Queues, not just mediocrity: Inefficiency in decentralized markets...

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International Journal of Industrial Organization998–1014

www.elsevier.com/locate/econbase
26 (2008)
Queues, not just mediocrity: Inefficiency in decentralizedmarkets with vertical differentiation

Clara Ponsatí a,⁎, József Sákovics b

a Institut d'Anàlisi Econòmica — CSIC, Barcelona, Spainb University of Edinburgh, United Kingdom

Received 6 November 2006; received in revised form 23 April 2007; accepted 16 September 2007Available online 26 September 2007

Abstract

We analyze a dynamic, decentralized market with endogenous entry, where in each period the active sellers supply one unit of anindivisible service at varying degrees of quality. The customers that have entered themarket are randomlymatchedwith the active sellersand prices are set by (complete information) pair-wise bargaining. In its unique steady state, the market leads to an excess diversity ofquality and customers may have to suffer costly delays. Notably, efficiency is not regained as per period delay costs disappear. We alsoshow that setting minimal quality standards, such as licensing rules by a professional college, will improve welfare (and even ConsumerSurplus), relative to the free market, whenever the inefficiency is caused by a large enough excess supply.© 2007 Elsevier B.V. All rights reserved.

JEL classification: C78; D43; D62; L44; L84

Keywords: Liberal professionals; Quality standard; Queuing externality

2 According to recent estimates, professional services employ about6.4% of the EU15 work-force and represent 10% of overall high skillemployment; in 2001 the sector had turnover of around 980 billionEuros and created around 500 billion Euros of total value added. See

1. Introduction

We model markets where sellers with limited capacitysupply goods, the quality of which is heterogeneous andobservable only upon entering the store/office. In thesemarkets reputation is not a major factor.1 This is typicallythe case for services provided at small establishments overdispersed locations like road restaurants, bed-and-break-fast accommodation, or car repair shops. It is also the casefor tradesmen, like carpenters, small builders or plumbers;or middle-of-the-road professional service providersthat are hired infrequently, like architects, accountants,lawyers and other liberal professionals. While small in-dividually, these activities play a major role in the per-

1 Otherwise, quality could be predicted.

⁎ Corresponding author.E-mail address: [email protected] (C. Ponsatí).

0167-7187/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.ijindorg.2007.09.003

formance of advanced industrialized economies. They arequantitatively important in the aggregate2 and, being anintermediate input for many other activities, improve-ments in their quality and competitiveness have majorspill over effects across the economy.3

These markets are often characterized by a very highlevel of regulation. While specific regulations vary acrosscountries, trades and professions, quality standards, ad-vertising constraints, price limits or other controls are

the report on Competition in Professional services COM (2004) 83.3 For example, according to a recent estimate by the Italian Antitrust

Authority, 6% of the costs in Italian exporting firms are due toprofessional services.

mailto:[email protected]

http://dx.doi.org/10.1016/j.ijindorg.2007.09.003

999C. Ponsatí, J. Sákovics / Int. J. Ind. Organ. 26 (2008) 998–1014

extensively imposed either by State regulation or in theform of self-regulation by professional associations. Therisk of regulatory capture has been a major concern sinceStigler (1971). It has been argued that these arrangementsraise prices and limit entry but fail to assure the ap-propriate quality of service, opening an intense debateamong policy-makers on (de-)regulation policies.4 In fact,since the landmark ruling of the US Supreme Court in1975 on Goldfarb v. Virginia State Bar there is an everincreasingly active application of antitrust laws in theprofessional services sector in all OECD countries.

The conventional wisdom argument about marketswith quality heterogeneity is based on asymmetricinformation: if at the time of contracting the quality isnot observable by the customer, it actually may be thehigh quality services that are driven out of the market.5

Consequently, the main focus of policy debates is thecorrection of market failures due to informationalasymmetries. Arguments are often based on thepresumption that in the absence of asymmetricinformation a free market would perform efficiently.The present paper shows that this presumption is notcorrect; we contend that even if customers couldascertain the quality of service before purchase,efficiency would not obtain. Asymmetric informationis an important problem indeed. However, argumentsabout (de-)regulation policies deserve further scrutiny;understanding the source of inefficiencies abstractingfrom asymmetric information considerations is anecessary first step.

We develop our arguments in the context of decen-tralized trade model based on a dynamic randommatching market. Matching is random to reflect thatthe customers cannot tell the quality offered by thedifferent sellers before they choose who to approach. Inorder to focus on the issues beyond asymmetric infor-mation, we assume that customers learn the quality ofthe service upon meeting a service provider. In short,we model service as a search good. We capture limitedcapacity by assuming that each supplier can serveonly one customer at a time (making agreements onprices also bilateral). We model the resulting two-person bargaining with complete information using asimple strategic model, which can also be interpreted

4 See the OECD DAFFE/CLP(2000) 2 report on “Competition inprofessional services” for a host of evidence on how anti-trustauthorities have approached the issue in twelve countries in additionto the EU. See also Paterson et al. (2003).5 Uncertainty about service quality (Arrow, 1963) and the resulting

adverse selection (Leland, 1979; Stiglitz, 1979; Wilson, 1979, 1980)and moral hazard (Shapiro, 1986) were modelled and understoodsome time ago.

as the — possibly asymmetric — Nash BargainingSolution.

We characterize the unique steady state of the market,and show that it leads to an inefficient outcome: there isan excess of active sellers leading to a sub-optimalaverage/marginal quality and in addition, for lowenough delay costs, a queue will form and customerswill suffer costly delay. A suboptimal provision ofquality and delays are standard equilibrium features inmarkets with costly search. The novelty of our analysisis that we provide a simple model where the removal ofthe per-period delay costs does not dissipate both theseinefficiencies. While the supply distribution doesconverge to the efficient one, as the delay cost shrinksthere is an ever larger queue of customers waiting to beserved and the aggregate waiting cost is increasing.

The unique market equilibrium admits a straight-for-ward characterization, thus comparative statics and policyissues are easily addressed. We highlight that regulatorypolicies using a single policy tool (such as settingminimalquality standards) can be helpful but they only lead tosecond-best outcomes. Finally, we show that when thesellers are allowed to self-regulate, welfare (and evenConsumer Surplus) is increased relative to the a freemarket, whenever the inefficiency was caused by toomany low quality providers in the market.

Helmut Bester's (1988) model of a decentralizedmarket with quality heterogeneity and bargaining6 is adirect forbearer of our analysis. Under positive searchcosts, market equilibria are similar in his formulationand ours. In his model comparative statics are prob-lematic because of the multiplicity of equilibria, so hedid not address issues of (self-)regulation. The mostimportant modelling difference is that Bester allows aseller to sell to several buyers simultaneously, with noexternalities between trades. As a result, there is norationing — and, therefore, queuing — in his set-up.Thus, unlike us, he concludes that under negligiblesearch costs any market equilibrium approaches theefficient, competitive, outcome. Chaim Fershtman andArthur Fishman (1994) examine the effects of aminimum wage in a labor market with costly search.They find that worker search activity may decrease as aresult of a higher minimum wage, leading to higherfirm bargaining power paradoxically leading to lowerwages at ultramarginal firms. While this exercise issomewhat similar to the introduction of minimumquality standards, both the intuition and the effects arerather different. In the more recent literature, Max

6 Bester (1993) extends this model to endogenous quality choiceand analyzes how trading rules affect this choice.

8 Since one of our claims is that too much of mediocre quality isproduced in equilibrium, by not giving mediocre producers a costadvantage, we actually strengthen our result.9 Alternatively, travel costs, either literally or in the sense of

Hotelling.10 The presence of a positive measure of buyers with (very) negativeopportunity costs is not necessary. However, otherwise for some

1000 C. Ponsatí, J. Sákovics / Int. J. Ind. Organ. 26 (2008) 998–1014

Blouin (2003) provides a simple model with experiencegoods and no new entry after the first period. For thelatter reason, waiting is again not an issue for negligiblewaiting costs. Rachel Kranton (2003) stresses the im-portance of competition between suppliers — and ingeneral externalities among them — as an importantfactor determining the choice of the quality produced.She shows that competition may lower the incentive toproduce high quality and thus finds that self-regulationmy serve as a way of guaranteeing product quality.Steven Davis (2001) looks at a labor market applicationwith investments in— job and worker— quality, withina search context. However, he assumes one-time entryand a single round of search, so again he does notencounter the queuing problem.

