Search, Common Knowledge, and Competition - Semantic Scholar

Search, Common Knowledge, and Competition Dmitri Kuksov University of California, Berkeley May, 2003 Abstract This paper analyzes the effects of buyer search costs and incomplete information, private and common knowledge of the sellers, on seller competition. It turns out that small changes in search costs may have large but quantifiable effects on pricing when the seller has good private knowledge about the buyer, but the common knowledge of the buyer valuations is lacking among the sellers. As an application, one can see how the effect of a firm’s information-gathering technology may depend on whether the implemented technology is expected to make the final knowledge of the firms more or less consistent across firms. JEL Classification: C72, D49, D80. Keywords: Common Knowledge, Search Cost, Information Technology

I would like to thank Dirk Bergemann, Richard Gilbert, Ganesh Iyer, Chris Shannon, and J. Miguel Villas-Boas for their comments on a previous version of this paper.

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Introduction

Buyers and sellers often lack complete information about the market variables relevant to their decisions. Private information of the market agents may be very precise, but common knowledge may still be limited. Specifically, sellers do not know buyer valuations with certainty, and buyers must incur search costs to learn the price of a product. Buyer search costs play an important role in the marketplace. On one hand, they keep the prices above marginal costs allowing firms to be profitable. On the other hand, they reduce social welfare due to waste on search and due to deadweight monopoly loss coming from the higher prices. At first glance, it may seem that if search costs are small, they should not have an economically significant effect. But in some situations, relatively small search costs seem to have a strong impact. Consider for example the market for minimum liability car insurance. This is the market of a homogeneous good, where the buyer valuation is very high compared to the marginal cost (most drivers can not afford to post a bond to cover the minimums and hence are required by the law to have the insurance). Without search costs, a market of a homogeneous good with more than one seller should yield pricing at marginal cost. However, given small but positive search cost for price by all buyers, one can argue that prices should rise to the monopoly level (Diamond 1971). In reality, we know that neither extreme happens. The apparent conclusion is that search costs matter, but they are not sufficient to bring prices up to the monopoly level. Furthermore, one may wonder how strong the effects of a particular level of search costs may be and what the magnitude of the effect of search costs depends on. With the advent of e-commerce, there has been a surge of interest, mostly empirical, in how changing search costs may affect the marketplace (e.g., see Baye and Morgan (2001), Baye et al. (2002) and references therein). This paper addresses the above issues through consideration of the implications of an information structure sellers lack common knowledge about buyer valuations. The uncertainty of market agents about payoff-relevant variables, such as demand for sellers and quality for buyers, may be fairly small. But the theoretical acceptance of at least a small incompleteness of private information leads to the possibility of a relatively small base of common knowledge. For example, if two sellers know the buyer valuation of the product up to a (possibly very small) error, then possibly each seller’s information about what the other seller knows about the buyer valuation may have twice the error of 2

the private information available to each of the sellers. Further, what each seller thinks the other seller knows about what the first seller knows about the demand, may have three times the error of the private information. Continuing like this iteratively, one obtains that the amount of public (common) knowledge available to both sellers about the demand may be very small, and the absence of such common information, as it will be shown below, may lead to a strong differences between the equilibrium behavior in this situation and a situation where agents have complete information. This paper demonstrates how considering lack of common knowledge in the context of buyer search costs and seller uncertainty about demand, allows us to better understand how changing search costs may affect market outcomes. In particular, we will see how even if the error with which the sellers know the demand is small, the price as a function of buyer search costs may change all the way from the perfectly competitive level to the perfectly collusive level as search costs change from zero to large. Also, we can see that the magnitude of the effect of a change in the search costs depends on the relationship between the common and private knowledge of sellers on the buyer demand. The rest of the paper is structured as follows. The next section (Section 2) provides motivation for using firm side uncertainty in a model with buyer search cost. Section 3 develops the model used. The model solution and implications are presented in Section 4, which is followed by the discussion of the results in Section 5 and some possible extensions of the model in Section 6. Section 7 discusses some market situations that can be thought of in terms of the ideas of this paper, and Section 8 concludes.

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Modelling Search Costs

A general difficulty with modelling search costs is that introducing even infinitesimally small search costs in a theoretical model may lead to drastic changes in the predicted equilibrium. In particular, no matter how small, search costs for price may change a perfectly competitive market to one that is not competitive. For example, consider a homogeneous good market with a number (possibly infinitely many) of sellers carrying the product and a continuum of buyers, each with valuation V for the product and a demand for one unit only. The seller’s marginal cost of the product is normalized to zero. In the case of full information and zero search cost, there is perfect competition and one obtains a Bertrand equilibrium price that is equal to the marginal

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cost. However, if each buyer has a cost s > 0 of obtaining every price quote, then as Diamond (1971) shows1 , the equilibrium price charged by the sellers will be equal to buyer valuation V , i.e., the price is at the monopoly level (no competition).2 This discontinuous jump of prices from one extreme of perfect competition to the other extreme of monopoly as search costs go from zero to slightly positive is known as the Diamond paradox. This result may be viewed as a theoretical problem, since it is difficult to believe that an infinitesimal change in one attribute of the market environment (search cost) will change a perfectly competitive market into one that is not competitive. As Stiglitz (1979) noted, the paradox of the above model is robust in many ways: it withstands a distribution of buyer valuations of the product, product differentiation, a distribution of search costs among buyers (as far as they are bounded away from zero by a positive constant), non-unitary demand, etc. Further, as long as the price is set by the seller and is not negotiable, the paradox remains if the number of buyers is finite (i.e., when each buyer is not infinitesimally small). The rest of this section outlines some of the approaches that were applied in the existing literature, and then, on the intuitive level, demonstrates how the difficulty will be approached in this paper. The formal model and its solution is presented in sections 3 and 4. Reinganum (1987) uses downward sloping aggregate demand in combination with marginal cost uncertainty to illustrate how price dispersion and prices lower than monopoly price may result when buyers have positive search cost. With that approach, the lowest price that can be charged in the market is equal to the monopoly price under the lowest possible realization of the seller cost. As search costs approach zero, the prices charged approach the above lowest-cost monopoly level, which can still be much higher than the price under perfect Bertrand competition (with zero search costs). Varian (1980) assumes mass points of buyers with zero search costs (the “informed buyers” in other interpretation), which leads to a mixed strategy equilibrium in which, as the number of buyers with zero search costs increases, prices decrease to the marginal cost. In that framework, however, if all the buyers had strictly positive search costs, no matter how small, the equilibrium would revert 1

See Butters (1977) for a simple argument Further, buyers have zero post search surplus from purchase, a net negative utility of search (−s), and therefore, will not search or buy the product. Hence, the market disappears. Downward sloping individual demand together with linear pricing can be used to ensure that buyers have a positive surplus with monopoly prices, and hence, solve the problem of no search/no demand. 2

