Dynamic Matching

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Econometrica, Vol. 75, No. 1 (January, 2007), 155200

DYNAMIC MATCHING, TWO-SIDED INCOMPLETE INFORMATION,AND PARTICIPATION COSTS: EXISTENCE AND CONVERGENCE

TO PERFECT COMPETITION

BY MARKS ATTERTHWAITE AND ARTYOM SHNEYEROV1

Consider a decentralized, dynamic market with an infinite horizon and participationcosts in which both buyers and sellers have private information concerning their valuesfor the indivisible traded good. Time is discrete, each period has length and, eachunit of time, continuums of new buyers and sellers consider entry. Traders whose ex-pected utility is negative choose not to enter. Within a period each buyer is matchedanonymously with a seller and each seller is matched with zero, one, or more buyers.Every seller runs a first price auction with a reservation price and, if trade occurs, boththe seller and the winning buyer exit the market with their realized utility. Traders whofail to trade continue in the market to be rematched. We characterize the steady-stateequilibria that are perfect Bayesian. We show that, as converges to zero, equilibriumprices at which trades occur converge to the Walrasian price and the realized alloca-tions converge to the competitive allocation. We also show the existence of equilibriafor sufficiently small, provided the discount rate is small relative to the participationcosts.

KEYWORDS: Matching and bargaining, double auctions, price formation, founda-tions of Walrasian equilibrium.

1. INTRODUCTION

THE FRICTIONS OF ASYMMETRIC INFORMATION, search costs, and strategic be-

havior interfere with efficient trade. Nevertheless economists have long be-lieved that for private goods economies, the presence of many traders over-comes these imperfections and results in convergence to perfect competition.Two classes of models demonstrate this. First, static double auction models in which traders costs and values are private exhibit rapid convergence to thecompetitive price and the efficient allocation within a one-shot centralized

1We owe special thanks to Zvika Neeman, who originally devised a proof that showed thestrict monotonicity of strategies. We also thank Hector Chade, Patrick Francois, Paul Milgrom,Dale Mortensen, Peter Norman, Mike Peters, Jeroen Swinkels, Asher Wolinsky, Jianjun Wu,Okan Yilankaya, and two very perceptive anonymous referees; the participants at the Electronic

Market Design Meeting (June 2002, Schloss Dagstuhl), the 13th Annual International Confer-ence on Game Theory at Stony Brook, the 2003 NSF Decentralization Conference held at Pur-due University, the 2003 General Equilibrium Conference held at Washington University, the2003 Summer Econometric Society Meeting held at Northwestern University, the 2004 CanadianEconomic Theory Conference held in Montreal, and Games 2004 held at Luminy in Marseille;seminar participants at CarnegieMellon, Washington University, Northwestern University, Uni-

versity of Michigan, Harvard and MIT, University of British Columbia, and Stanford; and themembers of the collaborative research group on Foundations of Electronic Marketplaces fortheir constructive comments. Adam Wong provided excellent research assistance. Finally, bothof us acknowledge gratefully that this material is based on work supported by the National Sci-ence Foundation under Grant IIS-0121541. Artyom Shneyerov also acknowledges support fromthe Canadian SSHRC Grant 410-2003-1366.

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156 M. SATTERTHWAITE AND A. SHNEYEROV

market. Second, dynamic matching and bargaining models in which traderscosts and values are common knowledge also converge to the competitive equi-librium. The former models are unrealistic in that they assume traders who fail

to trade now cannot trade later. Tomorrow (almost) always exists for economicagents. The latter models are unrealistic in assuming traders have no privateinformation. Information about a traders cost/value (almost) always containsa component that is private to him. This papers contribution is to formulate anatural model of dynamic matching and bargaining with two-sided incompleteinformation and to show that it converges to the competitive allocation andprice as frictions vanish.

An informal description of our model and result is this: An indivisible goodis traded in a market in which time progresses in discrete periods of length and generations of traders overlap. Each unit of time traders who are activein the market incur a participation cost and a discount rate Thus the per

period participation cost is the per period discount factor is e and theyboth vanish as the period length converges to zero. Each period every activebuyer randomly matches with an active seller. Depending on the luck of thedraw, a seller may end up being matched with several buyers, a single buyer, orno buyers. Each seller solicits a bid from each buyer with whom she is matchedand, if the highest of the bids is satisfactory to her, she sells her single unit ofthe good, and both she and the successful buyer exit the market. A buyer orseller who fails to trade remains in the market, is rematched the next period,and tries again to trade.

Each unit of time a large number of potential sellers (formally, measure 1 of

sellers) consider entry into the market along with a large number of potentialbuyers (formally, measure a of buyers). Each potential seller independentlydraws a cost c in the unit interval from a distribution GS and each potentialbuyer draws independently a value v in the unit interval from a distribution GBIndividuals costs and values are private to them. A potential trader enters themarket only if, conditional on his private cost or value, his equilibrium expectedutility of entry is at least zero. Potential traders whose discounted expectedutilities are negative elect not to participate.

If, in period t, trade occurs between a buyer and seller at price p, then theyexit with their gains from trade, v p and p c, respectively, less their partic-ipation costs accumulated at the discount rate from their times of entry on-ward. If is large (i.e., periods are long), then participation costs accumulatein a short number of periods and a trader who chooses to enter must be confi-dent that he can obtain a profitable trade without much search. If, however, is small, then a trader can wait through many matches looking for a good pricewith little concern that participation costs and discounting will offset his gainsfrom trade. This option value effect drives convergence and puts pressure ontraders on the opposite side of the market to offer competitive terms. As becomes small, the market for each trader becomes, in effect, large.

We characterize steady-state equilibria for this market in which each agentmaximizes his expected utility going forward. We show that, as the period


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EXISTENCE AND CONVERGENCE TO PERFECT COMPETITION 157

length goes to zero, all such equilibria converge to the Walrasian price andthe competitive allocation. The Walrasian price pW in this market is the solu-tion to the equation

GS(pW) = a(1 GB(pW))(1)i.e., it is the price at which the measure of entering sellers with costs lessthan pW equals the measure of entering buyers with values greater than pWIf the market were completely centralized with every active buyer and sellerparticipating in an exchange that cleared each periods bids and offers simulta-neously, then pW would be the market clearing price each period. Our preciseresult is this. Among active traders, let c and v be the maximal sellers typeand minimal buyers type, respectively, and let [p

p] be the range of prices

at which trades occur. Also let c be the smallest bid acceptable to any activeseller. As 0, then c c v, p and p all converge to the same limit p.In the steady state, the only way for the market to clear is for this limit p to beequal to the competitive price pW . That the resulting allocations give tradersthe expected utility they would realize in a perfectly competitive market fol-lows. Finally, we show that if the period length and, relative to the level ofparticipation costs , the discount rate are both sufficiently small, then a fulltrade equilibrium exists. Full trade equilibria are a special class of equilibriain which active sellers immediately trade on being matched with at least onebuyer.

This is a step toward a theory of how a completely decentralized, dynamicmarket with two-sided incomplete information and participation costs imple-ments, increasingly well, an almost efficient allocation as the speed with whichtraders are able to seek out potential trading partners increases. In makingthis step we assume independent private values, which means that all tradersa priori know the underlying Walrasian price in the market. In the theory thatwe ultimately seek, traders would have less restrictive preferences (e.g., cor-related private values or interdependent values) in which the Walrasian pricefollows some stochastic process. Efficient trade would therefore require thattraders equilibrium strategies reveal sufficient information not only to iden-tify the most valuable currently feasible trades, but also to reveal the underly-

ing, changing Walrasian price even as it simultaneously facilitates trade at thatprice. A complete theory would both identify sufficient conditions for whichconvergence to an efficient allocation and Walrasian price is guaranteed, andshow how, when those conditions are not met, the equilibrium may fail to con-verge to efficiency.

We hope that the insights and results here will contribute to the developmentof such a theory. This paper first shows that convergence to one price occursand is driven by option value: as , the length of a period, decreases, eachtrader becomes more willing to decline a merely decent offer so as to preservethe option to accept a really excellent offer in the future. With both buyers and


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sellers doing this, the price distribution in the market rapidly narrows as be-comes small. Second, it shows that the price distribution must converge to theWalrasian price because if an equilibrium exists in which it does not, then too

many traders accumulate on the long side of the market, which creates incen-tives for these traders to deviate from their equilibrium strategies by biddingmore aggressively. We would be surprised if either of these insights fails tocarry over to models with less restrictive processes for generating valuations.

This progression has been true for static double auctions. Almost all theearly papers assumed independent private values, e.g., Chatterjee and Samuel-son (1983), Myerson and Satterthwaite (1983), Gresik and Satterthwaite(1989), Satterthwaite and Williams (1989a, 1989b), Williams (1991), and Rusti-chini, Satterthwaite, and Williams (1994). Recently Cripps and Swinkels (2006)and Fudenberg, Mobius, and Szeidl (in press) have generalized rates of con-

vergence results from the independent private values environment to the cor-related private values case. Furthermore, Reny and Perry (2003), in a carefullyconstructed model with interdependent valuations, showed that the static dou-ble auction equilibrium exists and converges to a rational expectations equilib-rium as the number of traders on both sides of the market becomes large.

