Contractual Relationship
Chyi-Mei Chen
Department ofFinance,National Taiwan University
Ying-Ju Chen
Graduate Instituteof CommunicationEngineering,National Taiwan University
Ping-Chao Wu
Department ofFinance,National Taiwan University
December 2001
Abstract
Agencytheoristsconsidera rmasa nexusof contractual relationships.
Contract-ing parties such as shareholders, debt holders, managers, inputsuppliers, advertising
agenciesandretailersmayhaveinformationsuperiortooutsidersconcerningthe rm's
quality. In this paper, we show that the duration of contractual relationship within
a rm has important implications on the dynamics of the price and trading volume
of the rm's stock. If the duration of informationasymmetry corresponds to that of
contractual relationship, insiders with long-term relationship is shown to marginally
prefer less aggressive strategies of trading than those who can only access superior
Address for correspondence: Ying-Ju Chen, phone: 886-2-86657702, fax: 886-2-86657702, e-mail:
to infer theexistence of informedtraders. The market maker will henceforth provide
smaller bid-ask spreads, which attract discretionary liquidity traders to gather their
tradesinthat period.
Withthelong-termcontractualrelationship,uncertaintyisresolvedmoreslowly
be-causetheinformedtradertradesonhisinformationadvantagegradually. Thisstrategy
enlargestheliquiditytraders'lossanddiscourages themfromholdingthe rm'sstock.
We concludethat a rm's fundamental value willbe in uencedbyits choice between
changingpartners regularlyand committingto apartner. From theviewpointsof the
rm and uninformed traders,changingpartners regularlyis always an optimal policy
underthe topicsof marketeÆciency andinformationasymmetry.
1 Introduction
In Madhavan[1992], current trading mechanisms are categorized as two types, the
order-driven type and the quote-driven one. In an order-driven mechanism, traders submit their
orderswithoutspecifyingany price-relatedactions;the marketmakerwillaggregate market
ordersandset apricetoclearthemarket. Inaquote-drivenonesuchasNASDAQ,however,
eachtraderbasesonthebid andaskpricesset by themarketmakertodeterminehisorders.
The bid-ask spread has been a mainstream research topic in both the theoretic and
empirical papers. The literature on this topic can beclassi ed into two streams. Papers of
the rststreamdiscuss therelationshipbetween the bid-askspreadand inventorycost,such
as Garman[1976], Amihud and Mendelson[1980], Stoll[1978], Ho and Stoll[1981,83], Cohen,
et. al.[1981], and O'Hara and Old eld[1986]. They argue that the existence of the bid-ask
spread is to avoidthe marketfailure, re ect the market power of the monopolistspecialist,
and depictthe excess returntocompensatethe exposure ofthe market makerinrisk. They
wellasassetpositions,andthetruevalueoftheassetwilleventuallyturnout. However,these
papers cannot give a persuasive and unambiguousexplanation about the optimalinventory
policy of the market maker.
The other stream of literature starts from the adverse-selection problem, such as
Bage-hot[1971],CopelandandGalai[1983],GlostenandMilgrom[1985],andEasleyandO'Hara[1987].
They deem the existence ofthe bid-ask spread asthe consequence of the information
asym-metry between insiders and the market maker. Trading with insiders who have superior
informationcauses the marketmakertolose money since he willalways tradeonthe wrong
side, and therefore he has toset abid-ask spread toensure his breakeven.
This paper commences with a contract design within a rm, and bridges between the
bid-ask spread and the duration of contractual relationship. Jensen and Meckling [1976]
consider a rm as a nexus of contractual relationships, where contracting parties such as
shareholders,debtholders,managers,inputsuppliers,advertisingagenciesand retailersmay
have information superior to outsiders concerning the rm's quality, see alsoRajan [1992],
Dewatripont [1994], and Villas-Boas [1994]. The duration of contractual relationship thus
implies a period of time when these contracting parties hold their superior status of the
rms' information.
