High profit equilibria in directed search models∗ Gábor Virág† August 2010
Abstract We consider a model of directed search where the sellers are allowed to post mechanisms with entry fees. Regardless of the number of buyers and sellers, the sellers are able to extract all the surplus of the buyers by introducing entry fees and making price schedules positively sloped in the number of buyers arriving to their shops. This is in contrast to results that are achieved for large markets under the assumption that sellers cannot in‡uence the utility of any particular buyer (market utility assumption), in which case buyers obtain strictly positive rents. If there is a bound on the prices or on the entry fees that can be charged, then the equilibrium with full rent extraction does not exist any more, and the market utility assumption is restored for large markets.
∗ I am grateful to Paulo Barelli, Jan Eeckhout, Stephan Lauermann and various audiences for their useful comments and the University of Michigan for its hospitality. All remaining errors are mine. † University of Rochester, Economics Department, 222 Harkness Hall, NY14627, e-mail:
[email protected]
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1
Introduction
There is a large and growing literature on labor and good markets with search frictions. To capture competition in such markets, a recent literature studies directed search models where the sellers post mechanisms and each buyer, after observing those mechanisms, chooses which seller to visit. If more buyers visit a seller than his available quantity, then the seller has to ration the buyers, which introduces frictions into the model. In this framework Burdett, Shi and Wright (2001) allow firms to post only fixed prices, while Coles and Eeckhout (2003) allow sellers to post more general mechanisms. We investigate the set of equilibria if the sellers are allowed to use a wide set of mechanisms. More precisely, we study the equilibria that provides the highest profit for the sellers when they can charge entry fees, a tool not considered in the literature so far.1 The first main result of the paper states that if arbitrarily large entry fees and small prices are allowed, then there are equilibria where the buyers achieve zero utility regardless of the number of buyers and sellers. The intuition is that the sellers can find an equilibrium where attracting additional customers is very costly for each seller, because the utility of the buyers increases very quickly when a deviating seller attempts to induce extra traffic at his store. Suppose that the other sellers charge low prices when only a few buyers visit their store. If a single seller deviates, then each buyer hopes that the other buyers make the switch (to the deviator), and he makes a large utility gain by staying with a non-deviating seller. We show that a price schedule that increases in the number of buyers visiting sustains high profit equilibria by making deviations for each seller costly, but the profit itself is reaped by charging positive entry fees. To establish this result, we use a key technical contribution: Lemma 1 characterizes how equilibrium utilities of buyers respond to the deviation. This result provides a compact characterization result utilizing the fact that when a single seller deviates, his profit depends only on the traffic he induces at his own site, and the mechanism of the other sellers, but not on the exact mechanism he posted. This insight not only allows us to characterize the set of equilibria in a relatively straightforward way, but also stresses that by choosing the non-deviators’ mechanism in an appropriate way (as described below), any deviation can be made unprofitable, and high profits can be sustained in equilibrium. This result is also interesting in the context of the general search literature as it has repercussions for the Diamond paradox. Diamond (1971) shows that if sellers are contacted sequentially by the buyer, then each seller can use his 1 Charging an entry fee amounts to allowing that a buyer pays a positive entry free and not obtain the object at the same time. A shopping club membership fee is a prime example, if stockouts may prevent buyers to take advantage of membership discounts.
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temporal monopoly to actually charge a monopoly price, which becomes an equilibrium outcome. The search literature then concludes that sellers need to directly compete in the sense that the buyer has to have at least two offers at his disposal at the same time. Butters (1977) and Burdett and Judd (1984) are early examples that study simple price competition between sellers. The directed search literature has also studied the case where the sellers can only post prices, and showed that in this case the buyer’s side captures a positive share of the surplus (see for example Peters (1984,1991)). However, our result shows that the full rent of the buyers may be extracted even if they have several offers on their disposal at the same time, if the set of mechanisms to choose from is very large. As we show below, this result is overturned only if the set of available mechanism is restricted by some price or entry fee constraints. The second main result of the paper is a reinterpretation of the first: even in arbitrarily large markets there are equilibria that behave very differently than the ones identified in works using the market utility approach. In his seminal article, McAfee (1993) proceeds under the assumption that each deviating seller takes the market utilities (of the buyers) as given. Under this assumption he shows that if the sellers can post any mechanism, then it is an equilibrium that all sellers post second price auctions with reserve price equal to their marginal costs. Peters (1997) extends this insight to the case of heterogenous sellers relying on the assumption of large markets, but relaxing the market utility assumption. Our result is in stark contrast with these papers, since we show that the limit of some equilibria of finite markets may not correspond to the equilibrium identified by those articles. The reason for this discrepancy is that if the sellers may post mechanisms with unbounded prices, then regardless of the market size a deviating seller may have a non-negligible effect on the utility of the agents. In particular, if the sellers use mechanisms where the price is positively related to the number of buyers visiting (as described above), then as the slope of the price schedule becomes unbounded the utility effect of a single seller’s deviation becomes unbounded too, undoing the general tendency of large market size to dampen the utility effect of a single seller’s deviation. To explore this intuition further, we inquire the effect of constraints on the entry fees or prices that can be charged.2 We show that if there are such constraints, then the unique seller optimal equilibrium involves positive entry fees and price schedules that positively depend on the number of buyers who showed up at a given shop. More importantly, we confirm that if prices and entry fees are bounded, then as the market becomes infinitely large, the equilibrium payoffs converge to the one obtained by Peters (1997) for infinite markets (which is the same as the competitive outcome studied by Montgomery 2 The rationale for such constraints may also come from the fact that in many situations the buyers (or the sellers) cannot make too high a loss - if any - ex-post, so one may need to impose constraints on the entry fees (or prices charged).
