cost, and sets the price at which it offers the good (all firms set their prices simultaneously). Each household demands one unit of the good, up to a given willingness to pay (assumed greater than the cost). Without loss of generality we normalize the cost to 0 and the willingness to pay to 1 (the same for all firms, respectively households). Households differ in the information they have about the firms’ prices. A household of type k observes the prices of k firms and buys from the cheapest (randomizing with equal probabilities in case of ties), provided the price does not exceed its willingness to pay. We denote by qk the fraction of households of type k. The information structure N (or consumer search distribution) is represented by the vector q = (q1 , . . . , q N ), where of course qk 0 for all k and k=1 qk = 1. Equivalently, we can let qk stand for the number of type k households. We usually do this in the examples. A household of type k = 1 is called uninformed, households of types k 2 are called informed (also searchers or shoppers). We thus obtain a strategic market game among the N firms, where we can take without loss of generality the strategy set of each firm to be the unit interval [0, 1] (a price below 0 would generate losses, and at a price above 1 nobody would buy). Trivial cases apart, this game has equilibria only in mixed strategies, generating price dispersion. From now on, we assume always 0 < q1 < 1. The following known result is stated here for easy reference. A more precise formulation is in Appendix A. Proposition 2.1. Consider a market game with information structure q = (q1 , . . . , q N ) and assume 0 < q1 < 1. Then there exists a unique symmetric equilibrium in mixed strategies: each firm chooses its price at random according to a continuous distribution F ( p ) with support [ p min , p max ], where 0 < p min < p max = 1. Moreover, the equilibrium profit per firm is π = μq1 , and the average selling price (average household expenditure) is pav = q1 . The expected price pk paid by a household of type k is given by
1 pk =
(k = 1, . . . , N ),
p dF k ( p ) p min
where F k ( p ) = 1 − [1 − F ( p )]k is the distribution of the minimum of a sample of size k from the distribution F . The distribution F k shifts more and more mass near p min , as k increases, so that pk → p min for k → ∞. It is easy to see1 that the expected price pk can also be written as
1 pk =
k
1 − F ( p)
dp .
(1)
p min
This implies the well-known fact that pk is a strictly decreasing, convex function of k (cf. Burdett and Judd, 1983, p. 961). We write p = ( p 1 , . . . , p N ) for the list of expected prices paid by the various types. Note that if N > K , we have expected prices pk even for some types k > K , although there are actually no such consumers. Such a pk is simply what a (hypothetical) consumer would expect to pay if she sampled k prices from the given distribution F .
1
pk =
p . F k ( p ) dp =
pk[1 − F ( p )]k−1 F ( p ) dp =
pk[1 − F ( p )]k−1 dF ( p ) = [1 − F ( p )]k dp, where the last equality follows by partial integration.
M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
23
3. The informational divide The boundary between households who have access to the internet and those who do not is sometimes referred to as the “digital divide”, and there is evidence that households above the divide pay lower prices on average than those below (see e.g. Baye et al., 2003). This is trivially true in our model, too, wherever we put the “divide”, simply because the expected prices pk decrease with k. In what follows, we will demonstrate the existence of a much less obvious, but perhaps even more deplorable kind of divide which we call the “informational divide”: if some of those consumers who already search begin to search even more (in a certain well-defined sense, see Definition 1 below), then the equilibrium price distribution changes in such a way that all types below a certain threshold (our “informational divide”) face higher expected prices than before (in particular, the uninformed households always suffer), while those above the divide face lower prices than before (Theorem 3.2). That is, (a certain way of) increasing the information in the market favors only the high types, and harms the low types. Nobody here searches less than before, but the increased activity of some produces a negative (pecuniary) externality for the others (the low types who do not change their behavior). This is made precise in the following. Consider two different search distributions q = (q1 , . . . , q N ) and q˜ = (˜q1 , . . . , q˜ N ), and denote the associated equilibrium quantities by F , F˜ , pk , p˜ k , etc. Definition 1. q˜ has fewer low types and more high types than q if there exists a threshold type , 1 < N, such that
q˜ k qk
for k < ,
q˜ < q ,
q˜ k qk
for k > .