The rest of this article continues as follows. InSection 2, we lay out the details of our model, providethe tools to evaluate welfare in the free market andunder the different regulatory regimes and discuss thefirst best allocation. Section 3 contains the analysis ofthe market equilibrium and addresses comparativestatics (illustrated with numerical examples), measuringthe impact of changes in the level of frictions and inbargaining power. We analyze the effect of minimumquality standards (licensing) in Section 4. Section 5concludes. Appendix A discusses the interpretation ofthe concept of competitive equilibrium in our market.The demonstrations omitted from the text are inAppendix B.

2. The model

We consider a market for a single commodity ofheterogeneous quality, composed of a set of serviceproviders (sellers) and a set of customers (buyers). Themarket operates over (discrete) time. Agents are riskneutral and maximize their (undiscounted) expectedutility. Trade is carried out by decentralized agreementsbetween buyer–seller pairs that meet randomly andnegotiate the price to trade one unit of an indivisiblegood.

There is a continuum of buyers and sellers. Thepopulation of potential sellers is constant and is given bythe unit interval, where each seller is uniquely indexedby her type, θ∈ [0, 1]. Seller θ can produce a good ofquality q(θ), where q(.):[0, 1]↦ [0, 1] is assumed to bestrictly increasing.7 For simplicity, we also assume thatq(.) is differentiable, q(0)=0 and q(1)=1. Note that q(.)

7 Note that it would be without loss of generality to assume that it isnon-decreasing. We assume strict monotonicity to simplify theanalysis.

can also be thought of as the inverse of the probabili-ty distribution function of quality. We denote the aver-age quality in the market when only goods above qualityq(θ⁎) are produced by Pq ðh⁎Þ ¼ 1

1�h⁎R 1h⁎ qðxÞdx.

Each seller can produce a single unit—which cannotbe stored — in each period. The cost of production isindependent of the quality8 and it is normalized to zero.We also assume, for simplicity, that the sellers' humancapital is fully specific to the market, and thus theiropportunity cost of being in the market is zero. Sellerswho (rationally) expect not to be able to sell at a positiveprice are supposed not to be present in the (steady state)market. Since a higher quality seller who trades canalways do better than a lower quality one, the activesellers form an interval: (θ⁎, 1].

In every period a measure 1 of new potential buyersappear. They are heterogeneous in terms of their outsideopportunities,9 which are distributed according to the(strictly increasing and differentiable) probability distri-bution function E(.):[−∞, 1]→ [0, 1].10 They mustdecide whether to enter the market. If they don't, theytake up their outside option and exit the model. As aresult, the flow of buyers entering the market in eachperiod is E(.) evaluated at the expected profit of a buyerupon entering the market (conditional on the measure ofbuyers entering simultaneously). In order to inform ourintuition, it will be useful as we proceed to consider thehomogeneous case of completely inelastic entry as well,where a measure eb1 of buyers enter each period (andsuffer no opportunity cost). While in the market, allbuyers are identical: they wish to purchase a single unitof the good, which they value at its quality, q(θ). Inevery period that a buyer is in the market but does notget served he suffers a cost, c∈ (0, 1]. Following anunsuccessful period in the market, buyers can choosewhether to continue in the market next period or toreturn to their outside option.

In every period, traders seeking a match meet atrandom. We denote the probabilities of finding a tradingcounterpart by πb and πs for buyers and sellers, respec-tively. These probabilities depend (only) on the state of

parameter values a steady state equilibrium would fail to exist (seeSection 3.1). We assume that �R 0�l xdEðxÞ bl, that is, the aggregate“opportunity benefit” of the buyers entering in a given period is finite.Otherwise, welfare would be infinite, independent of the marketoutcome.


the market, denoted by (θ⁎, b⁎). θ⁎ is the marginalseller (so that 1−θ⁎ is the measure of active sellers inthe market), while b⁎ is the measure of buyers that seeka seller. There is efficient,11 uniform random matchingamong the active market participants: sellers receive a

customer with probability ps ¼ min 1; b⁎1�h⁎

n oand buyers

find an active seller with probability pb ¼ min 1;1� h⁎

b⁎

� �.

Once a buyer finds himself in a store, he learns thequality of the product offered by the seller. Next, they startbargaining over the price, p, where the buyer maximizesq−p while the seller maximizes p. Bargaining proceedsas follows. One of the parties is randomly selected tomake a proposal. The probability that the buyer (the seller)is selected is λ∈ (0, 1) (respectively, 1−λ). If theresponder accepts the proposed price, the transaction isconsummated. The buyer then leaves the market, whilethe seller can serve a new customer in the followingperiod. If the responder rejects, then they break upnegotiations and both traders search for a newmatch in thefollowing period (or the buyer may decide to take up heroutside opportunity).12

We are interested in characterizing the market in itssteady state, that is, when the composition of activetraders in equilibrium is constant over time. We willrefer to such a situation as the market equilibrium.

2.1. Welfare

In this subsection we develop the (utilitarian) welfarefunction, which will enable us to evaluate the efficiencyof our market. We also establish the aggregate profitfunction, which we will use to derive the optimal deci-sions of a (self-)regulatory authority.

In equilibrium,13 the measure of active sellers in eachperiod is 1−θ⁎. Given the matching technology, every-one on the short side of the market is matched in eachperiod, so— since, as we will see shortly, in equilibriumall matches end in trade — the measure of buyers en-tering the market in its steady state is the same as themeasure of trade, t⁎=min{b⁎, 1−θ⁎} in each period.

11 Considering the frictionless limit is for simplicity; our analysis canbe easily extended to environments where the matching technologydisplays search frictions.12 Note that this bargaining procedure is an equilibrium outcome in amuch more elaborate model, where following the random choice ofthe first proposer, there is an alternating offer bargaining game witheither player being able to leave the negotiating table following arejection. See Ponsatí and Sákovics (1998).13 For ease of exposition, we define the welfare function directly fora market equilibrium.

The aggregate opportunity cost incurred by the buyersentering in a given period is thus

Z E�1ðt⁎Þ

�lxdEðxÞ ¼

Z t⁎

0E�1ðxÞdx:

Welfare per period is the realized surplus (qualityweighted measure of trade) minus the opportunity andwaiting costs:14

W ¼ t⁎Pq ðh⁎Þ �Z t⁎

0E�1ðxÞdx� ðb⁎ � t⁎Þc: ð1Þ

When entry is inelastic, there is no opportunity costto worry about:

W ðeÞ ¼ ePqðh⁎Þ � ðb⁎ � eÞc:A profit-maximizing, self-regulating professional col-

lege's objective function can be derived along the samelines as the welfare function. Denote by p(θ) the expectedprice that prevails in the q(θ) bargaining pair. Then wehave that the aggregate profits are given by

P ¼ t⁎

1� h⁎

Z 1

0pðxÞdx ¼ t⁎Pp ðh⁎Þ:

2.2. The efficient outcome

As a benchmark, it is useful to characterize thewelfare-maximizing allocation — attainable undercentralized trade with complete information — in ourmarket.

Proposition 1. The unconstrained welfare maximizingallocation results in a balanced market. The marginaltrader, θc, is the solution of E(q(θc))=1−θc, and 1−θc

buyers enter in each period. For exogenous entry, themarginal trader in the optimal allocation is given byθc(e)=1−e.