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(1991) and McAfee (1993)). The intuition is that in large market it becomes costless for each seller to induce extra traffic, because a seller’s influence on the utility of the buyers is negligible in the limit when prices and entry fees are bounded. The paper is organized as follows: Sections 2 and 3 describe the model and provide basic analysis, while Section 4 deals with various price and entry fee constraints, and relates those to large market analysis. Section 5 concludes. 1.1
Literature review
This work belongs to the directed search literature initiated by Peters (1991) and built upon by Burdett, Shi and Wright (2001), Shi (2009) and Coles and Eeckhout (2003). Burguet and Sakovics (1999) consider competing second price auctions where the choice variable is the reserve price. Virag (2007) shows that when general mechanisms are allowed in the case of incomplete information, auctions could form an equilibrium (with an appropriate reserve price), but not posted prices. The closest paper to ours is Geromichalos (2008) who establishes results for the case of positive entry fees and large markets among others. He concentrates on (constrained) efficiency, entry and production, but not on rent extraction and equilibria that do not converge to the one obtained under the assumption of market utility. There is a growing literature on models of large markets, and how equilibria of finite markets behave in the limit. Besides the already described McAfee (1993) and Peters (1997) articles, Peters and Severinov (1997) show that if there is a pure strategy in a competing auctions model, then it must converge to the equilibrium of the infinite economy. Virag (2010) shows that this holds for all equilibria, possibly mixed, if the sellers are homogenous. Finally, Camera and Selcuk (2009), Kultti (1999) and Eeckhout and Kircher (2010) study infinite economies making a direct use of the fixed market utility assumption.
2 2.1
Model and preliminary analysis Setup and equilibrium
There are buyers and sellers, with ≥ 2. Each buyer has a unit demand and reservation price 0 and each seller has one unit of the indivisible good. Production costs are normalized to 0 without loss of generality. Each seller maximizes his expected revenue. A buyer’s utility is − if he obtains the object and pays and − if he does not obtain the object and pays . Each buyer maximizes his expected utility. There are two stages of the game considered. First, the sellers simultaneously post their pricing mechanisms that belong to a set described below. After observing these mechanisms each 4
buyer decides which one of the sellers to visit (mixed strategies are allowed). Finally, the goods are allocated and payments are made according to the rules of the mechanisms posted by the sellers. Now, we turn to the set of admissible mechanisms. We discuss why we concentrate on this particular set after the strategies and the equilibrium are defined. First, we assume that if buyer shows up at seller , then each buyer obtains the good with probability 1. We also assume that each seller posts a mechanism that consist of + 1 elements: = (1 2 ) where is the price paid by the buyer who purchases the good if buyers showed up at seller and is the entry fee to be paid to the seller by every buyer who showed up at seller . A buyer who visits seller and does not obtain the good pays only the entry fee . The only restriction on possible prices is the one that follows from the ex-post participation constraint of the buyers, i.e. that ≤ for all .3 (This also means that negative prices are allowed!) This restriction reflects a basic participation constraint: a buyer cannot be forced to pay more than the object is worth, otherwise he would walk away. Formally, define the (common) strategy space of each seller as Γ = {((1 2 ) : ≤ ∈ R}. The strategy of each buyer is described by function: : Γ →M where M is the dimensional unit simplex. The interpretation is that ( 1 2 ) ∈ [0 1] is the probability with which buyer visits seller = 1 2 given the announced mechanisms (1 2 ). Of course, it holds that 1 + + ≤ 1 because each buyer visits (at most) one seller. We concentrate on symmetric subgame perfect equilibria: Definition 1 A symmetric subgame perfect equilibrium is vector ∗ = (∗1 ∗2 ∗ ∗ ) and strategy , such that i) the buyers use symmetric strategies: = for all ii) given the buyers’ strategies , the mechanism ∗ is optimal for each seller if all other sellers announced ∗ , iii) it holds that ( ∗ ∗ ∗ ) = 1 for = 1 2 . 2.2
Discussion
Since this paper studies properties of equilibria in directed search models, it is in order to discuss the set of admissible mechanisms, and the equilibria we concentrate on in relation to the existing directed search literature. This literature introduces several anonymity and symmetry assumptions to emphasize the anonymity of the markets. First, the literature assumes that the mechanism posted by each seller must treat all buyers symmetrically. Second, it also assumes that the buyers follow identical strategies for any mechanisms posted 3 This assumption is imposed by the previous literature. In our model it is without loss of generality, as one can always decrease all the prices and increase the entry fee by the same amount, without changing the incentive properties of the mechanism.
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by the sellers. Third, most of the papers with homogeneous sellers concentrate on equilibria where the sellers post identical mechanisms. Fourth, most of the literature concentrates on the case where the sellers are not able to use mechanisms like the meet-the-competition-clause, that directly base the allocation on the mechanisms posted by the other sellers.4 Once one assumes that the sellers cannot distinguish between the buyers and cannot directly base their offered allocation on the mechanisms posted by the other sellers, it is without loss of generality to concentrate on the mechanisms that we focus on. Given the anonymity and symmetry assumptions summarized above, our definition of equilibrium arises naturally in the context of mechanisms studied. It is also important to note, that the equilibrium we concentrate on remains an equilibrium if the strategy space of the sellers is expanded to include arbitrary mechanisms. In particular, we show it later that if all the other sellers use the mechanism ∗ , and ∗ is a best response in the class of admissible mechanisms above (Γ), then there is no profitable deviation even if a very large set of mechanisms (even including mechanisms like the meet-the-competitionclause) is allowed. This follows, because the payoff of the deviating seller only depends on the visiting probability that seller receives after his deviation, but not on the exact form of the deviation, and we show that no visiting probability yields a higher profit, then inducing a visit with probability 1. 2.3
Preliminary analysis
Consider a deviation by seller to that induces the buyers to visit seller with probability ∈ [0 1]. First, it might be that after the deviation seller is visited with zero probability by each buyer ( = 0), but this cannot be profitable for that seller, since then he achieves no revenues. Therefore, we only need to consider deviations after which the seller is visited by the buyers with a positive probability, that is where 0. The Lemma below establishes that the mechanism offered by seller influences buyers’ utilities only through , but the details of the mechanism announced is otherwise unimportant. To formally state our result, define function as ( ) = 1 ( )( − ∗1 ) + + ( )( − ∗ ) − ∗ where ( ) = Then one obtains:
(1)
( − 1)! 1 1 − − 1 − −1 ( ) (1 − ) ( − 1)!( − )! − 1 −1
4 A notable exception is Epstein and Peters (1999). Coles and Eeckhout (2003) also considers this case, as we discuss it below.