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to the monopoly pricing. Also, the highest possible realization of the price in the mixed strategy equilibrium remains at the monopoly level. Stiglitz (1979) proposes a model that also assumes a mass of buyers with zero search cost. Other approaches include considering non-sequential search with buyers knowing something about the exact distribution of prices (Salop and Stiglitz, 1977), such as the average of the two prices in a duopoly or the whole array of prices without knowing which seller offers which price, and bargaining models (see Lippman and McCall, 1976), where the cost comes not from explicit attempt of search but from cost of delay in purchasing the product (such models often assume an exogeneous stream of bargaining possibilities, as in Diamond, 1987). Bundling search cost for price and product attributes has been used in the more recent Internet related literature (for example, Bakos, 1997). Such models, while showing what happens in an equilibrium given the cumulative search costs for price and product, do not make it easy to consider the effects of separately reducing search costs for price and the product, whereas the effects of the two kinds of search costs can be quite different (Lal and Sarvary, 1999). To make the motivation of the approach of this paper clear, let us first go back to the buyer and firm behavior when the search costs are small in the model described as an example at the beginning of this section and consider why one may think that the arguments (and the conclusion) may not hold up in practice. The argument was that given the distribution of prices (single price in this instance), buyers search until they find a price below the reservation price, and hence, firms will not price above the reservation price. Therefore, the theory predicts that the seller should price at the buyer valuation. Suppose sellers were to follow the equilibrium strategy. Then they would need to set the price equal to the buyer valuation V . This is based on the implicit assumption that sellers know V exactly. The probable case in actual markets is that sellers know V with (possibly very small) error. That error implies that if sellers were to follow the monopoly pricing strategy, they would set slightly different prices.3 Given that buyers expect firms not to know their valuation for sure, and having very small search cost, buyers have an incentive to search for the lowest price. Therefore, if the seller j has an estimate Vj of the valuation V , and if it knows that the other firms have good estimates of demand, it may expect that prices will be somewhere around Vj . Furthermore, not willing to risk being among those with too high prices, the seller should set the price slightly below Vj . But, 3

Communication between sellers may reduce the dispersion in their evaluations of demand, however it may not eliminate it completely. Hence, some price variability is still expected. Communication of prices themselves may be not credible because of secret price cuts. Furthermore, may be unlawful under antitrust laws.

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that strategy is now followed by all sellers. So, how much reduction in price by a given seller is sufficient to ensure that the seller does not lose the buyer? One may suggest that the price should be equal or greater than the valuation that all sellers are certain the buyers have. But, when a seller thinks the valuation is Vj , it will always think that there is some chance that another seller thinks that the valuation is below Vj , and no common knowledge on the bound from below may exist. Therefore, given small uncertainty about the buyer valuation, and given that the search costs are in the same order of magnitude, the resulting equilibrium may be well below the monopoly price (and well below what each seller knows with certainty about the valuation), and the exact equilibrium price distribution will be determined by the interplay between buyer search costs, seller uncertainty about buyer valuation, and common knowledge or lack thereof. Thus, to model the effects of search costs, this paper considers the effect of seller incomplete information, where not only is a seller uncertain about buyer valuation, but also uncertain about the information other seller(s) have about the buyers. This approach does not require a large degree of uncertainty of any of the sellers to show that the price can be forced to be lower than the monopoly price and close to a perfectly competitive one. It also allows to consider how competition changes when search costs uniformly decrease (i.e., when search costs decrease for all buyers alike), and how different structures of seller information differently affect the market environment.4

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The model

Consider a market consisting of two sellers selling a homogeneous good to a number of buyers. Each buyer knows exactly her valuation of the product. The valuation can also be interpreted as the expected valuation in the case when the valuation is uncertain and the buyer has no means of finding the exact valuation, or in the case when the cost of finding the exact valuation is prohibitive. 4

It is interesting to observe that considering the lack of common knowledge among traders has also been proven fruitful in the context of financial markets for solving the problem of multiplicity of equilibria with the possibility of a speculative attack (Morris and Shin (1998)). Shin (1996) also considers a model with a lack of common knowledge and shows that common knowledge about the benefits of trade may be important for the existence of markets. A similar form of information structure has been used in Biais and Bossaerts (1998) to consider equilibrium in asset markets where people value goods for their resale value. The current paper considers the effect of information and search costs on competition among sellers when buyers have intrinsic value for the product (i.e., when a buyer valuation of the product does not depend on how much other agents value the product).

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A buyer has search costs associated with finding any seller’s price, and can not buy the product from a seller without incurring the search cost associated with that seller. The latter assumption may be realistic as buyers may not be able to purchase goods without incurring the cost of finding out the price because they have to agree to pay the price to complete the transaction. Explained alternatively, the search for the price includes the search for location. On the other hand, no transaction can be completed before the price is known, and hence, the cost of obtaining the price can not be more than the cost of making the transaction. The fact that buyers may know their valuation of the product, but not the price comes from the reality that prices can be (and are) changed frequently, whereas product design that affects the valuation can not be changed quickly. I use the simplifying assumptions that buyers have a single-unit demand, all buyers and sellers are risk neutral, and each buyer knows one price to start with.5 The cost s of finding the second price is the same for each buyer. The sellers have the same constant marginal cost of the product, which is normalized to zero. Sellers do not know buyer valuations with certainty and can not discriminate between them. Alternatively, this setup is equivalent to one where sellers set different prices for each of the n buyers, as then the market reduces to n completely independent markets with one buyer each. The demand function is known to sellers up to a certain seller-specific error. A part of the information that the sellers use can be commonly known to both sellers, but sellers can also differ in the type or amount of information about the demand they may privately receive, or in the way they handle the information. Some imperfect information about the way different sellers use the knowledge they have can be modelled as a form of incomplete information. For example, if a seller tends to discard certain types of information, it behaves as if it doesn’t receive that kind of information. I model the above informational structure by assuming that demand is determined by the demand parameter V representing the level of buyer valuations (so that in the case of homogeneous buyers, V is the buyer valuation). Sellers have a common prior on V , which I assume to be uniform on [V , V ].6 Also, each seller j = 1, 2 receives individual information 5 The rationality of at least one search, resulting in one price being known, can be explained, for example, by a downward sloping buyer demand, so that the purchase of the optimal amount (further known as the unit of the good) at the linear monopoly price leaves the buyer with enough surplus to cover one search cost. It can also be that at least one price quote is discovered during the time the buyer is looking for the product related information – the stage preceding one that is being modelled. 6 One may consider a general prior G(V ) on V . The assumption of the uniformity of the prior is not essential in any way, and is made here for simplicity. Also, the uniform prior may be interpreted as the least informative prior, i.e., the assumption is of the “absence of prior knowledge” on the relative probability of

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about the demand (i.e., about the parameter V ). More precisely, it is assumed that the private information of the seller j is the signal xj , such that xj given the demand parameter V are i.i.d. Bernoulli variables for j = 1, 2, i.e., the signal is either that the demand is high or demand is low.7 Inversely, it means that V given x can attain only two values. Let us call the higher one Vh (x), and the lower one Vl (x). Hence, we have that before receiving the signal xj , the seller j assumes that the demand parameter V can be anywhere in [V , V ] with equal probability (uniform distribution). After receiving the signal x, since the prior was uniform, and assuming both Vl and Vh are within the range [V , V ], the seller believes that the demand parameter V is either high Vh (x) or low Vl (x) with equal probability (if the common prior was not uniform, the probabilities the seller assigned to Vl and Vh would be different). Assume that the mapping Vh (x) to Vl (x) is one-to-one. Then, since signal x only brings the information that the demand is Vh (x) or Vl (x), without loss of generality, we can assume (by renaming “signal x” to “signal Vh (x)) that the signal is the value of the high demand parameter, i.e. x = Vh (x). Denote, for future use, f (x) = Vl (x). In this notation, omitting in the notation but accounting for the dependance on the prior, ( x Prob. 1/2, (V | x) = V (x) ≡ f (x) Prob. 1/2.

(1)

Also, let g(V ) = f −1 (V ) be the inverse function of f (·), i.e., the value of the signal x that would imply that the demand is at least V . Note that the above implies that given the valuation V , the seller signal attains values V and g(V ) with equal probability: ( V Prob. 1/2, (x | V ) = x(V ) ≡ g(V ) Prob. 1/2.