Proof. See Appendix B. □

Quite intuitively, the first best results in no idletraders, and the marginal consumer having an opportu-nity cost which equals the quality (surplus) provided bythe marginal seller. See Fig. 1. Note that unlike in theusual graph, the vertical axis measures buyer's surplus,

14 Note that the expected number of periods a new entrant has to waitbefore getting served is 1�pb

pb. Since pb ¼ t*

b*, we have that per entrantthe expected waiting cost is b*�t*

t* � c. Since there are t⁎ new en-trants, the aggregate cost of waiting is (b⁎− t⁎) c. This is the same asthe cost of waiting incurred by the currently waiting buyers in anygiven period.

E-1 (1-ΘΘ)

q (Θ)

Θ

Z

Fig. 1. Determining the buyer's surplus in the competitive equilibrium.


z≔q(θ)−p(θ), rather than price. For a further elaborationon this, and on how we can derive this “competitive”allocation from first principles, see Appendix A.

3. The market equilibrium

The characterization of the market equilibrium re-quires the simultaneous determination of the price dis-tribution, p(.), the marginal quality, q(θ⁎), and themeasure of buyers active in the market, b⁎.

Consider a market equilibrium. Denote the expectedsurplus of a buyer upon entry by xb. This variable iscentral to the analysis, as it impacts on the outcome ofbargaining and the entry decisions of traders on bothsides of themarket. The first result wewant to establish isthat no buyer ever leaves the market empty handed andthat every match will result in a trade. Note that, despitethe fact that only sellers who (rationally) expect to beable to trade (eventually) participate in the matching, thelatter claim is not obvious as we could have a situationwhere some buyers only agree to trade if they are chosento make the first offer — and otherwise they prefer towait for a new match.

Lemma 1. In equilibrium every match results in tradeand no (unmatched) buyer leaves the market. Moreover,the marginal quality is given by

qðh⁎Þ ¼ maxfxb � c; 0g: ð2Þ

Proof. By stationarity, if a buyer expected more in themarket than her outside opportunity in the previous period,this will still be the case in the current period. Thus, nobuyer ever leaves the market empty handed. By the above(and stationarity) the buyers' continuation value upondisagreement is the same as their expected surplus uponentry, but “discounted” by one period: xb−c. Thisdetermines the marginal quality that is supplied inequilibrium, q(θ⁎), in a straightforward way. Note that no

buyer will ever purchase from a seller that offers a qualityof q(θ)bxb−c, since even trade at zero price— the lowestprice a seller would accept—would yield less thanwaitingfor a rematch. Thus, potential sellers of quality below xb−cwill not be active in a market equilibrium. On the otherhand, a situation where the marginal quality offered in themarket is q(θ⁎)Nmax{xb−c, 0} cannot happen in a marketequilibrium either. Otherwise, potential sellers of qualityq(θ), max{xb−c, 0}bq(θ)bq(θ⁎), would enter themarketand — if matched and chosen to make an offer, whichwould happen with positive probability— make positiveprofits by trading at a price slightly above q(θ)− (xb−c)N0,since any matched buyer is strictly better off trading atsuch price than not trading. Hence, in a market equi-librium q(θ⁎)=max{xb−c, 0}. This, in turn, implies thatall matches end in trade, since for θ′Nθ⁎, both traders arehappy to trade for any p∈ (0, q(θ′)−q(θ⁎)). □

The endogenous outside option of a buyer (not to beconfused with the exogenous outside opportunity whenshe does not enter the market) is equal to the expectedprofits from a future match, net of the expected cost ofwaiting for that match. For a buyer the expected valueattainable in future matches is independent of the qualityof the good that he is currently bargaining for. Instead, itis a function of the distribution of quality effectivelysupplied in the market, and it coincides with the “dis-counted” value of the expected payoff of a buyer uponentry, xb−c. Turning to the sellers, note that their out-side option is zero, since the future sales are independentof what happens in this period, and upon disagreementwith the buyer they are currently matched with, theyhave no further opportunity to sell in the current period.

In order to derive the (expected) outcome of bar-gaining, observe that in equilibrium all proposers offerthe price that leaves the responder indifferent to his/heroutside option. Therefore, the expected equilibriumprice (as a function of quality) is given by

pðhÞ ¼ ð1� kÞðqðhÞ � xb þ cÞ: ð3ÞNext, note that the expected surplus of a buyer uponentry satisfies

xb ¼ pb

Z 1

h⁎

qðhÞ � pðhÞ1� h⁎

dhþ ð1� pbÞðxb � cÞ: ð4Þ

To understand this equality, note that the buyer willeither be matched and then obtains the expected dif-ference between the quality of the good and the price shewill pay for it, or, she will be unmatched and will haveto wait until next period to find herself in the samesituation as now.


Substituting the price Eq. (3) into Eq. (4) andrearranging, we obtain15

xb ¼ Pq ðh⁎Þ � cð1� kpbÞkpb

: ð5Þ

Therefore, putting Eqs. (5) and (2) together, we haveidentified our first equilibrium condition:

qðh⁎Þ ¼ max Pq ðh⁎Þ � ckpb

; 0

� �: ð6Þ

Turning to the buyers' entry decision, recall that in asteady state the measure of buyers entering and leaving(trading) must be equal. By Lemma 1, all matches lead toan exit, so the flow of buyers through the market equalst⁎. Thus, in general, E(xb)= t⁎. Substituting in fromEq. (2), we obtain our second equilibrium condition

qðh⁎Þ ¼ maxfE�1ðt⁎Þ � c; 0g: ð7Þ

In order to simplify the analysis, we will assume thatthe distribution of quality is well-behaved: We say thatthe quality distribution is regular if q̄′(θ)bq′(θ) for allθ∈ [0, 1]. A sufficient16 condition for regularity isthat q(.) is concave. Since q(.) is the inverse of thedistribution function of quality, this roughly says that weshould not have too many low quality potential sellers.

Let θb(λ, c) be the unique (guaranteed by regularity)solution to our first equilibrium condition (6) for abalanced market:

k½PqðhÞ � qðhÞ� ¼ c; ð8Þ

when it exists (that is, when q̄(0)≥c /λ), and zerootherwise. This equation determines the marginal sellerif the sellers' entry decision turns out to be the bindingconstraint. Note that in a buyers' market (where buyersare the short side and, therefore, πb=1) the left-hand sideis the expected gain to a buyer matched with the marginalseller from refusing to trade and waiting for a rematch,while the right-hand side is the cost of switching. Ifthe gain were larger, then no buyer would be willing totrade with this seller, so she would not enter themarket. Similarly, define θs(c) as the unique solution

15 A more intuitive way of writing Eq. (5) is xb ¼ Pq ðh*Þ�cð1�kÞkpb

� cð1�pbÞpb

, where the second term on the right-hand side is theaverage price and the last term is the expected waiting cost of a newentrant.16 But, by far, not necessary: the uniform quality distribution satisfiesregularity by a wide margin, as PqðhÞ � qðhÞ ¼ 1�h

2 .

to17 the second equilibrium condition (7) for a balancedmarket:

EðqðhÞ þ cÞ ¼ 1� h: ð9Þ

This equation determines the marginal seller if thebuyers' entry decision turns out to be the binding con-straint. The left-hand side gives us the measure of buyerswilling to enter as a function of the marginal quality andthe waiting cost (c.f. Eq. (7)). The right-hand side isthemeasure of active sellers, which in a sellers'market hasto equal the measure of new buyers entering the market.

We are now ready to state our characterizationtheorem:

Theorem 1. Assume that q(.) is regular. Then thereexists a unique market equilibrium.

i) When θb(λ, c)≤θs(c), in equilibrium buyersare the short side, the marginal seller is θb(λ, c)and the measure of buyers in the market isbb ¼ E PqðhbÞ � cð1�kÞ

k

� �.

ii) When θb(λ, c)≥θs(c), in equilibrium sellersare the short side, the marginal seller is θs(c)and the measure of buyers in the market isbs ¼ ½Pqðhs � qðhsÞÞ� kð1�hsÞ

c .


