competition among middlemen when buyers and sellers can trade directly

COMPETITION AMONG MIDDLEMEN WHEN BUYERSAND SELLERS CAN TRADE DIRECTLY*

John Fingleton

This paper examines competition among middlemen when sellers andbuyers can trade directly. Direct trade alters the supply and demandfacing the middlemen, making them interdependent, and reduces themarket power of intermediaries. However, it does not alter the Stahl[1988] result that middlemen may have an incentive to ``corner'' themarket if demand is inelastic. The model is applied to market makingin ¢nancial markets, vertical integration in goods markets and to thequestion of bypass in utilities. This discussion suggests that corneringis most likely in markets for essential inputs and that it may enableseller collusion.

i. introduction

In many markets, sellers and buyers face a choice between trading with amiddleman and attempting to trade directly with someone on the otherside of the market. For example, a seller of a second-hand car or bicyclecan choose between a dealer or advertising in a buy-and-sell magazine.The seller may get a ¢rm o¡er quickly from the dealer, whereas a moreattractive price from direct trade may be delayed or completely elusive.Even if a buyer is easily found (say a friend), negotiation of a price mayentail sensitivities that are costly, especially if the ``market price'' orreservation prices are not known.

Intermediation has been widely analysed in recent years. Some authorsfocus on the analysis of intermediation where there is no direct trade: Stahl[1988] models Bertrand competition among middlemen and Bloch andRyder [1994] consider how a matchmaker can improve e¤ciency in adirect trade model. Others (e.g., Biais [1993], Rubinstein and Wolinksy[1987] and Shin [1996]) compare the performance of alternative marketmechanisms, with a particular focus on the determination of the bid-askspread and the distribution of the gains from trade. However, a relativelysmall amount of this research examines the situation where intermediatedtrade co-exists with direct trade.

ß Blackwell Publishers Ltd. 1997, 108 Cowley Road, Oxford OX4 1JF, UK, and 350 Main Street, Malden, MA 02148, USA.

405

THE JOURNAL OF INDUSTRIAL ECONOMICS 0022-1821Volume XLV December 1997 No. 4

* I am grateful to Jim Mirrlees, Meg Meyer, Candice Prendergast, Paddy Waldron, theeditors, anonymous referees and participants at various seminars for helpful suggestions andcomments and to the European Centre for Advanced Research in Economics, Universite Librede Bruxelles where I was a visitor during part of this research. This research has also beensupported by the Arts and Social Science Benefaction Fund, TCD. Any errors are mine.

Models in which sellers' and buyers' expectations about the liquidityof a middleman generate multiple equilibria have been analysed byGehrig [1993] and Yanelle [1989]. The question of how the existence ofa middlemen alters an equilibrium in a direct trade market has beenanalysed by Moresi [1991] in the context of sequential bargaining andYavas� [1996] with search intensities. A consistent theme of thisresearch (and one that emerges in this paper) is that traders withpotentially larger gains from trade will tend to value more the serviceof immediacy or liquidity provided by the middleman, leaving thosewith less strong potential gains from trade to seek bargains via directtrade.1

This paper focuses speci¢cally on the question of how direct trade alterscompetition among middlemen. The outcome from Bertrand competitionamong middlemen without direct trade has been established by Stahl[1988] who examines a two-stage model of duopolistic intermediation inwhich middlemen ¢rst compete for stocks of a homogeneous good andlater sell to buyers on the other side of the market.

The structure of Stahl's model makes two signi¢cant departures fromthe standard Bertrand model. First, the marginal cost (bid price) in themarket is determined by an increasing supply schedule rather than bytechnology. Because higher stocks cost more per unit, a type of decreasingreturns prevails. Moreover, these decreasing returns apply to the market,so that the appropriate analogy in terms of technology might be anessential input (such as a scarce natural resource or bottleneck). Thesecond di¡erence is that the stocks of the middlemen act as capacities inthe second stage, making this a very natural example of capacity-constrained price competition.

If the elasticity of demand, evaluated at the Walrasian price, pw, exceedsone and an equal-sharing tie-break rule prevails, Stahl ¢nds a uniqueequilibrium in which both middlemen set the Walrasian price in bothstages. The middlemen implement the Walrasian outcome and welfare ismaximised. However, if the demand is inelastic, then a monopolist sellerwould wish to set a higher ask price than the Walrasian price �pa > pw�.Such a monopolist would anticipate greater revenue and would thereforewish to bid more than pw in the ¢rst stage. Thus pb gets bid up as themiddlemen compete for the monopoly position. With an equal share tie-break rule, this is not a subgame perfect equilibrium, as both middlemenare plunged into destructive competition (i.e., the margin pa ÿ pb < 0).With a random tie-break rule, one middleman becomes the monopolyseller, but makes zero pro¢t because of the competition for that position.The equilibrium is characterised by the middleman dumping stock (see

1 There is additionally a literature on the role of middlemen where there is asymmetricinformation, viz. Biglaiser and Friedman [1994] and Garella [1989].

406 john fingleton

ß Blackwell Publishers Ltd. 1997.

Figure 1) and the monopoly rent that is extracted from the buyers is fullytransferred to the sellers.2

In Stahl's model, a higher bid price wins all the stock and, in the eventof a tie, the stock either gets divided equally among the middlemen or allof it is randomly allocated to one middleman. Thus the second stage ischaracterised either by monopoly or duopoly with equal capacities.3

Because it is not possible for a middleman to increase his stockincrementally, the Walrasian predictions of the model di¡er from theCournot result obtained by Kreps and Scheinkman [1983].

Stahl's model thus presents somewhat of a contrast. In the ¢rst case,the middlemen replicate the e¤cient allocation of the Walrasianauctioneer. In the second, either there is no equilibrium (with enormouswelfare costs) or middlemen engage in destructive rent-seeking to cornerthe market, with welfare costs and substantial redistribution from buyersto sellers. He does not speculate about the e¡ect that direct trade wouldhave on his results.