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Lemma 1 Suppose that all sellers other than follow strategy (∗1 ∗2 ∗ ∗ ) and that if two sellers use the same strategy, then buyers visit them with equal probability and thus for all 6= it holds that = . If seller is visited with probability ∈ (0 1) by each buyer and each buyer participates with probability 1, then the utility of the buyers can be written as ( ) = ( ). For = 1, the utility of the buyers satisfies (1) ≥ (1). Proof. Since the buyer is indifferent between visiting this seller and any other seller after any deviation, ( ) is equal to the utility of the buyer from visiting any seller 6= . By symmetry among sellers other than seller is visited . Let ( ) denote the probability of the event that with probability e = 1− −1 − 1 ∈ {0 1 2 − 1} buyers other than buyer showed up at seller and buyer is served. By construction, the utility of the buyer from visiting seller is equal to ( ) = 1 ( )( − ∗1 ) + + ( )( − ∗ ) − ∗
(2)
By anonymity, if − 1 other buyers showed up then buyer obtains the object with probability 1 and therefore, ( ) =
( − 1)! 1 e−1 (1 − e)− ( − 1)!( − )!
(3)
Since such a probability is a function of only, but not the price vector of seller directly, the proof is complete for the case where 1. When = 1 holds then the buyers weakly prefer to visit seller and thus the utility offered is weakly higher than the one offered by the other sellers. Lemma 1 is our main vehicle for calculating equilibria, and it is a useful technical contribution to the directed search literature. The result provides a tractable way to characterize how the utility of the agents (and consequently the revenues of the sellers) respond to efforts of a single deviating seller to increase the traffic at his site. Previous works like McAfee (1993) relied on the assumption that each deviating seller takes the market utility as given. Peters (1997) considers a general setup with seller heterogeneity, and allows general mechanisms without imposing the market utility assumption. However, instead of considering exact equilibria he works with profit functions that are exact only when the market becomes large. Only papers studying the case where the set of mechanisms were restricted5 (like Peters (2000), Burdett, Shi and Wright (2001), Coles and Eeckhout (2003) and Virag (2010)) could circumvent the difficulties arising from the fact that when a seller deviates the equilibrium utility changes. The following Lemma characterizes best replies: 5 As we remarked in Section 2.2, the generality of our result also allows us to study any deviations. Showing this is not considered formally, since the extra notation needed would be too burdensome and would distract us from the main points.
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Lemma 2 Suppose that all sellers other than seller post identical mechanisms (∗1 ∗2 ∗ ∗ ) such that ∗1 ≤ ∗2 ≤ ≤ ∗ ≤ Then it is optimal for seller to induce a visiting probability ∗ that satisfies ∗ ∈ arg max{[1 − (1 − ) ] − ( )}
(4)
Proof. In the Appendix we prove that for any deviation of seller all sellers other than must be visited with the same probability by the buyers, and thus one can use Lemma 1 to calculate the utility of the buyers. The risk-neutrality of all players pins down the expected revenue of seller , which can be written as the difference between the total expected surplus generated by seller and the total expected utilities the buyers reap from visiting seller . The total surplus generated by seller is the probability he sells his unit times , which becomes = [1 − (1 − ) ].
Note, that the sum of expected utilities of the buyers at seller is equal to ( ) where ( ) denotes the expected utility of buyer conditional on visiting seller , calculated above. Therefore, the expected revenue of seller simplifies to the following expression ( ) = [1 − (1 − ) ] − ( )
(5)
for all ∈ (0 1). For = 1 it holds by Lemma 1 that
( ) ≤ [1 − (1 − ) ] − ( )
Therefore, if seller maximizes {[1−(1− ) ]− ( )} and the solution satisfies ∗ 1, then he cannot profitable deviate to = 1, since such a deviation would yield a revenue of no more than [1 − (1 − 1) ] − ∗ 1 ∗ (1) which was revealed to be no more than [1−(1− ∗ ) ]− ∗ ( ∗ ) If the best response is such that ∗ = 1, then seller can make sure to actually reach the upper bound [1 − (1 − 1) ] − ∗ 1 ∗ (1) by setting an appropriate entry fee. Therefore, if ∗ = 1, then choosing = 1 is optimal. To complete the proof, we prove it in the Appendix that seller does not have a profitable deviation that induces buyers to stay at home with a positive probability. The following Corollary proves useful when checking sufficient conditions for a symmetric equilibrium: Corollary 1 Suppose that under the assumptions of Lemma 2 ∗ = 1, ( 1 ) ≥ 0 hold, and (1) ≥ 0 Then it is an equilibrium outcome in the game that all sellers post mechanisms (∗1 ∗2 ∗ ∗ ) and buyers visit them with equal probability, i.e. for all = 1 2 it holds that = 1 . 8
As it was noted at the end of Section 2.2, this result implies that the equilibrium we concentrate on remains an equilibrium if the strategy space of the sellers is expanded to include arbitrary (even including mechanisms like the meet-the-competition-clause) mechanisms.