Similar arguments lead to the characterization of themarket equilibrium when entry is exogenous:

Corollary 1. Let q(.) be regular and assume that buyersenter at a constant rate e. Then there is a unique marketequilibrium.

1. For small entry rates, e∈ (0, 1−θb], buyers are theshort side (b=e) and the marginal seller is θb(λ, c).

2. For larger entry rates, e∈ (1−θb, 1), sellers are theshort side (θ⁎=1−e) and the measure of buyers inthe market is b* ¼ Pqð1� eÞ � qð1� eÞð Þ kec .

3.1. The price distribution

In order to complete our description of the marketequilibrium we need to establish how the surplus

17 To be rigorous, if we do not want to use the additional informationthat q(θ)+c≤1 before solving the equation, we need to extend thesupport of E(.) up to 2: E(x)≡1 for x ∈ [1, 2].

Fig. 2. The Nash Bargaining Solution.


generated by trade is shared. Putting Eqs. (3) and (5)together, we obtain the price distribution:

pðhÞ ¼ 1� kð Þ qðhÞ � Pqðh⁎Þ þ ckpb

� �: ð10Þ

That is, expected prices are linearly increasing in quality,with zero — by Eq. (6) — being paid for the goodof the marginal producer (unless the marginal qualityis zero, in which case the lowest price is pð0Þ ¼ð1� kÞ c

k � Pqð0Þ� N0Þ.18 Consequently, the sellers do

obtain the share corresponding to their raw bargainingpower, but not of the full surplus, rather of the incrementalsurplus over the one provided by the marginal seller. Thisis consistent with the fact that our bargaining equilibriumis equivalent to the (asymmetric) Nash Bargaining Solu-tion, with the vector of outside options for the buyer andseller, Pqðh⁎Þ � c

kpb; 0

� �, as the threat point and q(θ) the

size of the bargaining surplus.19 See Fig. 2.Note that the price at all supramarginal sellers is

decreasing in the marginal quality, q(θ⁎). In otherwords, more competition — additional (though lowerquality) sellers active in the market — corresponds tohigher prices earned by the incumbents. This is a directconsequence of the fact that the sellers' entry is endog-enous, implying that the marginal seller must expect zeroin equilibrium, whoever she is. Of course, this does notimply that high quality sellers should prefer to maximize

18 For c /λ high enough to result in a negative buyer's optionPqðh*Þ � c

kpbb 0

� �, the price charged may exceed the quality

provided. This is a direct consequence of the fact that the benefit toa buyer from trade is not just q(θ) but also the avoidance of thewaiting cost. Of course, this can only happen in equilibrium because,by assumption, there are buyers with negative enough opportunitycosts who are still willing to enter the market. If this were not the case,the market would fail for c /λ high enough.19 See Binmore et al. (1986) for a discussion of this equivalence.

entry, since to achieve that they may have to decreasetheir matching probability. We will examine this ques-tion in more detail in Section 4, where we analyze theeffects of the introduction of a quality threshold — as aresult of, say, occupational licensing — on the marketequilibrium.

Condition (10) also allows us to make our firstempirically relevant/testable observation.

Proposition 2. If the price charged by the marginalseller is zero (positive), the “price-quality ratio”, pðhÞ

qðhÞ , isstrictly increasing (decreasing) in the quality of thegood traded.

Proof. From Eq. (10) it is immediate that

pðhÞqðhÞ ¼ ð1� kÞ 1�

Pq h⁎� �

� ckpb

qðhÞ

24

35; ð11Þ

which — for λ∈ (0, 1) — is strictly increasing (de-creasing) in q(θ), as long as Pq h⁎ð Þ � c

kpbN bð Þ0. From

(6) and (10), this corresponds to whether, the pricecharged by the marginal seller is zero or positive. □

That is, the proportion of the gains from trade ap-propriated by the sellers is increasing with the quality oftheir good as long as the buyers can always expect apositive surplus. Since the latter condition is usuallysatisfied, this implies that the mark-up is increasing withquality, as is commonly observed. On the other hand,when the buyers expect a loss in the market (even if asmaller loss than what they would suffer outside), theresult is the opposite, the mark-up is decreasing withquality, as it is driven by a fixed (independent of quality)effect. If customers are in dire straits, even low qualitysellers can extract significant surplus, so prices are less

Fig. 3. Marginal seller θ⁎ as a function of λ; θs is the flat line and θb isthe increasing curve.

Fig. 5. Welfare, consumer surplus and profits (top, middle and bottomcurves); the flat line displays first best welfare.


dispersed (c.f. the small(er) price dispersion observed,say, for emergency plumbing services).

At this point an example is useful to illustrate themechanics of the market equilibrium and to preview themain points that we will be making in the sequel.

3.2. An example

Consider a market where potential qualities aredistributed uniformly in [0, 1], i.e. q(θ) =θ, andassume that entry is given by EðxÞ ¼ 1

ð2�xÞ2. Theefficient quality provision is given by the solution to

1ð2�xÞ2 ¼ 1� x, that yields θc=.534. The first best levelof welfare is thus W fb ¼ ð1� hcÞPqðhcÞ � R 1�hc

0 E�1ðxÞdx ¼ð1� :534Þð1þ :534Þ

2�Z :466

:0

2� 1ffiffiffix

p� �

dx ¼ :357þ :434 ¼ :791.Let us fix the delay cost at, say, c=.1, and examine

the market equilibrium for different values of the bar-gaining parameter λ. Eq. (8) yields hbðc; kÞ ¼ 1� :2

k ,which is positive for λ≥ .2. Eq. (9) yields θs(c)= .494,and 1� :2

k V:494 if and only if λ≤ .395. Fig. 3 displaysθs and θb as a function of λ, and the determination of themeasure of active sellers, θ⁎. Thus, for c= .1, the marketequilibrium is as follows:

1. For λ≤ .2 all sellers are active, θ⁎=0 and the buyersare the short side; b⁎ ¼ 100k2

ð14kþ1Þ2, and ps ¼10ð1�kÞ15�14k b1.

2. For .2bλ≤ .395 not all potential sellers are active,h⁎ ¼ 1� :2

k N0, but buyers continue to be the shortside; b⁎ ¼ 10ð1�kÞk

1þ9ð1�kÞk ; and ps ¼ 50ð1�kÞk21þ9ð1�kÞk b1.

3. For λN .395 the marginal seller is θ⁎=.494, and sellersbecome the short side; b⁎ ¼ 1:28k and pb ¼ :395

k b1.

The market equilibrium measures of active agents foreach side of the market are displayed in Fig. 4. Withthese measures in hand, welfare, profits and consumersurplus as a function of λ are readily evaluated. Theresults of this exercise are displayed in Fig. 5, where

Fig. 4. Buyers (b⁎ continuous line) and sellers (1−θ⁎ dashed line) as afunction of λ.

values can also be compared to the first best welfare(top horizontal line). It is apparent that the efficientallocation of bargaining power is one where bothsides make the offer with positive probability.Furthermore, the maximum welfare (attained at thevalue of λ which balances the market) still falls shortof the first best, as the marginal quality is too low. Letus now fix the bargaining power at, say, λ=1/2, andexamine the market equilibrium for varying levels ofthe delay cost c. Eq. (8) yields θb(c, λ)=1−4c, whichis positive for c≤1/4. Eq. (9) is now the cubicequation

1

ð2� x� cÞ2 ¼ 1� x. Its unique real solution, x(c),is displayed in Fig. 6, along with 1−4c; observe thatx(0)=θc and x(c) crosses 1−4c from below at c= .129.Thus, the market equilibrium for λ=1/2 is as follows:

1. For c≤ .129 sellers are the short side and the mar-

ginal seller is h⁎ ¼ xðcÞ; b⁎ ¼ ð1� xðcÞÞ22c

and pb ¼2c

1� xðcÞ b 1.

2. For .129bc≤ .25 buyers are the short side andthe marginal seller is h⁎ ¼ 1� 4c; b⁎ ¼ 1

ð1þ3cÞ2 andps ¼ 1

ð1þ 3cÞ24cb1.