In this paper, a rm chooses tochange its counterpart regularly or to have a long-term
cooperation with a speci c partner. Here a long-term cooperation doesn't mean to simply
choose a long-term collaborator, but to sign a long-term contract. As the rm resigns
a contract with a speci c partner again and again, we classify this as a short-term case.
Within the duration of contractual relationship, the contracting party will have superior
information with a positive probability lower than 1. If this party does not receive the
private information,he willact asa liquiditytrader; otherwise he will tradeon hissuperior
discuss an insider 's trading behavior with short-term information advantage and suggest
that there existtwo di erent equilibriums,the separating equilibriumand the pooling one.
In the former equilibrium,insiders trade aggressively to exploit hisprivate information. In
thelatterone, theydisguisethemselvesbytradingrandomlyasliquiditytraders. Easley and
O'Hara alsoclaim that if this model is extended to be a multi-periodone, insiders in each
period willrepeat their single-periodoptimal strategy.
While the insider possesses a long-term position of acquiring superior information,it is
no longeranoptimal strategyto simplymaximizehis pro tin each period even though the
acquired signal is valid for only one period. If a rm signs a long-term contract with his
partner, this contracting party intentionally turns to trade randomly in the early stage of
the contractual relationship, which makes the market maker more diÆcult to identify him,
and therefore adjust downward the belief of hisexistence. As a resultthe bid-ask spreadin
the latter stage becomes narrower, whichcreates pro table roomfor the insider.
Wehavealsointroducedthediscretionaryliquiditytraders 1
todiscussthedynamic
inter-action ofthe price and tradingvolume. We nd that, if the rmsigns a long-termcontract,
the discretionary liquidity traders willmove forward to the latter stage of the current
con-tractual relationship due to the relatively favorable prices. The moving-forward action of
discretionary liquidity traders increases the market depth in the latter stage, which further
enhances the incentive of insiders to camou age the uninformed in the initial stage and
1
Inpresent,manylarge-sizeinstitutions oftensplittheirtradesamong severalmarketsor inround lots.
These traders can choose when to trade but they shall ful ll their liquidity demand before some speci c
date. Theyarecalled"discretionarynoisetraders"referredtoAdmatiandP eiderer[1988,1989],Fosterand
Viswanathan[1990], Seppi[1990], and Speigel and Subramanham[1992]. Admati and P eiderer[1988,1989]
arguethat,inanorder-drivenmarket,thediscretionarynoisetraderswillaggregatetheirtradesinaperiod
oftimeundertheirassumptionthatthesetraderscannotsplittheirtrades,whichincreasesthemarketdepth
increment of liquidity traders also robs the trading opportunities from incumbent traders
due to the quote-driven trading mechanism. Insiders may bene t from the narrow bid-ask
spread, but this bene t realizes with lower probability 2
.
Wedemonstrate thata rm'sdecisionof contractualrelationshipwillin uence the
mar-ket eÆciency, the uninformed traders' welfare, and the rm's value. If the rm regularly
changes his contractualpartners, the insider will trade aggressively, reveal rapidly not only
the content of information but also the fact that the contractual relationship has brought
private information tothe counterpart, and henceforth reduce future insiders' pro t. If the
rmsigns along-termcontract,themarketmakercannotbyordersdistinguishfrominsiders
and liquidity traders, the market eÆciency will be lower. Since the bid-ask spread in each
stage will be a ected by the duration of contractual relationship, investors' expected loss
will alsobe a ected due to their risk attitudeorthe timing of liquidity demand, and hence
the capital collected willbechanged when the rm makes nancing.
Under the long-term contractual relationship, insiders' aggregate pro t among all
pe-riods will be higher than the summation of short-term insiders' due to the informational
externality. Since the stock turnover is a zero-sum game, the expected loss of uninformed
traders willbehigher under along-termcontractualrelationship. Notethat this conclusion
con icts the bindinge ect of the long-termcontract pointed out by Hartand Moore[1988].
They demonstrate that when one party constructs a relationship with a speci c party, the
long-termcontractpreventsoutsidecompetitorsfromharmingeithersideofthem. However,
wefocus neitheronthe incompletecontract,nor onthe contractingpoweragainstoutsiders.