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Full rent extraction
Our focus is to characterize the equilibria where full rent extraction occurs, that is the buyers make zero utility in equilibrium. In this case ( 1 ) = 0 and the first order condition of seller ’s problem (see Lemma 2 above) | 1= 0 =
(6)
can be written as
1 1 0 1 (1 − )−1 = ( ). (7) By choosing (∗1 ∗2 ∗ ) appropriately it is always possible to satisfy (7) and then one can choose the entry fee ∗ so that the utility of each buyer (as defined by equation (1)) satisfies ( 1 ) = 0. Therefore, an equilibrium with full rent extraction exists for any and . Moreover, it is sufficient to consider auctions with reserve price and entry fees to obtain such an equilibrium: Theorem 1 For any 1 there exists an equilibrium with sure buyer participation where the buyers obtain zero utility in equilibrium and each seller charges the same prices (∗1 ∗2 ∗ ∗ ) with ∗2 = = ∗ = and ∗1 = (1 − and ∗ =
( − 1)2 ) −1
( − 1)2 1 (1 − )−1 −1
(8)
(9)
Proof. Using (7) and ∗2 = = ∗ = implies that in a zero utility equilibrium it holds 1 1 1 0 1 1 (1 − )−1 = ( ) = 01 ( )( − ∗1 ) Similarly, by the zero utility condition we obtain 1 1 ( ) = 1 ( )( − ∗1 ) − ∗ = 0 Using
1 1 1 ( ) = (1 − )−1 9
(10)
(11)
−1 1 1 (1 − )−2 01 ( ) = −1 implies formulas (8) and (9). Those two equations are also sufficient for a zero utility equilibrium if the second order condition for the optimization problem 2 of seller holds. For this it is sufficient to show that ( )2 ≤ 0 for all ∈ [0 1]. To calculate this value, note that = [1 − (1 − ) ] − ( ) and Therefore6 ,
( ) = 1 ( )( − ∗1 ) − ∗
2 0 00 = −( − 1)(1 − )−2 − 2 ( ) − ( ) = 2 ( ) = −( − 1)(1 − )−2 − 201 ( )( − ∗1 ) − 001 ( )( − ∗1 ) 0
after using (3), which implies that 01 ( ) 001 ( ) ≥ 0 for all . The specific (auction) mechanism used by the other firms is chosen not only to provide zero utility to the buyers if the first order conditions for optimality hold. The auction mechanisms also satisfy global optimality condition for the seller’s optimization problem, that is convexity of function . As we will see in Proposition 1, one can choose any mechanism such that crowding is costly for the sellers (that ≥ for all ), and global optimality conditions would still hold, but a proof is much harder (see Claim 2 in the Appendix). To illustrate this result, consider the case of 2 buyers and 2 sellers, i.e. let = = 2. The above equilibrium then becomes: ∗1 = 0 ∗2 = and ∗ = 2 . In contrast, the equilibria considered by Coles and Eeckhout (2003) are of the form: ∗1 = 2 ∗2 ∈ [0 ] and ∗ = 0 and in all of those equilibria the buyers achieve positive utility. When the sellers can charge entry fees it becomes possible to extract the full surplus of the buyers without explicitly colluding. To build intuition, let us consider the first order condition of the sellers problem, 1 0 1 1 1 (12) (1 − )−1 = ( ) + ( ). This equation implies that to generate the lowest possible equilibrium utility of the buyers ( ( 1 )) one must maximize the sensitivity of that utility to the 0 strategy of seller ( ( 1 )). Intuitively, the larger this sensitivity is, the more extra utility seller must provide to the buyers if he wants to achieve a higher traffic . This makes deviation less profitable, increasing equilibrium revenues. The only question remaining then is how to maximize this sensitivity. 6 We
assume below that ≥ 3, but the proof goes through with a minor modification for = 2.
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If the probability that seller is visited changes, then the utility of each buyer 0 (− ) changes according to ( ) = (−1 )− 2 2 when = = 2. By choosing 0 a small 1 and a large 2 , one makes ( 1 ) large and achieves an equilibrium with high revenues. It is a general feature that the slope of the utility function 0 ( ( 1 )) is more positively sensitive to prices charged when only a few buyers visit a seller. The reason is that when the probability that seller obtains a visit increases, it increases the probability that seller 6= obtains only a few visitors. Adopting a price schedule that has a low value for 1 implies that when seller increases then a buyer finds it much more profitable to visit seller 6= , since it becomes more likely that he obtains a stellar deal being the only buyer there. Thus seller can increase his visiting probability only at a high cost, which then supports any allocation with high profits. Theorem 1 has important consequences for large markets, as it implies that regardless of how large the market is, there is always an equilibrium where the sellers extract all the surplus of buyers. This non-competitive equilibrium is in stark contrast with the results of McAfee (1993) and Peters (1997, 2000). McAfee (1993) considers a large market where the market utility assumption is directly applicable, that is when a seller deviates he can take the utility of the buyers as given. He shows that if the sellers are allowed to post any mechanism, then the equilibrium is such that all sellers post a second price auction with a reserve price equal to their marginal cost. Peters (1997) obtains a similar result with heterogenous sellers, without explicitly imposing the market utility assumption, but using an approximation when calculating utilities that is not fully exact for finite markets. Our result shows that the approximations used by these articles are not valid, if the sellers can post mechanism with unbounded prices (or entry fees). Formally, consider the equation (12) and let 0 go to infinity. In this case as long as ( 1 ) remains bounded, the utility of the agents converges to (1 − 1 )−1 , which is equivalent to the market utility assumption. However, if prices and entry fees can be chosen at arbitrary 0 levels, then ( 1 ) may approach infinity,7 and the utility of the agents can be depressed all the way to zero. The analysis of McAfee (1993) and Peters (1997) then implicitly assumes that the mechanism space does not allow unbounded prices. Peters (2000) considers the case of price posting only, and shows that the market utility assumption holds in the limit. This is not surprising, since 0 posted prices are always between 0 and , and thus ( 1 ) remains bounded. Finally, Virag (2010) obtains a similar result for competing auctions, which can then be explained similarly as the Peters (2000) result. In general, if a single price is chosen, then it must stay bounded and thus the market utility assumption must hold in the limit. Similarly, if negative prices are not al7 For
example, if 1 is chosen arbitrarily small, while is chosen arbitrarily high.
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lowed, or more generally the direction of transfers is always in one way (from the buyer to the seller), then the prices remain bounded and convergence to the "competitive" outcome occurs.