Fig. 6. Marginal seller θ⁎ as a function of c.

Fig. 8. Welfare, consumer surplus and profits (top, middle and bottomcurves) as a function of c; the flat line displays the first best welfare.


3. For cN .25 buyers are the short side with all potentialsellers active; b⁎ ¼ 1

ð1:5þcÞ2 and πs=b⁎b1.

The measures of agents active on each side of themarket as a function of c are displayed in Fig. 7. Notethat, as c→0, θ⁎→θc, b⁎→∞ and πb→0. Hence, asdelay costs vanish, the provision of quality approachesthe efficient level. However, the measure of buyers thatkeeps the market at a steady state grows without bound.In fact, this increase in the queue is faster than thedecrease in the search cost. Fig. 8, displays welfare,consumer surplus and profits as a function of c, andrelates them to the first best welfare; it is clear thatefficiency is not attained in the limit. Indeed, at c=0,welfare, profits and consumer surplus are all increasingin c! Again, the maximum welfare — attained at thevalue of c which balances the market — still falls shortof the first best.

Before turning to the analysis of regulatory tools thatmight constrain and perhaps correct the marketperformance, in the next subsections we show that theequilibrium features and comparative statics displayedin this example hold in general, and we shed light on thereasons behind them.

3.3. Delay costs

Let us first investigate the behavior of the marketequilibrium in general as the per period cost of delay, c,varies.

Let us first look at the imperfection caused indirectlyby this friction: the deviation of the quality provided bythe market from the efficient provision (given by thecompetitive allocation derived in Section 2.2). Asexpected, since frictions “lock buyers in”, the marketequilibrium always has too low marginal — and

Fig. 7. 1−θ⁎, and b⁎, as a function of c.

therefore average — quality. However, as (per period)frictions disappear, we obtain the competitive allocation:

Proposition 3. For any λ∈ (0, 1), the marginal qualityin the market equilibrium is decreasing in c,20 and inthe limit as (per period) frictions disappear, c→0, itcoincides with the competitive one.

Proof. Implicitly differentiating their defining equal-ities, it is easy to see that both θb(λ, c) and θs(c) aredecreasing in c. Since limc→0 θb(λ, c)=1 for all λN0(even if q(.) is not regular!),21 as c→0 we always havea sellers' market, and the marginal seller is θs(0), whichcoincides with θc. □

It is important to observe that the convergence tothe Walrasian outcome is not complete, though.Notably, the price distribution in the “frictionless”market does not match that of the competitive equilib-rium. As we show in Appendix A, the latter leads to aconstant buyer's surplus: q(θ)−pc(θ)≡q(θc), while theformer is given by (7): p(θ) = (1−λ) (q(θ)−q(θc)).Thus, the competitive price distribution correspondsto the frictionless market price distribution where al-most all the bargaining power is given to the producers(λ≃0).

For λN0, the buyer's surplus is increasing in quality inthe “frictionless”market equilibrium: q(θ)−p(θ)=λq(θ)+(1−λ) q(θc). This is at first blush surprising: if there areindeed no frictions, why a consumer matched with themarginal seller does not refuse to trade and wait for abetter match? The explanation can be extracted fromTheorem 1: the measure of buyers in the market, b⁎, isincreasing without bound as the buyers become more and

20 Until it reaches zero. From then on it is, obviously, constant.21 Note that this result does not require q(.) to be regular, since asc → 0 all the roots of q̄ (θ)−q(θ)=c converge to 1 anyway.


more patient. As a result, the probability of gettingrematched in the next period tends to zero as c tends tozero.

Consequently, we do not achieve an efficient outcomeeven in the limit as (per period) delay cost disappear:while the quality distribution and the measure of newbuyers entering the market are the same as in thecompetitive equilibrium, there exists an infinite queueand the buyers incur positive delay costs on average.

Actually, the limiting outcome is not only inefficient,but it is not even second best! To see this, start byobserving that once c is low enough to put us in a sellers'market, the expected payoff of a buyer upon entry, xb, isincreasing(!) in c.

Corollary 2. In a sellers' market, the expected payoff ofa buyer upon entry, xb, is increasing, while the expectedpayoff upon opting out from a match is decreasing, inthe delay cost: dxb

dcN0; dðxb � cÞ

dcb 0.


That is, as c decreases, the effect on buyers who arein a match and on those who are not are divergent: whilea buyer considering leaving the seller he is matched withwill find it easier to switch (as expected), a buyer justentering the matching process will actually face a morecostly wait! This latter phenomenon opens the possibil-ity of welfare being increasing in the delay cost. That is,contrary to intuition, the more patient traders are in adecentralized market the lower welfare they mightenjoy! This is exactly what is displayed in Fig. 8.

The explanation of the “paradox” has a taste of thetragedy of the commons: when buyers decide to enterthe queue they do not take into account the negativeexternality imposed on the rest. As the cost of waitingdecreases,22 during a transitory phase even more buyersenter the market. However, as we have seen above, thisis accompanied by a decrease in the number of sellers,since now it is easier to switch away from a low qualityseller. As a result the queue grows larger, eventuallydissuading entry sufficiently so that it stabilizes at thelevel of the sellers present in the market. This means thatwe end up with fewer buyers entering per period thanbefore, implying that the expected profit upon entry hasdecreased. Since prices have also decreased, the sellersare worse off as well. Thus — since by Proposition 3and the fact that b (and thus the amount of trade) isdecreasing in c, welfare is always decreasing in c in abuyers' market as well — we have proved that

22 In the classic commons example this would correspond to a lower(private) cost of maintaining a goat.

Proposition 4. The delay cost that maximizes welfare isnot 0. Rather, it is cw=E−1(1−θb)−q(θb), the delay costthat equates the number of buyers and sellers in themarket.

At first glance, one might think that the above resultis driven by the endogeneity of the buyers' entryprocess. However, this is not the case, as we show next.

Consider a balanced market. When entry is exoge-nous, a decrease in the delay cost will, obviously, haveno effect on entry. Instead, initially it will decrease thenumber of sellers, since now it is cheaper to switch awayfrom them. Since entry stays constant, this will result inmore unmatched buyers. Eventually, the queue will besufficiently long, so that the expected payoff uponswitching is the same as before and the marginal seller isthe θ=1−e, once more. What has happened to welfare?Well, if xb−c stayed constant while c decreased, it mustbe the case that xb has decreased. Since production andprices are unchanged, this implies that welfare has alsodecreased. Consequently, Proposition 4 holds forexogenous entry as well:

Corollary 3. For exogenous entry the welfare optimaldelay cost is c=λ (q̄ (1−e)−q(1−e))N0, which balancesthe market.

Pulling all these results together, we can see that ifthe free market is a buyers' market then a regulatorwould like to decrease waiting costs, since that wouldhave two beneficial effects: it would increase themarginal quality and decrease the measure of idlesellers. On the other hand, if the free market is onewhere the consumers are queuing, then the trade-offbetween improving the quality distribution and reducingwaiting time is clearly decided in favor of the latter. Theimportance of the cost of waiting in the evaluation ofsocial welfare is a recurrent and key observation in ouranalysis.

3.4. Bargaining power

When analyzing the effects of the distribution ofbargaining power, the key observation to make is thatfor any interior values of the parameters, changes in λcan move the equilibrium between a sellers' and abuyers' market. In a buyers' market the marginal selleris increasing in λ — since the higher the buyers'bargaining power the better is their outside option —,while no queues are formed. Consequently, since theoriginal level of marginal quality was too low, anincrease in λ will always lead to a welfare improvement.However, at some level of λ the market will turn into a

Fig. 9. Welfare as a function of c and λ.


sellers' market. In a sellers' market we have the oppositesituation. The marginal seller is independent of λ, whilethe queue length is increasing in it (since the better deala buyer can get the more waiting he is willing to put upwith). Consequently, welfare is decreasing in λ. As aresult, we have the following:

Proposition 5. The welfare maximizing (second best)bargaining power, λw, is the one that leads to an equalnumber of buyers and sellers entering the market,if feasible:kw ¼ min 1;

c�qðθsÞ � qðθsÞ

� �or min 1;

c�qð1� eÞ � qð1� eÞ

� �;

�for exogeneous entryÞ.