A substantial literature exists that investigates the noncooperative founda-tions of perfect competition using dynamic matching and bargaining games.2

Most of the work of which we are aware has assumed complete informa-tion in that each participant knows every other participants values (or costs)for the traded good. The books of Osborne and Rubinstein (1990) andGale (2000) contain excellent discussions of both their own and others contri-butions to this literature. Papers that have been particularly influential includeMortensen (1982), Rubinstein and Wolinsky (1985, 1990), Gale (1986, 1987),and Mortensen and Wright (2002). Of these, our paper is most closely relatedto the models and results of Gale (1987) and Mortensen and Wright (2002).The two main differences between their work and ours are that (i) when twotraders meet, they reciprocally observe the others cost/value rather than re-maining uninformed and (ii) the terms of trade are determined as the out-come of a full information bargaining game rather than an auction. The firstdifferencefull versus incomplete informationis fundamental, because thepurpose of our paper is to determine if a decentralized market can elicit pri-vate valuation information at the same time it uses that information to assignthe available supply almost efficiently. The second difference is natural givenour focus on incomplete information.

The most important dynamic bargaining and matching models that in-corporate incomplete information are Wolinsky (1988), De Fraja and Sko-

2There is a related literature that we do not discuss here that concerns the microstructure ofintermediaries in markets, e.g., Spulber (1999) and Rust and Hall (2002). These models allowentry of an intermediary who posts fixed ask and offer prices, and is assumed to be large enoughto honor any size buy or sell order without exhausting his or her inventory or financial resources.


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vics (2001), and Serrano (2002).34 To understand how our paper relates tothese papers, consider the following problem as the baseline. Each unit oftime, fixed measures of sellers and buyers enter the market, each of whom

has a private cost/value for a single unit of the homogeneous good. The sellersunits of supply need to be reallocated to those traders who most highly valuethem. Whatever mechanism that is employed must induce the traders to revealsufficient information about their costs/valuations so as to carry out the reallo-cation. The static double auction results of Satterthwaite and Williams (1989a)and Rustichini, Satterthwaite, and Williams (1994) show that even moderatelysized centralized double auctions held once per unit time solve this problem es-sentially perfectly by closely approximating the Walrasian price and then usingthat price to mediate trade.5

Given this definition of the problem, the reason why Wolinsky (1988), Ser-

rano (2002), and De Fraja and Skovics (2001) do not obtain competitive out-comes as the frictions in their models vanish is clear: the problems their modelsaddress are different and, as their results establish, not intrinsically perfectlycompetitive even when the market becomes almost frictionless. Thus Wolin-skys model relaxes the homogeneous good assumption and does not fully ana-lyze the effects of entry/exit dynamics. Serranos model embeds a discrete-pricedouble auction mechanism in a dynamic matching framework. There are, how-ever, no entering cohorts of traders. Consequently, the option-value effectsbecome progressively smaller as the most avid buyers and sellers leave themarket through trading and are not replaced. As the market runs down and

becomes small, necessarily it becomes less and less competitive. Not surpris-

3Butters (circa 1979) in an unfinished manuscript that was well before its time consideredconvergence in a dynamic matching and bargaining problem. The main differences between ourmodel and his are (i) he assumes an exogenous exit rate instead of a participation cost, (ii) traders

who have zero probability of trade participate in the market until they exit stochastically due tothe exogenous exit rate, and (iii) the matching is one-to-one and the matching probabilities do notdepend on the ratio of buyers and sellers in the market. We thank Asher Wolinsky for bringingButters manuscript to our attention after we had completed an earlier version of this paper.

4In a companion paper, we (Satterthwaite and Shneyerov (2003)) considered a dynamic match-ing and bargaining model that has no participation costs, but instead has the alternative friction

of a fixed, exogenous, per unit time rate of exit among active traders. For this model we showconvergence to pW as the period length becomes short, but have not been able to show existence.There are two main effects of substituting an exit rate for participation costs. First, traders whoenter with positive expected utility do not necessarily trade; they may spontaneously exit withzero utility prior to making a successful match. This makes market clearing more subtle and, asa consequence, demonstrates more clearly than this papers model the power of supply and de-mand to force price to converge to pW Second, the structure of equilibrium strategies is differentthan in the participation cost case. In particular, full trade equilibria, which play a leading role inthis papers existence proof, are easily shown not to exist.

5Another example of a centralized trading institution is the system of simultaneous ascending-price auctions, studied in Peters and Severinov (2006). They also found robust convergence tothe competitive outcome.


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ingly Serrano finds that equilibria with Walrasian and non-Walrasian featurespersist.

Closest to our model is De Fraja and Skovics model. Traders search for

the best price in a market similar to the market we study. Option value drivesthe market to one price, much as in our model. Its entry/exit specification,however, is quite different than our specification in that it does not specifythat fixed measures of buyers and sellers enter the market each unit of time asin our baseline problem. Instead, whenever two traders consummate a tradeand exit the market, then two traders of identical types replace them. Thismeans that, no matter what limiting price the market converges to as searchbecomes cheap, the distribution of traders valuations in the market remainsconstant. Consequently, if the limiting price is above the Walrasian price, thensellers do not accumulate in the market and, unlike in our model, sellers have

no incentive to reduce the prices that they ask. Supply and demand does notaffect price in De Fraja and Skovics model. Instead, the exogenously specifiedbalance of bargaining power determines the limiting price.

The next section formally states the model and our convergence and exis-tence results. Section 3 derives basic properties of equilibria. Section 4 provesconvergence for all equilibria and Section 5 proves that, for sufficiently small and the special class of equilibriafull trade equilibriaexist. Section 6concludes.

2. MODEL AND THEOREMS

We study the steady state of a market with two-sided incomplete informa-tion and an infinite horizon. In it heterogeneous buyers and sellers meet onceper period (t= 1 0 1 ) and trade an indivisible, homogeneous good.Every seller is endowed with one unit of the traded good and has cost c [0 1]This cost is private information to her; to other traders it is an independent ran-dom variable with distribution GS and density gS Similarly, every buyer seeksto purchase one unit of the good and has value v [0 1]. This value is private;to others it is an independent random variable with distribution GB and den-sity gB Our model is therefore the standard independent private values model.

We assume that the two densities are bounded away from zero: a g > 0 existssuch that, for all c v [0 1] gS( c ) > g and gB(v)>g

The strategy of a seller, S : [0 1] R {N} maps her cost c into either adecision N not to enter or a minimal bid that she is willing to accept. Similarlythe strategy of a seller, B : [0 1] R {N} maps his value v into either adecision N not to enter or the bid that he places when he is matched with aseller. A trader only enters if his expected discounted utility from doing so isnonnegative; if he elects not to enter, he receives utility zero.

The length of each period is > 0. Each unit of time, measure 1 of poten-tial sellers and measure a of potential buyers consider entry, where a > 0. This


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means that each period, measure of potential sellers and measure a of po-tential buyers consider entry. A period consists of five steps:

1. Entry occurs. A type v potential buyer becomes active only if B(v) = Nand a type c potential seller becomes active only ifS(c) =N2. Every active seller and buyer incurs participation cost , where > 0 isthe cost per unit time of being active.

3. Each active buyer is randomly matched with one active seller. Conse-quently, every seller is equally likely to end up matched with any activebuyer. The probability k that a seller is matched with k {0 1 2 }buyers is therefore Poisson,

k() = k

k!e(2)

where is the endogenous ratio of active buyers to active sellers.6 Conse-quently, a seller may end up being matched with zero buyers, one buyer,two buyers, etc. These matches are anonymous, i.e., no trader knows thehistory of any trader with whom he or she happens to be matched.

4. Each buyer simultaneously announces a bid B(v) to the seller with whomhe is matched. We assume that, at the time he submits his bid, each buyeronly knows the endogenous steady-state probability distribution of howmany buyers with whom he is competing. After receiving the bids, theseller either accepts or rejects the highest bid. Denote by S(c) the minimalbid (i.e., reservation price) acceptable to a type c seller. If two or morebuyers tie with the highest bid, then the seller uses a fair lottery to choosebetween them. If a type v buyer trades in period t, then he leaves themarket with utility v B(v) If a type c seller trades at price p, then sheleaves the market with utility p c, where p is the bid she accepts.

5. All remaining traders carry over to the next period.

Traders discount their expected utility at the rate 0 per unit time; e istherefore the factor by which each trader discounts his utility per period oftime.

To formalize the fact that the distribution of trader types within the markets

steady state is endogenous, let TS be the measure of active sellers in the marketat the beginning of each period, let TB be the measure of active buyers, let FSbe the distribution of active seller types, and let FB be the distribution of activebuyer types. The corresponding densities are fS and fB and, establishing usefulnotation, the right-hand distributions are FS 1 FS and FB 1 FB Theratio is therefore equal to TB/TS

6In a market with M sellers and M buyers, the probability that a seller is matched with k

buyers is Mk =

M

k

( 1

M)k(1 1

M)Mk. Poissons theorem (see, for example, Shiryaev (1995))

shows that limM Mk=

k.