This paper extends Stahl's analysis to the case where direct trade is alsopossible. Our analysis exploits the fact that supply and demand incompeting markets will usually be interdependent, taking the form S�pb; pa�and D�pb; pa� (instead of ~S�pb� and ~D�pa�), where pb and pa are bid andask prices in one market. To illustrate, consider a middleman competingwith a direct trade market. An increase in the middleman's ask-price will,with downward sloping demand, switch some buyers to direct trade. Thisis good for sellers, attracting some of them to the direct market. In thisway, the middleman's supply falls without any change in the bid price.Interdependence is a consequence of the coordination between buyers andsellers about which market to choose, and can occur where sellers andbuyers have access to more than one market for a good. Interdependencearises if the payo¡ of a seller (buyer) from trading directly depends on theparticipation decisions of buyers (sellers). This is the case with manydecentralised trading institutions.4

Interdependence can exist in ¢nancial or goods markets. The case ofspot and forward markets provides an example of the former.5 Supposethat the bid price in the forward market increases. This induces moresellers to sell forward, reducing the spot supply. The reduction in the spot

2With forward selling (competing for demand in the ¢rst period and then buying stock inthe second stage), the problem does not arise.

3 Thus the tie-break rules avoid the rationing problems discussed by Davidson andDeneckere [1986] that arise when sellers have unequal capacities.

4 Fingleton [1991] ¢nds interdependence in a model where sellers and buyers bargainbilaterally, Moresi [1991] in a model of sequential bargaining, and Yavas� [1996] in a modelwhere sellers and buyers choose search intensities. Explicit demand and supply functions arenot obtainable in any of these models.

5 An example is provided by de Jong et al. [1995] who compare trading of French shareson exchanges in Paris and London that exhibit di¡erent liquidity and trading costs.

competition among middlemen with direct trade 407


supply causes more buyers to buy forward, even if there is no change inthe forward ask price. Thus the demand in the forward market woulddepend on the forward bid price as well as on the forward ask price.Another example is that of a trade magazine (say, for second-hand cars) inwhich sellers advertise to buyers who purchase the magazine. Alongsidethere is a dealership market. Suppose there was an exogenous increase inthe bid price in the dealership market. Fewer sellers would advertise in themagazine, and hence more buyers would visit dealers. Thus the supplyand demand facing dealers are interdependent. Next suppose the magazineincreased the fee charged to its advertisers without changing either itsselling price or its perceived quality. More sellers would go to dealers andthe buyers' payo¡s at the auction would improve, so fewer would buy the(unchanged) magazine. Thus the supply and demand for the magazine arealso interdependent. If the quality of the magazine was seen to decline,its readership would fall: with a parallel alternative market, the size of anysuch change would be greater.

This paper analyses Bertrand competition among middlemen whensupply and demand are interdependent. Section II analyses a speci¢cmodel in which a middleman competes against a direct trade alternativefor buyers and sellers. This model is used to illustrate and investigate theinterdependence in the supply and demand functions for the intermediatedmarket. The interdependence e¡ects (e.g., bid price on demand) arestronger, the more e¤cient the direct market, but are always weaker thanthe direct e¡ects (e.g., bid price on supply). In the limit, if the direct tradeprocess is completely e¤cient, a monopolist middleman would replicatethe Walrasian auctioneer.

In Section III, a general formulation of interdependent supply anddemand functions is used to examine two-sided Bertrand competitionbetween middlemen when direct trade is possible. We obtain conditionsrelating to the elasticities for existence of equilibrium that di¡er from thoseof Stahl in several respects. However, the general e¡ect of Stahl's resultremains intact and for su¤ciently elastic demand, a Walrasian outcomewill prevail.

In Section IV, applying these conditions to the model from Section II,we ¢nd that Walrasian equilibrium exists for identical parameter values,regardless of whether or not there is direct trade. However, in the non-Walrasian equilibrium, direct trade reduces the extent of the distortion inthat the amount of stock dumped is not as great. Thus direct trade doesnot alter the incentive to outbid for the rent, but by reducing the size ofthat rent, it lessens the distortions. The more e¤cient the direct trademechanism, the smaller will be the distortion. However, the incentive tooverbid the Walrasian price will persist as long as there is some ine¤ciencyin the direct market. The incentive to engage in rent-dissipating behaviouris not a¡ected by the existence of direct trade, even if its consequences

408 john fingleton


are. An extensive concluding section discusses applications andimplications of the model in both ¢nancial and goods markets.

ii. a model illustrating interdependence

This section illustrates and investigates the phenomenon of inter-dependence using a model in which sellers and buyers choose betweentrading with a middleman, trading directly or not trading. The modelhas two stages. In the ¢rst stage, a middleman6 moves bysimultaneously setting a bid price pb and an ask price pa. These areobserved by all agents. In the second stage, sellers and buyers decidewhether to trade with the middleman, to trade directly, or not to trade.These are mutually exclusive activities, and there is no second-chancetrading. Trade takes place at the prices determined by these decisions.This gives rise to a coordination problem among sellers and buyersabout which market to select.

There is a continuum of risk-neutral sellers and buyers with linearpreferences who may trade 0 or 1 unit of a homogeneous good. Each sellerhas at least one unit of the good and each buyer has at least one unit ofmoney. Sellers' valuations, y, are distributed uniformly on 0; 1a

� �giving a

cumulative distribution function ay. Buyers' valuations, f, are distributeduniformly on

�1ÿ 1

b ; 1�

giving a cumulative distribution function1ÿ b� bf. The total number of sellers equals the total number of buyerswhich equals measure 1. Payo¡s are

�1� p�y� �yÿ p if buys 1 unit at price p

0 if no trade at price p

pÿ y if sells 1 unit at price p

8<:The parameters of the model, the distributions of valuations and the pricesof the middleman are common knowledge.