From the viewpoints of the rm and the uninformed traders, changingpartners regularlyis
always anoptimalstrategyunderthe topicsofmarketineÆciencyand information
asymme-try.
2
Theresultis di erentfrom theorder-drivenmarket. InKyle[1985],theliquidity traderswillcoverthe
Section3listssomeresultsofthesingle-periodmodelalreadyobtainedinEasleyandO'Hara
[1987]. We derive the market equilibrium of our model in Section 4, and summarize main
resultsinSection5. Finally,we concludethis paperinSec. 6. Detailedproofs arepresented
in the Appendix.
2 The Model
We consider an economy with 4 periods, period 0, 1, 2, and 3. There is a rm doing an
investment plan whosepayo isexpected inperiod3. In the initialperiod, the rmchooses
to cooperate with three di erent managers to execute its plan in each period, or with the
same onewithinthe rsttwoperiodsand withanotheroneinthe lastperiod 3
. Aslongas it
signs contracts with its partners, the duration of contractualrelationship becomes common
knowledge. The contractualrelationship willbring informationsuperior tooutsiders with a
probability strictly lower than 1, denoted . 4
If information asymmetry occurs, it will last
for three periods, i.e., managers in all periodswill enjoy informationadvantage; otherwise,
they knownothing and act likeliquiditytraders 5
. Ifthe rm chooses tochangeits partners
regularly,I 1 ;I 2 andI 3
denote thesemanagers inperiods1,2,and3,respectively. Ifthe rm
hires the same manager forthe rst two periods, wecall this person I
L .
Two assets are held: the stock and the cash. The value of the stock is (
1 + 2 + 3 ), where i
is acquired by the informed trader at the beginning of period i and will become
3
This paper only depicts hiring a manager with di erent duration of contractual relationship as an
illustrativeexample. Actually,ourdeductioncanalsoapplytoothercontractingparties.
4
Wemustclarifythatinformationadvantagedoesn'tguaranteeamanagertoacquireprivateinformation;
itonlyrepresentsanaccessandthemanagers"may"beinformedthroughtheaccess.
5
Our conclusion is still valid while extending to the multi-period economy. The existence of the last
period represents that there will bemanagement turnovers for at least one time, and there will be other
the occurrence of information nor see the content of information. Suppose that
i
has two
equally probable outcomes, the bad one and the good one. Let L and H denote these two
outcomes, and have respectively value 0 and 1 for simplicity. Hence the stock price in the
initialperiod shall be1/2.
There are nnondiscretionary liquiditytraders ineachperiodtradingonlyin thatperiod
for ful llingtheir liquidity demands, which consists of buying and selling with equal
prob-abilities. Moreover, the liquidity demand will be small-quantity orders with probability
and large-quantity ones with probability 1 . For simplicity, the quantities of orders are
assumed tobe respectively one unit and two units.
The discretionary liquiditytrader, calledn
D
, shall ful llhisliquidity demand inperiods
2 and 3. If the liquidity demand isnot satis edbefore the expirationof period 3,there will
be some cost, or say penalty, denoted C per unit of stock. We assume that the penalty is
large enough for the nondiscretionary liquidity traders so that they will not give up their
trading opportunities.
The market maker sets the bid-ask spread to clear the market and trades with only
one person in each period. Since a liquidity trader will submit two di erent-sized orders,
the market maker should set prices for a trader when he buys a small or large quantity
(denoted B 1
andB 2
),orsellsasmallorlarge quantity(denoted S 1
and S 2
). All tradersare
risk-neutral.
Forconveniencewesettheproportionoftheinformedtradertonondiscretionaryliquidity
traders is:1 . It is easytoverify that = 1
1+n .
3 A Review of the Single-Period Equilibrium
In this section we review the single-period model depicted by Easley and O'Hara [1987].
the informed trader willtrade aggressively to exploit his superior information; while in the
latter one,he turns totraderandomlyso thathe willnot beidenti edbythe marketmaker
and henceforth facepoorprices.