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By a steady-state equilibrium we mean one in which every seller in everyperiod plays a time invariant strategy S() every buyer plays a time invari-ant strategy B() and both these strategies are always optimal. Let WS(c) andWB(v) be the sellers and buyers interim utilities for sellers of type c and buy-ers of type v, respectively, i.e., they are beginning-of-period steady-state equi-librium net payoffs conditional on their types. Given the friction a marketequilibrium M consists of strategies {S B}, traders masses {TS TB}, and dis-tributions {FS FB} such that (i) {S B}, {TS TB}, and {FS FB} generate {TS TB}and {FS FB} as their steady state and (ii) no type of trader can increase his orher expected utility (including the continuation payoff from matching in futureperiods if trade fails) by a unilateral deviation from the strategies {S B} and(iii) equilibrium strategies {S B} masses {TS TB} and distributions {FS FB}are common knowledge among all active and potential traders. We study per-fect Bayesian equilibria of this model. (See Definition 8.2 in Fudenberg andTirole (1991, p. 333).)

Four points need emphasis concerning this setup. First, because within agiven match buyers announce their bids simultaneously and only then does theseller decide to accept or reject the highest of the bids, the subgame perfec-tion aspect of perfect Bayesian equilibria implies that a seller whose highestreceived bid is above her dynamic opportunity cost of c + eWS(c) acceptsthat bid. In other words, a sellers strategy is her dynamic opportunity cost,

S(c) = c + eWS(c)

and is independent of the number of buyers who are bidding, i.e., S(c) is herreservation price. Second, beliefs are simple to handle because our assump-tions that there are continuums of traders, that all matching is anonymous, andthat traders values and costs conform to the standard independent private val-ues model imply that off-the-equilibrium path actions do not cause inferenceambiguities. Third, step 5 within each period requires every trader who entersto stay in the market until he eventually succeeds in trading. Obviously, giventhat our goal is modeling a decentralized market, this is inappropriate; tradersshould be free to exit. However, given independent private values and a steady-state equilibrium, forbidding exit has no loss of generality. The reason is thata trader only enters the market if his expected utility is nonnegative. Being

in a steady state implies that if he had nonnegative expected utility when heentered, then at the beginning of any subsequent period after failing to tradehe has the same nonnegative utility going forward. Even if exit without tradewere permitted, he would not do so. Fourth, an uninteresting no-trade equilib-rium always exists in which all potential buyers and sellers decline to enter. Weanalyze equilibria in which positive trade occurs, i.e., equilibria in which eachperiod positive measures of buyers and sellers enter and ultimately trade.

Let c and v be the maximal seller and minimal buyer types that choose toenter. These are the marginal participation types. Let cp and p be, respec-

tively, the lowest bid that is acceptable to the cost zero seller, the lowest bid any


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active buyer makes, and the highest bid that any active buyer makes. Formally,define AS [0 1] and AB [0 1] to be the sets of active seller and buyer types,respectively. Then

c sup{c|c AS} (maximum active seller type)(3)v inf{v|v AB} (minimum active buyer type)c = inf{S(c)|c AS} (minimum acceptable bid)

p = inf{B(v)|v AB} (maximum bid)p = sup{B(v)|v AB} (minimum bid)

Subsequent Figures 1 and 3, among other purposes, illustrate how these de-scriptors summarize an equilibriums structure.

Given an equilibrium M, we index with its components S, B, FS , FB,TS , TB, and and its descriptors c, v, c, p, and p. This notation allows

us to state our convergence result:

THEOREM 1: Fix > 0 and 0 Suppose that a > 0 exists such thatfor all (0 ) a market equilibrium M exists in which positive trade occurs.Let {c v c p p} be the descriptors of these equilibria, and let WS(c) andWB(v) be traders interim expected utilities. Then

lim0 c = lim0 v = lim0 c = lim p = lim0 p = pW(4)In addition, each traders interim expected utility converges to the utility he wouldrealize if the market were perfectly competitive:

lim0

WS(c) = max[0 pW c](5)

and

lim0

WB(v) = max[0 v pW](6)

Sections 3 and 4 prove this theorem. In Section 5, given > 0 we prove,for sufficiently small and that a special class of equilibriafull tradeequilibriaexists and results in positive trade. Formally a full trade equilib-rium is an equilibrium in which c = v Figure 1 diagrams such an equilibriumand shows each of the descriptors c, v, c, p, and p. We call these equilibria

full trade because if, in a particular period, a seller is matched with at least onebuyer, then that seller for sure trades no matter what her cost ci is and what thematched buyer(s) value vj is. In a full trade equilibrium, trade occurs as fast aspossible among active sellers and buyers. The theorem is this:


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FIGURE 1.Strategies in full trade equilibrium.

THEOREM 2: For given > 0a neighborhood X of the point (0 0)exists such

that for all nonnegative and positive in X, an equilibrium M exists in whichpositive trade occurs.

Two points deserve emphasis. First, the restriction that must be small rel-ative to implies that our existence theorem concerns situations in which par-ticipation costs, not delay per se, are the issue. Section 5.3, which discussesthe existence result, develops intuition about why the ratio is important.Second, we do not know if all equilibria are full trade or not. Conceivably forsome parameter values, a sequence of full trade equilibria may not exist, but asequence of non-full-trade equilibria in which c > v may exist. If so, conver-gence of price to pW is guaranteed because Theorem 1 applies to all sequencesof equilibria.

The intuition for our convergence result can be understood through the fol-lowing logic. In a match in which a buyer is bidding for an object, the typethat is relevant is not his static type v but rather his dynamic opportunity value

IB(v) = v eWB(v)Similarly the cost that is relevant to a type c seller is not c but her dynamicopportunity cost

IS(c)

=c

+eWS(c)


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When the buyers bid in the auction, they act as if their types were drawn fromthe density hB() ofIB(v) and the sellers types were drawn from the densityhS() ofIS(c). Because lim0 WS(c) = max[0 pW c] and lim0 WB(v) =max[0 v pW], the convergence theorem indicates that as the time periodlength 0, the distributions of the dynamic types IS(c) and IB(v) ap-proach pW and become degenerate: the distribution of sellers dynamic oppor-tunity costs concentrates just below c and the distribution of buyers dynamicopportunity costs concentrates just above v Viewed this way, as 0, thedynamic matching and bargaining market in equilibrium progressively exhibitsless and less heterogeneity among buyers and sellers until there is none andthe incomplete information vanishes. The underlying driver that causes theheterogeneity to vanish as 0 is the option value that each traders optimalsearch generates. This is the same pathway that drives convergence in the fullinformation matching and bargaining models of Gale (1987) and Mortensenand Wright (2002). In their models, as in our model, the option value that op-timal search creates causes the distributions of buyers and sellers dynamicopportunity costs to become degenerate as the friction goes to zero. Once thisis understood, our result that incomplete information does not disrupt conver-gence is natural.

Figure 2 is a table of graphs that illustrate the general character of theseequilibria and the manner in which they converge. These computed examplesassume that the primitive distributions GS and GB are uniform on [0 1] equalmasses of buyers and sellers consider entry each unit of time (i.e., a = 1), theparticipation cost is

=1 and discount rate is

=1.7 The left column shows

an equilibrium for = 010, while the right column shows an equilibrium for = 002 Traders costs and values, c and v are on each graphs abscissa. Thetop graph in each column shows strategies: sellers strategies S(c) are to theleft and above the diagonal while buyers strategies B(v) are to the right andbelow the diagonal. Because masses of entering traders are equal and theircost/value distributions are uniform, the Walrasian price is 0.5; this is the hor-izontal line cutting the center of the graph. Observe also that B(v) and S(c)are not defined for nonentering types. Comparison of the strategies in the toptwo graphs illustrates the convergence of all the descriptors {c v c p p}toward pW as decreases from 0.10 to 0.02.

The middle graph in each column shows the endogenous densities, fS and fBof active traders in the equilibrium. The density fS for active sellers is on theleft and the density for active buyers is on the right. Note that to the rightof c the density fS is zero because sellers with c > c choose not to enter. Simi-larly, to the left ofv the density fB is zero. These two densities show that bothhigh value, active sellers and low value, active buyers often have to wait before

7In computing these equilibria, we numerically solve Equations (44) and (45) from Section 5.1that characterize a full trade equilibrium. See also the discussion in Section 5.3. The Mathematicaprogram that was used is available upon request.


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FIGURE 2.Equilibrium strategies and associated steady-state densities. The left column ofgraphs is for = 010 and the right column is for = 002 The top row shows buyer and sellerstrategies S and B. The middle row graphs the densities fS and fB of the traders types, and thebottom row graphs the densities hS and hB of the traders dynamic opportunity costs and values.


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trading and, therefore, accumulate in the market. The bottom graph in eachcolumn shows the equilibrium densities hS and hB of the dynamic opportu-nity costs and values, and dramatically demonstrates how, as decreases, the

heterogeneity of traders dynamic opportunity costs and values narrows.The ex ante expected utility of a trader is

W =1

1 + a1

0

WS(c)gS(c)dc + a1 + a

10

WB(v)gB(v)dv

where the weights on sellers and buyers ex ante utilities depend on the mea-sure a of potential buyers who consider entry each period relative to the mea-sure 1 of potential sellers who consider entry each period. For the two equi-libria graphed in Figure 1, W01 = 00564 and W002 = 01088 In the limit, when

0 and the market is perfectly competitive, W00

=01250 Define the rel-

ative inefficiency per trader of an equilibrium M to be (W0 W)/W0 Therelative inefficiency of the graphed = 010 equilibrium is 0.548 and the rela-tive inefficiency of the graphed = 002 equilibrium is 0.129. This convergencetoward 0 is driven by two distinct mechanisms. First, the direct effect of cut-ting the period length, from 0.10 to 0.02 is that even if traders strategiesremained unchanged, then the decrease in reduces by a factor of 5 the waitbefore trading for all traders who are not lucky enough to trade immediatelyupon entry. Second, the gap between buyers and sellers strategies reduces,resulting in fewer traders who should tradesellers for whom c < 05 and buy-ers for whom v > 05but do not trade. Thus, when = 010 sellers whosecost c is in the interval (0347 0500) do not trade even though they would inthe competitive limit. When = 002 however, this interval shrinks in lengthby approximately a factor of 5 to (0470 05000) which results in a secondefficiency gain.