II(i). Trade with a Middleman

The middleman, knowing the distribution of valuations, sets bid and askprices. We assume that liquidity is the core service provided by themiddleman. Thus a seller expects that the middleman will always beprepared to buy at pb and a buyer expects that the middleman will be ableto supply the good at pa (or compensate him otherwise). It would not alterthe insights obtained from this model if we included an explicit cost to

6At this stage, we are only interested in characterising market supply and demand. Thiscan be done e¡ectively by considering a monopolist middleman.



the middleman of providing liquidity: ¢xed costs would simply alter thecritical value of pro¢t required for middleman participation, and variablecosts would reduce the scale of the middleman's operation. The idea thatmiddlemen provide liquidity, enabling sellers and buyers to trade withgreater speed or security, goes back to Demsetz [1968]. Its assumption hereis made to focus on the coordination problem between di¡erent tradechannels.7

It is useful to outline the monopoly benchmark case where there is nodirect trade (we use the notation ~x to distinguish it). In the second stage,all sellers with y � pb and all buyers with f � pa trade with the middleman.This gives supply and demand

�2� ~S�pb� � apb

�3� ~D�pa� � b�1ÿ pa�:The Walrasian price and quantity are ~pw � b

a�b and s � aba�b. If the market

is supplied by a monopolist, the prices set are ~p�b � 12

ba�b and ~p�a � 1

2a�2ba�b . The

middleman can charge a double mark-up, ~pw ÿ ~p�b � 12

ba�b on the sellers' side

and ~p�a ÿ ~pw � 12

aa�b on the buyers' side. The middleman's margin is

~p�a ÿ ~p�b � 12 and only trades with half the Walrasian quantity of buyers and

sellers.Next consider Bertrand duopoly. Stahl shows that a Walrasian

equilibrium exists if

�4� ~ed� ~pw� � 1;

where ~ed � ÿ d ~Ddpa

pa~D. For this example, ~ed�pw� � pw

1ÿpw� b

a so Bertrand duopolyresults in a Walrasian outcome for

�5� b � a:

This is the case if the support of buyer types is less than the support ofseller types (steeper demand). If (5) does not hold, a non-Walrasianequilibrium may exist (with a random tie-break rule) in which onemiddleman gets all of the supply and must dump some in order to extractthe monopoly revenue (see Figure 1).

7 If middlemen are not liquid, sellers and buyers would face an additional coordinationproblem within the middlemen market. This scenario has been analysed comprehensively byGehrig [1993] who shows that multiple equilibria are supported by sellers' and buyers' beliefsabout which middleman will be chosen by agents on the other side of the market. Yanelle[1989] explores a similar theme for scale economies in intermediation.

410 john fingleton


II(ii). Direct Trade

Direct trade is modelled as a bilateral matching market with a centralisedprice coordinator who is not a trader. Sellers and buyers pay a cost c (eithera fee to cover the coordinator's costs or a cost of reaching the market8)and then make a report to the coordinator about their reservation price.The coordinator uses these reports to determine a market clearing price pd

(with continuous types, as here, this is simply the intersection of the supplyand demand schedules reported). The coordinator publicly announces pd

and identi¢es and excludes any sellers (buyers) whose report exceeds (is lessthan) pd. Buyers and sellers are then matched randomly. We assume thatthe costs of renegotiating are su¤ciently high that all matched pairs agreeto trade at pd. This is an equilibrium strategy because each knows thepartner is prepared to trade at pd.The incentive structure created by the co-odinator means that it is an

equilibrium strategy for each seller and buyer to make a truthful report. Anyseller reporting a higher reservation price has no e¡ect on the market priceunless her report exceeds pd but this would exclude her from the matchingprocess, giving a negative payo¡. Exclusion is not strictly necessary: it issu¤cient that her identi¢cation by the coordinator reduces her probability oftrading discretely, as the marginal e¡ect on the price is continuous. A similarargument applies for a buyer.Note that buyers and sellers who are not willingto trade at pd will not enter the market. This means that, in equilibrium, thenumbers of sellers and buyers entering the directmarket will be equal.

Figure 1Non-Walrasian Equilibrium of Stahl. The middlemen bid pm > pw and one becomes a

monopolist. To break even, he must charge pa > pm and dump stock as indicated.

8 If such a cost exists also for trade with the middleman, c can be reinterpreted as thedi¡erence in the two costs.



We assume that sellers and buyers who remain in the market arematched with an opposite type agent with probability s � 1. Thisexogenous parameter s represents the e¤ciency of the market and willbe used for comparative static analysis below. A market withconventions that encourage sellers to meet buyers instead of meetingcompeting sellers or non-traders would be characterised by a high valueof s. Thus a centralised exchange in which sellers wore red and buyerswore blue and non-traders were excluded would be characterised by avalue of s close to 1. On the other hand, a market place open to thegeneral public might result in some sellers and buyers being matchedwith non-traders.9

This direct trade mechanism has the property that the payo¡ to a seller(buyer) depends on the participation of the buyers (sellers) on the otherside, as is common in most mechanisms of decentralised or direct trade(see Footnote 4). The advantage of the particular mechanism chosen hereis that it gives linear supply and demand functions.

II(iii). Trading Decisions of Sellers and Buyers

The payo¡s to sellers and buyers from trading with the middleman are

�6� pms �y� � pb ÿ y

�7� pmb �f� � fÿ pa

and for direct trade are:

�8� pds �y� � �pd ÿ y�sÿ c

�9� pdb�f� � �fÿ pd�sÿ c:

These payo¡ functions are shown in Figure 2 for the relevant case, wheredirect trade is active.10 The slopes of the payo¡ functions from visiting themiddleman are steeper than those from entering the direct market andagents with higher payo¡s from trade (extreme valuations) visit themiddleman as these have proportionately more to lose from theuncertainty.

9 This could provide a rationale for an endogenous entry fee c. Fingleton [1991] shows thata small but positive entry fee in a matching-bargaining market can improve e¤ciency. Byexcluding low-value traders, it ensures that a higher proportion of matches result in trade. Interms of the design of markets, this raises the question of whether s should be increasing inc, a prospect that we do not explore here. Many auctions and trade fairs charge such entryfees, often in the form of a catalogue cost.

10 For all parameter values, there will still exist an equilibrium with no direct trade. If eachseller expects that all buyers trade with the middleman, no seller would attempt to tradedirectly. With no sellers trading directly, no buyers would wish to trade either. The outcomein this situation is identical to that where direct trade is prohibited, as presented in SectionII(ii) above.