This definition is the discrete analogue of the rotation ordering of Johnson and Myatt (2006) for continuous functions.2 We say simply that households search more under q˜ than under q, or that q˜ has more search than q. We want to see what happens to the expected prices pk if households search more in the sense of Definition 1, and consider the following two cases in turn. First, we consider the case when uninformed consumers start to compare more prices, and second, we consider the case when already informed consumers compare more prices. Uninformed Consumers begin to search. Consider first the case that only the uninformed households (type 1) search more. This means that q˜ 1 < q1 and q˜ k qk for all k 2. Here we find, in line with the previous literature: Theorem 3.1. Assume q˜ 1 < q1 and q˜ k qk for all k 2. Then p˜ k < pk for all k. That is, if some uninformed households begin to search, the expected prices for all types go down. The average selling price, pav = q1 also goes down. Intuitively, there is more search and the market becomes more competitive. Informed Consumers search more. Consider next the case that only the informed households (types k 2) search more. This means that q˜ 1 = q1 and q˜ has more search than q. Here the situation is not so transparent: there is also “more search”, but the average selling price pav = q1 = q˜ 1 remains the same. This suggests (but does not prove) that not all prices pk can go down. Our main result is the following. Theorem 3.2. Assume that households search more under q˜ than under q, and q˜ 1 = q1 . Then there exists a number d, d > 1 such that p˜ k > pk for k < d and p˜ k < pk for k > d. Sketch of proof. The proof depends on a mathematical result about the order statistics of distributions with the singlecrossing property (cf. Chateauneuf et al., 2002).3 First we show that expected prices (the order statistics) can be written as integrals of the quantile function with respect to a certain log-supermodular density (Lemma 4.2). Then we show that the two distributions F and F˜ have the single-crossing property (Fig. 1). This follows from Eq. (2) and Lemma 4.3. Finally we show that this property implies the existence of a divide d. This follows from Lemma 4.4 and Lemma 4.5 (essentially, the single-crossing property is preserved under integration with respect to a log-supermodular density, cf. Athey, 2002). Details are in Appendix A. 2 We call d the informational divide. All types below the informational divide pay higher expected prices, and all types above it pay lower expected prices. In particular, the uninformed households (k = 1) always suffer because d > 1. Loosely speaking, the distribution F˜ has less weight “in the middle” (near the crossing point), and is in some sense more “spread out” than F . It has a higher mean and a lower minimum price. The higher mean hurts non-searchers, the lower minimum price helps consumers who search a lot.
2
We thank the advisory editor for making us aware of this relationship. Loosely speaking, if F˜ is more “spread out” than F in the sense of single-crossing, the expectations p˜ k of its order statistics are also more spread out in the sense of the “divide”. One can check that, in general, the two equilibrium distributions are not comparable in terms of other usual dispersion measures like variance, second-order stochastic dominance, mean-preserving spread, monotone single-crossing, right or left monotone single-crossing. 3
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M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
Fig. 1. The solid line is the graph of the function x = F ( p ) which is the same as the graph of p = F −1 (x). The dashed line is the same for F˜ . A resp. B is the area to the left resp. right of the solid curve. A 1 resp. B 1 is the area between the two lines. The thick bars below [resp. above] the horizontal axis indicate the bottom and top decile of prices under the distribution F [resp. F˜ ]. Clearly the latter are more extreme. The picture is drawn for q = (20, 20, 10, 0), q˜ = (20, 0, 10, 20).