One final comment concerning the model and theorems is important. In set-ting up the model, we assume that traders use symmetric pure strategies. Wedo this for simplicity of exposition. At a cost in notation we could define trader-specific and mixed strategies, and then prove that they in fact must be symmet-ric and (essentially) pure because of independence, anonymity in matching,and the strict monotonicity of strategies. To see this, first consider the impli-cation of independence and anonymous matching for buyers. Even if different

traders follow distinct strategies, every buyer would still independently draw hisopponents from the same population of active traders.8 Therefore, for a givenvalue v every buyer will have the identical best-response correspondence. Sec-ond, we show below that every selection from this correspondence is strictlyincreasing; consequently, the best response is pure apart from a measure zeroset of values where jumps occur. These jump points are the only points wheremixing can occur, but because their measure is zero, the mixing has no conse-quence for the maximization problems of the other traders.

8This is strictly true because we assume a continuum of traders.


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3. BASIC PROPERTIES OF EQUILIBRIA

In this section we derive formulas for probabilities of trade and establish thestrict monotonicity of strategies. These facts are inputs to the proofs in the

next two sections. We separate them out because they apply for all > 0 Weemphasize that they apply to all equilibria, not just full trade equilibria.

3.1. Discounted Ultimate Probability of Trade and Participation Cost

An essential construct for the analysis of our model is the discounted ulti-mate probability of trade. It allows a traders expected gains from participatingin the market to be written as simply as possible. In the steady state, let S()be the probability of trade in a given period of a seller who chooses reservationprice and, similarly, let B() be the probability of trade in a give period of a

buyer who chooses bid . Also, let S() = 1 S() and B() = 1 B()Define PB() recursively to be a buyers discounted ultimate probability of

trade if he bids :

PB() = B() + B()ePB()Therefore,

PB() = B()1 e + eB()

(7)

Observe that this is, in fact, a discount factor because every active trader ul-timately trades. Its interpretation as a probability follows from formula (9),which follows. The parallel recursion for sellers implies that

PS() = S()1 e + eS()

(8)

This construct is useful within a steady-state equilibrium because it convertsthe buyers dynamic decision problem into a static decision problem. Specif-ically, the discounted expected utility WB of a type v buyer who follows thestationary strategy of bidding is

WB(v) = B()(v ) + B()eWB(v)Solving this recursion gives the explicit formula

WB(v) = PB()(v ) KB()(9)where

KB() = 1

e

+eB()

= PB()B()

(10)


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is the buyers expected discounted participation costs over his lifetime in themarket. To get intuition for the last equation, note that 1/B() is the expectedlifetime of a trader in the market, so that /B() is the expected participa-

tion cost over the lifetime. The discounted participation cost KB() equals theexpected participation cost over the lifetime times the discount factor PB().Similarly, the discounted expected utility WS of a type c seller who follows thestationary strategy of accepting bids of at least is

WS(c) = PS()( c) KS()(11)where

KS() =

1 e + eS()(12)

is the discounted participation cost of a seller who asks In accord with ourconvention for nonentering types, we assume that

B(N) = S(N) = KB(N) = KS(N) = 0In Section 3.3 we derive explicit formulas for B() and S()

3.2. Strategies Are Strictly Increasing

This subsection demonstrates the most basic property that our equilibria sat-

isfy: strategies are strictly increasing. We need the following preliminary result.LEMMA 3: In equilibrium, PB[B()] is nondecreasing and PS[S()] is nonin-

creasing over[0 1]. The buyers for whom v > v elect to enter, while the buyers forwhom v < v do not:

(v 1] AB [0 v) ABThe type v is indifferent between entering and not entering. Similarly,

[0 c) AS (c 1] ASand the type c is indifferent between entering or not.

Equation (3) and Lemma 3 define the sets AB and AS and the descriptors vand c.

PROOF OF LEMMA 3: The buyers interim utility,

WB(v) = supR{N}

(v )PB() KB()

=(v

B(v))PB(B(v))

KB(B(v))


16/46


is the upper envelope of a set of affine functions. It follows by the envelope the-orem that WB() is a continuous, increasing, and convex function. Because WBis continuous, the definition of v = inf{v : v AB} implies that (i) WB(v) = 0and v is indifferent between entering or not, and (ii) the types v < v prefernot to enter. Furthermore, convexity implies that WB() is nondecreasing. Bythe envelope theorem, WB() = PB[B()]; PB[B()] is therefore nondecreasingat all differentiable points. Milgrom and Segals (2002) Theorem 1 implies thatat nondifferentiable points v [0 1],

limvv

WB(v) PB(B(v)) limvv+

WB(v)

Thus PB[B()] is everywhere nondecreasing for any best response B. Further-more, Milgrom and Segals Theorem 2 implies that

WB(v) = WB(v) + vv

PB[B(x)] dx for v v.(13)

Because v is indifferent between entering or not, we can choose a best re-sponse B in which v is active, while B(v) = B(v) for v = v. Response B maydifferent from B at v = v, because in B, the type v is active, while in B he maynot be. Importantly, the function WB() is the same for both B and B, becauseby Milgrom and Segals Theorem 2, the envelope condition (13) holds for any

selection from the best-response correspondence. Now PB[

B(v)] > 0, because

otherwise the active buyer v would not be able to recover his positive partici-pation cost. Because PB[B()] is nondecreasing, PB[B(v)] > 0 for all v v, andthe envelope condition (13) then implies that the buyers for whom v > v electto enter. The argument for the sellers is parallel and is omitted. Q.E.D.

Recall that we assume if a potential traders expected utility from enteringis at least zero, then he or she enters. Thus types v and c enter, which keepsour notation simple. Because {v} and {c} have measure 0, all our results wouldhold in substance under the alternative assumption that entry occurs only ifexpected utility is positive.

LEMMA 4: Response B is strictly increasing on [v 1].

PROOF: Pick any v v [v 1] such that v < v. Because PB[B()] is non-decreasing, PB[B(v)] PB[B(v)] necessarily. We first show that B is nonde-creasing on [v 1]. Suppose, to the contrary, that B(v) > B(v). The auctionrules imply that PB() is nondecreasing; therefore, PB[B(v)] PB[B(v)]. Con-sequently, PB[B(v)] = PB[B(v)] > 0 However, this gives v incentive to lowerhis bid to B(v), because by doing so he will buy with the same positive prob-ability but pay a lower price. This contradicts B being an optimal strategy andestablishes that B is nondecreasing. If B(v)

=B(v) (

=) because B is not


17/46


strictly increasing, then any buyer with v (vv) will raise his bid infinitesi-mally from to > to avoid the rationing that results from a tie. This provesthat B is strictly increasing on [v 1]9 Q.E.D.

LEMMA 5: Response S is continuous and strictly increasing on [0 c].

PROOF: Any active seller will accept the highest bid she receives, providedit is above her dynamic opportunity cost:

S(c) = c + eWS(c)(14)

Milgrom and Segals Theorem 2 implies that WS() is continuous and can bewritten, for any active seller type c, as

WS(c) = WS(c) +c

c

PS(S(x))dx =c

c

PS(S(x))dx(15)

where the second line follows from the definition of c and the continuityofWS(). Combining (14) and (15) we see that

S(c) = c + ec

c

PS(S(x))dx

for all sellers who are active. This also implies that S(

) is continuous. There-

fore, for almost all active sellers c [0 c],S(c) = 1 ePS[S(c)] > 0(16)

because WS (c) = PS[S(c)] Because S() is continuous, it is sufficient to es-tablish that S() is strictly increasing for all active sellers c [0 c]. Q.E.D.

LEMMA 6: We have c < B(v) < v, S(c) = c < p, and B(v) c.

PROOF: Given that S is strictly increasing, S(0)

=c is the lowest reservation

price any seller ever has. A buyer with valuation v < c does not enter the mar-ket because he can only hope to trade by submitting a bid at or above c, i.e.,above his valuation. In equilibrium, any buyer who enters the market must sub-mit a bid below his valuation and above c, because otherwise he is unable torecover a positive participation cost. It follows that c < B(v) < v. Similarly,a seller who is only willing to accept a bid at or above p never enters themarket, because she is unable to recover her participation cost. This implies

9This proof uses the same argument that Satterthwaite and Williams (1989a) used to provetheir Theorem 2.2.


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S(c) < p. Any active seller has acceptance strategy given by (14), so in partic-ular S(c) = c.

Finally, suppose that B(v) > c. Then the buyer for whom v = v bids morethan necessary to win the object: he can only be successful if there are no rivalbuyers, and when this is the case, bidding c is sufficient to secure acceptance ofthe bid by the seller. Q.E.D.