412 john fingleton


Sellers with valuations between 0 and sm have a higher payo¡ fromvisiting the middleman than trading directly and so the supply to themiddleman is asm. Those with valuations between sm and st have a higherpayo¡ from direct trade and so the supply in the direct market is a�st ÿ sm�.Sellers with valuations between st and 1

a also have a higher payo¡ fromtrading directly, but this payo¡ is negative so they do not trade. Thesituation for buyers is symmetrically reversed with demand from themiddleman given by b�1ÿ bm� and demand in the direct market ofb�bm ÿ bt�. For parameter values where st ÿ sm � 0, there is no direct tradeand st is not well-de¢ned. In this case, �sm ��bm� will be used to distinguishthe position of the seller (buyer) indi¡erent between trading with themiddleman and not trading.

II(iv). Supply and Demand

In order to ¢nd the supply and demand in the intermediated market, it isnecessary to solve for sm; st; bm; bt; �sm and �bm. De¢ning the payo¡s of theindi¡erent agents gives

pds �st� � 0 ) �pd ÿ st�sÿ c � 0

pdb�bt� � 0 ) �bt ÿ pd�sÿ c � 0

pds �sm� � pm

s �sm� ) �pd ÿ sm�sÿ c � pb ÿ sm

pdb�bm� � pm

b �bm� ) �bm ÿ pd�sÿ c � bm ÿ pa:

The positions of the indi¡erent agents are given by:

Figure 2Seller's valuations are uniform on

�0; 1a

�. The payo¡ function from trade with the

middleman, pms is steeper than that from direct trade, pd

s . Sellers in �0; sm� have higherpayo¡ from trade with the middleman, those in �sm; st� do better from direct trade, and

those in�st;

1a

�do not trade.



�10� st � pd ÿc

s

�11� bt � pd �c

s

�12� sm �pb ÿ spd � c

1ÿ s

�13� bm �pa ÿ spd ÿ c

1ÿ s

subject active direct trade, that is sm < st and bm > bt, which hold forpb ÿ spd � c

1ÿ s< pd ÿ

c

s, pd ÿ pb >

c

s

pa ÿ spd ÿ c

1ÿ s> pd �

c

s, pa ÿ pd > �

c

s:

Otherwise, we have no direct trade and sellers and buyers trade with themiddleman if their valuations are covered by his prices, giving

�14� �sm � pb

�15� �bm � pa

With active direct trade, the coordinator, observing a supply ofa�st ÿ sm� and a demand of b�bm ÿ bt� sets a market-clearing price at theintersection of these so that

�16� pd �a

a� bpb �

ba� b

pa �aÿ ba� b

c

s

which gives

�17� sm �1

1ÿ s

��1ÿ sa

a� b

�pb ÿ

sba� b

pa �2b

a� bc

�

�18� bm �1

1ÿ s

��1ÿ sb

a� b

�pa ÿ

saa� b

pb ÿ2a

a� bc

�:

The middleman's supply and demand are now

�19� Sm�pb; pa� �asm for pd ÿ pb >

c

s

a�sm for pd ÿ pb � c

s

(

�20� Dm�pb; pa� �bÿ bbm for pa ÿ pd >

c

s

bÿ b�bm for pa ÿ pd � c

s

8<:

414 john fingleton


We focus on the case where direct trade is active.11 This equilibriumexists when frictions in direct trade are low, namely

�21� c <s21ÿ s2ÿ s

;

and is characterised by

p�b �12

ba� b

�1� sÿ 2c�

p�a � 1ÿ 12

aa� b

�1� sÿ 2c�

S � B � 12

aba� b

�1ÿ s� 2c�1ÿ s

p � 14

aba� b

�1ÿ s� 2c�2�1ÿ s�

The middleman's margin is

�22� pa ÿ pb �1ÿ s2� c

so this solution exists for pa ÿ pb > 2 cs which gives (21) above.

Each of the supply and demand functions with active direct trade has bothbid and ask prices as arguments, that is they are interdependent. Thesupply function is increasing in the bid price (with slope a

1ÿsÿ1ÿ sa

a�b�) and

the demand is decreasing in the ask price (with slope ÿ b1ÿsÿ1ÿ sb

a�b�) . The

interdependence e¡ects take the form anticipated: the supply decreases inthe ask price and the demand increases in the bid price. The inter-dependence e¡ects are always weaker than the direct e¡ects if there is apositive probability of not trading in the direct market (note the entry cost,c, plays no role here).

�23� @Dm

@pb

�@Sm

@pb

� sba� bÿ sa

< 1 for s < 1

�24� @Sm

@pa

�@Dm

@pa

� saa� bÿ sb

< 1 for s < 1

11 The result without direct trade is di¡erent to that considered above in that the middlemanis constrained to set a maximum margin of pa ÿ pb � 2 c

s. The solution, exists for c � s21ÿs2ÿs

and is characterised by pa ÿ pb � 2 c

s ; Sm � Dm � aba�b �1ÿ 2 c

s� and p � aba�b �1ÿ 2 c

s� 2cs . It may beveri¢ed that the pro¢t with direct trade is greater than that without direct trade in the regionc < s

21ÿs2ÿs and vice versa in the region c � s

21ÿs2ÿs.



The magnitude of the interdependence e¡ects is increasing in s, indicatingthat as the direct market is more e¤cient, changes in the middleman'sprices have greater e¡ects on both sides of the market. Another way ofexpressing this is that as coordination among sellers and buyers becomesless costly, interdependence e¡ects become stronger.

As matching in the direct market becomes more e¤cient and less costly,the market power of a middleman is eroded. The middleman's margin,pa ÿ pb � 1ÿs

2 � c, decreases and his volume of trade, ab2�a�b� � 1

1ÿsabc

�a�b�,increases as the direct market is more e¤cient. It can be shown that themonopoly pro¢t is decreasing in s and increasing in c.

iii. intermediation with direct trade

III(i). Outline of the Model

Competition between the middlemen is presented as a two-stage gameof complete information identical to that of Stahl [1988] except thathere direct trade is allowed. In the ¢rst stage, the middlemen choosebid prices. Sellers and buyers observe pbi and pbj and form expectations,pe

ai and peaj, about ask prices. Sellers choose whether to sell to the

middleman or to wait to trade directly. In the second stage, themiddlemen choose ask prices. Sellers and buyers observe pai and paj.Buyers choose whether to buy from the middleman or to attempt totrade directly with those sellers who did not sell to the middleman.There is no discounting.