A more precise intuition is as follows. For a given search structure q, a market equilibrium is given by a mixed pricing strategy F ( p ): each firm chooses its price at random from the distribution F , and all prices in the support of F give the same expected profit. This profit is ‘price times market share’. The market share of a firm charging price p is an increasing function of 1 − F ( p ) (the number of more expensive firms). If we now introduce “more search” (by already informed consumers), this market share function changes: it becomes smaller for expensive firms (small values of 1 − F ( p )) and larger for cheap firms (large values of 1 − F ( p )), so that no longer all prices give the same profit. To restore equilibrium, we must increase 1 − F ( p ) where it is small (i.e. for p above a certain p¯ ) and decrease 1 − F ( p ) where it is large (for p < p¯ ). Therefore the new equilibrium distribution function F˜ , corresponding to “more search”, crosses the old F in one point p¯ < 1, such that F˜ ( p ) lies above [resp. below] F ( p ) to left [resp. right] of p¯ (see Fig. 1). Moreover, F˜ has a higher mean than F , because now more consumers buy from (relatively) cheap stores, but the total (hence also average) profit remains the same (it is anchored by the profit μq1 a firm can make by selling only to the uninformed, charging p max = 1). Finally, since F˜ has a higher mean than F , consumers who do not search expect to pay a higher price under F˜ ; the same may also be true (depending on parameters) for consumers who search only little so that they have a high probability of buying from a firm in the upper deciles. By contrast, consumers who search a lot have a high chance of finding a firm in a lower decile, and will expect to pay lower prices under F˜ than under F . It may be that d > K , i.e. there are actually no consumers above d. In this case, all types k of consumers with qk > 0 will expect to pay a higher price, see Example 3.3. It is important to realize that firms’ profits and hence also total consumer welfare depend only on the fraction of uninformed consumers. More search by already informed consumers does not affect total consumer welfare. However, it amounts to a redistribution of consumer welfare from relatively uninformed consumers to relatively informed consumers. In this sense, more search by informed consumers imposes a negative externality on relatively uninformed consumers and may impose a positive externality on highly informed consumers. We will now present an example where some consumers who previously compared two prices start to compare three prices. Example 3.3. Consider the case where consumers only sample at most three firms, i.e. K = 3 and consider the consumer search distribution q = (5, 5, 10). Using (1) we can numerically compute the expected prices paid by the different types of consumers, obtaining p 1 0.3761, p 2 0.2408, and p 3 0.1915. Suppose now that one consumer who previously sampled two firms samples now three firms, obtaining a new consumer search distribution q˜ = (5, 4, 11). Computing expected prices under this new consumer search distribution we now obtain that p˜ 1 0.3832 > p˜ 1 , p˜ 2 0.2434 > p 2 , and
M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
25
p˜ 3 0.1919 > p 3 . In addition, we find that a consumer of type 4 would actually pay a lower price (i.e. p˜ 4 < p 4 ), but there are no such consumers represented in the population. At a first glance, the previous example seems to be in contradiction with the observation that as the number of uninformed consumers is the same under both consumer search distributions also the expected selling price has to be constant. This contradiction is however easily resolved if one takes into account that the consumers who switched from sampling two firms to three firms now pay a lower price, i.e. p˜ 3 < p 2 .4 It is clear from the above that if some shoppers search more they can exert a negative externality on other households. Perhaps surprisingly, it is even possible that all consumers who search more hurt themselves: Example 3.4. Starting from the initial search distribution q = (1, 20 000, 0, 0, 0, 10 000), suppose that all type 2 households switch to become type 3 (begin to compare three prices instead of two), resulting in the new search distribution q˜ = (1, 0, 20 000, 0, 0, 10 000). Clearly q˜ has more search than q and q˜ 1 = q1 , i.e. the average selling price pav = p˜ av remains the same, and one can check that p˜ k > pk for k 5 and p˜ 6 < p 6 (the informational divide is d = 5). But even more is true: the expected price p˜ 3 paid by the new type 3 households under distribution q˜ is not only higher than p 3 , but even higher than p 2 , the expected price paid by the same households before the change. Thus the increased search by these former type 2 households benefits only the highest type 6 in the population, and actually makes those who search more worse off than before. 4. Concluding remarks We conclude by discussing an alternative interpretation of our model in the context of networks, providing a short discussion on endogenous search, and by drawing some policy implications. As regards the first point, our model specification can also be interpreted as a model of price competition on a network in the following sense5 : Firms and consumers represent nodes in a bipartite network, i.e. a network where there are only links between firms and consumers. A link between a consumer and a firm in this network indicates that a consumer observes the price of this firm and may buy from this firm and that the firm may sell to this consumer. If we assume that firms only know the degree distribution of consumers, i.e. the probabilities that a given consumer has one link, two links, and so forth, we obtain a network game of incomplete information.6 Alternatively, we could also assume that the network is symmetric in the sense that every firm faces the same consumer degree distribution and