4 4.1
Constraints on prices and large markets Price and entry fee constraints in finite markets
The analysis in the previous Section shows that an equilibrium with full rent extraction exists for any market size. However, we informally argued that in such an equilibrium the sellers need to use unbounded prices and entry fees when the market gets large. Take for example the full extraction equilibrium suggested in Theorem 1. It requires that ( − 1)2 ), 1 = (1 − −1
(13)
which means that if converge to infinity at the same rate, then 1 converges to minus infinity. To formally show that unbounded prices and entry fees need to be allowed for a zero utility equilibrium to exist in large markets, we study different types of constraints. We consider the price constraint ≥ , and the entry fee ( ≤ ) constraints. Finally, we consider the case where both price and entry fee constraints are in place. To emphasize that our results carry through even to the case where no entry fee is allowed and prices must be positive, we assume that the minimum price and the maximum entry fee allowed are both zero, that is we set = = 0. To avoid trivial results, we assume throughout that −1 (−1)2 otherwise a zero utility equilibrium exists for all , because the market is very imbalanced favoring the sellers. (This can be confirmed by referring to formula (13) that describes the full rent extraction equilibrium suggested in Theorem 1.) The following Proposition collects the results: e Proposition 1 i) If − 1 ( − 1)2 then ∃( ) such that if a minimum e price of ≥ ( ) is imposed, then the unique seller optimal equilibrium is such that ∗1 = = ∗ = and ∗+1 = = ∗ = , where = (1 + −1 ). ii) There exists a threshold e( ) , such that if 0 ≤ e( ) then the buyers earn positive utility in the seller optimal equilibrium and 2 = = = = and 1 . iii) When = = 0 the seller optimal equilibrium is such that 2 = = (−1) = and 1 = (−1) 2 +(−1) . e Proof. Let us start with result i). Let ( ) be such that a zero utility e equilibrium exists if and only if ≤ ( ). To find the equilibrium that 12
offers the maximum revenues for the sellers (and lowest utility for the buyers) e when ( ), one must maximize 0 ( 1 ) as we argued above. Inspecting (2), this implies that one must choose a low price ∗ if 0 ( 1 ) 0 and a high price if 0 ( 1 ) 0. From the definition of it follows that −1 1 0 ( ) ≥ 0 ⇔ ≤ 1 + This observation suggests the strategy to the sellers that prescribes ∗1 = = ∗ = and ∗+1 = = ∗ = , where is the integer value value of 1 + −1 . Result i) follows from the above except for the second order condition for seller optimization, which follows from Claim 2 in the online Appendix. The proof of the other two results are in the Appendix. Proposition 1 implies that when the minimum price constraint binds ( ≥ e ( )) the sellers should introduce a price schedule that is positively sloped in the number of buyers at a given seller. Moreover, the entry fee in the e optimal equilibrium when ≥ ( ) is also positive.8 If there is a binding maximum entry fee constraint, then the unique seller optimal equilibrium is when sellers offer auctions, i.e. they charge a low price if and only if a single buyer visits, otherwise the price is equal to the buyers’ valuations. The result is starker than under price constraints, because there is only one low price. The cause of the divergence is that when there was a minimum price constraint once 1 reaches the lower bound, the sellers may need to use the second most effective tool they have, i.e. to start lowering 2 as well. Result iii) implies that if entry fees cannot be charged ( = 0), then full rent extraction is not possible and the unique seller optimal equilibrium is the one in auctions. 4.2
The case of large markets
Next we consider the case of large markets, where → ∞ and → ∞ We show that in any symmetric equilibrium where each seller posts a price vector (1 2 ) the utilities (( 1 )) and revenues () are independent of the equilibrium played: Proposition 2 For any fixed lower bound or maximum entry fee and any equilibrium played, the limiting utility of the buyers and the limiting revenues of the sellers are given as 1 lim ( ) = [−] →∞ → and lim
→∞ → 8 This
= {1 − [−] − [−]}
is shown in the online Appendix as Claim 3.
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Proof. See the Appendix. This result formalizes our intuition that if the prices (or the entry fee) are bounded, then the sellers cannot achieve an equilibrium with full rent extraction.9 The intuition behind Proposition 2 is that each individual seller loses his market power as the market becomes large, and the sensitivity of utilities to unilateral deviations are constrained by price and entry fee constraints. The utility characterized here is consistent with the one obtaining under the market utility assumption as in McAfee (1993) restricting his results to homogenous buyers. Peters (2000) considers the same setup as ours with price posting, and obtains the exact same result as our Proposition 2. His result can be considered as a special case of ours after observing that the prices posted in equilibrium are always bounded (between 0 and ).
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Conclusions
We showed that in a model of directed search the sellers can extract all the surplus of the buyers in a static equilibrium. Such equilibria involve a positive entry fee and positively sloped price schedules in the number of buyers arriving to their shops. As a consequence the fixed market utility assumption for large markets is invalid, if the sellers can use unbounded prices and entry fees. Only if sellers face constraints on the set of mechanisms can we arrive at the standard result that every equilibrium converges to the one under the market utility assumption. If there is a lower bound on the prices or an upper bound on the entry fee that can be charged, then the equilibrium with full rent extraction does not exist any more if the market is large. The seller optimal equilibrium involves a positive entry fee and a positively sloped price schedule implying that if the firms can coordinate on an equilibrium, then they would not choose a fixed price one (studied by Burdett, Shi and Wright (2001)) or one without entry fees (as in Coles and Eeckhout (2003)).
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Appendix
The rest of the proof of Lemma 2: Proof. First, we show that for all 6= the utility of the buyers from visiting seller is decreasing in . The probability that a buyer obtains the object at (−1)! ( )−1 (1 − )− . seller and − 1 other buyers visited is e ( ) = 1 (−1)!(−)! 9 Since the equilibria do not allow full rent extraction anymore, therefore it is not without loss of generality to concentrate on equilibria of this form. Indeed, for any market size there is an equilibrium where sellers post meet-the-competition-clause mechanisms that extracts all the rent of the buyers, even if there are constraints on the prices at which transactions take place.