All these findings are summarized as follows. In reading the theorem, recallthat the descriptors (c cvp p) are defined in (3).

THEOREM 7: Suppose that {B S} is a stationary equilibrium. Then, over[v 1]and [0 c], B and S are strictly increasing, and S is continuous and almost every-where on [0 c] has derivative

S(c) = 1 ePS[S(c)]Finally, B and S have the properties that c < p < v, S(c) = c < p and p =B(v) S(c) = c.

The strict monotonicity of B on [v 1] and S on [0 c] allows us to define Vand C, their inverses over [B(v)B(1)] and [S(0)S(c)]:

V () = inf{v [0 1] : B(v) > }C()

=inf

{c

[0 1

]: S(c )>

}

Finally, that B(v) S(c) is a weak inequality, not a strong inequality, makespossible the existence of full trade equilibria.

3.3. Explicit Formulas for the Probabilities of Trading

Focus on a seller of type c who in equilibrium has a positive probability oftrade. In a given period she is matched with zero buyers with probability 0 andwith one or more buyers with probability 0 = 1 0 Suppose she is matchedand v is the highest type buyer with whom she is matched. Because by Theo-

rem 7 each buyers bid function B() is increasing, she accepts his bid if and onlyif B(v) , where is her reservation price. The distribution from which vis drawn is FB(). For v [v 1]

FB(v) =1

0()

i=1

i()[FB(v)]i(17)

where FB() is the steady-state distribution of buyer types and {0 1 2 }are the probabilities with which each seller is matched with zero, one, two,or more buyers. Note that this distribution is conditional on the seller being


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matched. Thus if a seller has reservation price , her probability of trading ina given period is

S()=

01 FB(V ())(18)

This formula takes into account the probability that she is not matched in theperiod.

A similar expression obtains for B(), the probability that a buyer submit-ting bid successfully trades in any given period. To derive this expression,we need a formula for k(), the probability that the buyer is matched with krival buyers. IfTB is the mass of active buyers and TS is the mass of active sell-ers, then k()TB the mass of buyers who participate in matches with k rivalbuyers, equals k + 1 times k+1()TS , the mass of sellers matched with k + 1buyers:

k()TB = (k + 1)k+1()TSSolving, substituting in the formula for k+1() and recalling that = TB/TSshows that k() and k() are identical:

k() =(k + 1)

k+1() =

(k + 1)

k+1

(k + 1)!e = k()(19)

The striking implication of this, which follows from the number of buyers in agiven meeting being Poisson, is that the distribution of bids that a buyer mustbeat is exactly the same distribution of bids that each seller receives when sheis matched with at least one buyer.10

Turning back to B a buyer who bids and is the highest bidder has proba-bility FS(C()) of having his bid accepted. This is just the probability that theseller with whom the buyer is matched will have a low enough reservation priceso as to accept his bid. If a total of j+ 1 buyers are matched with the seller withwhom the buyer is matched, then he has j competitors and the probability thatall j competitors will bid less than is [FB(V ())]j Therefore, the probabilitythat the bid is successful in a particular period is

B()=

FS(C())

j=0 j()FB(V ())j

(20)

= FS(C())

j=0j()

FB(V ())

j= FS(C())

0 + 0F(V ())

10Myerson (1998) studied games with population uncertainty and showed that the Poissonassumption is both necessary and sufficient for players beliefs about the number of other playersto be equal to the external observers beliefs.


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We are now in position to prove Theorem 1 on convergence and Theorem 2on existence. The next section proves convergence to pW , the Walrasian price,in two steps. Convergence to one price follows from each traders search for

a better price becoming cheaper as the period length, becomes shorter.Traders who do not offer a good price to the opposite side of the market fail totrade and therefore revise their price, narrowing the range of prices realized inthe market. That convergence is to pW is a consequence of supply and demand.Traders who enter stay in the market until they trade. If the one price to whichthe market converges is not Walrasian, then either more buyers will enter thansellers or more sellers will enter than buyers. Either way the market will notclear, traders on one side of the market will accumulate without bound, andthe market will not be in a steady-state equilibrium. Therefore, if a sequenceof steady-state equilibria exists as 0, the equilibria must converge to pW .

Section 5 proves existence in three steps. In the first step we identify a classof equilibriafull trade equilibriaand show that a necessary condition for afull trade equilibrium to exist is that it satisfy a system of two equations that the discount rate, and the period length parameterize. In step 2 we showthat at () = (0 0) a solution to these equations always exists and we applythe implicit function theorem to establish that a unique solution always existsfor all () in a neighborhood around (0 0) Finally, in step 3 we show thatif is sufficiently small relative to the per unit time participation cost, thenthe solution to the two equation systemwhich existsdefines an equilibrium,i.e., a solution to the system is sufficient for an equilibrium to exist. Section 5then concludes with a discussion of two issues: why it is important that be

small relative to and what we are able to say about the rate at which equilibriaapproach full efficiency as 0

4. PROOF OF CONVERGENCE

Theorem 1 consists of two parts: the law of one price part, which, giventhe characterization in Theorem 7, reduces to

lim0

c = lim0

p = lim0

v = pW

and the efficiency part

lim0

WS(c) = max[0 pW c], lim0

WB(v) = max[0 v pW]

These parts are dealt with separately in Theorems 8 and 12. All the proofsin this section apply to all equilibria, not only to full trade equilibria. Fig-ure 3 shows the structure of a non-full-trade equilibrium. The difference be-tween this figure and Figure 1 (a full trade equilibrium) is that the equalityp = B(v) = S(c) = c which defines an equilibrium to be full trade, is changedto the inequality p = B(v) S(c) = c which allows sellers and buyers strate-gies to overlap.


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FIGURE 3.Structure of an equilibrium that is not full trade. The figure also shows the con-struction ofb b c, and c

that are used in the proof of Lemma 10.

THEOREM 8: We have lim0 c = lim0 p = lim0 v = pW

The proof of this theorem relies on three lemmas.

LEMMA 9: We have lim0(p c) = 0

PROOF: Suppose not; i.e., there exists an > 0 such that

p

c > along

a subsequence. Let

b = p /2v = sup{v : B(v) b}

Let the probability be the sellers equilibrium belief that the maximum bidin a given period is greater than or equal to b. If lim0 = > 0 along asubsequence, the seller for whom c = (c + b)/2 would prefer to enter forsmall enough The reason is this. By definition, b c > /2 and, therefore,b

c > /4 Consequently, if seller c sets her offer to be

=b then the


22/46


gain b c she realizes if she trades is at least /4 and her per period prob-ability of trade is S() > Inspection of formulas (8) and (12) establishes

that as 0, her discounted probability of trade goes to 1 and her discountedparticipation costs goes to 0. Therefore, her expected utility, as given by ( 11),is

WS(b c) = PS()(b c) KS() > 4

> 0

as 0 a contradiction.If, on the other hand, lim0 = 0 along all subsequences, then the buyer

for whom v = 1 would prefer a deviation to b. If he deviates, then in the limit,as 0, his probability of trading in a given period, B(b), approaches 1.This is an immediate implication of the observation that follows ( 19): isnot only the probability that the maximum bid a seller receives in a given pe-riod is greater than or equal to b but it is also the probability that the maxi-mum competing bid the type 1 buyer must beat is greater than or equal to bTherefore, 0 implies that deviating to b results in his discounted prob-ability approaching 1 and discounted participation cost approaching 0. Conse-quently, this buyer deviates and secures the lower price b, which completesthe proof. Q.E.D.

LEMMA 10: We have lim0(p p

) = 0

PROOF: The proof is by contradiction: Pick a small suppose

p

p

>

> 0 along a subsequence, and define

b = p 1

3(21)

b = p 2

3

Note that b p >

3 Select a buyer and let

= FS(C(b))be the equilibrium probability that the seller with whom he is matched in agiven period would accept a bid that is less than or equal to b. Lemma 9 guar-antees that the seller for whom S(c) = b exists, at least for small enough .Select a seller and let

=

k=0k

FB(V(b))

k = 0 + 0FB(V(b))be the equilibrium probability that, in a given period, she receives either no bidor the highest bid she receives is less than or equal to b. Observe that is


23/46


the equilibrium probability that a buyers bid b is maximal in a given match;this follows directly from formula (20) for B() Given these definitions, thislemmas proof consists of three steps.

STEP 1: The fraction of sellers for whom S(c) b does not vanish as 0, i.e., lim0 > 0 Suppose not. Then 0 along a subsequence.Fix this subsequence and fix some period, say period 0. Let N be a sequenceof integers whose values are chosen later in the proof. Define, without loss ofgenerality, the time segment of length N periods that begins with period 0and ends with period N Define three masses of sellers: Mass m+N is the mass of sellers who enter the market within time segment

and for whom b S(c) b.

Mass mN is the mass of sellers who both enter and exit the market withintime segment and for whom b

S(c)

b.