analyze this game of complete information. Within this network interpretation, the consumer degree distribution essentially plays the role of the consumer search distribution in the consumer search model. In this sense, the equilibrium we find in the search context and the comparative static exercise of varying the consumer search distribution translate into our network context. In particular, the comparative static exercise of introducing more search can be interpreted as increasing the density of the network. We then have that adding links to consumers who are only linked to one firm decreases expected prices for all consumers, whereas adding links to consumers with already more than one link amounts to a redistribution of social welfare from consumers who just have a few links to well-connected consumers. In the present model, the search distribution (the network) is exogenous. A natural question is what happens if consumer types are endogenous, derived from some form of optimal search?7 Assume, for example, that the search costs of some consumers who already have low search costs (and compare prices) become even lower. Then these consumers will search more, and this will lead firms to adjust their pricing policy so that we observe an informational divide, by Theorem 3.2. Let’s call this the first-round effect. But with endogenous search there is a second-round effect: the new price distribution may be such that it now becomes worthwhile for other consumers (whose search cost has not changed) to search more (or less). This will in turn change the equilibrium price distribution again, and it seems very hard to say in general where this process will end. Clearly, a sufficient condition for our results to hold is that the second-round effects are negligible. It is easy to give examples where this is the case, so that a lowering of the search cost for some consumers who are already searching leads to a new equilibrium (after optimal adjustment of all search behavior) which exhibits an informational divide. Second-round effects will be negligible if consumer types are robust, in the sense that a consumer does not change her search behavior in response to very small price changes. This may happen with linear search costs that take only a few, discrete values; it may also happen with nonlinear search costs: for example, when a consumer goes to a certain shopping mall every weekend, she can compare the prices of all the shops there, at a very low marginal search cost, but she might not
4
Similar observations are known as Simpson’s paradox in statistics. This second interpretation of our model is related to the literature of competition on networks. See e.g. Kranton and Minehart (2001), Corominas-Bosch (2004), Blume et al. (2009), Piccione and Spiegler (2012), or Galeotti (2010). 6 This incomplete information setup stipulates the use of the Bayesian-Nash equilibrium concept for networks, as e.g. in Jackson and Yariv (2007) or Galeotti et al. (2010). 7 Such models have been studied extensively in the literature, see e.g. Burdett and Judd (1983), Janssen and Moraga-González (2004), Burdett and Smith (2010) and the survey of Baye et al. (2006). 5
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M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
drive to the other end of town to check the prices at another shopping mall.8 Such a structure of search costs implies that, at least for a certain price range, a consumer’s type is essentially independent of prices, i.e. exogenous. A similar argument can be made for consumers who do, or do not, have access to the internet, etc. These examples clarify that allowing for endogenous search would entail assumptions on how individual search distributions change with the number and location of firms. Lach and Moraga-González (2012) state that since all the existing theoretical studies in this field provide results under very specific assumptions, they prefer to follow a reduced form approach to the problem. Our conclusion is that assuming the search distribution to be exogenous is not worse than the existing viable alternatives. Finally, we note that in our model, search is not socially productive. Thus, if search is costly, the first best outcome is that consumers do not search and trade at the monopoly price. This need not be the case in a more general model: for example, when consumers or products are heterogeneous, search may be socially productive. In such a model, an increase in search intensity might not only lead to redistribution, but also increase the social surplus.9 Our final considerations start from the fact that it has become widely accepted in economics that consumer welfare can be increased by making the comparison of prices easier. If previously uninformed consumers start to compare prices this is definitely the case. However, if only already informed consumers compare more prices total consumer welfare stays unaffected. Moreover, it will result in a redistribution of consumer welfare from relatively uniformed to informed consumers. The internet – and with it the onset of price comparison sites – has been largely praised by economists and policy makers as a way to ease consumer search and thereby increase competition. Consequently, policies to improve internet accessibility, and to promote the use of price comparison sites, are also seen as tools to increase consumer welfare. Our findings suggest that it may be more important to make the internet accessible to a broader audience than to make existing connections faster. Appendix A First we introduce some more terminology and notation. The average number of searches (or links) per household is
κ = E q [k] = k kqk ; it is a measure for the intensity of search in the market (density of the network); the number of links per firm is μκ . For q = (q1 , . . . , q N ), q1 > 0, define the auxiliary function ϕ (x) := kqk xk−1 for x ∈ [0, 1] k
and if q1 > 0
ψ(x) :=
1 q1
kqk xk−1
for x ∈ [0, 1].
