Kultti, Klaus

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öMmföäflsäafaäsflassflassflasffffffffffffffffffffffffffffffffffffDiscussion PapersInefficiency caused by random matching andheterogeneityKlaus KulttiUniversity of Helsinki and HECERDiscussion Paper No. 90December 2005ISSN 1795-0562HECER – Helsinki Center of Economic Research, P.O. Box 17 (Arkadiankatu 7), FI-00014University of Helsinki, FINLAND, Tel +358-9-191-28780, Fax +358-9-191-28781,E-mail info-hecer@helsinki.fi, Internet www.hecer.fiHECERDiscussion Paper No. 90Inefficiency caused by random matching andheterogeneityAbstractI study an economy with sellers and buyers with unit supplies and unit demands. Bothparties have valuations uniformly distributed on a unit interval. I quantify the inefficiency,compared to the Walrasian market, caused by a market where the agents meet randomly.There are several causes of inefficiency that I deal with separately. First, even if there isperfect information about valuations it makes a difference whether all agents participate inthe markets or whether only those who would trade in the Walrasian market participate.The same applies when there is private information about valuations.JEL Classification: C78, D40, D50, D60, D80Keywords: Imperfect information, markets structure, random matching.Klaus KulttiDepartment of EconomicsUniversity of HelsinkiP.O. Box 17 (Arkadiankatu 7)FI-00014 University of HelsinkiFINLANDe-mail: klaus.kultti@helsinki.fi1 IntroductionThe purpose of this article is to quantify the e¢ciency loss associated withimperfect or non-Walrasian markets. Economies where the agents meet ran-domly constitute one of the best behaved classes of non-Walrasian models. Aparticularly applicable model of random matching is the urn-ball model where,say, buyers contact randomly sellers; when there is a continuum of buyers andsellers a seller may meet any number of buyers, and the number is given by aPoisson-distribution. This environment allows for an interesting price forma-tion mechanism: When there are several buyers who want what the seller has anatural way to resolve the allocation problem is by auction. In a meeting wherethere is only one buyer and asymmetric information about the value of the ob-ject between the buyer and the seller there is no trading mechanism that resultsin an e¢cient outcome (Myerson and Satterthwaite, 1983). For simplicity, Iassume that the buyer makes a take-it-or-leave-it o¤er.Auction is the archetypical competitive price formation mechanism and Iam interested only in market imperfections that are not too di¤erent from theWalrasian markets; it is very easy to come up with markets and price mech-anisms that are so di¤erent from the Walrasian ones that they have nothinginteresting in common. For instance, the standard search theoretic marketswith pairwise meetings with Nash-bargaining is of no great interest (for anotherexample see Gale and Sabourian, 2005). Further, it is not applicable in situ-ations with uncertainty about the valuations of the parties. Non-cooperativebargaining protocols under uncertainty feature a multitude of equilibria and aretechnically unnecessarily di¢cult for my purposes.This set-up allows me to quantify, i.e., to calculate, the exact percentagesas to the ine¢ciency that is just due to giving up the Walrasian frameworkbut retaining perfect information. This is accomplished by assuming that allthe buyers are homogeneous and that all the sellers are homogeneous. Then Iintroduce heterogeneity in the valuations and study two cases. In one case onlythe buyers and sellers who would trade in the Walrasian markets participate inthe markets where the agents are randomly matched. In the other case all agentsparticipate in the market. To do this I cannot work with general distributionsof valuations. Not too surprisingly I use the uniform distribution; it is amenableto calculations. But it is also the distribution with maximum entropy, and myconjecture is that it results in maximum ine¢ciency, too. This I do not, however,attempt to prove.2 The modelAssume that there is a unit interval of both buyers and sellers. Their valuationsare uniformly distributed between zero and unity. The economy lasts only forone period, meaning that the agents have only one chance to trade. This sim-pli…es things greatly since in a dynamic economy I should calculate the agents’expected future utilities. Then the agents’reservation values should be deter-1mined in equilibrium and this is very hard. I think that limiting the study tothe static case is not a serious drawback since it can still be shown that thehigher a seller’s valuation the higher his expected utility in both the dynamicand static model. And the same applies to the buyers, too.The buyers contact sellers randomly in a symmetric fashion, and the numberof buyers a seller meets is given by the Poisson distribution with rate 1 =#bu yers#sellers. Thus, the probability that a