(2)

The degree of seller uncertainty is represented by the difference between the possible values of the demand parameter V that can exist given the firm’s knowledge, i.e., by δ(x) = Vh (x) − Vl (x) ≡ x − f (x). V within the interval [V , V ] (with V may be 0 and V may be “close” to infinity). 7 One may consider possibility of sellers exchanging their signals. This exchange may be imperfect or even impossible at all due to the legal antitrust restrictions and incentive incompatibility to reveal the true signal. Partial exchange of information and the process of generation of xj may cause the actual signals to become correlated. This possibility does not undermine the intuition behind the main results, and will be discussed later. Also, instead of the Bernoulli distribution, one can consider a general distribution H(V |x) of the demand parameter V given the signal x. This is easy to see through the model setup, but would greatly complicate the tractability of the solution.

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The mass of all buyers is normalized to one (i.e. maximum aggregate demand is 1). I further assume that if a buyer expects the same utility from either seller, they will go to either one with probability 1/2. I will first consider the case of homogeneous buyers, where each buyer i has valuation V of the product from either seller, and hence, has a net utility of Uij = V − pj of the product of seller j. I will then consider the case of heterogeneous buyers. The timing of the game is as follows. At the beginning, sellers have the common prior on buyer valuations. The number of sellers is also common knowledge.8 Next, sellers receive the individual signals xj of demand (private knowledge to seller j, but there is common knowledge on distribution of xj ), and simultaneously set prices for their products. Finally, buyers decide when to stop searching (or whether to search) and which product to buy (if any). The buyers decision on purchase defines the final buyer payoffs (buyer surplus) and seller payoffs (profit) of the game. To simplify calculations, assume that the uncertainty is not too large, namely that δ(x) < x/2, and that x − δ(x) is a continuous and non-decreasing function of x: x δ(x) < , (x − δ(x)) % continuously (3) 2 These assumptions are not essential, but they are intuitive: the first assumption (δ(x) < x/2) means that the uncertainty is not too large and yields that a monopoly would serve buyers of all possible valuations in the case of homogeneous buyers. I have considered an example of uncertainty schedule that violates this assertion: the case of constant absolute uncertainty, i.e., when δ(x) = δ does not depend on x and the results of the paper are preserved. The assumption that x − δ(x) is non-decreasing means that as signal increases, the demand increases in both high- and low-demand possibilities. All the assumptions simplify the proofs, but are not essential for the main results. An example of the uncertainty schedule to keep in mind throughout the paper is the case when δ(x) = ax until some point xb , and then δ(x) = δ = const for x > xb . That is when the relative uncertainty is constant for small V and absolute uncertainty is constant for larger x.9 Without loss of generality, assume f (x) > V for all x ∈ [V , V ]. 8

One can also consider that at this point, sellers may decide on the technology that will allow them to receive the private signal of buyer valuation x. This decision (choice of distribution of x) is assumed to be simultaneous across firms, but then known to all the firms and buyers (since the distribution of xj is assumed to be common knowledge). The assumption that the distribution of x is known to all parties involved is similar to the statement that “all sellers have the same or known possibilities of market research”, and therefore, may be realistic. For now, let us consider that the distribution of xj is exogenously given. 9 This is until x = V − δ. One may think that if the valuation of the product is larger, then the

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An important feature of this model is that once the value V of the demand parameter is settled, the distribution of seller types (a type is identified by the signal received) may be narrow, but a seller of an extreme type does not know that it is of an extreme type (as far as x ∈ (V , V )). If a seller receives signal x of V , it knows that V is either x or f (x) and therefore all the other signals in the game are either f (x), or x, or g(x). Therefore, for example, this seller would know that another seller is of the low type if that seller receives the signal f (x). But the seller who actually receives the signal f (x) still does not know that it is of the low type. This contrast between private information about the possible seller types (x ± δ) available to the seller with signal x and common knowledge about the types that may be in the market ([V , V ]) is the key to understanding the informational structure of the market. In general, buyers may use the price quotes they know to update the expected distribution of the other prices in the market if the seller signals (given V ) are correlated or if the aggregate demand is unknown to the buyer, i.e., when given a buyer’s knowledge (xj | {Vij }j=1,2 ), the signals are correlated. This may lead to an additional incentive for a seller to set a higher price (since a higher price seen by a buyer may imply that the buyer will expect higher prices in the market, and therefore may reduce the benefit of search that the buyer expects, and hence, reduce the probability that the buyer that receives the price quote from this seller will search and so, increase demand). However, in the model, buyer knowledge of their own valuation, the fact that buyers are homogeneous and independence of seller signals means that buyers know the valuation of all buyers, and since there is also no correlation between seller signal error term, the price at one seller does not bring any information to the buyer about the price at the other seller. Let us consider an example of a marketplace where the above model directly applies. Consider a B2B market with several producers and N industrial buyers, such as, for example, the market for airplane engines. In such a market, the price is set on an individual level, i.e., sellers have the ability to discriminate between buyers. Assume that the marginal cost is constant in the possible demand range, so that the supply is not limited, and therefore, buyers are not competing for products, but rather sellers are competing for each buyer. Such a market is equivalent to N separate markets with several sellers and one buyer and price is normally set after a one-on-one negotiations of a buyer and a seller. uncertainty about the product leads to relatively larger dollar uncertainty. However, when the payoff is large enough, additional investment in market research reduces relative uncertainty so that the absolute uncertainty remains approximately constant. The derivation of the equilibrium, however, does not use any assumptions on the exact form of f (x) except (3).

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When the final price is set by the seller, the negotiations can be thought of as the process whereby the seller receives the signal xj of the buyer valuation. Naturally, the buyer knows its valuation, and the seller needs to use its best guess of buyer’s valuation to offer a price. During negotiations, by suggesting high prices, the seller may try to elicit buyer valuation, while the buyer may respond in such a way that the seller thinks the buyer is of low valuation. In any case, since the negotiations do not oblige either buyer or seller to act, they can only be used by the seller as information on the basis of which to set the final take it or leave it offer. Hence, all the negotiations reduce to the signal xj of buyer valuation that the seller receives. As it takes negotiations to discover a seller’s price, which will depend on the signal xj that the seller receives during these negotiations, sellers can not determine with certainty the price other sellers will set. In this example, the search cost is the buyer’s cost of negotiating. The concept of equilibrium used in this paper is perfect Bayesian equilibrium. I now turn to the solution and implications of the model.

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Model Solution and Implications

4.1

The Case of Identical Buyers

In this case, each buyer i utility of the product of seller j is Uij = V − pj where V is buyer valuation of the product and pj is the price set by the seller j. Also, unless otherwise stated, assume that there are only two sellers. Given a signal x, a seller expects at most three values of the signal of valuation received by the other sellers: f (x), x, and g(x). Assume first that the equilibrium is in pure strategies and is given by the pricing strategy pe (x), i.e., given the signal x, a seller sets the price pe (x).10 Then, the seller expects that the price set by the other seller is pe (f (x)), pe (x), or pe (g(x)). It can only lose some demand because of a higher price if a) the price is above the valuation of some sellers, or b) the other price is lower and buyers search. The equilibrium price is determined by buyer valuations and, therefore, it is the monopoly price, unless b) provides a sufficient incentive to reduce the price. Consider the buyer problem. Having valuation V , they expect the seller signal to be V or g(V ), and therefore, prices pe (V ) or pe (g(V )). If they first see the lower price, they have 10

The possibility of mixed strategy pricing is also considered in the Appendix.