Proposition 1 If 2 >
+(1 )(1 )
(1 )(1 )
, there exists a separating equilibrium, in which the
informed trader will buy 2 units whenthe signal is H, and sell 2 units when the signal is L.
The stock price of buying or selling a small quantity is 1/2 and that of a large quantity are
described below: b(S 2 )= 1 2 (1 )(1 ) +(1 )(1 ) ;a(B 2 )= + 1 2 (1 )(1 ) +(1 )(1 ) :
The expected pro t of the informed trader
s is
(1 )(1 )
+(1 )(1 ) .
Making good use of his information advantage, the informed trader trades aggressively
inthe separating equilibrium. The marketmakerabsorbsa buyingorselling small-quantity
order at price 1/2 because it is certainly submitted by a liquidity trader. The price of a
large-quantity order is set according to the proportion it comes from the informed trader
or from liquidity ones. The necessary condition for the separating equilibrium represents
thatonlywhenthenumberofliquiditytradersislargeenoughtocoverthe informedtrader's
actionwillhefollowsapurestrategy. Theinformedtrader'sexpectedpro twhilesubmitting
a large-quantity order is largerthan that while submitting a small-quantity one.
Proposition 2 A pooling equilibrium exists whenno separating one isable to sustain. Ina
poolingequilibrium,theinformedtraderisindi erentbetweensubmittingasmall-quantity
or-der anda large-quantityone. Hewillrandomlysubmit asmall-quantityorderwithprobability
b(S 1 )= 2 (1 ) +(1 ) ;a(B 1 )= + 2 (1 ) +(1 ) ; b(S 2 )= 1 2 (1 )(1 ) +(1 )(1 ) ;a(B 1 )= + 1 2 (1 )(1 ) +(1 )(1 ) :
The expected pro t of the informed trader
p is
(1 )
+(1 )
=(1 )(2 ).
If the number of liquidity traders is relatively small, there exists a pooling equilibrium,
inwhichitisindi erentforthe informedtradertosubmit asmall-quantity orderora
large-quantity one. The market makerwillset fairbid and askprices considering the ratio of the
submitted order fromthe informedtrader to liquidityones.
4 Equilibrium Analysis under Di erent Contractual
Re-lationships
In this section, we discuss the manager's optimal strategies under short-term contractual
relationship and long-term cooperation; and evaluate if other market participants will be
in uenced by these di erent strategies.
Equilibrium Concept: PBE
First we de ne the PBE (perfect Bayesian equilibrium),which isa bundle of strategies and
beliefsheldbyplayers. InaPBE,allparticipantsinthemarketwillupdatetheirinformation
sets period by period. Thus, in each period the market maker willset an updated bid-ask
spread to ensure his breakeven. I
1
and I
2
will maximize their one-period pro ts based on
the signalsthey have received, andI
L
maximizeshistotal pro ts,i.e., those ofperiod1and
period 2. Nondiscretionaryliquidity traders willtrade for ful llingtheir liquidity demands,
and the discretionary liquidity trader trades to maximize his utility in considerationof the
Market Maker
In the rst period as an order is submitted, the market maker will take into consideration
the proportionthatthis ordercomes fromthe informedtraderandset a fairbid-askspread.
After the rst-period trading, the market maker examines how likely this order comes up
when the information event occurs, and thus updates his posterior belief of the existence
of informationevent. In the second period the market maker does what he did in the rst
period.
Short-Term Informed Trader
If the information advantage lasts for only one period, the manager in each period will
follow the optimal single-period strategy since his object function is the same as in Easley
andO'Hara[1987]. Let
i
denotetheposteriorbeliefoftheexistenceoftheinformationevent
held by the market maker in period i. If 2 > i +(1 i )(1 ) (1 i )(1 )
, the informed trader willbuy
two units of stock when he receives H, and sell two units when L; otherwise, he willfollow
a mixed strategy and hence apooling equilibriumcomes out.