k
We have ψ(0) = 1, ψ(1) = κ /q1 , and ψ is continuously differentiable and strictly increasing in x. Clearly
ϕ (x) = q1 ψ(x).
Proposition 4.1. Consider a market game with information structure q = (q1 , . . . , q N ) and assume 0 < q1 < 1. Then there exists a unique symmetric equilibrium in mixed strategies: each firm chooses its price at random according to a continuous distribution F ( p ) with support [ p min , p max ], where
0 < p min =
1
ψ(1)
=
q1
κ
< 1,
p max =
1
ψ(0)
=1
and the distribution is given implicitly by the equilibrium condition
p ψ 1 − F ( p ) ≡ 1,
p min p p max
(and of course F ( p ) = 0 for p p min and F ( p ) = 1 for p p max ). Moreover, the equilibrium profit per firm is average selling price (average household expenditure) is pav = q1 .
π = μq1 , and the
The equilibrium condition gives us an explicit formula for the inverse p = F −1 (x) (also known as the quantile function) of the distribution function x = F ( p ) (restricted to its support, of course), viz.
p = F −1 (x) =
1
ψ(1 − x)
.
(2)
Proposition 4.1 is known at least since Burdett and Judd (1983) (see especially the proof of Lemma 2 there); for the reader’s convenience, we give a brief sketch of the proof. Sketch of proof of Proposition 4.1. First one establishes that the equilibrium price distribution F must be continuous and strictly increasing on a support of the form [ p min , p max ] with 0 < p min < p max = 1 (otherwise, one can easily find profitable 8 9
Such a strongly convex cost structure is perhaps no less realistic than the assumption of linear costs frequently found in the literature. This was pointed out by a referee.
M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
27
deviations). Given this, we observe that every price in the support must give the same (expected) payoff to a firm. If the firm charges p max = 1 it gets only its share of uninformed consumers, so π = p max .μq1 = μq1 is the equilibrium profit. If it charges p min it gets all households to which it is linked, so π = p min .μκ , hence p min = q1 /κ . Total profits are N π and must be equal to total household expenditure Mpav , so pav = ( N / M )μq1 = q1 . A firm charging any price p ∈ [ p min , 1] is observed by μqk k households of type k and makes a sale to such a household iff the other (k − 1) prices observed by the household are higher than p, which occurs with probability [1 − F ( p )]k−1 . Thus the firm’s profit is
p.
N
k−1
μqk k 1 − F ( p )
= p .μq1 .ψ 1 − F ( p ) = π = μq1 .
k =1
This implies
p .ψ 1 − F ( p ) ≡ 1 or F ( p ) = 1 − ψ −1 (1/ p ).
(3)
2
The following lemma gives an explicit formula for the expected prices pk . It enables us to compute the expectations pk of the many distributions F k , k = 1, 2, . . . in terms of the single function ψ . Lemma 4.2. Let F be a price distribution with associated prices pk . Then, for k = 1, 2, . . . :
1 pk =
F −1 (1 − x) dxk =
0
1
1
1
ψ(x)
dxk =
0
kxk−1
ψ(x)
dx.