We assume that the middlemen's supply Sm�pb; pa� and demandDm�pb; pa� are di¡erentiable and that total revenue, paDm�pb; pa� is concavein pa and that there exists p0 such that for pa � p0;D�pa� � 0. TheWalrasian outcome is a unique price, pw satisfying

�25� Sm�pw; pw� � Dm�pw; pw�:With an equal-share tie-break rule, the supply to an individual middlemanis

�26� si �Sm�pbi; p

eai; p

eaj� pbi > pbj

12 Sm�pbi; p

eai; p

eaj� if pbi � pbj

0 pbi < pbj

8>><>>:The intermediated supply function Sm�pb; p

eai; p

eaj� is increasing in pb and

non-increasing in each of peai and pe

aj and strictly decreasing in min�peai; p

eaj�.

The demand12 realised by an individual middleman is

12Demand rationing means that both Pei and Pe

j enter Sm�pi;Pei ;P

ej � but all sellers trade

at the same bid price, max�pi; pj�, hence Dm�Pi;max�pi; pj��.

416 john fingleton


�27� di �Dm�pai;max�pbi; pbj�� pai < paj

Dm�pai;max�pbi; pbj�� if pai � paj.

RDm�pai;max�pbi; pbj� j paj� pai > paj

8><>:The market demand function Dm�pai;max�pbi; pbj�� is decreasing in pa andincreasing in max�pbi; pbj�. With e¤cient rationing, the residual demandis13

�28� RDm�pai;max�pbi; pbj�jpaj� � maxfDm�pai;max�pbi; pbj�� ÿ sj; 0g

III(ii). Competition in Ask Prices

We solve the game by backward induction, starting with competition inask prices, given the middlemen's stock levels si and sj. An equal share tie-break rule implies either si � sj or one of si � 0 or sj � 0. Given theassumptions above, there exists a uniquely de¢ned market-clearing askprice �pa

�29� si � sj � Dm� �pa;max�pbi; pbj��;and a uniquely de¢ned revenue-maximising ask price pa

�30� pa � arg maxfpaDm�pa;max�pbi; pbj��jDm � si � sjg:

Lemma 1. The equilibrium strategy in the second stale is:

for si � sj > 0;

pai � paj � �pa for ed� �pa� � 1pai � paj � pa for ed� �pa� < 1;

for si > sj � 0;

pai � �pa for ed� �pa� � 1pai � pa for ed� �pa� < 1:

Proof. Equal Stocks: Let paj � �pa and consider pai � �pa. For ed� �pa� � 1,MR � pa

@Dm

@pa� Dm < 0. An increase in pai would reduce revenue. As the

stock constraint binds, pai < �pa would reduce revenue. Hence, pai � �pa is abest response to paj � �pa if ed� �pa� � 1. Uniqueness follows because theargument holds only for �pa. For ed� �pa� < 1, pai � pa is a best response topaj � pa.

13 Stahl [1988] shows that the results hold for convex combinations of e¤cient andproportional rationing and this carries over here. E¤cient rationing is used for theexposition.



Monopoly: For si > sj � 0, the middleman is a monopolist in stage 2 andthe optimal ask price is pai � �pa�si� for ed� �pa�si�� 1 and pai � pa�si� fored� �pa�si�� < 1. It is unique because pa�pb� and �pa�pb� are single-valuedfunctions. &

III(iii). Competition in Bid Prices

We next consider the choice of bid prices by middlemen in the ¢rst stage.De¢ne p�pb� as the pro¢t of a middleman who obtained all of the stock atprice pb in the ¢rst stage

�31� p�pb� � maxpa

fpaDm�pb; pa� ÿ pbSm�pb; pa�jSm � Dmg

De¢ne pm as the highest bid price consistent with pro¢table monopoly

�32� pm � maxfpbjp�pb� � 0g:In the ¢rst stage, the middlemen choose bid prices. They will bid up to

the zero monopoly rent price. Either, pm � pw and the Walrasian bid priceis set in the ¢rst stage, or pm > pw and the Walrasian price is outbid.

Lemma 2. pm � pw if and only if (33) and [(34) or (35)] hold.

�33� 1ÿ ed�pw� � esd�pw� � 0

�34� pa

@Dm

@pb

ÿ Sm ÿ pb

@Sm

@pb

ÿ pb

@Sm

@pa

dpa

dpb

� 0

�35� @Sm

@pb

ÿ @Dm

@pb

ÿ @Dm

@pa

ÿ @Sm

@pa

� �dpa

dpb

� 0

Proof. We determine the conditions under which pro¢t is increasing ineither pa or pb, evaluated at pa � pb � pw. If these conditions do not prevail,then pm � pw. Only strategies with pa�pb� are relevant becausep�pw; pw� � 0 and p�pb; �pa�pb�� < 0 for pb > pw. The pro¢t of themiddleman, seen in the ¢rst stage, is p�pb; pa�pb�� pa�pb�Dm� pa�pb�; pb� ÿpbSm�pb; pa�pb��:

Pro¢t is increasing in pa if Dm�pa; pb� � pa@@pa

Dm�pa; pb� ÿ pb@@pa

Sm�pb; pa� > 0. Thus a necessary condition for pm � pw is (33). A secondpossibility is that pro¢t is increasing in pb. This requires either thatdp� pb;pa� pb��

dpb� 0 (which, given ed � 1 from the envelope theorem, gives (34))

or that an increase in pb would violate the stock constraint, that isdSm� pb;pa� pb��

dpb< dDm� pa� pb�;pb�

dpbgiving (35). &

418 john fingleton


Proposition 1. There exists a perfect equilibrium consisting of strategiespbi � pbj � pw and pai � paj � pw, if (33) and [(34) or (35)] hold.