Propositions 4 and 5 are two sides of the same coin.Together they demonstrate that the welfare functionalways has a type of contract curve (that is, a locus ofmaxima) in c−λ space that forms a “ridge:” at any pointalong it welfare is maximized simultaneously23 in bothc and λ. This feature would indicate that the two “tools”are interchangeable. However, the “ridge” also has theparticularity that it can only “spend time” at one edge,the λ=1 one. In other words, for any λ, we can find alow enough c that puts us in a sellers' market, but notvice versa. This is illustrated in Fig. 9 that displays the3-dimensional picture of welfare as a function of c andλ for the parameters of the example of Section 3.2.

Having fully characterized the free market, we turn tothe possibility of (self-)regulation next.

4. Licensing

In this section we analyze the distributive and welfareeffects of a minimal level of quality, q⁎, directly imposedby a (self-)regulator. Note that this quality threshold actsas a constraint on themarket and therefore it necessitates a

23 Note that this does not imply that the level of welfare achievedalong this curve is constant.

re-examination of the equilibrium behavior of the tradersrather than “just” being a comparative statics exercise.

In Section 3 we have seen that not even the“frictionless” free market leads to the first best welfare.Since it does lead to the competitive level of qualitysupplied, this implies that, as a second best, the mostefficient outcome in a “frictionless” market shouldinvolve a somewhat different marginal quality from thecompetitive one. Whether higher or lower, depends onhow the expected cost of waiting varies with themarginal consumer. This is not obvious, since there aretwo effects at work— in opposite directions. On the onehand, increasing the marginal quality lowers the numberof available sellers, so it increases the wait for a match.On the other hand, such an increase also decreases thesurplus, which makes it possible to maintain theequilibrium entry rate with a shorter queue, whichdecreases the wait for a match. How this trade-off isdecided is the first question we investigate.

Note that the expression for the buyers' expectedpayoff upon entry, Eq. (5), is not affected by the qualitythreshold. On the other hand the marginal quality is nowgiven by

qðh⁎Þ ¼ maxfq⁎; xb � c; 0g:Assume that q⁎Nmax{xb−c, 0}, so that the qualitythreshold is binding (otherwise, it has no effect on themarket equilibrium), and let θ⁎=q−1(q⁎). With abinding quality threshold the identity of the sellersactive in the market is directly driven by the constraint.Consequently, the measure of buyers in the market issufficient to characterize the equilibrium allocation:

Proposition 6. When the quality threshold is binding,the measure of buyers in the market is given by

b⁎ ¼ E Pqðh⁎Þ � cð1� kÞk

� �; if E Pqðh⁎Þ � cð1� kÞ

k

� �V1� h⁎

kð1� h⁎ÞPqðh⁎Þ � E�1ð1� h⁎Þ

c1

� �; otherwise:

8>><>>:


Turning to the price distribution, notice that it is stilldetermined by Eq. (10). Using the queue length given byProposition 6 we obtain:

Corollary 4. In a sellers' (buyers') market, the prices atsupramarginal sellers are increasing (decreasing) in abinding quality threshold.

In a buyers' market, a rise in the marginal — andtherefore the average — quality improves the buyers'outside option, resulting in lower prices. In a sellers'market there is a countervailing effect, which turns out

Fig. 10. The profit-maximizing minimal quality can lead to animprovement for both sides.


to be the stronger one: the waiting time increases as aresult of the restriction in supply. Consequently, pricesincrease.

When entry is exogenous we cannot have a bindingquality constraint in a sellers' market, since there wouldbe too much entry for the market to stabilize. In abuyers' market the trades are completely determined bythe two constraints:

Corollary 5. When the quality threshold is binding andentry is exogenous, the measure of buyers in the marketis b⁎=e, while the measure of sellers is 1−θ⁎≥e.

Turning to the socially optimal quality constraint, wehave that:

Proposition 7. The welfare maximizing (binding)quality threshold is the one that equates the measureof buyers and sellers, determined by:

E�1ð1� h⁎Þ ¼ Pq ðh⁎Þ � cð1� kÞk

;

for endogenous entry and θ⁎=θc(e)=1−e, for exoge-nous entry.


In other words, when the unregulated market is asellers' market an imposed minimum quality cannotimprove welfare.24 This has the important implicationthat licensing cannot alleviate the inefficiency caused bycongestion (in a sellers' market). On the other hand, ifthe unregulated market is a buyers' market then there isroom for improvement.

When entry is endogenous, the optimal threshold willbe strictly below θc. The distortion from the competitivelevel of marginal quality is increasing in the friction (c)and decreasing in the buyers' bargaining power (λ).When entry is exogenous, we can obtain (full) efficiencyby imposing the optimal quality constraint.

In either scenario, if the regulator were able to setboth the bargaining power and a quality threshold, thefirst best outcome could be guaranteed:25

Corollary 6. For any c∈ (0, 1], there exists a λ∈ (0, 1),such that setting the quality threshold at the efficientlevel the market equilibrium leads to the competitiveoutcome.

24 Interestingly, a constraint imposing a maximum on the marginalquality would help, but it would be rather difficult to implement(possibly with a lower bound imposed on number of licensees?).25 Because of our restriction that the per period cost cannot exceedthe maximum quality, setting the delay cost (and the marginal quality)may not be sufficient to achieve the first best.


When the (aggregate) profit-maximizing seller'scartel can set the quality standard, we have that thequality threshold is determined independently of thebargaining powers:

Proposition 8. When entry is endogenous, the profit-maximizing quality threshold always results in a sellers'market. If interior, this marginal quality satisfies

qðh⁎Þ þ c ¼ E�1ð1� h⁎Þ � ð1� h⁎Þ dE�1ð1� h⁎Þdh⁎

:


Note that the second term on the right-hand side ispositive, and that if it were zero, the solution of the first-order condition would coincide with the unregulatedoutcome. That is, when the solution is interior, we havethat self-regulation will be welfare improving as longas — the unregulated outcome is a buyers' marketand — entry is not too inelastic.26 In fact, for someenvironments the minimal quality set for the benefit ofactive sellers can be beneficial for the buyers as well.Fig. 10 displays welfare, consumer surplus and profits(top, middle and bottom curves) as a function of theminimum quality standard for an environment27 wherelicensing by the college increases Consumer Surplus.

As we have seen above, exogenous entry is notcompatible with a binding quality constraint in a sellers'market. Since in a buyers' market prices are decreasing,while the amount of trade stays constant, we have that

26 In this latter case, the marginal quality may be set so high thatwelfare is below the one that would result in the unregulated market.27 The corresponding parameters are: c=.2, λ=.5, E(x)=1/(2−x)5,q(θ)=θ.


Corollary 7. When entry is exogenous, a sellers' cartelprefers not to impose a quality threshold.


5. Conclusions

We have shown that in a market with verticaldifferentiation, if sellers can sell only one unit per periodand buyers do not observe quality prior meeting a seller,there is a need for public intervention even in the absenceof ex post asymmetric information about the quality at thepoint of service. The equilibrium outcome involves excesssupply (too low marginal quality) and possibly excessdemand (queuing customers) relative to the first best. Asexcess demand is inversely proportional to the unit cost ofdelay, the inefficiency is robust to the elimination offrictions (delay costs): while excess supply disappears, wehave permanent excess demand in the limit.

While it is convenient to think of this inefficiency asthe result of an externality, it is important to note thatwhen the cost of delay tends to zero the externalitycaused by a new buyer entering the queue is alsoapproaching zero. Consequently, the existence ofexternalities for positive costs does not directly implythat the inefficiency continues to be present at the limit.