(4)
0
Proof of Lemma 4.2. 10 The expected price pk is the mean of the distribution F k ( p ) = 1 − [1 − F ( p )]k . Since F 1 = F , the expected price p 1 is given by (integrating by parts)
1
1 p dF ( p ) = p F ( p )
p1 =
p min
p min
1 −
F ( p ) dp .
(5)
p min
In Fig. 1, we have A + B = 1, where A is the area in the unit square to the right (under) the curve x = F ( p ) and B is the area to the left of this curve. In Eq. (5) the first term on the right is equal to 1 and the second is the area A. Therefore p 1 is equal to the area B:
1 p1 = B =
F −1 (x) dx,
0
where we compute the area B by integrating along the x-axis. Clearly, this can also be written as
1 p1 =
F −1 (1 − x) dx,
0
which proves the assertion for k = 1. By the same logic, the expected price pk is given by the area to the left of the curve x = F k ( p ):
1 pk =
F k−1 (1 − x) dx.
0
−1
we re-write x = F k ( p ) = 1 − [1 − F ( p )]k as F ( p ) = 1 − (1 − x)1/k or p = F −1 [1 − (1 − x)1/k ] = F k−1 (x). This
To find F k implies
1 pk =
F −1 1 − x1/k dx.
0
10 It may well be that this formula is known. But as we do not know a reference, we give a proof. It is also easy to see that it generalizes to any support of the form [a, b].
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M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
The substitution g (t ) = t k transforms this to
1 pk =
F
−1
1
(1 − t ) g (t ) dt =
0
F −1 (1 − t )kt k−1 dt .
0
The result follows by formula 2.
2
Using this lemma, we can check that the average of the pk ’s is indeed equal to the average selling price pav = q1 :
q·p=
1
k qk ku
qk pk =
k
k −1
ψ(u )
du = q1 = pav .
0
Proof of Theorem 3.1. Obviously the assumption implies that q˜ k /˜q1 qk /q1 for all k, with strict inequality for some k, so ˜ x) > ψ(x) for all x ∈ (0, 1). This in turn implies p˜ k < pk for all k by Eq. (4). 2 that ψ( Lemma 4.3. (“Single Crossing”) Assume households search more under q˜ than under q, and q˜ 1 = q1 . Then there is a number b ∈ (0, 1) ˜ x) < ψ(x) for x ∈ (0, b) and ψ( ˜ x) > ψ(x) for x ∈ (b, 1]. such that ψ(
˜ b) = ψ(b), i.e. ψ˜ and ψ cross only once in (0, 1], at the point b. Of course, by continuity, ψ( Proof of Lemma 4.3. Since q˜ 1 = q1 > 0, it suffices to prove the assertion for ϕ = q1 ψ . By assumption, qk − q˜ k 0 for k < , q − q˜ > 0, and qk − q˜ k 0 for k > . We know that ϕ (0) − ϕ˜ (0) = q1 − q˜ 1 = 0 and ϕ (1) − ϕ˜ (1) = κ − κ˜ < 0. For 0 < x 1 we have
ϕ (x) − ϕ˜ (x) =
k(qk − q˜ k )xk−1 +
k =1
N
k(qk − q˜ k )xk−1 = x−1 A (x) − B (x) ,
k=+1
where
A (x) =
−1
k(qk − q˜ k )
k =1
1 xl−k
+ (q − q˜ )
is nonincreasing in x and (qk − q˜ k ) > 0 ∀x, and N
B (x) =
k(˜qk − qk )xk−l
k=+1
is strictly increasing in x (because at least one coefficient k(˜qk − qk ) must be positive), and tends to zero for x → 0. Therefore the function f (x) := A (x) − B (x) is strictly decreasing on (0, 1], positive for x near zero, and negative for x = 1 (because ϕ (1) − ϕ˜ (1) < 0). This implies the assertion. 2 The following lemma is the mathematical core of the proof. Lemma 4.4. Let g (x) be a nonnegative continuous function on the interval [0, 1] which is strictly positive except in at most finitely many points, and choose b with 0 < b < 1. For i = 1, 2, . . . define
b
b i
A i :=
g (x) dx = 0
g (x)ix 0
Then
Bi Ai
<
B i +1 A i +1
for i = 1, 2, . . .