The proof follows from Lemmata 1 and 2. Three further results of Stahlcarry over. First, if the conditions in Proposition 1 are not satis¢ed, thereis no perfect equilibrium with bid price competition in the ¢rst stage.Middlemen bid up to pm > pw. The second stage equilibrium results in�pa�pm� < pa�pm� which yields negative pro¢ts. Second, replacing the equal-share with a random tie-break rule, there exists a non-Walrasianequilibrium. Each bids pm, one randomly gets all of the stock and sets askprice or demand so that pa�pm� > pm > pw. Pro¢t is zero and stock isdumped since D� pa�pm�� < S�pm� (as in Figure 1). Third, with forwardselling (setting ask prices in the ¢rst stage and bid prices in the second) anddefault penalties (for non-delivery by the middleman), the Walrasianequilibrium emerges for all parameter values.

iv. the effect of direct trade

IV(i). General Comparison

Given that a non-existence result holds with direct trade, the next issue isthe e¡ect of direct trade on the likelihood of a non-existence result. FromProposition 1, direct trade alters the conditions for existence of aWalrasian equilibrium by modifying Stahl's condition (4) to the condition(33) and by introducing an additional condition [(34) or (35)] .Consider ¢rst the di¡erences between (4) and (33). The change in

revenue from a change in the ask price is measured by 1ÿ ~ed without directtrade and by 1ÿ ed � esd with direct trade. The esd term represents thereduced cost of supply as fewer sellers choose intermediation. However, ed

will generally exceed ~ed because, in addition to the direct e¡ect of anincrease in the ask price, there is an indirect e¡ect in the same directionbecause more sellers also trade directly, and this further reduces themiddleman's demand. As the size of this second order e¡ect is directlyrelated to the interdependence factor, we would expect ed ÿ ~ed to bepositively related to esd suggesting that the conditions (33) and (4) arecorrelated.

Second, consider the condition [(34) or (35)]. Equation (35) can be re-expressed as

ÿ@Sm

@pbÿ @Dm

@pb

��ÿ@Dm

@paÿ @Sm

@pa

�> dpa

dpb, that is, the ratio of the di¡erence

between the direct and indirect e¡ects of a change in the bid price to thesame di¡erence for the ask price exceeds the e¡ect on the revenuemaximising price of a change in the bid price. This is a relatively easycondition to check for any speci¢c trade mechanism, if supply and demandfunctions are known. For cases where the interdependence e¡ects aresymmetric, the condition becomes dpa

dpb< 1.



Examination of (34) suggests that it is unlikely to hold. An increase inthe bid price results in a direct increase in cost

ÿpb

@Sm

@pb� sm > 0

�. On the

other side, there is an indirect reduction in costÿpb

@Sm

@pa

dpa

dpb< 0 if dpa

dpb> 0

�and

an indirect increase in revenueÿpa

@Dm

@pb> 0

�. Even with strong inter-

dependence e¡ects, the greater magnitude of the direct e¡ect would seemlikely to outweigh the indirect e¡ects in the opposite direction. Onlyextremely strong interdependence e¡ects (low frictions in direct trade)coupled with a very sensitive ask price

ÿdpa

dpb> 1

�could negate (34). Overall,

therefore, the composite condition [(34) or (35)] appears more of atheoretical curiosity.

IV(ii). With a Speci¢c Model

In the model in Section II, the Walrasian price is pw � ba�b regardless of

whether there is direct trade. The relevant elasticities are ed � b1ÿsÿ1ÿ sb

a�b�

pa

Sm

and esd � sab�1ÿs��a�b�

pa

Sm. Evaluating (33) at pw

�36� ed ÿ esd �b

1ÿ s1ÿ sb

a� b

� �pa

Dm

ÿ sab�1ÿ s��a� b�

pa

Sm

� ba� 1

which is identical to (5) without direct trade. To see if (35) holds, considerSm�pb; pa� ÿ Dm�pb; pa� � apb � bpa ÿ b. Di¡erentiating with respect to pb

gives a� sab2�a�bÿsb� > 0. It is not necessary to know whether equation (34)

holds and the computation is di¤cult. Hence, for this speci¢c model, directtrade has no e¡ect on whether an equilibrium exists.

However, it does a¡ect the nature of the non-Walrasian equilibrium ifa random-tie break rule enables one middleman to become a monopolist(cornering the market). In particular, the magnitude of the distortion inthe non-Walrasian equilibrium is smaller the more e¤cient the directtrade. In particular, it can be shown that direct trade reduces the revenue-maximising ask price

�37� pa �1ÿ �2� �1ÿ ��b� �apb � 2ac

2�a� �1ÿ ��b� <12:

Moreover, dpa

d�< 0 if pb < 1ÿ 2bc

a�b so that for su¤ciently low c the distortionin the non-Walrasian case is decreasing as direct trade becomes moree¤cient. This suggests higher welfare with direct trade even if direct tradedoesn't actually occur.

This discussion raises a competition policy question, which we merelyraise at this stage. Does monopoly or (non-Walrasian) duopoly givehigher welfare if demand is inelastic? If monopoly gives higher welfare,this would support the case for restricting entry in markets where

420 john fingleton


middlemen are observed to corner the market, buying up all the stockto eliminate competitors, and then not selling it all so as to maintainhigher ask prices.

IV(iii). Discussion of the Results

The robustness of Stahl's result to the introduction of direct trade is dueto the fact that direct trade lowers but does not eliminate the monopolyrent, leaving a possible incentive for middlemen to corner the market. Ifdemand is elastic, duopoly means that all trade goes through themiddlemen, even when direct trade is possible. This will be true as long asthere is a positive friction in direct trade. If demand is inelastic, themiddlemen o¡er even higher prices to the sellers, so it is not likely that anyof them will wish to trade directly.14 The only e¡ect direct trade has onthe duopoly is via the change in demand function which becomes moreelastic (reducing the likelihood of cornering) but shifts out if the supply tothe middleman increases (the interdependence e¡ect). For our speci¢cmodel these two e¡ects exactly cancel each other out. However, the lowerrent (slope of the demand function) means that the extent of the distortioncaused by cornering the market will be smaller.

Our results are not sensitive to assumptions about the middlemanoperating costlessly or there being no discounting within the tradingperiod. The only essential assumption is that there is a payo¡ linkageacross the direct market, that is that the payo¡ to sellers (buyers) dependson the participation of buyers (sellers) on the other side. Many directmarkets share this feature, and so we would expect the analysis ofSection III to apply generally to them.