seller meets exactly k buyers is e¡1 1k!.Before going to calculations let me note that in the Walrasian equilibriumthe price would be one half, and the maximum number of trades would be com-pleted. In the decentralised market there are sellers who do not meet any buyers.Further, not all meetings result in a trade as either the buyers’valuations arenot as high as the sellers’valuations, or the buyers strategically o¤er less thantheir own valuation trading o¤ the high probability of trade for high gains fromtrade. Also trades that would not take place in the Walrasian market happen:Consider, for instance, a buyer with a low valuation (less than one half) whomeets a seller alone. With positive probability the seller is a low valuation typewho accepts the buyer’s o¤er.Auction is straightforward but I have to determine a buyer’s optimal take-it-or-leave-it o¤er when he meets a seller alone and the valuations are privateinformation. Consider a buyer with valuation v. Denote his o¤er by b. Itmaximises his expected utility(v ¡ b)F (b) (1)where F is the distribution function of the sellers’valuations. With a uniformdistribution the …rst order condition yields the optimal o¤erb =v2(2)Note that the meeting probabilities, i.e., the ratio of buyers to sellers remainsthe same in both cases.2.1 All buyers and sellers in the market or the large mar-ket2.1.1 Private information about valuationsAssume then that a seller with valuation w meets k buyers. Now the buyersengage in an auction where the winner is the buyer with the highest valuationand he pays the second highest valuation. But there is a twist since the secondhighest valuation can be less than half of the highest valuation. Then thewinning buyer raises his bid until it is in accordance with (2). This means thatif a buyer’s valuation is not at least twice the seller’s valuation or if the secondhighest valuation of the buyers is not above the seller’s valuation trade does nottake place.I need the joint density function for the highest and second highest orderstatistics. Remember that the valuations are assumed uniform with density2f(x) = 1 on [0; 1]. Assume that a seller meets k ¸ 2 buyers. Denote the highestvaluation by v1 and the second highest by v2.1 It is given byg(v1; v2) = k(k ¡ 1)f(v1)f(v2)Fk¡2(v2) = k(k ¡ 1)vk¡22 (3)where the second equality follows from the assumption about uniform distribu-tion.When k ¸ 2 there are three di¤erent cases to consider. First, the seller mayhave any valuation w between zero and one half and the highest valuation ofthe buyers is larger than twice the seller’s valuation. Then the expected gainsare given by Z 1=20Z 12wkvk¡11 (v1 ¡ w)dv1dw = ¡18+2k + 14 (k + 2)(4)Second, the seller may have any valuation w between zero and one half andthe highest valuation of the buyers is less than twice the seller’s valuation andthe second highest valuation is larger than the seller’s valuation. Then theexpected gains are given byZ 1=20Z 2wwZ 2wv2k(k ¡ 1)vk¡22 (v1 ¡ w)dv1dv2dw=k ¡ 14 (k + 1) (k + 2)¡ k ¡ 1k + 112k+3(5)Third, the seller may have any valuation w between one half and unity. Thentrade takes place only if the second highest valuation is higher than the seller’svaluation. The expected gains are given byZ 11=2Z 1wZ 1v2k(k ¡ 1)vk¡22 (v1 ¡ w)dv1dv2dw=2k + 12 (k + 1)+1k + 112k+1¡ 78+k ¡ 1k + 112k+3(6)Summing (4), (5) and (6) I get the expected gains in a k-buyer meetingk ¡ 12(k + 1)+1k + 112k+1(7)If a seller meet exactly one buyer trade only takes place if the buyer’s val-uation is at least twice the seller’s valuation. The expected gains are givenby Z 1=20Z 12wkvk¡11 (v1 ¡ w)dv1dw =18(8)1These should be indexed by k, too, but I omit this as there should not arise any confusion.32.1.2 Perfect information about valuationsWhen a seller with valuation w meets k buyers there is trade if the highestvaluation of the buyers is greater than w. The density of the highest valuationv1 is given by h(v1) = kf(v1)Fk¡1(v1) = kvk¡11 . Thus, the expected gains fromtrade in a k-buyer meeting areZ 10Z 1wkvk¡11 (v1 ¡ w) dv1dw =12¡ 1k + 1+1(k + 1) (k + 2)(9)2.2 Agents with certain gains from trade in the market orthe small market2.2.1 Private information about valuationsLet me now assume that only the agents who ’should’be in the market arethere. Namely, all the buyers whose valuation are above one half and all thesellers whose valuation is less than one half. Given that there are k buyers thedensity function, i.e., the probability that any buyer’s valuation is x is given byd(x) = 2 and the distribution function by D(x) = 2x ¡ 1. In the same vein thedensity of the highest valuation is given byh(v1) = kf(v1)Fk¡1(v1) = 2k (2v1 ¡ 1)k¡1Given that the number of buyers is k ¸ 2 there are two cases to consider.First, when the seller’s valuation w is less than one fourth. Then the buyers’valuations are necessarily at least twice as high and trade always takes place.If the seller’s valuation w is between one fourth and