Long-Term Informed Trader
For the long-term manager the information advantage lasts for two periods, and therefore
he will choose to maximize his two-period pro t. The informed trader will also buy when
he receives H, and sell when L But in the rst period, how many units he shall buy or sell
dependsonthepro tinthe currentperiodplusthe expectedpayo inthe secondperiod. In
most cases the large-quantity orderbrings aworse
2
, and hence reduces his second-period
no longerthat one in Easleyand O'Hara.
In the second period, given
2
, he will act as an informed trader with single-period
informationadvantage since there is noconsequent period.
Discretionary Liquidity Trader
Since the bid-ask spreadof a small-quantity order is narrower than that of a large-quantity
one,thediscretionarytraderhasincentivetosplithisdemandintotwoperiods. Whichorder
he willsubmitsinperiod2 depends onthe bid-ask spreadsinboth periods, the opportunity
totradeinperiod3,and the penalty he shalltake whilehisliquiditydemand isnot ful lled.
If the penalty C islarge, he willspeed up histradingin case he can catch more chances
to trade successfully. If C is so large that its e ect outweighs the expected loss originated
from the bid-ask spread, he will submit a two-unit order in period 2 when he seizes the
trading opportunity. Another pulling force to large-quantity trade is the bid-ask spread. If
the bid-ask spread of a two-unit order in period 2 is smallenough while compared to that
in period 3, a large-quantity order is expected. Otherwise, if the penalty C is small or the
spread of the two-unit order is large, he'd better tosplit hisorders totwo periods
4.1 The Economy Without the Presence of the Discretionary
Liq-uidity Trader
We rstconsidertheeconomywithoutthe presenceofthediscretionaryliquiditytrader,and
provideseveral staticcomparisons. In the next sectionwe willintroduce this player and see
what willhappen.
Lemma 1 If there exists a separating equilibrium in period 1, (S ) > (S ); (B ) >
(B 1
).
Lemma1demonstratesthatinaseparatingequilibrium,theprobabilitythattheinformation
asymmetry exists will be higher when a large-quantity order is submitted. In our model,
(S 1 ) = (B 1 ); (S 2 ) = (B 2
). We therefore in the following mention only the left-hand
side for simplicity.
Lemma 2 Thesingle-period expected pro t of the informed trader decreasesin .
Intuitively, the expected pro t of the informed trader will be lower if the market maker's
belief of the existence of the informationasymmetry.
Lemma 3 If increases,the pooling equilibrium will come out more likely.
Ifthemarketmakerbelievesthattheinformationasymmetryexistswithahigherprobability,
theadverseselectioncostincreases. Henceforthiftheinformedtraderfollowsapurestrategy,
his order will be priced more unfavorably. The optimal strategy for himis to disguise as a
liquidity trader.
Proposition 3 Given that the rst-period equilibrium is separating, if in the rst period a
small-quantity order is submitted, the market maker will revise downward his belief and set
a favorable bid-ask spread in the next period, which makes the separating equilibrium stand
more rmly. In contrast, if the trading counterpart submits a large-quantity order in one
period, the market maker will interpret that the information asymmetry exists more likely,
and thus set worse prices in the next period. In such a case, the equilibrium in the next
period turns out to a pooling one.
term contract is strictly lower than that under a short-term one. Henceforth there exists
a range that a separating equilibrium sustains under a short-term contract and a pooling
equilibriumsustainsunderalong-termcontract. Inthe followingdiscussion wewillfocus on
the innovations within this range.
Here we introduce a condition originated fromProposition4:
Condition C: +(1 )(1 ) (1 )(1 ) <2< +(1 )(1 ) (1 )(1 ) + +(1 )(1 ) 1=2(1 )(1 ) [( s (S 1 ) ( s (S 2 )]
Thisconditionensures thatthe rst-periodequilibriumsundertheshort-termandlong-term
contractualrelationship are, respectively, separating and pooling ones.
Lemma 4 Suppose Condition C holds.
s (S 1 )< p (S 1 ); p (S 2 )< s (S 2 ).
Lemma 4 demonstrates that if there exists aseparating equilibrium inthe rst period, the
belief of the information event held by the market maker when a large-quantity order is
submitted will be higher than that as the rst-period equilibrium is a pooling one; if the
rst-periodorder is a small-quantity one, the information event willbe more unlikely when
there is apooling equilibriuminthe rst period.