The model makes some clear predictions about the distribution andoverall level of welfare. When there is no direct trade, welfare is maximisedin the Walrasian equilibrium, but otherwise there is a welfare loss eitherbecause there is no equilibrium with zero welfare or a non-Walrasianequilibrium in which there is a deadweight loss and redistribution frombuyers to sellers who ultimately capture all the rent in the market.

The e¡ect of adding direct trade depends on which of these threeoutcomes prevails. With the Walrasian equilibrium, direct trade has noe¡ect as welfare is already maximised. However, if the competitionbetween the middlemen were less intense (as with Cournot competition ormonopoly), then direct trade would unambiguously increase welfare. Withthe non-existence of an equilibrium, direct trade would enable some sellersand buyers to trade despite the absence of middlemen and hence welfare

14 There is the slight possibility that the presence of many rationed buyers might create adirect trade opportunity for a small number of sellers and buyers. This could only happen ifpa�pm� ÿ pm > 2 c

�.



would be improved. In the dumping case, although it is not possible toestablish it formally, even for the example above, it appears likely thatdirect trade also increases welfare.15

This paper models intermediation as providing certainty of deliveryand thus it predicts that the sellers and buyers who have the most inelasticsupplies and demands will choose the services of a middleman. A minorre-interpretation of the model of direct trade in Section II would suggestthat it is the relatively impatient traders that choose to trade with amiddleman. Suppose that direct trade is managed by a matchmaker whomatches sellers and buyers e¤ciently (and in order of their reportedreservation prices) until she ¢nds a seller and buyer with equal reservationprices, which she then sets as the market clearing price. The parameter scould be interpreted as a discount factor, where s close to 1 indicates thatthe matchmaker works quickly. All sellers and buyers manage to trade atpd but not as soon as they would trade with the middleman. Thisinterpretation of direct trade is closer still to Demsetz's [1968] idea ofimmediacy from trading with a middleman.

Characteristics of sellers and buyers other than their willingness to paymay determine whether they are likely to go to a middleman. If sellers andbuyers are di¡erentiated according to their information about the locationof the sellers and buyers on the other side of the market, then themiddleman would provide a similar matching service to that discussedabove.16 If they are di¡erentiated according to the volume they wish totrade, then there are two possibilities. If direct trade involves ¢xed costsand no liquidity costs, then the larger traders would trade directly. This ismore likely to happen in markets where a £ow of goods continues overtime so that there are few liquidity costs. It might explain why large buyersand manufacturers vertically integrate. If on the other hand, the cost ofdirect trade is per unit and there are liquidity costs (as on ¢nancialmarkets), then the larger traders might prefer to trade with amiddleman.17 Although our model is not directly applicable to suchgeneral cases, it does indicate that the extent to which sellers and buyersview di¡erent trade channels as substitutes is essential to competitionbetween these trade channels.

15 From equation (37), consumer surplus would be higher, even without direct trade. Anadditional positive welfare e¡ect would occur if direct trade existed (if the condition infootnote 14 holds), as such direct trade would not crowd-out indirect trade.

16 Another reason for trading with a middleman is that he provides information about theproduct (rather than its location). Thus the middleman provides additional retail serviceswhich may be di¤cult to distinguish from the matching service that he also provides.

17 Handa and Schwartz [1966] show that in securities markets institutional investors andimpatient traders prefer to deal with middlemen whereas limit orders (direct trade) are chosenby those trading smaller quantities or who are more patient.

422 john fingleton


v. predictions of the model

The possibility of competing middlemen or dealers trying to corner amarket will only arise in speci¢c circumstances. First, it is necessary thatthe demand be inelastic. Second, it appears unlikely that a middlemanwould attempt to corner a market for durable goods, regardless of theelasticity, as the incentive to establish a monopoly position would bedissipated by Coasean arguments.18 Third, it is necessary that a slightlyhigher bid price will attract all of the stock on the market. On the otherhand, it may not matter whether direct trade is possible. Thus the modelwould be applicable in auction markets, but not in markets whereindividual middlemen had buying power, as this would prevent theaccumulation of a monopoly position in stocks.

In what type of market settings might such cornering of a market arise?The model in this paper applies to markets in which intermediated trade

competes with direct trade. The distinguishing feature of intermediatedtrade here is that middlemen buy and sell rather than match buyers andsellers (as an estate agent or travel agent does on commission). The purestapplication of this is in ¢nancial markets where dealers or market-makersbuy and sell stocks as on the London and New York stock exchanges. Onthe NYSE dealers compete with a public order book in which sellers andbuyers trade directly with the assistance of matchmakers. On the LondonSE, direct trade is possible but accounts for only a small proportion oftrade. A recent report, O¤ce of Fair Trading [1995], argued that the rulesof the LSE should be changed to permit more direct trade in order toimprove competition against the dealers, especially in thinly traded stockswhere margins are high.

The presence of futures markets and the intensity of price competitionin such markets may make it possible for an intermediary to acquire alarge position in a stock, but this might not be a pro¢table strategy for tworeasons. First, the value of the stock has a high common value (as opposedto private value) element and the demand is likely to be ``elastic'' in thesense that traders will not wish to pay above the common value. Second,there are rules speci¢cally designed to prevent such a secret accumulationof stock. That such a strategy might be unpro¢table is illustrated by theattempt by the Hunt family to corner the market for silver in 1980.19

Despite the use of future contracts, the Hunts pushed up the price of silverbut there was no long-run inelastic demand for them to exploit.

The model also applies to goods markets, and particularly to verticalintegration. A manufacturer who faces a choice between selling directly or

18 See Coase [1972].19Williams [1995] provides a detailed account of the incident and discusses the question

of market manipulation generally.



using a dealer is to some extent choosing between direct and intermediatedtrade. A similar choice faces a buyer who decides between out-sourcinginputs and producing them internally. In the PC market, for example,companies such as Dell and Gateway opt for direct sales, whereas otherssuch as Toshiba and IBM sell mostly through dealers. In these markets,what distinguishes the sellers and buyers who by-pass the intermediatedchannel is a high level of knowledge about the product or the location ofthe product, and thus they have access to lower direct trade costs thanother sellers and buyers. A middleman or dealer might still be able tocorner such a market for the less informed buyers if direct trade was not agood substitute for this cohort of demand. What prevents cornering inthe computer market is the impossibility of buying up all computers, andeven if this were possible, new entry to the market would occur in thelonger term.