Mass m is the steady-state mass of active sellers for whom b S(c) b.The assumption that 0 implies that m 0 We show next that m 0entails c b This establishes a contradiction because Theorem 7 states thatc < p

and by construction p+

3< b

The fraction of sellers in the mass m+N that do not exit during the time seg-ment is

m+N mNm+N

mm+N

because the surviving mass m+N mN of sellers who entered in time seg-ment cannot exceed the total, steady state of the mass m of sellers withreservation prices in the interval [b b]. Therefore, the fraction of the sellersin m+N who have traded within time segment is at least

1 mm+N

(22)

In the mass m+N , pick a seller c who enters in period 0 and for whom

S(c)

=b

Such a seller c always exists because S is continuous (see Theorem 7) and g isa lower bound on the density of entering sellers. This sellers reservation priceis as low as any other seller in m+N and has the full time segment in whichto consummate a trade. Her probability of trading within is therefore ashigh as any other seller in m+N . Let r be her probability of trading within thetime segment It is, therefore, at least as great as the average probability oftrading across all sellers in m+N :

r 1 m

m+N(23)


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Now, because the slope ofS is at most 1 (see the formula in Theorem 7), itfollows that

m+N

3 gN(24)

because m+N is minimized when the slope ofS is the largest (i.e., equal to 1)and the density gS is minimal. Substituting this lower bound on m

+N into (23)

gives

r 1 m

3gN

For seller c , her discounted probability of trading PS(b) from setting reser-

vation price b is bounded from below by

PS(b) eN

1 m

3gN

(25)

The right-hand side understates the discounted probability of trade because,literally, the lower bound is the discounted probability that trader c waitsthe full N periods before attempting to trade, having only probability 1 m(

3gN) of succeeding in that period, and then never trying again.

Set the period length to be

N = min

k : k is integer, k m

Substitution of this choice into (25) and taking the limit as 0 shows that thediscounted probability of seller c trading approaches 1 from below becausem 0:

lim0

PS(S(c)) lim

0exp(m)

1

m

3g

= 1

Recall from Theorem 7 that, for almost all c [0 c)

S(c) = 1 ePS[S(c)]

Because PS(S(c)) PS(b) for c c and S is increasing on [0 c), it followsthat, for all seller types c [0 c], PS[S(c)] 1 and

lim

0

S(c) = 0


25/46


Consequently, because S is continuous,

c = S(0) b

This is in contradiction to c < p < b /3 Therefore, it cannot be that

0

STEP 2: If the ratio of buyers to sellers is bounded away from 0, then theprobability that the highest bid in a given meeting is less than b

is also

bounded away from 0. Proof of this step stands alone and is not based on theresult in Step 1 of this proof.

Formally, if lim0 > 0, then lim0 > 0 Suppose not. Then 0 and > 0 along a subsequence. Fix this subsequence and recall that byconstruction

b > p +

3(26)

First, we show that the seller with cost c such that S(c) = b prefers to enter.

Because and 0, for all sufficiently small, the probability thathe meets a buyer for whom B(v) b = b + /3 is at least 12 (1 e). This isbecause, with 0 (i) almost every bid she receives is greater than b and(ii) her probability of getting at least one bid is approaching 1 e Therefore,as 0, her discounted probability of trading with a buyer for whom B(v) b

approaches 1 even as her discounted participation costs, given by formula

(10), approach 0. Consequently, the profit of the c seller, in the limit as 0,is at least /3 and she will choose to enter.

Second, because she chooses to enter, it must be that c c. Therefore, theslope ofS for c [0 c) satisfies

S(c) = 1 ePS(c) 0because PS(S(c)) PS(S(c)) and PS(S(c)) 1. Therefore c b, acontradiction of (26) and Theorem 7s requirement that c < p.

STEP 3: For small enough , a buyer for whom v = 1 prefers to deviate tobidding b instead ofp. There are two cases to consider.

CASE 1lim0 > 0: We show, using both Steps 1 and 2 of this proof, thatbidding p cannot be equilibrium behavior for a type 1 buyer. Recall that is the probability that a seller will accept a bid less than b and that, accord-ing to Step 1, = lim0 > 0 Additionally, recall that is the probabilitythat the maximal rival bid a buyer faces in a given period is no greater than band that, according to Step 2, lim0 = > 0 For small enough > 0, thissecond probability is bounded from below by (1/2). It follows that, for small


26/46


enough , the buyer who bids b (i) wins over all his rival buyers with probabil-ity greater than (1/2) and (ii) has his bid accepted by the seller with probabil-

ity greater than (1/2). Therefore, as 0 the buyer who bids b trades witha discounted probability approaching 1 and a discounted participation cost ap-proaching 0 Consequently, deviating to b gives him a profit of at least 1 bwhich is greater than 1 p the profit he would make with his equilibrium bidB(1) = p Therefore, deviation to b is profitable for him.

CASE 2lim0 = 0: Fix a subsequence such that 0. The proof ofthis case relies only on the result in Step 1 of this proof. The probability ofmeeting no rival buyers in a given period is e and, because 0, thisprobability is at least 1/2 for sufficiently small . In any given period, for atype 1 buyer and for all small , (i) the probability of meeting no rivals is at least

1/2 and (ii) the probability of meeting a seller who would accept the bid b isat least (1/2). It follows that as 0, his discounted probability of tradingapproaches 1 and his discounted participation cost approaches 0. Therefore,deviating to b gives him a profit of at least 1 b > 1 p, which proves thata deviation to b is profitable for him.

Step 3 completes the proof of the lemma because it contradicts the hypoth-esis that lim0(p p

) = > 0. Q.E.D.

LEMMA 11: We have lim0(v p) = 0

PROOF: Suppose not. Recall that B(v) p and that Theorem 7 states thatv > p

Pick a subsequence such that v p > 0 along it. Define =12

(p+ v) and observe that p /2 and < v. The latter inequality

implies that a type buyer does not enter the market because his expectedutility is nonpositive. However, suppose to the contrary that a type buyerenters and bids p Bidding p guarantees that he wins the auction in whatevermatch he finds himself, i.e., B(p) = 1 Therefore, in the first period after heenters he earns profit of

p = p

+ p

p

2

+ p p

2

because Lemma 10 states that, as 0 p p 0. This contradicts the

equilibrium decision of the type buyer not to enter. Q.E.D.


27/46


PROOF OF THEOREM 8: Consider any sequence of equilibria n 0 Thedescriptors p and v converge because

lim0

(

p

v

)=

lim0

(

p

v

)

lim0

(p

v

)(27)

= lim0

(p p

)

= 0 where lim0(p

v) = 0 (from Lemma 11) implies the first equality and

lim0(p p

) = 0 (from Lemma 10) implies the third equality. Theorem 7establishes that c [p

p); therefore, Lemma 10 implies

lim0

(

p

c)

=0(28)

Pick a convergent subsequence of (v p c p) and denote its limit as

(p p p p).Traders who choose to become active in the market exit only by trading.

Therefore, in the steady state the mass of sellers entering each period mustequal the mass of buyers entering each period:

GS(c) = aGB(v)(29)Taking the limit in (29) along the convergent subsequence as

0, we get

GS(p) = aGB(p)This is just Equation (1) that defines the Walrasian price; therefore, p = pW .Because pW is the common limit of all convergent subsequences, it follows thatthe original sequence (v p c p) converges to the same limit:

lim0

p = lim0

p= lim

0c = lim

0v = pW(30)

All that remains is to show that c also converges to pW The type c seller

who is on the margin between participating and not participating must in ex-pectation be just recovering his participation cost each period. Recall thatS(c) = c. Because the price this seller receives is no more than the highestbid, p, it follows that

S[S(c)](p c) Therefore,

S[S(c)]

p

c


28/46


by (28). The discounted probability of trade may be written as

PS

[S(

c)

] =

S[S(c)]

1 e

+ e

S[S(c)](31)

= 11e

/ S

[S(c)]

+ e

It follows that lim0 PS[S(c)] = 1 because lim0((1 e)/) = andlim0(S[S(c)]/) = Furthermore, for all c [0 c]

lim0

PS[S(c)] = 1(32)

because PS[S(

)]

is decreasing. Therefore,

S(c) = 1 ePS(S(c))

the slope ofS on [0 c] converges to 0. Together with the continuity ofS thisimplies that c c which completes the proof. Q.E.D.

Next we prove the second part of Theorem 1.

THEOREM 12: We have lim0 WS(c) = max[0 pW c] and lim0 WB(v) =max

[0 v

pW

]

PROOF: Equation (32) establishes that lim0 PS[S(c)] = 1, for all c [0 c] The same argument, slightly adapted, shows that, for all v [v 1]lim0 PB[B(v)] = 1 Thus the buyer for whom v = v must just recover itsparticipation cost each period:

B[B(v)](v B(v)) = B(p)(v p) =

Therefore,

B(p)

= v p

by Lemma 11 and, exactly as with (31),

lim0

PB(p

) = lim0

B(p

)

1 e + eB(p

)= 1

Because PB() is increasing, this establishes that lim0 PB[B(v)] = 1 for allv

[v 1

]


29/46


The envelope theorem (see (13) and (15)) implies that

WS(c)

= c

c

PS[S(x)

]dx

WB(v) =v

v

PB[B(x)] dx

Passing to the limit as 0 gives lim0 WS(c) = max[0 pW c] andlim0 WB(v) = max[0 v pW] because c pW and v pW. Q.E.D.