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The utility of a buyer when visiting seller is e 1 ( )( − ∗1 ) + + e ( )( − ∗ ). ( ) =
e 1 ( ) + + e ( ) First, we show that 0 ( ) 0. By construction, = is the probability that a given buyer obtains the good from seller if each e 01 ( ) + other buyer visits that seller with probability . Obviously, 0 = + e 0 ( ) 0, since it is harder to obtain the object when the other buyers visit more often. Moreover, there is a cutoff value such that for all ≤ it holds that e 0 ( ) 0 and for all it holds that e 0 ( ) ≥ 0. Therefore, ∗ ∗ ∗ using that 1 ≤ 2 ≤ ≤ ≤ implies that 0 ( )(−∗ )+e 0+1 ( )(−∗+1 )++e 0 ( )(−∗ ) = e 01 ( )(−∗1 )++e ≤ [e 01 ( ) + + e 0 ( )]( − ∗ ) + [e 0+1 ( ) + + e 0 ( )]( − ∗+1 ) =
e 0 ( )]( − ∗ ) + [e 0+1 ( ) + + e 0 ( )](∗ − ∗+1 ) 0 = [e 01 ( ) + +
and thus 0 ( ) 0. This implies that for any deviation of seller and for all 6= it must hold that = , otherwise the indifference condition for the buyers does not hold. Therefore, the probability of buyer participation is + ( − 1) ≤ 1. In the rest of the proof we show that it cannot be profitable for seller to deviate in a way that induces a situation where + ( − 1) 1, i.e. where buyers opt out with positive probability. It is obviously not a profitable deviation for seller to induce an equilibrium where = 0, because then his revenue is 0, while he made a positive expected revenue in the candidate equilibrium. Moreover, the fact that buyers stay at home with positive probability implies that they make zero expected utility. Therefore, for all 6= it holds that ( ) ≤ 0. Then the fact that is decreasing and that ( 1 ) ≥ 0 implies that ≥ 1 0. Therefore, the buyers visit each seller with positive probability after any potentially profitable deviation of seller . Moreover, there is a cutoff ∗ such that for all 6= it holds that ( ∗ ) = 0 1 and ( ) 0 for all ∗ . If ∗ ≥ −1 , then no matter how low becomes it still holds that ≤
1 ∗ −1
and thus it is still profitable for the buyers to visit all the sellers other than regardless of what mechanism posted. 1 Therefore, in what follows we may assume that ∗ −1 . Now, suppose ∗ that seller posts a mechanism such that = holds for all 6= and +(−1) = 1, i.e. where buyers participate for sure but are also indifferent
15
between participating and staying home. This is equivalent to inducing a traffic for himself that is equal to e = 1 − ( − 1) ∗
The revenue of seller in this case is given as
e = (1 − ( − 1) ∗ ) = (1 − (1 − e ) ).
Now, take any deviation such that + ( − 1) 1. In this case the buyers achieve zero utility in equilibrium and thus = ∗ must hold for all 6= regardless of the exact value of . Then 1 − ( − 1) ∗ = e and the revenue of seller can be written as e . = (1 − (1 − ) )
But the requirement (4) that
e = (1 − ( − 1) ∗ ) ≤ ( 1 )
and thus ≤ ( 1 ) holds as well, which concludes the proof. Proof of result ii) in Proposition 1: Proof. If e( ), then it is easy to see that the constraint on the entry fee is binding and thus the seller optimal equilibrium has = . Then the condition for seller optimization implies that 1 1 0 1 1 (1 − )−1 = ( ) + ( ) = 1 1 = 1 ( )( − 1 ) + + ( )( − ) − + 1 0 1 1 + [1 ( )( − 1 ) + + 0 ( )( − )] (14) Consider a local deviation with 1 and ( 1) that leaves (14) satisfied. Change 1 by − and to offset this change. Then the change in is 1) 01 ( 0 ( 1 ) 1 ( )+
1 1 ( )+
. Then the change in utility is 1 ( ) = 1 ( 1 ) ( 1 )( ( 1 ) +
It can be shown that
1 01 ( ) 1 1 ( )
1 0 ( ) 1 ( ) 1 0 ( )
−
depending on whether ( 1 ) +
01 ( 1 ) 0 ( 1 ) ( 1 1 − 1 ). 0 ( ) ) 1 ( ) ( )
(15)
≥ 0. There are two cases to consider
is positive or negative. One can show that 16
0 ( 1 )
( 1 ) + ≥ 0 if and only ≤ , where ≥ . If 1 ≤ and then one can increase and decrease 1 without violating (14) and obtain a higher revenue by (15). Therefore, = for all 1 ≤ . Next, we note that in the seller optimal equilibrium 1 must hold. Otherwise, if 1 = then for all 1 ≤ ≤ it must be that = But since 0 ( 1 )
( 1 ) + 0 for all and then for some would imply using (14) that 0 a contradiction. Proof of result iii) in Proposition 1: Proof. Ignore the minimum price constraint first and use result ii) of the
Proposition to solve the relaxed program. Then after using that = = 0 one obtains that 1 1 0 1 1 (1 − )−1 = ( ) + ( ) = 1 1 0 1 = 1 ( )( − 1 ) + 1 ( )( − 1 ) = (1 − 1 )−2 1 −1 1 = (1 − ) ( − 1 ) + ( − 1 )( − 1) −1 or ( − 1) 1 = 0 ( − 1)2 + ( − 1) establishing the result. Proof of Proposition 2: Proof. The proof when the constraint is on the entry fee is very similar to the case when a price constraint is introduced and thus it is omitted. We show that each individual seller loses his market power, i.e. 1 0 1 ( )=0 →∞ → lim