If there were barriers to entry, the cornering of a goods market couldbe relatively stable, even in a repeated setting (provided the good isperishable) if the buyers were poorly coordinated and small, as the sellerswould have no incentive to change the arrangement. In general, it wouldbe the buyers rather than the sellers who would wish to integrate vertically.Dual-sourcing by ¢rms, such as taking a small proportion of their stockdirectly from upstream suppliers, might not prevent the cornering of themarket, but would mitigate some of its adverse e¡ects. Thus it might be asensible insurance strategy even if the second source was less reliable.

A related issue is that of service provision in markets such as electricityand telephony. Technological and regulatory advances have made itpossible to bypass the service provider (who may be thought of as amiddleman). In this manner, bypassing resembles direct trade. The natureof ¢xed costs in such markets makes them qualitatively di¡erent, and thewelfare predictions discussed above may not apply. In particular, if thereare returns to scale in intermediation, then bypassing by larger buyers mayreduce welfare by pushing up the average cost of intermediation for thosethat remain. In addition, there may be a duplication of ¢xed costs, if thebypass technology involves ¢xed costs.20

As noted in the introduction, the model is particularly relevant tomarkets with essential inputs, where the ¢rms that use this input bid foraccess. Examples might include ferry companies or airlines bidding forlanding rights and service providers bidding for network access. With suchbottlenecks, the good (here a service) is generally perishable and the model

20 Curien et al. [1994] examine the issue of bypass in telecommunications in the context ofnon-linear pricing. They ¢nd that bypass will enable larger customers to do better, but thiswill exclude some smaller customers from the market. Optimal access prices could be used tointernalise such externalities, although this requires regulatory intervention. Armstrong etal. [1996] outline the determination of optimal access prices when bypass is possible.

424 john fingleton


applies, even in a repeated setting. If the owners of the essential input wereto auction the entire supply to the highest bidder, this would be an e¡ectiveway of obtaining the entire monopoly rent. Thus cornering of the marketmight occur and our model predicts that the existence of a direct trade orbypass possibility might not prevent such behaviour.

To some extent, this may be borne out in the context of the UKelectricity industry, although the analogy is not perfect.21 The market ischaracterised by a type of intermediated trade via the pool and severalauthors have argued that the sellers manage to extract substantial rentfrom the buyers.22 Although there is some direct trade (private generation)in this market, this has not seriously competed with the electricity in thepool. Our results predict that direct trade is unlikely to disciplinecompetition between the generators unless it were feasible for a very highproportion of buyers, and suggest that increased competition in generationmight be a more e¡ective means of improving market performance.

Of course, direct trade may improve welfare, not because it reduces theincentive to corner a market, but because it provides competition to themarket power of intermediaries. In the Stahl model, competition betweenmiddlemen is perfect when it exists. With imperfect competition amongmiddlemen, direct trade increases welfare by reducing the market power ofthe middlemen.23 This suggests that the proposals by the O¤ce of FairTrading to remove barriers to direct trade on the London Stock Exchangeshould see a reduction in the margins of the market makers, especially inthinly traded stocks where margins are highest. The impact on marginswill be greater the higher the proportion of sellers and buyers that cancredibly threaten to trade directly. Factors that add to the e¤ciency of analternative trade mechanism, such as increased liquidity in a ¢nancialmarket or better organisation of direct links between upstream anddownstream ¢rms, will also improve competition. Thus as more directtrade occurs in a ¢nancial market, liquidity may increase and attract evengreater levels of direct trade. In markets where direct trade or bypassimposes costs on others (because of returns to scale in intermediation, forexample), direct trade that was not properly priced might sometimes havea negative welfare e¡ect.

21 The pool is not an intermediated market in the sense of this paper but the generatorsannounce supply schedules that might have a similar e¡ect to a middleman's margin. The factthat capacity is ¢xed and that the product is non-storable conform closely with theassumptions of our model. Dumping of stock in this context would represent the systematicunder-utilisation of capacity.

22 See Green [1996], Green and Newbery [1992] and von der Fehr and Harbord [1993].23 Fingleton [1997] characterises these welfare results for the monopoly case. An earlier

working version of this paper, Fingleton [1993] examines the case of imperfect competition. Ifmiddlemen set quantities on the supply side, then the outcome is Cournot regardless ofwhether there is price competition or quantity on the demand side. Adding a direct tradepossibility reduces the margins of the imperfectly competitive middlemen.



Finally, the model focuses attention on the question of how stocks orinputs are sold. If sellers opt for intermediation as a means of sale, thenthis might be tantamount to a commitment to collusion. Moreover, ourmodel suggests such collusion would be robust to the introduction of directtrade, because the intermediary as opposed to the sellers would have theexcess capacity. In such cases, public policy should be directed to theorganisation of an intermediated market for the input. In particular, theselling mechanism should make it impossible for one intermediary toacquire all or most of the stock.

An example of the design of a market mechanism to prevent corneringmight be the ban on ticket touts that accompanies many sporting andother events. If the event's objective function is to maximise attendanceand ensure broad access to the event (that is, a social welfare goal), thenthey would wish to prevent bulk purchases for resale. Otherwise, a tout orgroup of touts could corner the market for tickets and might ¢nd ifpro¢table to leave tickets unsold in order to keep the price high.

v. conclusion

In summary, this paper highlights the issues that are relevant to analysingwhether an intermediary could corner the market, even in a setting ofintermediaries competing with each other and with direct trade. Goodsmarkets and particularly those for essential inputs are more likely than¢nancial markets to present pro¢table opportunities for cornering. Ifcornering a market can occur, then direct trade would not prevent this,although it would mitigate the negative welfare consequences, and inproportion to the ease with which buyers and sellers can substitute directtrade for intermediated trade. The latter occurs because direct trade has agenerally positive welfare e¡ect in disciplining the overall level of marketpower in intermediated trade.

JOHN FINGLETON, ACCEPTED JUNE 1997Trinity College,Dublin 2,Irelandemail: [email protected]

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competition among middlemen when buyers and sellers can trade directly

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