one half trade again takesplace if the buyers’highest valuation is at least double the seller’s valuation.The expected gains areZ 1=40Z 11=22k (2v1 ¡ 1)k¡1 (v1 ¡ w) dv1dw +Z 1=21=4Z 12w2k (2v1 ¡ 1)k¡1 (v1 ¡ w) dv1dw=3¡2k2 + 4k + 1¢16 (k + 1) (k + 2)(10)Second, if the seller’s valuation w is between one fourth and one half tradetakes place even if the highest valuation is below double the seller’s valuationsince the second highest valuation is necessarily above the seller’s valuation.The expected gains areZ 1=21=4Z 2w1=22k (2v1 ¡ 1)k¡1 (v1 ¡ w) dv1dw = 2k + 116 (k + 1) (k + 2)(11)4Expressions (11) and (12) sum to3k + 18(k + 1)=38¡ 14(k + 1)(12)When there is only one buyer and the seller’s valuation is less than one fourthtrade takes place for certain. If the seller’s valuation is between one fourth andone half trade takes place only if the buyer’s valuation is at least double theseller’s valuation. The expected gains areZ 1=40Z 11=22 (v1 ¡ w) dv1dw +Z 1=21=4Z 12w2 (v1 ¡ w) dv1dw = 732(13)2.2.2 Perfect information about valuationsAll buyers’valuations are higher than all sellers’valuations and consequentlythe agents always trade. The expected gains from trade in a k-buyer meetingare given byZ 1=20Z 11=22k (2v1 ¡ 1)k¡1 (v1 ¡ w) dv1dw = 38¡ 14 (k + 1)(14)3 AnalysisIn the previous section the expected gains were calculated given that k buyerscome to a seller with valuation w, and then I calculated the expectation ofthese gains over all possible values of w, i.e., I integrated over w. To get the…nal results it is necessary to sum over k, too, and this is what is done next.But before that let me remind of the case with homogeneous buyers and sellers.In this case it does not matter whether valuations are private information or notsince all the buyers have the same valuation and all the sellers have the samevaluation. Trade takes place whenever a seller meets at least one buyer and thishappens with probability 1 ¡ e¡1 ¼ 0:632.The expected gains in the large market can now be calculated to be withprivate informatione¡118+ e¡11Xk=21k!1k + 112k+1+ e¡11Xk=21k!k ¡ 12 (k + 1)= e¡1=2 ¡ 12¼ 0:107 (15)and with perfect informatione¡11Xk=11k!µ12¡ 1k + 1+1(k + 1) (k + 2)¶=12¡ e¡1 ¼ 0:132 (16)The expected gains in the small market with private information are givenbye¡1732+ e¡11Xk=21k!38¡ e¡11Xk=21k!14(k + 1)=332e¡1 +18¼ 0:159 (17)5and with perfect informatione¡11Xk=11k!µ38¡ 14 (k + 1)¶¼ 0:171 (18)Now I am in a position to compare the e¢ciency of the two market structuresand the two information structures. In the Walrasian markets all the possibletrades are executed at price one half. The price is just a transfer that does nota¤ect the total gains which add up to 0:375. To quantify the ine¢ciency causedby heterogeneity and not by private information the appropriate comparison isto the small market with perfect information. There 0:171=0:375 = 45:6% ofthe possible gains are realised. When the valuations are private information therealised gains are 0:159=0:375 = 42:4%. In this case the di¤erence is quite small.In the large market under perfect information 0:132=0:375 = 35:2% of thepossible gains are realised. When valuations are private information 0:107=0:375 =28:5% of the gains are realised.Proposition 1 Both under perfect information and private information thesmall market is more e¢cient than the large market Under perfect informa-tion the small market yields 45:6% of the gains of the Walrasian market and thelarge market yields 35:2% of the gains of the Walrasian market . Under privateinformation the corresponding gains are 42:4% in the small market, and 28:5%in the large market.Notice that the drop in e¢ciency in far more dramatic in the large marketthan in the small market when comparing perfect information and private infor-mation cases. This is because private information leads to strategic behaviour,i.e., solitary buyers o¤ering just half of their valuation, but in the large marketthis leads to no trade more often than in the small market where it is knownthat each buyer values the goods more than any seller.4 ConclusionI study a parameterised situation of decentralised markets where the di¢culty ofco-ordinating the agents contacts keeps the economy from realising all potentialgains. On top of that I introduce heterogeneity into the agents’valuations.Depending on which agents participate in the market there may be signi…cantdi¤erences in e¢ciency.To my knowledge these e¢ciency losses have not beenspelled out before, and they may be instructive even though the results arebased on uniform distribution of valuations and there being equal numbers ofbuyers and sellers.ReferencesGale D. and H. Sabourian, 2005, Markov equilibria in dynamic matchingand bargaining games, Games and Economic Behavior, forthcoming.Myerson R. and M. Satterthwaite 1983, E¤cient Mechanisms for BilateralTrading, Journal of Economic Theory, 28, 265-281.6

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