holds and then (12) implies that the equilibrium buyer utilities are as claimed. To see this, note that ( ) = 1 ( )( − ∗1 ) + + ( )( − ∗ ) − ∗ and thus
0
( ) = 01 ( )( − ∗1 ) + + 0 ( )( − ∗ ). 0
Since ∗ ∈ [ ] it follows that to maximize ( ) one needs to choose ∗ = if 0 ( ) 0 and ∗ = if 0 ( ) ≤ 0, i.e. to choose ∗1 = = ∗ = and ∗+1 = = ∗ = . Therefore, 0
0 ≤ ( ) ≤ [01 ( ) + + 0 ( )]( − ) 17
(16)
To prove that lim
→∞
yields
1
lim
→∞ →
1 1 01 ( )++0 ( ) →
0
( 1 ) = 0 it is then sufficient to prove that
= 0. By formulas (17) and (19), taking derivatives
1 1 1 1 = 01 ( ) + + 0 ( ) = [e 01 (1 − ) + + e 0 (1 − )] =
(−1) −2 2
+ −3 (1 − ) (−1)(−2) + + (−1)(−) −−1 (1 − )−1 6 ! = −1 with = 1 − 1 . Substituting = 1 − 1 yields −3 (1 − ) (−1)(−2) 6 (−1) −2 2
=
−2 1− −2 = → 3 3( − 1) 3
as → . Similarly, one can establish that the next such ratio converges to 4 and so on. Then can dominate with a geometric series with 1 + 1 −2 (−1)(1− ) members and a quotient 3. Thus if the limit of the first term 2(−1) is finite, then the whole sum is finite (in the limit) and thus lim = 0. The limit of the first term
1 −2 (−1)(1− )
2(−1)
→∞ →
is
( − 1)(1 − 1 )−2 = [−] →∞ → 2( − 1) 2 lim
concluding the proof of the first statement. The equilibrium expected revenue of a seller is lim
→∞ →
=
1 1 (1 − (1 − ) ) − ( ), →∞ → lim
since in expectations there are buyers for each seller in the limit. The last formula yields the second result after calculating the limits on the right side.
References [1] Burdett, K. and K. L. Judd (1983): “Equilibrium Price Dispersion”, Econometrica 51 (4), 955-969 [2] Burdett, K., Shi, S. and R. Wright (2001): "Pricing and matching with frictions", Journal of Political Economy, 109 (5), 1060-1085 18
[3] Burguet, R. and J. Sakovics (1999): "Imperfect Competition in Auction Designs", International Economic Review 40 (1), 231-247 [4] Butters, G. (1977): "Equilibrium distributions of sales and advertising prices", Review of Economic Studies 44 (October), 465-92 [5] Camera, G. and C. Selcuk (2009): "Price Dispersion with Directed Search", Journal of the European Economic Association, 7(6), 1193—1224 [6] Coles, M. and J. Eeckhout (2003): "Indeterminacy and directed search", Journal of Economic Theory, 111, 265-276 [7] Eeckhout, J. and P. Kircher (2010): "Sorting vs Screening — Search Frictions and Competing Mechanisms", Journal of Economic Theory, 117(5), 861- 913 [8] Geromichalos, A. (2008): "Directed Search and Optimal Production", Mimeo, University of Pennsylvania [9] Kultti, K. (1999): "Equivalence of Auctions and Posted Prices", Games and Economic Behavior, 27 (1), 106-113 [10] McAfee, P. (1993): "Mechanism design by competing sellers", Econometrica 61, 1281-1312 [11] Montgomery, J. (1991): "Equilibrium wage dispersion and inter-industry wage differentials", Quarterly Journal of Economics, 163-180 [12] Peters, M. (1984): "Bertrand Equilibrium with Capacity Constraints and Restricted Mobility" Econometrica, 52(5), 1117-27 [13] Peters, M. (1991): "Ex-ante price offers in matching games: non-steady states", Econometrica, 59, 1425-1454 [14] Peters, M. (1997): "A competitive distribution of auctions", Review of Economic Studies, 64 (1), 97-123 [15] Peters, M. (2000): "Limits of Exact Equilibria for Capacity Constrained Sellers with Costly Search", Journal of Economic Theory, 95, 139-168 [16] Peters, M. and S. Severinov (1997): "Competition among sellers who offer auctions instead of prices", Journal of Economic Theory, 75, 141-179 [17] Shi, S. (2009): "Directed search for equilibrium wage-tenure contracts", Econometrica, 77(2), 561-584 [18] Virag, G. (2007): "Buyer heterogeneity and competing mechanisms", mimeo, University of Rochester 19
[19] Virag, G. (2010): "Competing auctions: finite markets and convergence", Theoretical Economics, 5 (2), 241-274
20
Online Appendix (not for publication) Claim 2 If − 1 ( − 1)2 , then the second order condition for seller optimization holds if ∗1 = = ∗ = ≤ and ∗+1 = = ∗ = where ) and ∗ is chosen such that the first order condition for seller = (1 + −1 optimization holds, i.e. equations (2) and (12) are satisfied. If −1 ≥ (−1)2 , then the second order condition for seller optimization holds if ∗1 0 and ∗2 = = ∗ = . Proof. Let us start with the observation that when = 1 it is very easy to show that the second order condition holds. This also implies that when −1 ≥ (−1)2 the suggested equilibrium satisfies the second order conditions for seller optimization. Therefore, for what follows we assume that ≥ 2 and − 1 ( − 1)2 . Since ≤ 1 + −1 and − 1 ( − 1)2 it follows that ≤ holds. Let =1− and
−2 1 − = + −1 −1 −1
(17)
() = (( − 1) − ( − 2))()
where () is the utility of each buyer if = . Rewriting the revenue function (5), using now as the choice variable yields = ( ) = (1 − (1 − ) ( − 1) ) − ( ).