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no incentive to search. If they first see the higher price, their expected benefit of search is (pe (g(V )) − pe (V ))/2, where 1/2 is due to probability 1/2 that the other seller has signal V . Therefore, given the higher price, buyers will search if and only if pe (g(V )) − pe (V ) > 2s. If the low-valuation buyers search, then the seller has the probability 1/4 of losing demand (probability 1/2 that the price is high, i.e., that the buyers have the lower valuation, and probability 1/2 that the other price is lower given that the valuation is lower). This leads to the following proposition. Proposition 1. The following iterative formula defines an equilibrium price schedule pe0 (x) for x ∈ [V , V ]: pe0 (x)

 =

f (x), if δ(x) = 0, e min{f (x), p0 (f (x)) + 2s}, if δ(x) 6= 0.

Further, in the equilibrium, buyers do not search. Proof. See Appendix. The Appendix also proves that the above equilibrium price strategy is a unique symmetric equilibrium price strategy. Let x0 be the smallest x such that f (x) − f (f (x)) = 2s. Figure 1 depicts the equilibrium price pe (x) and the monopoly price pm (x), which is also the price the sellers would set under a collusive regime, on the example where V = 0 and δ(x) % in the range on the graph. Now, for transparency of argument and model implications, let us restrict our attention to the range of x where δ(x) %. We assume this holds on [V , V1 ], where V1 ≤ V ). All further discussion will be restricted to this range. Going back to the main questions of the paper, we are interested to see how search costs (s) and the structure of the uncertainty (δ(x) and [V , V ]) affect the market equilibrium. We are now ready to answer the first question for this model. Proposition 2. If search cost s decreases in the above model then 1. The expected equilibrium prices and profits decrease. 2. The buyer surplus increases. 3. The changes are continuous, and as search costs decreases from infinitely large to infinitesimally small, the equilibrium outcome continuously changes from monopoly level to one in the perfect Bertrand competition if x − f (x) > 0 for all x > 0. 12

p

pm(x) = f(x)

pe(x) 2s/δ

x x0 : f(x0 )−f(f(x 0))=2s

Figure 1: Equilibrium price: homogeneous product and buyers. The last point of the above proposition is perhaps the most intriguing property of the model. It shows that allowing the lack of common knowledge resolves the Diamond paradox, i.e., it reconciles our belief that a continuous change in search costs (s > 0 and s = 0) should result in a continuous change in the outcomes (prices do not jump as s becomes slightly positive). This result emphasizes the importance of considering the lack of common knowledge in the context of the economic theory of search in markets. Proof. As s decreases, the point x0 = x0 (s) after which search becomes a constraint on the price (see Proposition 1) continuously decreases, since f (x) − f (f (x)) is monotonically increasing in x. Also, the slope of the price schedule after x0 (s) continuously decreases as s decreases. Therefore, when s decreases, the equilibrium price (given the signal x) decreases continuously. Further, firm’s demand does not change, since in a no-search equilibrium, when the firm sells to all buyers, the demand per firm is always 1/2. Therefore, profits decrease due to a decrease in the equilibrium price, and Point 1 is proven. Since the expected buyer surplus V −E(pe (x)|x) is inversely related to price paid, buyer surplus increases as s (and pe (x)) decreases, which proves Point 2. The continuity of the equilibrium price dependance on the search cost s has already been noted. When s is very large, then x0 (s) is larger than V , and therefore, the equilibrium price is at the monopoly level. When s is very small, then both x0 (s) and the slope of the equilibrium price schedule pe (x) become close to zero. Therefore, the equilibrium price 13

pe (x) becomes close to zero. Since the price under perfect Bertrand competition in nondifferentiated markets equals marginal cost, which is zero in this model, we have that as s decreases, price decreases continuously from the monopoly level to one under perfect competition. Some additional insights that the model reveals are related to the way the search cost interacts with private and common knowledge of the seller. The next proposition summarizes some of such properties of the equilibrium in this model. Proposition 3. In the above model: 1. The equilibrium price is below the monopoly level when the uncertainty of the seller’s demand signal is large enough compared to the buyer search cost. 2. For sufficiently large signal x (namely, for x > x0 ), the increase in price due to increase in expected valuation is increasing in buyer search cost s and decreasing in the uncertainty of the private signal of buyer valuation (∂pj /∂xj increases in s and decreases in Var(x) or δ(x)). 3. The increase in price due to the common knowledge increase in buyer valuation V is equal to the increase in V in the following meaning. For a given ∆V > 0, the equilibrium price given the signal x + ∆V and the uncertainty function δ 0 (x) = δ(x − ∆V ) (with δ 0 (x) = 0 if δ(x−∆V ) is not defined) is by ∆V higher than the equilibrium price given the signal x and the uncertainty function δ(·). 4. The sellers weakly benefit (i.e., they are not worse off and possibly better off ) from increasing V . 5. As the private uncertainty delta(·) decreases uniformly to zero, the equilibrium price increases to the monopoly price. Proof. 1. As it follows from Proposition 1, the equilibrium price becomes lower than the monopoly level when x > x0 . The left hand side in the equation defining x0 is the uncertainty f (x) − f (f (x)) the seller has about the buyer valuation after receiving signal f (x). The right hand side is twice the search cost. In the region where the price is the monopoly level (x < x0 ), the uncertainty is smaller than 2s. When the uncertainty becomes large enough (higher than 2s), the price becomes lower than the monopoly level .

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2. The increase in price due to increase of the signal is larger when s is larger, because for every δ(x) = x − f (x) the signal increases, the price goes up by 2s (for sufficiently high x). By the same reason, the increase in price is smaller when the uncertainty (δ(x)) is higher (and V ar(x|V ) = δ 2 (g(x))/2 is increasing in δ(x)). 3. The transformation on the signal x and the function δ(·) shifts the graph of the equilibrium price schedule up and to the left along the diagonal p = x by ∆V in a coordinate dimension (when uncertainty is absent, the price p(x) is at the monopoly level x). Therefore at the correspondingly increased private signal x + ∆V , the price will increase by ∆V as compared to the previous equilibrium price pe (x). 4. Sharing information will reduce the uncertainty each seller has about the other seller in the market, and hence will reduce δ(x). Consider an extreme case: suppose sellers are able to (credibly) communicate their signals in full. Then their expectation of demand is the same, and therefore, they will set the same price, which then has to be the monopoly price. 5. It suffices to note that as the private uncertainty decreases to zero, f (x) − f (f (x)) = δ(f (x)) decreases to zero, and hence becomes less than 2s for all x, which by Proposition 1 implies monopoly price. In fact, the common level of uncertainty does not change the price level more than the amount of uncertainty itself, since the prices are then constrained only by the possible (expected) valuations of buyers in the market. The level of individual uncertainty increases the chance that the other seller sets price below the first seller’s price, and hence (by propagating to sellers of other types) may decrease the prices substantially more than by the magnitude of the range of the possible valuations that is privately known to every agent. Note the difference, implied in points (2) and (3) of the above proposition, of how the private and common knowledge affects the division of surplus between buyers and sellers: if the surplus is considered as a function of V , then the precision of private seller information (the function δ(·)) affects the ratio of buyer and seller marginal surplus, and the amount of common knowledge (V or, in other notation, the minimal x such that δ(x) = 0) affects the lump sum of the surplus that the sellers receive. The expected seller surplus given V , as a function of V , can be roughly decomposed in a constant that depends on the amount of common knowledge (V in this case) and the term linear in the amount of private information (δ = Vh − Vl ). 15