(obviously A i , B i are always positive).
i −1
1 dx,
1 i
B i :=
g (x) dx = b
b
g (x)ixi −1 dx.
M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
29
Intuitively, the distribution H i (x) = xi on the interval [0, 1] has more weight on the right if i increases, hence B i should increase relative to A i (one might also think that B i − A i increases in i, but one can show by examples that this is not true in general).11 Proof of Lemma 4.4. Fix b ∈ (0, 1). We note that for positive x the function xα is defined for all
b Aα =
1
g (x)α xα −1 dx,
Bα =
0
Bα
g (x)α xα −1 dx,
b
where α is now a continuous variable. For we obtain
Aα
α ∈ R and write
b
= 01 b
g (x)( bx )α −1 dx g (x)( bx )α −1 dx
α 1 these integrals are certainly finite and positive. Dividing A α , B α by αbα −1
.
If α increases, the numerator decreases because bx < 1 for 0 < x < b; and the denominator increases because b < x < 1. Therefore A α / B α is strictly decreasing in α . 2
x b
> 1 for
In Lemma 4.5 below we prove a result about the order statistics of single-crossing distributions which is valid not only for distributions F of the form given in Proposition 4.1, but more generally. For purposes of exposition only, let us now use the term “price distribution” for any continuous distribution F which has a support of the form [ p min , 1] with 0 < p min < 1, and is strictly increasing and differentiable on its support (the equilibrium distributions of Proposition 4.1 have these properties). We denote by F −1 (x) the inverse of the function x = F ( p ) (restricted to its support, of course). Clearly, this inverse exists and is strictly increasing and differentiable on [0, 1]. Finally, we denote by pk the expected value of the minimum of a sample of size k from the distribution F (the kth-order statistic), and call the numbers pk (k = 1, 2, . . .) the associated (expected) prices for F . Lemma 4.5. (The “Divide”) Consider two price distributions F and F˜ , with p˜ min < p min and assume that there exists a p¯ ∈ ( p min , 1) such that F˜ ( p ) > F ( p ) for p ∈ ( p˜ min , p¯ ), and F˜ ( p ) < F ( p ) for p ∈ (, p¯ , 1). Then there exists a “divide” d 1 such that p˜ k > pk for k < d and p˜ k < pk for k > d. Proof of Lemma 4.5. By continuity, F˜ ( p¯ ) = F ( p¯ ) and F˜ (1) = F (1) = 1. Write b := 1 − F ( p¯ ) = 1 − F˜ ( p¯ ) and define the function h(x) := F −1 (1 − x) − F˜ −1 (1 − x). By Lemma 4.2
1 pk − p˜ k =
b k
h(x) dx = 0
1 k
h(x) dxk .
h(x) dx + 0
b
The single-crossing assumption for F and F˜ implies a similar property for the inverse functions, viz. h(x) > 0 for 1 > x > b and h(x) < 0 for 0 < x < b. Therefore,
pk − p˜ k = B k − A k , b 1 where A k = 0 |h(x)| dxk , B k = b |h(x)| dxk . The numbers A k , B k correspond to the size of the areas A 1 , B 1 in Fig. 1, computed w.r.t. the measure dxk (after replacing x with (1 − x)) on the vertical axis. The function g (x) := |h(x)| satisfies the assumptions of Lemma 4.4, hence B k / A k increases strictly with k. Thus, if pm − p˜ m = B m − A m > 0 for some m, so that B m / A m > 1, we have also B k / A k > 1, hence pk − p˜ k = B k − A k > 0 for all k m. In other words, if the expected price decreases for some type m, then also for all higher types. We know that p˜ min < p min , therefore p˜ N < p N for N sufficiently large (because a household who searches long enough must find a price arbitrarily close to the minimum). The divide d is the first index such that p˜ d pd . 2 Proof of Theorem 3.2. Let households search more under q˜ than under q and q˜ 1 = q1 Denote by F , F˜ the corresponding equilibrium price distributions with associated expected prices pk , p˜ k . By Eq. (2),
h(x) := F −1 (1 − x) − F˜ −1 (1 − x) =
1
ψ(x)
−
1
˜ x) ψ(
.