Let E
2
(YjX) denote the equilibrium when the equilibrium in the rst period is X, and
theexecuted orderisY,whereX 2fS(Separating);P(Pooling)gandY 2fB 1 ;B 2 ;S 1 ;S 2 g.
With the help of these parameters weintroduce the following proposition:
Proposition 5 If bothE 2 (S 1 jS)andE 2 (S 2
jS)areseparating,then E
2 (S 1 jP)and E 2 (S 2 jP)
shallalsobeseparating. IfbothE
2 (S 1 jS) andE 2 (S 2 jS)are pooling, E 2 (S 1 jP)and E 2 (S 2 jP)
shallbe pooling, too. However, theagreement of E
2 (S 1 jP)and E 2 (S 2
jP)does notguarantee
thatE 2 (S 1 jS)andE 2 (S 2
jS)areof the sametype. Inother words,thedegree of disagreement
between E 2 (S 1 jP) and E 2 (S 2
jP) is higher than thatbetween E
2 (S 1 jS) and E 2 (S 2 jS).
more volatile whenaseparating equilibriumshows up inthe rst period. Since aseparating
equilibriumbrings more informationcontent tothe market maker, the magnitude by which
he adjusts hisposterior belief ishigher.
Proposition 6 Suppose Condition C holds. The rst-period expected pro t of I
L
is higher
than that of I
1 .
Proposition 6 is somewhat weird and against our intuition. The idea that long-term
insidershall camou agealiquiditytrader startsfromgiving-upsome informationadvantage
to protect himselffrom being identi ed. Here we interpretit in another way. Other things
being equal,inaseparatingequilibriumthe ratiothatalarge-quantityordercomesfromthe
informedtraderis higherthanthat inapoolingone, andhence thelarge-quantitypriceina
separating equilibriumshall be less favorable totraders. Notealsothat the existence of the
second period providesa stronger support toa rst-period pooling equilibrium because the
insider's single-periodpro ts of submitting a small-quantityorder and a large-quantity one
need not bethe same.
Proposition 7 Suppose Condition C holds. Given that the rst-period order is submitted
by the informed trader, the expected second-period pro t of I
L
is higher than that of I
2 .
This propositiondepictsthat, ifthe rst-periodinformedtraderconcerns onlyhis
single-period pro t,his tradingstrategy willbring externalityto hisfollower.
Proposition 8 Suppose Condition Cholds. Withina wide rangethe two-periodpro tof I
L
is higher than the aggregate pro ts of I
1 and I
2 .
Intuitively,a person maximizingthe two-periodpro t willgain morethan the one
ted by the insider, the expected second-period pro t of the long-term insider is higher, the
long-term insider will be worse o if the rst-period opportunity is occupied by a liquidity
trader.
Lemma 5 Suppose thatapooling equilibrium sustainsin the rst period. If < 1 2 , p (S 1 )< p (S 2 ).
Proposition 9 Suppose that 2<
+(1 )(1 ) (1 )(1 ) . Let and 0
denote respectively the
prob-ability with which in the rst period I
1 and I
L
will submit a small-quantity order. If 0 < 1 2 , 0 > . Let L p (Y) denote p (Y) if I L exists and 1;2 p (Y) denote p (Y) if I 1 and I 2 exist. If 0 < 1 2 , L p (S 1 )> 1;2 p (S 1 ) and L p (S 2 )> 1;2 p (S 2 ).
4.2 The Economy with the Discretionary Liquidity Trader
From now onwe introduce the discretionary liquidity trader.
Proposition 10 Suppose that the rm chooses the long-term contractual relationship. If
there exist discretionary liquidity traders and their population is not too large, in the rst
period a pooling equilibrium will come out with a higher probability.
The reason that a pooling equilibrium will exist more likely is that the presence of the
discretionaryliquiditytraderbringsmorepro tforthelong-terminsiderinthesecondperiod,
and henceforth the threshold for a pooling equilibrium to sustain is lower. The restriction
on the populationof the discretionary liquidity trader is due to the tradingopportunity of
the long-terminsider in the second period.