Point 4 may be interpreted as saying that the sellers are interested in increasing common knowledge, i.e. in sharing their beliefs and/or information about buyer valuation. Note that this result may be seen as counter-intuitive since one could hypothesize that sellers may be afraid of revealing their intentions due to fear that the competitor would undercut their price. However, in view of the Diamond paradox, it is intuitive, since if there is no uncertainty, the price becomes the monopoly price in the presence of search costs. The intuition is that in the presence of buyer search costs, sellers do not want to undercut each other, but rather want to ensure that they are not being undercut by a seller that reduces price based on her belief that the buyer valuation is low. As a corollary to Point 5 of the above proposition and Point 4 of Proposition 2, we have the following result Proposition 4. The amount of equilibrium price increase as search costs s increase from zero to a positive level may be equal to any part of the difference between the monopoly price and the price under perfect Bertrand competition. In other words, if one thinks of the effect of a change ∆s in search costs on the change ∆p of prices as ∆p = α∆s, where α is the price response to search costs parameter, then the magnitude of α may be arbitrarily large (depending on the structure of the uncertainty). Proof. It is sufficient to notice that we have a perfect Bertrand competition when s = 0 and monopoly price (according to Proposition 3, point 5) for any s as far as δ(·) is small enough. In conclusion, it can also be noted from the proof of the equilibrium in the Appendix that the equilibrium price schedule in Proposition 1 remains an equilibrium price schedule if the number of sellers in the market is not restricted to two.

4.2

The Case of Heterogeneous Buyers

The model of the previous section can easily be extended to the case when buyers have a distribution of valuations, and therefore generate a downward sloping demand.11 As it is 11

This model still assumes that buyers know the distribution of seller signals and therefore do not update their beliefs on the equilibrium price at the other seller after they find the price at one seller. The case when buyers do not know the distribution of seller signals is discussed among the extensions later and one would expect that, though providing additional incentives for the seller to increase price, this uncertainty will not change the qualitative results of dependance of the equilibrium price on search costs both in the case of exogenous and endogenous product differentiation.

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shown in the Appendix, Proposition 1 in the case of any downward sloping demand has to be reformulated as follows. Proposition 5. The following iterative formula gives the equilibrium price schedule in a pure strategy equilibrium of the model with heterogeneous buyers and homogeneous product for x ∈ [V , V ]: peh (x)

 =

pm (x), if δ(x) = 0, min{pm (x), pe (f (x)) + 2s}, , if δ(x) 6= 0.

where pe (x) is the equilibrium price set by a seller with signal x, and pm (x) is the price a monopoly seller would set given the signal x. Furthermore, when pe (x) = pe (f (x)) + 2s, buyers of the lower possible valuation are indifferent between searching and not searching, and in the case of the deviation p > pe (f (x)) + 2s by the firm, the buyers of valuation f (x) search. Proof. See Appendix. For example, consider the case of buyer heterogeneity as in the Hotelling model. In that model, buyers are uniformly distributed along the segment [0, 1], and the valuation of buyer i of the product at the firm j is Vij = V − yij t, where yij is the distance between buyer i location on the line and the location of the firm j, and t is the parameter of heterogeneity. The products are not differentiated if the locations of both firms are the same. Consider for example, the case when both firms are located at 0. Proposition 5 then implies that the equilibrium price will be as in Figure 2 (the graph is drawn assuming that 2s/δ < 1/2).

5

Discussion

As it was noted above, sellers benefit from an increase in common information about the demand, but the technology for receiving private signals of demand may be detrimental to profits of each firm if such information increases the uncertainty one firm faces about beliefs of other firms. If one thinks of the total information that a firm can gather about demand as a finite space, from which firms sample information at a cost, one can notice the following pattern. 17

pm(x) = f(x)

p

pe(x) 2s/δ

x x0 : pm(x0 )−pm(f(x0 ))=2s

Figure 2: Equilibrium price: heterogeneous buyers, homogeneous product. When firms sample small amount of information, the lack of knowledge about demand may keep prices down. As firms sample more information, private knowledge may allow them to know the demand better, but they may be unable to set high prices due to competition. However, if all firms sample all the information available, all firms have the same knowledge of the demand, and price may go back to the monopoly level. Hence, the benefit of investing in an information technology depends on how it will affect both the information available to the firm and beliefs of all the firms on how commonly known their information is. It may be interesting to also consider the above effect in the context of buyer access (buyer reach). A firm may invest to be able to access a fraction of buyers from the total demand pool. This set of buyers that the firm can access (sell products to) is called reach (Chen and Iyer (2002)). Since buyers are heterogeneous, different reach results in different beliefs of firms on the demand they face. If reach of firms is small, then the set of buyers for which firms compete may be relatively small as well, and hence competition level may be low. As reach increases, so does competition (Chen and Iyer (2002)). Without considering the effects of both search costs and incomplete information, one concludes that as reach becomes total, prices decrease to marginal costs. However, as reach approaches total, the difference in information that different firms have becomes smaller. The greater the amount of common knowledge, the lower the fear each firm has about the other firm’s actions, and the lower the competitiveness of the marketplace. Therefore in the presence of buyer search costs, as reach increases, we have two opposing effects on firm competition, and competition 18

may decrease again: in particular, when both (all) firms have knowledge/access to all buyers, there is no uncertainty, and therefore, with strictly positive buyer search costs, there is no competition. In the model, the buyer search for the first price was assumed to be costless. The problem of the search of buyer for the first price can be solved in several ways. The rationality of the first search can be explained, for example, by the fact that at first, the buyer is searching not only for price, but for product quality information as well, and hence, due to large uncertainty about the utility prior to the first search, the first search cost is justified for all buyers in the market. In the case of such behavior, the equilibrium price provides an expected surplus for buyers and hence, the at front expenditure on search for quality is justified. Alternatively, each buyer may have a downward sloping demand for the product, and sellers may not be able to force a minimum purchase requirement (for instance, because of having to follow linear pricing), and hence they know that they have positive net utility of buying the product even under the monopoly price. Also, one can consider search costs depending on the valuation, so that the smaller the valuation (and the expected surplus) is, the smaller the search costs are. While considering the amount of uncertainty that a seller has given a signal x as a function of x, it was assumed that the search costs s are independent of the buyer valuation. This assumption is easily relaxed. In fact, Proposition 1 remains valid when s = s(V ) depends on the true buyer valuation V , and s is replaced by s(f (x)).

6

Extensions

The model assumed two sellers. However, it is easy to extend the model setting to the case of N sellers. In fact, as it was noted above, the equilibrium price strategy of the Proposition 1 remains an equilibrium price strategy with any number of sellers. It is also not difficult to see that this equilibrium price strategy is the lower bound on the possible equilibrium price given the signal x. However, with sufficient uncertainty, when the above equilibrium price is much lower than the monopoly price, mixed strategy equilibria with higher pricing are possible. This indicates that in the presence of buyer search costs, as the number of sellers increases, prices may increase as well, which is consistent with findings of Stiglitz (1987). The model assumed independence of demand signals across sellers. If seller signals of