From a welfare maximizing point of view, theavoidance of delay costs turns out to be more importantthan the optimal quality mix being provided. Thus, theoptimal quality threshold will never increase the queueseven if that would result in a better quality distribution.Since the queue increases in the quality threshold, this alsomeans that if there is a queue in the unregulated market,minimum quality standards will not be able to improvewelfare. Under self-regulation by an association of thesellers, since the active sellers are only indirectly affectedby the queuing costs, entry is restricted excessivelycompared to the social optimum (thereby, in general,creating queues), however not necessarily relative to theunregulated outcome. The more elastic is the consumers'entry condition (distribution of opportunity costs) the betteris the seller's association optimal decision for welfare.28

For brevity's sake, we have not analyzed a numberof additional regulatory tools. For example, we havenot analyzed the effects of price fixing. It is ratherstraightforward that if it were feasible to set a priceschedule then p(θ)=q(θ)−q(θc) would guarantee the first

28 Note that, if the bargaining took place under asymmetricinformation, the association would have an additional reason toincrease licensing standards, so the social value of self-regulationwould be further increased.

best. However, in practice it is not realistic to assume thatsuch a price schedule can be implemented. A constantprice imposed on the entire market should still lead to animprovement, as forcing the price to be the same at allsellers gives a stronger incentive to switch away from alow quality seller, thereby increasing the marginal quality.

Other interesting generalizations include the casewhere the opportunity cost of the sellers is positive —capturing the effect of a costly investment (training) —or the effects of a license fee either imposed on everyoneor on newcomers only. Our model is clearly well-suitedas a workhorse for such further analyses.

Finally, an interesting question is to look at howmuch of the intuition arrived at in this paper can becarried over to a “standard” goods market, where thebuyers have heterogeneous valuations and the sellershave heterogeneous costs for one indivisible unit of agood. As shown by Anwar and Sákovics (2007), thelimiting inefficiency result persists in the standardmodel if and only if the good is perishable; i.e. thereis a non-negative probability that it perishes overnight.29

Acknowledgements

We are grateful to Santi Sánchez-Pagés, JonathanThomas and seminar audiences at the Universities ofBonn, Cambridge, Dundee, Edinburgh, Exeter, Leicesterand Stirling, as well as at the Stockholm School ofEconomics and University College London, for theiruseful comments. Inés Moreno has provided invaluablehelp with the illustrations. Financial support fromMinisterio de Educación y Ciencia (SEJ2006-02079 andCONSOLIDER-INGENIO 2010-CSD2006-00016) andCREA-Barcelona Economics is gratefully acknowledged.

Appendix A

In order to provide further insight, we would like toarrive at the competitive allocation from first principles aswell. In the current context, this is non-trivial for tworeasons.

First, since our set-up is dynamic — where in eachperiod new participants enter the market –, in order toconstruct supply and demand we cannot just take asnapshot of the stock of traders present in the market.Rather, we have to look at all the traders who couldpotentially enter the market at any point in time.30 By

29 In the current model, a service can be thought of as certainlyperishing: it disappears overnight with probability one.30 Strictly speaking, what we need to look at is the (discounted)“sum” of potential entrants.


stationarity, and following Gale (1987), the maximizerof the discounted sum of surplus is the same as that ofthe (per-period) flow of potential traders. In our modelthe demand flow is given by the new buyers whocould potentially enter in each period, while thesupply flow is given by all the sellers who couldpotentially produce in each period. That is, we canidentify the competitive allocation by looking at thedemand and supply that is newly generated in eachperiod.

The second difficulty stems from the fact that we arenot dealing with a homogeneous good, rather with a(vertically) differentiated one. As a result, the Law ofOne Price turns into the Law of One Surplus: Since theproducts are differentiated, consumers do not simplycompare prices when deciding which seller to buy from.Rather, they evaluate different products according tothe surplus they can obtain from their consumption.31

Thus, in a competitive market it is the buyer's surplusavailable, which determines both whether a buyer enters

31 Given the indivisibility inherent in a service being provided, we cannot32 Note that if q(θ⁎) were 0, then Eq. (7) would imply that E−1(1)−c≤0. Swe need not take the positive parts in Eq. (7).

the market, and if yes, from which seller does hepurchase. Then in a competitive equilibrium, we musthave — for the usual reasons — the buyer's surplus,z(θ) =q(θ)−p(θ), constant across sellers. Thus, thequantity demanded (per period) — as a function ofthe “market buyer's surplus” — is E(z). The supplyfunction — again as a function of z — can beconstructed from the constraint that no seller would bewilling to sell below cost, that is, we must have q(θ)≥ zfor each active seller. This implies that if they have tooffer z to the consumers, only sellers above θ=q− 1(z)will be willing to sell, leading to a supply of 1−q− 1(z).Inverting supply and demand the Marshallian way, weobtain that the quantity traded (per period) in thecompetitive equilibrium, Q, solves q(1−Q)=E−1(Q),which yields the marginal trader, θc, as the solution toq(θc)=E−1(1−θc), just as when we directly maximizewelfare (in Proposition 1). Finally, note that theWalrasian buyer's surplus equals the marginal quality,z=q(θc). See Fig. 1.

Appendix B

Proof of Proposition 1. Assume t⁎=b⁎. Then (1) is clearly increasing in θ⁎, so at the optimum b⁎=1−θ⁎. Similarly,if t⁎=1−θ⁎ (1) decreasing in b⁎ so at the optimum b⁎=1−θ⁎. Consequently, at the optimum we must havet⁎=b⁎=1−θ⁎. Substituting in and differentiating (1) with respect to θ⁎ we obtain the first-order condition

�Pq ðh⁎Þ þ ð1� h⁎ÞPq Vðh⁎Þ þ E�1ð1� h⁎Þ ¼ 0:

Noting that Pq Vðh⁎Þ ¼Pq ðh⁎Þ � qðh⁎Þ

1�h⁎yields

E�1ð1� h⁎Þ ¼ qðh⁎Þas claimed (the second-order condition is trivially satisfied).

When entry is exogenous, we have the feasibility constraints that e≤b⁎ and e≤1−θ⁎. Since W(e) is increasing inθ⁎ and decreasing in b⁎ the result follows. □

Proof of Theorem 1. Let us look at the two possible market scenarios— buyers' and sellers' markets — separately.Sellers are the short side: b≥1−θ⁎= t⁎.In this scenario Eq. (7) implies that32 E(q(θs)+c)=1−θs, which — as we have seen above — has θs(c) as its unique

solution. Using this equation and substituting Eq. (5) into E(xb)= t⁎=1−θ⁎ we have

qðhsÞ þ c ¼ E�1ð1� hsÞ ¼ Pq ðhsÞ � cð1� kpbÞkpb

:

In a sellers' market, buyers are not always matched: pb ¼ 1�hsbs

b1. Substituting in, we obtain

qðhsÞ þ c ¼ Pq ðhsÞ � cðbs � kð1� hsÞÞkð1� hsÞ :

remedy this problem by simply converting to a unit price of quality.ince E−1 (1)=1 this is equivalent to c≤1. Therefore, 1≥c implies that


Solving for bs, we obtain

bs ¼ Pq ðhsÞ � qðhsÞð Þ kð1� hsÞc

: ð12Þ

Recall that the sellers' market solution obtains if and only if bs≥1−θs. It is easy to see that λ[ q̄(θs)−q(θs)]=c is the same asEq. (12) with bs replaced by 1−θs. Therefore, since by the regularity of q(.), q̄(θs)−q(θs) is decreasing in θs, bs≥1−θs if andonly θs≤θb.