11 Lemma 4.4 can be seen as a ‘strict’ version of Lemma 5 in Athey (2002), using special properties of the log-supermodular density because direct appeal to Athey’s lemma would not give us the strict monotonicity needed for the proof of Theorem 3.2.
α xα −1 . We state it
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M. Nermuth et al. / Games and Economic Behavior 78 (2013) 21–30
By Lemma 4.3, there is a point b, 0 < b < 1, such that h(x) < 0 for 0 < x < b and h(x) > 0 for b < x < 1. These are exactly the properties used in the proof of Lemma 4.5, so that there is a divide d 1. It remains to show that d > 1, i.e. not all prices go down. Using vector notation, write q · p = k qk pk . Then the expected average prices satisfy
q˜ 1 = p˜ av = p˜ · q˜ = q1 = pav = p · q.
(6)
Moreover, the components of p = ( p 1 , . . . , p N ) are strictly decreasing and q˜ FOSD q. Therefore q˜ · p < q · p. If all prices p˜ k were less than pk , we should have p˜ p, hence q˜ · p˜ q˜ · p < q · p, contradicting (6). 2 References Athey, S., 2002. Monotone comparative statics under uncertainty. Quart. J. Econ. 67 (1), 187–223. Baye, M.R., Morgan, J., Scholten, P., 2003. The value of information in an online consumer electronics market. J. Public Policy Mark. 22 (1), 17–25. Baye, M.R., Morgan, J., Scholten, P., 2006. Information, search, and price dispersion. In: Hendershott, T. (Ed.), Handbook on Economics and Information Systems. Elsevier, pp. 323–376. Blume, L.E., Easley, D., Kleinberg, J., Tardos, V., 2009. Trading networks with price-setting agents. Games Econ. Behav. 67 (1), 36–50. Burdett, K., Judd, K.L., 1983. Equilibrium price dispersion. Econometrica 51 (4), 955–969. Burdett, K., Smith, E., 2010. Price distributions and competition. Econ. Letters 106, 180–183. Chateauneuf, A., Cohen, M., Meijlison, I., 2002. Comonotonicity-based stochastic orders generated by single crossings of distributions, with applications to attitudes to risk in the Rank-dependent Expected Utility model. Universitè de Paris I, CERMSEM. Corominas-Bosch, M., 2004. Bargaining in a network of buyers and sellers. J. Econ. Theory 115 (1), 35–77. Galeotti, A., 2010. Talking, searching and pricing. Int. Econ. Rev. 51 (4), 1155–1174. Galeotti, A., Goyal, S., Jackson, M.O., Vega-Redondo, F., Yariv, L., 2010. Network games. Rev. Econ. Stud. 77 (1), 218–244. Jackson, M.O., Yariv, L., 2007. Diffusion of behavior and equilibrium properties in network games. Amer. Econ. Rev. (Pap. Proc.) 97 (2), 92–98. Janssen, M.C.W., Moraga-González, J.L., 2004. Strategic pricing, consumer search and the number of firms. Rev. Econ. Stud. 71 (4), 1089–1118. Johnson, J.P., Myatt, D.P., 2006. On the simple economics of advertising, marketing, and product design. Amer. Econ. Rev. 96 (3), 756–784. Kranton, R.E., Minehart, D.F., 2001. A theory of buyer–seller networks. Amer. Econ. Rev. 91 (3), 485–508. Lach, S., Moraga-González, J.L., 2012. Heterogeneous price information and the effect of competition. Mimeo. Piccione, M., Spiegler, R., 2012. Price competition under limited comparability. Quart. J. Econ. 127, 97–135.