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buyer valuation are in fact correlated then each buyer may use the price at one seller to update its prior on the distribution of prices at the other sellers. In this sense, price then has a signaling value to the buyer. In this situation, when a buyer sees a higher price, she may infer that the valuations of other buyers are higher, and therefore the other prices are higher than previously believed. Exactly the same thing happens when buyers do not know the aggregate demand, since then seller signals given knowledge of an individual buyer are correlated (since each signal is correlated with the aggregate demand, a variable unknown to the buyer). This situation is more realistic in retail markets. I expect that even though this creates an additional incentive for a seller to raise the price (since high price may be a signal for a buyer of higher competitor prices, and hence raising the price does not hurt the seller as much), the incentive to lower price due to uncertainty about the demand and competitor prices remains, and hence, the qualitative results of the model will be preserved. Furthermore, if the equilibrium is with seller pure pricing strategy (depending on the signal it receives), buyers are able to correctly derive the sellers signal from its price. One may expect that a smooth distribution of the seller signal and buyers will, in fact, result in a pure strategy equilibrium. Let us consider in more detail buyer behavior under more general conditions. Given that a seller’s price depends on the expected demand curve that the seller faces, each buyer uses the following information to determine her optimal reservation price (which is based on the beliefs of the buyer on the aggregate demand and on the estimate of that demand by the sellers): a) her prior on the possible distributions of the aggregate demand; b) her own utility, as it gives the information about the utility of other buyers, since the demand from self and from others is correlated (that buyers use information about oneself to infer behavior of others is supported by buyer behavior literature: see Dawes (1989) on false consensus bias); c) each subsequent price quote, as it gives the information on both the aggregate demand and on the seller’s belief of the aggregate demand (Stigler (1969)). Accordingly, the buyer decision process is as follows: first, using her prior on the possible distributions of the aggregate demand and distributions of the seller demand signal error terms, the buyer guesses the prices that sellers are likely to set. Given the guess on the distribution of seller prices, she finds the reservation price rule (equating the benefit 20

of searching to the cost of search of the next price). Then (actually this step occurs simultaneously with the previous one) she uses her own utility to update the prior on the possible distribution of the aggregate demand, and together with her prior on the distribution of seller signal error terms, she updates her beliefs on the distribution of prices, and hence on the optimal reservation price.12 Then she updates the prior on the distribution of aggregate demand and seller signals every time she sees the next price. Therefore, the reservation price is updated upward following a high price, and downward following a low price. Sellers have to account for this effect of price, which gives them an additional incentive to keep prices higher. Also, buyers have to take into account the strategic firm behavior while setting the price. The resulting equilibrium price level may then be higher as opposed to the equilibrium price level when buyers knew the aggregate demand. However, the incentive to reduce prices due to uncertainty about the competitor’s price and the incentive to differentiate seems to remain in such a case.

7

Anecdotal Examples

The following are just a few situations which may be interpreted as consistent with the proposed theory. Auto Purchases. In a car buying experience, search for valuation usually precedes price search. However, prices are neither perfectly competitive, nor that of a monopoly (among dealers of a particular brand). When a customer comes to a dealer for a price quote, the dealer receives a private signal of customer valuation during the interview. Some part of the signal may be common among different dealers, but some is dealer-specific (dealers use some general rules to guess demand, and some rules that each dealer finds by oneself). Therefore, when offering a price to the buyer, the dealer has to keep in mind the possible assessment that other dealers will have of this buyer valuation of the product. Let us speculate in terms of the framework of this paper, how the possibility of email and phone price quote requests may affect the price quotes. Email or phone correspondence may seem to increase competitiveness through reducing the search costs (easier to talk online or send and receive emails than come to talk in person), but at the same time they 12 This decision process, in particular, rationally explains why people exhibit “belonging to majority” bias when people overestimate the number of other people who conform to their ideas.

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reduce the individual (dealer) specific part of the valuation signal (as dealers see less of a personality from which different people may draw different conclusions). Therefore, webassisted commerce in the presence of bargaining does not have to be more competitive than off-line.13 Tourist shops. The shops (souvenir, clothes, and other non-essentials) are often grouped together, so that the search costs are fairly small. Furthermore, the incentives of tourists to search are also larger (costs of search smaller) since, while searching, they also enjoy the local life, architecture, etc. However, one often sees little competition among the stores. Two reasons within the framework of this model can explain this phenomenon. First, there is some common knowledge about tourists likes and dislikes, and there is little private signals of demand/valuation, since the stores do not maintain a core of loyal customers or do an independent market research. Therefore the amount of private (independent across stores) uncertainty is also small. Therefore, even though the search costs are low, the price does not need to decrease. The second reason is that some of the stores are highly differentiated (apparel stores attracting tourists, for example, usually have a very specific style). The insurance market, noted in the introduction, seems to support the idea that the equilibrium prices may be well below buyer valuations even if search costs are high. Insurance market is a market of a homogeneous good, where the valuation by buyers is very high14 compared to the marginal cost (most drivers can not afford to post a bond to cover the minimums and hence are required by the law to have the insurance). Without search costs, a market of a homogeneous good with more than one seller should, according to Bertrand equilibrium idea, yield pricing at marginal cost (price competition may make sense in the above example); at the same time, given small but positive search cost for price by all buyers, prices should become close to the monopoly level (according to the Diamond paradox). In reality, we know that neither extreme happens. The apparent conclusion is that search costs matter, but are still not sufficient to bring prices up to the monopoly level. 13

Of course, another important aspect not considered here is the simultaneous existence of the off-line market for cars. 14 Some states may have regulations against too high pricing of the minimal insurance policies. Such regulations may set the upper bound on the possible price offered by the firms. However, one finds a wide distribution of prices for the minimum insurance. The model in this paper, in fact, illustrates how an upper bound can force prices to be lower than the bound by cutting off the possibility of high price equilibria.

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8

Conclusion

Search costs are often viewed as introducing friction in the marketplace, due to which prices and other equilibrium outcomes do not exactly match theoretical predictions. The possibility of interaction of search costs with other parameters of the industry structure is often overlooked. This paper considers one possibility of such interaction, namely, the interaction between incomplete information sellers have about the demand and buyer search costs. This paper develops a model of the effect of search costs on the equilibrium that captures the important aspects of the phenomenon and is analytically tractable. The constructed model predicts realistic effect of search costs on equilibrium prices. As one would expect, prices smoothly increase from perfect Bertrand competitive prices to monopoly prices as buyer search costs increase from zero to infinity. At the same time, the model shows that there is an important interaction of search costs and the type and amount of information sellers have. This interaction allows one to better understand certain market phenomena, such as failure of prices to decrease when search costs decrease in the case when informational structure of the market changes as well. The paper also considers the different effects that common and private knowledge of sellers have on the equilibrium outcomes and shows why and how small changes in search costs may yield large changes in the market price; changes, which are nevertheless, (loosely speaking) proportional to the changes in search costs. In particular, it shows how the effects of buyer search costs and seller uncertainty about demand may magnify under limited common knowledge.

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Appendix: Derivation of the Equilibrium Price Strategies Proof of Proposition 1 Proof. (existence.) To prove that the presented strategy is an equilibrium strategy, consider first the buyer problem, given that the sellers follow the equilibrium strategy p(·) = pe0 (·) defined in the statement of the proposition. A buyer with valuation V expects the seller signal to be V or g(V ) with equal probability, and therefore, expects to see price pe0 (V ) or pe0 (g(V )) with equal probability. The benefit of search for the buyer with valuation V who sees the price p = pe0 (g(V )) is then BS =

pe0 (g(V )) − pe0 (V ) , 2

since the buyer has probability 1/2 of finding the lower price pe0 (V ). The cost of searching is s. Consider two cases: 1. V such that pe0 (g(V)) < V. Then the buyer benefit of search given the buyer sees the higher possible in the equilibrium price pe0 (g(V )) is BS = s. It is optimal for the buyer not to search, but if the seller with the high price raises price at least a little bit, then it would benefit the buyer to search. 2. V such that pe0 (g(V)) = f (x). Then the benefit of search for the buyer is less then s. Hence, it is optimal for the buyer not to search even if the seller following the equilibrium strategy sets the high price pe0 (g(V )). Hence, it is optimal for the buyer not to search if the price she sees is one of the equilibrium prices (pe0 (V ) or pe0 (g(V ))). Furthermore, if the buyer sees the price at least slightly above pe0 (g(V )) in the first case, she will search. If she searches in that case, she will find a lower price as long as the other seller follows the equilibrium strategy. Therefore, if the other seller and the buyers follow the equilibrium strategies, a seller’s   1/2, e 1/4, ED (p) =  0,

expected demand is if p ≤ pe0 (x), if p ∈ (pe0 (x), min{pe0 (x) + 2s, x}) otherwise.