Buyers are the short side: b= t⁎≤1−θ⁎.Now all the buyers get matched in every period, so πb=1. Substituting back into Eq. (6), we obtain

qðhbÞ ¼ max Pq ðhbÞ � ck ; 0

� �, which — as we have seen above — has θb(λ, c) as its unique solution. In the buyers'

market, substituting Eq. (5) into E(xb)= t⁎=bb we obtain

bb ¼ E Pq ðhbÞ cð1� kÞk

� �: ð13Þ

This equilibrium prevails if and only if bb≤1−θb. When θb=0, the inequality is satisfied, since E(.) is a probabilitydistribution function. Otherwise, by Eq. (6), Pq ðhbÞ � cð1�kÞ

k ¼ qðhbÞ þ c; so bb ¼ EðqðhbÞ þ cÞ. Recall that θs(c)denotes the unique solution to 1−θ=E(q(θ)+c). Then, by the increasing nature of E(.) and q(.), we have that thebuyers' equilibrium exists if and only if θb≤θs(c). □

Proof of Corollary 2. From Eq. (5) we have that in a sellers' market

xb ¼ Pq ðhsÞ � cðb⁎=ð1� hsÞ � kÞk

:

Substituting in for b⁎ from Theorem 1 and simplifying, we obtain

xb ¼ qðhsÞ þ c:

Differentiating and calculating dusdc by implicitly differentiating E(q(θs)+c)=1−θs, we obtain

dxbdc

¼ �q VðhsÞE VðqðhsÞ þ cÞ1þ q VðhsÞE VðqðhsÞ þ cÞ þ 1N0:

Similarly,

dðxb � cÞdc

¼ �q VðhsÞE VðqðhsÞ þ cÞ1þ q VðhsÞE VðqðhsÞ þ cÞ b0: □

Proof of Proposition 6. Sellers are the short side: b⁎N1−θ⁎=E(xb).

In this scenario, pbu 1�h⁎b⁎ . Moreover, xb=E

−1(1−θ⁎). Eq. (5) translates into

E�1ð1� h⁎Þ ¼ Pq ðh⁎Þ � cðb⁎ � kð1� h⁎ÞÞkð1� h⁎Þ

:

Solving for b⁎, we obtain

b⁎ ¼ kð1� h⁎ÞPq ðh⁎Þ � E�1ð1� h⁎Þ

cþ 1

!: ð14Þ

Then we have that the sellers' market solution obtains if and only if

1� h⁎ bkð1� h⁎ÞPq ðh⁎Þ � E�1ð1� h⁎Þ

cþ 1

!: ð15Þ

Buyers are the short side: b⁎=E(xb)≤1−θ⁎.


In this case all the buyers get matched in every period, so πb≡1. Substituting back to Eq. (5), we obtain

E�1ðb⁎Þ ¼ Pq ðh⁎Þ � cð1� kÞk

: ð16ÞFor this to be the equilibrium we need

E�1ð1� h⁎ÞzPq ðh⁎Þ � cð1� kÞk

: ð17ÞNotice that this inequality is the complement of Eq. (15). □

Proof of Proposition 7. Note that there exists a threshold value of θ⁎, such that below it we are in a buyers' marketwhile above it we are in a sellers' market. Assume first, that we are in a buyers' market. From Eq. (1) we have that

dW

dh⁎¼ b⁎Pq Vðh⁎Þ þ db⁎

dh⁎Pq ðh⁎Þ � E�1ðb⁎Þh i

: ð18Þ

The first term is clearly positive, since the average quality is increasing in the marginal one. Since we are in a buyers'market E−1(b⁎) is the expected surplus of a buyer, which is less than the average quality as well (c.f. Eq. (16)).Consequently, all we need in order to complete this part of the proof is to show that in a buyers' market db⁎du⁎z0. Thattrivially follows form Proposition 6, since both E(θ⁎) and the average quality are increasing in θ⁎. Consequently,welfare is increasing in a (binding) quality constraint. That is, the planner would like θ⁎ as large as possible, while itstill results in a buyers' market.

Turning to the sellers' market, we have

dW

dh⁎¼

d ð1� h⁎ÞPq ðh⁎Þ � R 1�h⁎

0 E�1ðxÞdx� kð1� h⁎ÞPq ðh⁎Þ � E�1ð1� h⁎Þ

c þ 1

� �� ð1� h⁎Þ

� �c

� �dh⁎

¼ ð1� h⁎ÞPq Vðh⁎Þ � Pq ðh⁎Þ þ E�1ð1� h⁎Þ � cð1� kÞ þ k Pq ðh⁎Þ � E�1ð1� h⁎Þ� �

� kð1� h⁎Þ Pq Vðh⁎Þ � dE�1ð1� h⁎Þdh⁎

!

¼ ð1� h⁎ÞPq Vðh⁎Þ � Pq ðh⁎Þ � cþ E�1ð1� h⁎Þh i

ð1� kÞ � kð1� h⁎ÞE VðE�1ð1� h⁎ÞÞ

¼ E�1ð1� h⁎Þ � qðh⁎Þ � ch i

ð1� kÞ � kð1� h⁎ÞE VðE�1ð1� h⁎ÞÞ

;

where we have used that Pq Vðh⁎Þ ¼Pq ðh⁎Þ � qðh⁎Þ

1�h⁎ . This derivative is clearly negative, since the fact that the qualityconstraint is binding implies that the term in the square brackets is negative. □

Proof of Corollary 6. It is straightforward that if we set the quality threshold at the efficient quality and the markethappens to clear in such a way that the measure of sellers and buyers in the market is the same— and therefore there is noinefficiency caused either by waiting or by too low demand—we have the first best outcome. Consequently, all we needto prove is that it is possible to set λ in order to ensure that the market indeed clears in that way. Recall that the producerof the efficient marginal quality— given by the solution to q(θ)=E−1(1−θ),— is independent of both λ and c. On theother hand, by Proposition 6, we have a symmetric market when E�1ð1� h⁎Þ ¼ Pq ðh⁎Þ � cð1�kÞ

k . Using the equalitydefining the optimal marginal producer, means that we need qðhcÞ ¼ Pq ðhcÞ � cð1� kÞ

kor Pq ðhcÞ � qðhcÞ ¼ cð1� kÞ

k. The

left-hand side of this last equation is a number in (0,1). Thus the proof boils down to showing that for any c∈ (0, 1], anynumber in (0,1) can be achieved by a judiciously choosing a λ∈ [0, 1]. This is indeed the case, since 1�k

k is decreasingfrom ∞ to 1 in λ. □

Proof of Proposition 8. If we are in a sellers' market, by Eq. (3), the aggregate profits are given byp ¼ ð1� kÞ R 1h⁎½qðhÞ � xb þ c�dh. Recall that xb=E−1(1−θ⁎). Substituting into the profit function and differentiating,we obtain the first-order condition.


In a buyers' market not all sellers get matched, so p ¼ b⁎1� h⁎

ð1� kÞZ 1

h⁎qðhÞ � xb þ c½ �dh ¼ E Pq ðh⁎Þ�cð1�kÞ

kð Þ1�h⁎

ð1� kÞ R 1h⁎ qðhÞ � Pq ðh⁎Þ þ ck

h idh, where we used xb=E

−1(b⁎) and b⁎ ¼ E Pq ðh⁎Þ � cð1�kÞk

� �. Integrating and

simplifying, we get p ¼ E Pq h⁎ð Þ � cð1�kÞk

� �ð1� kÞc=k. Since the average quality is increasing in the marginal

producer, the seller's cartel will never want to be in a buyers' market. □

Proof of Corollary 7. Aggregate profits are given by

e

1� h⁎ð1� kÞ

Z 1

0qðhÞ � Pq ðh⁎Þ þ c

k

h idh ¼ eð1� kÞ c

k;

which is clearly constant in the marginal quality. □

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http://europa.eu.int/comm/competition/liberal_professions

http://europa.eu.int/comm/competition/liberal_professions

http://www.oecd.org/dataoecd/35/4/1920231.pdf

http://www.oecd.org/dataoecd/35/4/1920231.pdf

http://europa.eu.int/comm/competition/publications/publications/index.html#liberal

http://europa.eu.int/comm/competition/publications/publications/index.html#liberal

Queues, not just mediocrity: Inefficiency in decentralized markets...

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