(18)
The rest of the proof is conducted in two steps. First, we show that the revenue function is concave when ≤ e and characterize e. In the second step we show that any strategy such that ≥ e yields a lower revenue than the equilibrium strategy ( = 1 − 1 ) by inspecting the revenue function directly. Concavity of the revenue function requires that for all ≤ e 2 ≤ 0 2
and thus it is sufficient to have that for all ≤ e 2 ≥ 0 2
Under the assumption that ∗1 = = ∗ = , ∗+1 = = ∗ = one obtains that ( ) = (( − 1) − ( − 2)) ( ) = 21
= (( − 1) − ( − 2)){[e 1 () + + e ()]( − ) − ∗ },
where - rewriting (3) -
Therefore,
e () = ( ) =
( − 1)! 1 − (1 − )−1 ( − 1)!( − )!
(19)
X 2 X e 2 e ] = ( − )[(( − 1) − ( − 2)) + 2( − 1) 2 2 =1 =1
Because ≤ holds, for
2 2
≥ 0 to hold it is sufficient to prove that
(( − 1) − ( − 2)) Let =
X =1
X 2 e
=1
2
+ 2( − 1)
X e =1
≥ 0.
e . Then after sum algebra it follows that =
= +
−1
− (1− )
where
µ ¶ µ ¶ −2 − 2 (1 − ) + (1 − ) + + (1 − ) . 2
It is a well known result that µ ¶ Z −−1 (1 − ) . = ( − ) 0
(20)
Then taking derivatives yields that (( − 1) − ( − 2))
X 2 e
+ 2( − 1)
X e
= µ ¶ 1 −2 −1 = [2{− − (1−)−((−1) −(−2)) (1−)2 )}+ 3 (1 − ) 2 +2
=1
2
=1
2 (1 − ) + (( − 1) − ( − 2)) 2 (1 − )2 ]
Note that
µ ¶ −2 − − (1 − ) − (( − 1) − ( − 2)) (1 − )2 ) ≥ 2 µ ¶ −2 −1 (1 − ) − (1 − )2 ) ≥ 0 ≥ − − 2
−1
22
as long as ≥ 2. 2 Therefore, 2 ≥ 0 if it holds that 2
2 + (( − 1) − ( − 2)) 2 (1 − ) ≥ 0,
which is - using (20) - equivalent to 2 + (( − 1) − ( − 2))( − − 1 − ( − 1)) ≥ 0 , but fails if = 1 and ≥ 2. This equation obviously holds when = −2 −1 e such that this equation holds if and only Moreover, there exists a threshold , e if ≤ . 1 √ To conclude the first step we show that e ≥ −2 + (−1) holds. Let −1 1 −2 1 = ( − 1) − ( − 2). If = √ then = −1 + (−1)√ and −−1−(−1) = (−1)(1− )− =
√ −1 1 1 (1− √ ) −(1− √ ))− = −1
√ √ 1 1 −1 −1 −1 − )(1 − √ ) − ≥ ( − − 1)(1 − √ ) − = −1 −1 √ √ 1 −1 − 1)(1 − √ ) − ≥ −1 − . =( ( − 1) Therefore, √ 1+ = 2 + (( − 1) − ( − 2))( − − 1 − ( − 1)) ≥ 2 − √ =(
1 1 −2 √ )−1− √ ≥0 + − 1 ( − 1) if ≥ 4 and ≥ 2. Therefore, one only needs to check the cases when = 2 3 and ≥ 2. But ≥ reduces this requirement to three cases: ( = 2 = 2), ( = 3 = 2) and ( = 3 = 3) which can be checked individually to see that 1 √ + (−1) the second order condition indeed holds. This shows that e ≥ −2 −1 holds. Now, we turn to the second step. As a consequence of the above calcula1 √ + (−1) is not a tions, it is sufficient to show that choosing ≥ ∗ = −2 −1 1 profitable deviation from the equilibrium choice = 1 − . The revenue from choosing = 1 − 1 is given as = 2(
1 1 = [1 − (1 − ) ] − (1 − ). 23
Using (18), the revenue for any ≥ ∗ is such that () ≤ − ( ). Therefore, for any ≥ ∗
1 1 () − ≤ (1 − ) − [( ) − (1 − )] ∗ We showed above that is convex if ≤ . Therefore, 1 1 1 ( ∗ ) − (1 − ) ≥ ( ∗ − (1 − )) 0 (1 − ) = 1 √
−
1
1 1 1 0 (1 − ) 0 (1 − ) ≥ √ −1 ( + 1) ∗ ∗ because ≥ . For all ≥ it holds that ( ) ≥ ( ) since is increasing in . Therefore, for all ≥ ∗ =
1 1 () − ≤ (1 − ) − [( ∗ ) − (1 − )]
The first order condition for revenue maximization at = 1 −
1
becomes
1 1 0 (1 − ) = (1 − ) . Therefore, (1 − 1 ) 1 ( ∗ ) − (1 − ) ≥ √ ( + 1) and thus for all ≥ ∗
(1 − 1 ) 1 ( ) − ≤ (1 − ) − √ ( + 1)
Note, that
√ +1
≥ 1 as long as = (1 +
−1 )
≥ 2 and thus for all ≥ ∗
() − ≤ 0 which concludes the proof. Proof of Claim 3: We need to show that in the seller optimal equilibrium with price constraints (as characterized by result i) in Proposition 1) the optimal entry fee is strictly positive, that is ∗ 0. This holds, because 1 1 1 ∗ = 1 ( )( − ∗1 ) + + ( )( − ∗ ) − ( ) = 24
1 1 1 1 0 1 = 1 ( )( − ∗1 ) + + ( )( − ∗ ) − [(1 − )−1 − ( )] ≥ 1 1 1 ≥ 1 ( )( − ∗1 ) + + ( )( − ∗ ) − (1 − )−1 ≥ 1 1 1 ≥ 1 ( )( − ∗1 ) − (1 − )−1 = (1 − )−1 ∗ (−) ≥ 0 The first two equalities come from (2) and (12), while the inequalities follow 0 from the fact that ( 1 ) ≥ 0 and the restriction that ∗ ≤ . The last equality follows from the fact that ∗1 = holds in the seller optimal equilibrium and that 1 ( 1 ) = (1 − 1 )−1 .
25