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The price pe0 (x) is optimal, since in the first case, we have pe0 (x) = f (x) > x/2 (and therefore p = f (x) with demand 1/2 is preferable to p = x with demand 1/4), and in the second case, we have pe0 (x) = pe0 (f (x)) + 2s > 2s, and therefore it is not beneficial for the seller to raise price above the equilibrium price if half of the demand is lost. Hence, the stated price strategy and the buyer behavior is a part of an equilibrium. Note that the proof of the equilibrium above does not use that the number of sellers is two. Therefore the price strategy remains equilibrium for any number of sellers. Proof of Uniqueness. To prove that the equilibrium is unique, I prove a serious of lemmas that show restrictions on possible equilibria until the number of possible equilibria is narrowed down to one. Let pe (x) be an arbitrary equilibrium price strategy given the signal x. As a convention, a statement about pe (x) as if it were a single price means a statement about each price in pe (x) that is set with a positive probability. Also, let pe (x) = inf pe (x), andpe (x) = sup pe (x). Lemma 1. An equilibrium price is strictly positive (pe (x) > 0) and provides a strictly positive profit for x > 0. Proof. A price p < 0 can not be optimal since it provides a negative profit, whereas profit of 0 can be guaranteed by p = 0. Further, a price p = 0 can not be optimal, since it ensures profit of at most 0, whereas price p = min{f (x), s/2} ensures profit of at least min{x/2, s/2}/2 > 0 no matter what is the other sellers strategy is, as far as it has p ≥ 0), since buyer expected benefit of search can not be more than s/2, and therefore buyers will not search. Lemma 2. In any realization of the valuation and signals, the firm with higher signal charges higher price: pe (x) ≥ pe (f (x)). Proof. Proving by contradiction, suppose the contrary. Suppose there is a point x at which pe (f (x)) > pe (x). Then in the increasing sequence of f (x), x, g(x), g(g(x)), . . . there is a pair of neighboring points where pe (x) < pe (g(x)) + s (otherwise in contradiction to the above Lemma, the price would become negative). Then let x1 , x2 = g(x1 ), and x3 = g(x2 ) be points such that pe (x1 ) > pe (x2 ), and pe (x2 ) < pe (x3 ) + s, where x1 = f (x2 ), and x2 = f (x3 ). Consider the seller decision given signal x2 . The seller knows that the valuation of buyers can be either x2 or f (x2 ) = x1 . Neither of them will search if price is slightly 25

above pe (x2 ) (since the benefit of search is then for either of the buyer types not more than slightly more than s/2). Hence, the only reason the seller may find it optimal not to raise price is if the buyer valuation is a constraint, i.e. if pe (x2 ) ≥ f (x2 ). But that would mean that pe (x1 ) > f (x2 ) = x1 , and hence, pe (x1 ) is above buyer valuations (as known to the firm) and provides 0 (expected by the firm) profit. This is a contradiction with the above lemma. The following lemma strengthens the previous one. Lemma 3. pe (x) ≥ min{pe (f (x)) + 2s, f (x)}. Proof. Again, the proof is by contradiction. Suppose there is a point x at which pe (x) < min{pe (f (x)) + 2s, f (x)}. We also know from the previous lemma that pe (g(x)) ≥ pe (x). Therefore the seller having the signal x knows that the buyer’s benefit of search is less then s, and therefore a slight increase of price would not yield search. That is the only reason not to increase the price is due to the possible constraint on buyer valuation (which can be x or f (x)). We have now bounded a possible equilibrium to the one in Proposition 1 from below. Namely, we have the following. Corollary 1. The equilibrium strategy pe0 (x) in the Proposition 1 above has the lowest price of any price given signal x in possible in any equilibrium. Proof. Since, the equilibrium strategy pe0 (x) in defined the Proposition 1 is such that pe0 (x) = min{pe0 (f (x)) + 2s, f (x)}, the conclusion follows by induction. An interesting thing to notice at this point is that in the lemmas above, and therefore, in the above corollary, we did not use the fact that there are two sellers. That is, the above corollary holds for an arbitrary number of sellers, and therefore, the equilibrium of Proposition 1 is the equilibrium with the lowest prices regardless of the number of sellers. I will further prove that with two sellers, it is a unique equilibrium. For the uniqueness result, it remains to prove that an equilibrium price pe (x) satisfies pe (x) ≤ pe (f (x))+2s to prove that the equilibrium strategy pe0 (x) is a unique pure strategy. I proceed with the following lemma. 26

Lemma 4. A pure strategies equilibrium price schedule pe (x) satisfies: pe (x) ≤ pe (f (x)) + 2s. Proof. Suppose pe (x) > pe (f (x)) + 2s. Consider two cases: pe (x) ≤ f (x) and pe (x) > f (x). In the first case, buyers with valuation f (x) when seeing price pe (x) will search for a better price, and hence the expected demand from them is 1/8 (their expected mass is 1/2 and with probability 1/2 they will find better price, and if they don’t, their demand will be split between the two sellers). But undercutting the price by an infinitesimal amount, will bring the expected demand from buyers with valuation f (x) to 1/4 (in the case the other seller has signal x, and the buyer valuation is f (x), all of them will buy from this seller). Therefore, it is beneficial to undercut the price, and hence, the price pe (x) ∈ (pe (f (x)) + 2s, f (x)] can not be a part of a pure strategy equilibrium.15 In the second case, the expected demand from buyers with valuation f (x) is 0, since the price is above their valuation, and the expected demand from buyers with valuation x is 1/2 (since by the price condition from Lemma 2, they will not search if they see price pe (x) or a little more), hence pe (x) = x. The total expected demand can be increased to 1/2 by lowering the price to f (x). The increase by 1/4 is due to the expected mass 1/2 of buyers of low valuation (f (x)) and probability 1/2 that the price at the other seller is high. Hence the seller with signal x given the equilibrium strategy with pe (x) = x faces profit of x/4 if chooses price pe (x) = x, and profit f (x)/2 > x/4 if it chooses price f (x). Therefore, pe (x) = x is not a part of an equilibrium. The last lemma together with the corollary right before it conclude the proof of uniqueness of the pure strategy equilibrium. Note that if the number of sellers is large (or infinite), then there could be equilibria where the price follows one in Proposition 1 for some time, then jumps to x, then continues to follow the iterative formula of Proposition 1. That is there could be a discontinuous increase in the price schedule at some point x. This leads to the observation that the price may increase as the number of sellers increases due to the opening of the possibility of equilibria with higher prices. Lemma 5. An equilibrium price in pe (x) can not be such that buyers with valuation x search. The possibility of a mixed strategy equilibrium with highest price pe (x) < f (x) will be inspected and rejected later. 15