5. EXISTENCE OF FULL TRADE EQUILIBRIA

Recall that Theorem 7 shows that every equilibrium must satisfy B(v) S(c). The intuition for this is that the type v buyer can only trade if there isno rival buyer. Consequently, he should certainly not bid more than S(c), thelowest bid that every seller accepts. In a full trade equilibrium, B(v) = S(c)and the type c seller, who is the highest cost active seller, always trades if sheis matched with at least one buyer, even if he is the lowest value active buyer.This, of course, means that any seller with cost less than c also trades if she ismatched and that a buyer fails to trade only because he is beaten in the biddingby another buyer. In this section we characterize these equilibria and, given > 0 prove their existence for each sufficiently small pair of nonnegative

and positive Specifically, Lemma 13 proves that, given > 0 and 0then for each sufficiently small and the vector of equilibrium descriptors(c v ) exists and is unique. Theorem 14 then shows that, given > 0, thereis a neighborhood X of(0 0) such that if() X, then a unique full tradeequilibrium exists that the vector (c v ) characterizes. Theorem 2 thenfollows immediately as a corollary.

5.1. Preliminaries

Before introducing the equations that determine (

cv) we derive sellers

and buyers probabilities of trade as a function of their types c and v and thebuyerseller ratio . As a consequence of the equilibrium being full trade, buy-ers trade probabilities are independent of sellers equilibrium strategy S Thatthe sellers strategy does not feed back and affect the buyers trade probabil-ities and strategy implies that the market fundamentalsGS GBa andfully determine the equilibrium. This fact drives both the uniqueness andthe existence results of this section.

Given that the market is in a steady state, within every period the cohort ofbuyers who have the highest valuations in their matches and therefore trade isreplaced by an entering cohort of equal size and composition. Therefore, FB,


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the distribution function of the maximal valuation within a match, is equal tothe distribution ofv in the entering cohort conditional on v v:

FB(v) = GB(v) GB(v)1 GB(v) (33)

Let B(v) be the equilibrium probability that a type v buyer trades in anygiven period. As reference back to (20) and its derivation explains, it is equalto the probability that he bids against no rival buyers (0 = e) plus the com-plementary probability (0 = 1 e) times the probability that the maximalvalue among the rival buyers in his match is no greater than v11:

B(v) = e + (1 e)FB(v)(34)

The discounted equilibrium trading probability for a type v buyer is, therefore,

PB(v) =B(v)

1 e + eB(v)(35)

The carets (hats) on B() and PB() emphasize that these equilibrium proba-bilities are functions of the buyers value v, not of his bid B(v)

With this notation in place, we can introduce the equations that determine(cv) in a full trade equilibrium. First, because every meeting results in a

trade, the mass of entering buyers must equal the mass of entering sellers inthe steady state:

GS(c) = a[1 GB(v)](36)= aGB(v)

Second, the buyer for whom v = v must be indifferent between being activeand staying out of the market. The type v buyer only trades in a period whenthere are no rival buyers; his probability of trading is 0 = e. Because in a fulltrade equilibrium B(v)

=S(

c) and Theorem 7 states that S(

c)

= c, indifference

necessarily implies that his expected gains from trade in any period, (v c)eequals his per period participation cost:

(v c)e = (37)

11Formula (20), B() = FS (C())[0 + 0F(V ())] illustrates why the full trade case isdifferent than the general case. In the full trade case the factor FS (C()) is degenerate: for all [p p], FS (C()) = 1 As a consequence the sellers inverse strategy, C() does not affectB(). In the general case, an interval [p ] [p p] may exist such that, for all [p ],FS (C()) < 1 and C() does affect B(

)


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Third, parallel logic applies to any seller for whom c = c. This seller alwaystrades in any period in which he is matched with at least one buyer; the prob-ability of this event is 1 0 = 0 = 1 e. Denote the expected price thatany seller receives as p Note that this expected price is not a function of thesellers type c; it is the same for all active sellers in a full trade equilibrium.Then, because the c seller is indifferent between trading and staying out of themarket, it must be that

(p c)(1 e) = (38)

To find the price p, we use the envelope theorem to solve for the biddingstrategies of the active buyers (i.e., those buyers for whom v v) as

WB(v) = (v B(v))PB(v) K0(v)(39)=v

v

PB(x)dx

where PB(v) is the type v buyers discounted probability of trading and

K0(v) =

1 e + eB(v)

is his discounted participation cost. Solving Equation (39) for B(v) gives

B(v) = v B(v)

1PB(v)

vv

PB(x)dx(40)

Observe that this formula calculates B(v) directly; it is not a fixed point condi-tion. The expected price p that a seller receives is the expected value of B(v)for that buyer who has the highest valuation,

p=

1

v

B(v)dFB

(v)=

1

1 GB(v) 1

v

B(v)dGB(v)(41)

where the second equality follows from (33).Equations (36)(38) form a system of three equations in the three unknowns

(cv) that, for given and must hold in any full trade equilibrium.In Theorem 14, which follows, we prove that the converse claim is also true:given > 0 if is nonnegative, is positive, and they are in a sufficientlysmall neighborhood of (0 0) then a unique full trade equilibrium exists thatcorresponds to a solution (cv) of the system of equations. The three char-acterizing equations (36)(38) therefore identify a full trade equilibrium.


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It is useful to reduce (36)(38) to two equations. Substitute

c = v e(42)

from (38) into (36) to obtain

GS(v e) a(1 GB(v)) = 0(43)This eliminates c Equation (36) can be rewritten as

p = c + 1 e

= v e + 1

e

Given the new, two equation system in the two variables (v) and the twoparameters () is then

GS(v e) a(1 GB(v)) = 0(44)

p v + e 2

1 e = 0(45)

where Equations (41), (40), and (35) together imply that p is a function of and

5.2. Proof of Theorem 2

The method of proof we use has five steps. First, we fix > 0 and tediouslysubstitute (41) into the system (44)(45) to eliminate p Second, we changethe domain of the system from the economically meaningful set (v) D = (0 1) (0 ) [0 1) (0) to the mathematically more convenientset D1 = (0 1) (0 2) (01 1) (01 1). Third, we prove that at = 0,the system has a unique solution, (v) = (pW ), where = 114619 is theunique positive solution of the equation e 2 = 0 Fourth, we apply theimplicit function theorem in a neighborhood of (v)

=(pW

0 0) toestablish the existence of a unique, differentiable solution (v()()) for thesystem of equations. Fifth, by construction, we show that if both and aresufficiently small, then the solution (v()()) characterizes the unique, fulltrade equilibrium of the market. Lemma 13 accomplishes the first four stepsand Theorem 14 accomplishes the last step.

LEMMA 13: Given > 0 a neighborhood X of the point (0 0) exists suchthat the system (36)(38) has a unique, differentiable solution (c() v()()) for all ( ) X At = 0 v( 0) = pW and (0 0) = = 114619,

where is the unique positive solution of the equation e

2

=0.


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PROOF: By (41),

p

v

=1

1 GB(v) 1

v

(B(v)

v)dGB(v)(46)

and, by (40),

B(v) v = v v v

v

PB(x)

PB(v)dx

B(v)

=v

v

1 PB(x)

PB(v)

dx

B(v)

Algebra shows that

1 PB(x)PB(v)

= 1 e

B(v) B(x)B(v)(1 e + eB(x))

(47)

Substituting this into (40) gives

B(v) v

=

1 e

v

v

B(v) B(x)B(v)(1 e

+ e

B(x))

dx

B(v)Inserting this expression for B(v) v into (46) and substituting the resultingexpression for p v into (45) gives

L(v) = 0(48)

where

L(v)(49)

= 11 GB(v)

1v

1 e

v

v

B(v) B(x)B(v)(1 e + eB(x))

dxdGB(v)

11 GB(v)

1v

1

B(v)dGB(v)

+ e 2

1

e


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The second term ofL can be written as

1

1 G

B(v)

1

v

1

B(v)

dGB(v)

= 11 GB(v)

1v

1

e + (1 e) GB(v)GB(v)1GB(v)

dGB(v)

= 11 e log

1

e

= 1 e

Substituting this into (49) results in

L(v)

= 11 GB(v)

1v

1 e

v

v

B(v) B(x)B(v)(1 e + eB(x))

dxdGB(v)

+ e 21

e

Simplifying further, L(v) becomes

L(v)(50)

= 1(1 GB(v))2

1v

1 e

v

v

(1 e)(GB(v) GB(x))B(v)(1 e + eB(x))

dxdGB(v)

+e

2

1 e

Given this work, the system (44)(45) is, for > 0 equivalent to the system

GS(v() e()) a

1 GB(v( ))= 0(51)

L(v()()) = 0(52)where we index v and with and to indicate that we solve this system forthem as functions of and Note that we have divided the second equationthrough by

;this is essential to ensure a nonzero Jacobian.


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We apply the implicit function theorem to this system in a neighborhoodof the point (v) = (pW 0 0) To do so the system must be contin-uously differentiable in a neighborhood of (pW

0 0) Therefore, change

the domain of the system from D = (0 1) (0 ) [0 1) (0) toD1 = (0 1) (0 2) (01 1) (01 1). Equation (51) is obviously contin-uously differentiable on D1 To see that this is also so for (52), recall formula(34) for B and observe that B(v) B(x) and (1 e)/ are continuouslydifferentiable functions of v , and on D1.