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Proof. Buyer of valuation x knows that a firm’s signal can be only x or g(x), and therefore, may search only if it hopes to find a lower price in pe (g(x)). Therefore, by Lemma 2, pe (x) can not be a single price. Consider a seller setting price pe (x). Why isn’t it optimal to set a bit higher price? It must be that due to search or valuation constraints, some buyers will be lost. Since buyers of valuation x can not be lost at pe (x) (again by Lemma 2), it must be buyers of valuation f (x). Now, suppose there is a price p ∈ pe (x) after which buyers with valuation x will search. Then at that price and above, all the buyers that consider buying, will search. As they search, they are informed about the prices, and hence by the perfect Bertrand competition argument, sellers will undercut each other. Hence such a price is not set at all. Lemma 6. If pe (x) is not a single price, then for all p ∈ pe (x) that are above pe (x), buyers of the low valuation will not buy without searching. Proof. Since the possible buyers of high valuation do not search, if the buyers of low valuation would not search at a price slightly above pe (x), the price could be raised without a decrease in the expected demand. The following lemma strengthens Lemma 2 using Lemma 5 Lemma 7. pe (g(x)) ≥ pe (x). Proof. Suppose the contrary: for a certain x, we have pe (g(x)) < pe (x). It was proven that buyers of valuation x will not search at any price in pe (x), and therefore will not search at a price a little bit above pe (g(x)), and buyers of valuation g(x) will not search at pe (g(x)), since all the prices in pe (g(g(x))) are at least pe (g(x)). Hence, it is beneficial to raise the price above pe (g(x)). A contradiction. Lemma 8. If any pe (x) is not a single price, there is x such that pe (x) is not a single price, but pe (g(x)) is a single price. Proof. At the high price pe (x) of a mixed strategy equilibrium, the expected demand is 1/4+α, where α is the expected demand from higher valuation (x) buyers who searched first at the competitor, saw a price higher than pe (x) (i.e. the competitor must have received signal above x), and decided to search. α is either zero (if buyers don’t search) or 1/8 (if they do). The demand at the lowest price is 1/2 + α. Hence, at the higher price, the demand is (1/4 + α)/(1/2 + α) ≤ 3/5 lower, which means the higher price must be at least 28

5/3 times the lower one. With the previous lemma, it means that if the statement of this lemma is not true, the price must multiply by at least 5/3 in each step of the sequence of prices for x, g(x), g(g(x)), . . . . However, since as x −→ V , f (x) has to decrease, we have that for sufficiently large x, f (x)/x > 2/3. This means that the sequence of points x, g(x), g(g(x)), . . . does not increase as fast as a geometric sequence of rate 5/3. This is a contradiction with the obvious necessary seller optimality condition pe (x) < x. I now prove that there are no mixed strategy equilibria by contradiction. Suppose there is one. Let pe (x) be an equilibrium price strategy with mixed pricing (more than one price with positive probability) for at least one x. Further, let x be such that pe (x) is mixed strategy, whereas pe (g(x)) is a single price. This means that buyers of valuation x that see price pe (g(x)) do not search. Also, buyers that see (after or before search) price in pe (f (x)), will not buy at a price in pe (x) (since pe (x) ≥ pe (f (x))). The distribution, by Lemma 5 pe (x) must be also such that buyers with valuation x would not search. First, consider the case when pe (x) ≤ f (x). Then the expected demand from the high valuation (buyers with valuation x) buyers is always 1/4 (they don’t search and they have mass of 1/2 per seller with probability 1/2). The total expected demand from the low valuation buyers is 1/2, and it goes to the seller with signal f (x) if such exists (probability that the other seller has signal f (x) given that the buyer with valuation f (x) is in the market is 1/2), and otherwise it goes to the seller with signal x that has the lower price. Denote the CDF of pe (x) by F (p). Then the expected demand from low valuation buyers at price p ∈ pe (x) is (1 − F (p))/4. Let p = pe (x), then the indifference principle applied to the mix pe (x) now yields16 π(p) =

p (1 + (1 − F (p)))p = . 4 4

Hence, p F (p) = 2 − , for p ∈ p



 p ,p . 2

Given the distribution above, the buyer with valuation x facing the price p has the following benefit of search: Z 1 − ln 2 1 p (p − p)p dp = p. 2 p 2 2 p/2 16

This is exactly like in the model in Varian (1980) with mass of loyal buyers 1/4 per seller, and total mass of 1/4 of switchers.

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Hence, the buyer with valuation x will search if p>

2s , 1 − ln 2

which means that p ≤ 7s. Also note that the lower price of the distribution (p/2) must be at least pe0 (x). This is a contradiction, since pe0 (x) ≥ 4s (since x0 ≥ 9s and a mixed strategy equilibrium with p ≤ f (x) is, obviously, not possible for x ≤ x0 ). It remains to consider the case when pe (x) = x. Then, similarly to the above, we have a mass of 1/4 of “loyal” buyers with valuation x and a mass of 1/4 of “switchers” with valuation f (x). The difference of this case from the previous one is that the switchers have lower valuation than the “loyal” buyers. Similar derivation yields that the CDF F (p) of pe (x) is 2 − x/p for p ∈ (x/2, f (x)). The probability of p = x in this distribution is (x/f (x) − 1). The expected benefit of search by buyers with valuation x that see the price x is f (x)

 x (x − p) d 2 − p p=x/2 Z f (x) x 1 (x − p) 2 dp = 2 x/2 p   x 2f (x) − f (x) ln 2fx(x) − x = . 2f (x)

1 EB(x) = 2

Z



Equating the above benefit of search to the cost of search, one finds that x at which the mixed pricing is possible are such that EB(x) ≤ s. Let f (x) = bx by the definition of b. Then the above formula for EB(x) can be rewritten as   1 EB(x) = x ln2b + − 2 /2. b Therefore, the necessary condition that the buyers of high valuation would not search given the above distribution of prices is  −1 1 x ≤ 2s ln2b + − 2 . b Since, by assumption, b > 2/3, we have that the above inequality on x implies x < 11s (since the value of the right hand side is below 10s for b = 2/3 and is decreasing in d). However, if x < 10s then the lower bound on the equilibrium price (Corollary 1) pe0 (x) is larger than x/2, and therefore, the lower price (x/2) in the above mixed price strategy 30

is suboptimal. This is a contradiction. Therefore, no x exists such that pe (x) is a mixed strategy. This completes the proof of uniqueness. Note, if the number of sellers was very large, the average price in a mixed strategy (according to the Varian model with many sellers) would increase to the high price, therefore buyers of high valuation would have a lower benefit of search. Therefore, with sufficiently many sellers, a mixed strategy equilibrium would exist. By Corollary 1 it would then imply prices higher than in the case of two sellers. That is with a higher number of sellers, the price may increase.

Proof of Proposition 5 The proof that the price strategy is an equilibrium strategy is very similar to the proof that pe0 (x) was the equilibrium strategy in Proposition 1. Indeed, the proof that the buyer strategy is optimal given the seller pricing word for word repeats the corresponding part in the proof of Proposition 1, with f (x) replaced by pm (x) and pe0 (·) replaced by peh (·). The proof that the seller pricing is optimal for the seller is similar to the corresponding part in the proof of Proposition 1, except that the expected buyer demand is such that the demand is equal to half the expected monopoly demand if p ≤ peh (x), and is equal then half the expected monopoly demand from the lower valuation consumers only if p > peh (x) (since in the latter case, and if the valuation is lower of the possible two, buyers will search and not come back, whereas no consumers that see the price of the other seller to start with search if the other seller follows the equilibrium strategy). That is, the demand is not more than half the monopoly demand if the price is set higher than the monopoly price. Therefore, the monopoly price is optimal.

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