12 The function L as a com-position of these functions, is continuously differentiable on D1 provided thefactor (1 e + eB(x)) in the denominator of the inner integral is al-ways positive. This in fact is the case because B(x) is increasing for x v andB(v) = 0 = e (see (33) and (34)). Therefore

1

e

+e

B(x)

1

e

+ee

= e(e 1 + e)This last expression is positive ife 1 + e > 0 or, equivalently, whenever

> ln(1 e)(53)Inspection establishes that all (v) D1 satisfy (53) becausesup(02) ln(1 e) = 0145 Therefore, L() is continuously differentiableon D1 Finally, also note that within its interior D1 includes all points in

{(v)

D1 :

=0 and

=0

}

At () = (0 0), L(v 0 0) takes the simple form

L(v 0 0) = e 21 e

This is a function that increases in ,

d

d

e 21 e =

e(+ e2 3e + 3)(1 e)2 > 0

because e2

3e

+3

=(e

3

2

)2

+3

( 3

2

)2 > 0 Therefore,

L(v 0 0)

> 0(54)

We now claim that (v) = (pW ) is the unique solution to the system(51)(52) at () = (0 0) This is seen in two steps. First, if = 0, (51) re-duces to GS(v(0)) a(1 GB(v(0))) = 0 which is just (1) defining pW the

12The expression (1 e)/ is indeterminate at = 0 but selecting lim0(1 e)/to be its value at that point makes it continuous and differentiable over all of D1


36/46


Walrasian price. Therefore, at () = (0 0), pW is the unique solution to(51), the systems first equation. Second, inspection shows that the equationL(pW 0 0) = 0 reduces to e 2 = 0 at () = (0 0); its unique, pos-itive solution is = = 114619 Thus, as claimed, (v) = (pW ) is theunique solution to the system at () = (0 0)

The Jacobian J of the system (51)(52) at (v) = (pW 0 0) is notzero,

J=gS(pW) + agB(pW) 0 L(pW 00)

= (gS(pW) + agB(pW)) L(pW

0)

> 0

where denotes some expression and the last line follows from inequality (54).Consequently, the implicit function theorem applies: in some neighborhood Xof () = (0 0) a unique solution (v()()) to the system (44)(45)exists and is differentiable in and . The remaining function c() then isrecovered from formula (42). Q.E.D.

It remains to show that the equilibriums characterizing equations are alsosufficient; this is accomplished in Theorem 14. We have already shown that(36)(38) must hold in any full trade equilibrium, and that there exists aunique solution to these equations when and are sufficiently small. Westart with the solution values (

c()v()()) construct the unique

steady-state densities fB and fS and strategies B and S, compute the masses TBand TS that these densities and strategies imply, and check that the ratio ofthese masses equals the solution value The key insight to our construction isthe observation that the strategy for buyers can be constructed separately fromthe strategy for sellers; the solution (c()v()()) is a sufficientlink between the two.

A difficulty in the construction of the buyers strategy B is that the lowestactive buyer type v() (and hence also the neighboring types) may have anincentive to deviate, by lowering his bid into the support [cc) of the distrib-ution of the sellers offers.13 We show that such a deviation is not profitableif is sufficiently small.

THEOREM 14: Given > 0, a neighborhood X of(0 0)exists such that, for allpairs of nonnegative and positive in X a unique full trade equilibrium exists.

PROOF: Consider sellers first. We already know the marginal participationtype, c among sellers; it is a component of (cv) A seller trades in any

13We are grateful to a referee for drawing our attention to the possibility of such a deviation,thereby discovering a flaw in the previous versions proof.


37/46


period in which she is matched with at least one buyer; this probability is0 = 1 0 = 1 e and is independent of her type. Using formula (7), herdiscounted probability of trade is

PS =1 e

1 e + e(1 e) (55)

For active sellersthose with costs c cformula (15) gives their continua-tion values

WS(c) =c

c

PS dx(56)

= PS(c c)and formula (14) gives their equilibrium strategies

S(c) = c + eWS(c)(57)= c + ePS(c c)

Note that this is a linear function of c with slope 1 ePS . Sellers for whomc > c best respond by not entering.

To construct the unique, steady-state distribution of seller types, FS , observethat the sellers best-response strategy S is increasing on [0 c] and satisfiesS(c) = c. Because all active sellers trade with the same probability in any pe-riod, the distribution of their types in the market FS is just the distribution ofthe entering cohort conditional on c c:

FS(c) = GS(c)GS(c)

(58)

To complete the construction of the sellers part of equilibrium, we must findthe steady-state mass of active sellers TS . Mass balance holds every period in asteady state, therefore TS(1 e) = GS(c) and, solving,

TS = GS(c)1 e (59)

Turning to the buyers, our first step is to show that buyers unique, symmetric,mutual best-response strategy for v v is in fact given by (40),

B(v) = v B(v)

1PB(v)

vv

PB(x)dx(60)

where v denotes v(), a component of the characterizing equations solutionof the (36)(38). For v < v, the best response is not to enter. Observe that


38/46


the formula implies, as it should, that B() is an increasing function and thatB(v) = c.

Standard auction-theoretic arguments (e.g., the constraint simplification

theorem in Milgrom (2004, p. 105)) imply that the envelope condition (39)is sufficient to rule out any deviation from B(v) to a bid c. For devia-tions from B(v) downward into the region [c c), the restriction that () iscontained in a sufficiently small neighborhood of 0 is sufficient. To see this, fix > 0, restrict () [0 1](0 ln 2) and observe that this restriction impliese ( 1

2 1) Recall that the buyers payoff function (eq. (9)) may be written

as WB(v) = (v )PB() KB(), where KB() is his expected discountedparticipation costs from following the stationary strategy of bidding BecauseWB(v) is continuous at = c a sufficient condition for ruling out profitabledownward deviations is, for all and v such that v c > ,14

WB(v)

= (v )PB() PB() KB() > 0(61)

This condition is met if KB() > 1, because (v ) > 0, PB() > 0,PB() < 1 and K

B() < 0

Differentiate KB() as

KB() = e

(1 e + eB())2B()(62)

1

2B()

because e [ 12

1] B() [0 1] and, consequently, e/(1 e +eB())2 12 . The probability B() of a seller successfully trading is givenby formula (20),

B() = FS(C())

0 + 0F(V ())= 0FS(C())

where the second equality follows from the fact that, for deviations [c c)F(V ()) = 0 because the deviating buyers bid loses whenever he faces arival buyer.

Differentiation gives

B() = efS(C())1 e + e(1 e)

1 e (63)

where e = 0 fS(C()) is the density of sellers types at C() and1 e + e(1 e)

1 e = C() = 1

S(C())

14However, WB(v) is not a differentiable function at

=c.


39/46


from (57). Recall the mathematical inequality that, for all and 1 e Also recall that g > 0 bounds the density gS from below on [0 1] whichimplies that fS(c)

=gS(c)/GS(c)

g. Therefore, again using e

[12

1

]

B() eg(1 e)

Then

KB() 1

2eg(1 e)

1

4

ge

1

e

> 1

where the first inequality follows from substituting the bound for B() into(62), the second inequality follows from writing the result from Lemma 13 as() = + o(()) for some neighborhood of(0 0) and the third in-equality follows from choosing sufficiently small relative to This showsthat, for () in a sufficiently small neighborhood of (0 0) that is containedin [0 1] (0 ln 2) W (v) > 0 for (cc) and, therefore, no deviation to < c is profitable for active buyers. Buyers unique symmetric mutual best-

response strategy for v v is therefore given by (40). For v < v, the best re-sponse is not to enter. Observe that the formula implies, as it should, that B isan increasing function and that B(v) = c.

The steady-state distribution, FB of the maximal rival buyers type is givenby formula (33). The distribution FB is uniquely recoverable from F

B, the distri-

bution of the maximal value. Equations (34) and (33) imply that, in the steadystate,

B(v) = e + (1 e) GB(v) GB(v)1

GB(v)

(64)

On the other hand, direct computation shows that

B(v) =

k=0e

k

k! FB(v)k(65)

= e[1FB(v)]

k=0eFB(v)

[FB(v)]kk!

=e[1FB(v)]


40/46


Equating the right-hand sides of (64) and (65) and then solving, we obtain theunique steady-state distribution FB that corresponds to and v in the charac-terizing equations solution:

FB(v) = 1 + 1

log

e + (1 e) GB(v) GB(v)1 GB(v)

(66)

To complete the construction of the buyers part of equilibrium, we mustcompute TB the steady-state mass of buyers. Mass balance of buyers implies

TBFB(v)B(v) = agB(v)(67)

Substitution of (64) and the derivative of (66) into (67) and solving gives theformula

TB = a(1 GB(v))1 e =

aGB(v)

1 e (68)

A review of this construction shows that strategy B the distribution FB and themass TB depend only on the fundamentals GB a and the solution (cv)to the characterizing equations, but not the sellers strategy S This insulationof the buyers optimal actions from the sellers actions is, we again emphasize,the key insight behind this construction and the proofs overall design.

We need to check that, in addition to being mutual best responses and in-ducing their own steady-state distributions FB and FS , the strategies result in

steady-state masses of buyers and sellers that have the required ratio thatwas computed as a component of the solution to the characterizing equations.This is confirmed by dividing (68) by (59),

TB

TS= a(1 GB(v))

GS(c)= aGB(v)

GS(c)

= where the last line follows from market clearing, (36). Q.E.D.

5.3. Discussion

Lemma 13, Theorem 14, and their proofs raise two issues. First, even if is small, why does have to be small relative to to insure that a solution(cv) to the equations (36)(38) in fact characterizes a full trade equilib-rium. Second, our theorems d

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