Proposition 11 Suppose Condition C holds. Given that the rst-period order is submitted
by the informed trader, in the second period more likely the discretionary liquidity traders
willparticipatein themarketwhenthe rmchoosesalong-term contractualrelationship than
by the informed trader, the second-period price will be more favorable if the rst-period
equilibrium is a pooling one, and the discretionary liquidity trader willtherefore bewilling
to participatein. The next twopropositions are obvious observations.
Proposition 12 If there exists a separating equilibrium in period 2, the discretionary
liq-uidity trader will for certain compete the second-period trading opportunity.
Proposition 13 If only the discretionary liquidity traders with small-quantity demand are
inducedtothesecondperiod,thetypeofsecond-periodequilibrium isthe sameasthatwithout
the existence of the discretionaryliquidity traders. Conversely, the rst-periodequilibrium is
a ected by the presence of them when the rmhires a long-term manager.
5 Discussion
Withthe introductionofthelong-termcontractualrelationship,the single-periodseparating
equilibriuminEasleyandO'Hara[1987]willturnsintoapoolingone. Thelong-terminsider's
trading behavior will a ect the market maker's belief, induce the discretionary liquidity
trader to move forward, and henceforth change the equilibriums in both periods. If the
bid-ask spreadinthesecond periodisfavorableenoughsothat thediscretionary tradersare
willing to submit large-quantity orders, the sustainability of the separating equilibrium is
more strongly supported.
Due tothe single-tradersetupineachperiod,the discretionaryliquiditytraderwilldefer
his order to the third period if he cannot make his deal in the second period. Roughly
speaking, the traders' distribution will not vary apparently and the equilibrium in the last
period willhold the same.
that the expected pro t of this long-term manager is worse o , he is not willing to induce
the discretionary liquidity traders and hence a separating equilibrium will turns out in the
rst period.
We summarize our main results inthe followingfour theorems.
Theorem 1 Theduration of contractual relationship changes the bid-ask spread.
Under the long-termcooperation, the manager may abandon his informationadvantage
in the early stage of contractual relationship even though this advantage is short-lived.
Hence anadverse-selection problemoccurs whenasmall-quantityorder issubmitted, which
makes the small-quantity spread broader and the large-quantity spread becomes narrower
concurrently. In the latter stage, the market maker will reduce the bid-ask spread more
likely.
Theorem 2 Theduration of contractual relationship changes the rm's value.
Considering how risk-averse they are, when their liquidity demand will occur and what
prices in that moment will be, liquidity traders may adjust their expectation of their loss
based upon the rm's choice, and henceforth their evaluation of the rm's stock varies. If
the rm proposes to raise funds by IPO and its contracts do not coincide with investors'
preferences, it has to turn its eyes on debt or other nancing approaches to execute its
positive-NPVplan, and eventuallythe rm's value willbe changed.
Theorem 3 Long-term cooperation reduces liquidity traders' welfare.
Since the aggressivetradingofthe managerintheinitialstagewilldisclosetheexistence
of information asymmetry, the short-term contractual relationship will do injury to the
managerin the latter stage, and hence their overall pro t isless than that of the long-term
manager. Due tothe zero-sumnature, liquiditytraderswillonaverage lose moreif the rm
Asthelong-termmanagercamou ageshimselfasaliquiditytraderintheearlystage,the
marketmakercannottellifthiscontractualrelationshiphasbroughtinformationasymmetry.
Here the de ciency of market eÆciency does not mean that the market is unaware of the
result of investment plans but of the existence of informationasymmetry.
6 Conclusion
This paper conducts the connection between the rm's contractual relationship and the
capital market. We point out that the duration of contractual relationship will coincide
withthe periodofinformationadvantageheld bythe contractingparty, andexplainchanges
of equilibriums induced by the informed trader's strategic behavior. We suggest that the
long-termcontractualrelationshipofa rmwillmakeinsiderstocamou ageatinitialstages,
inducediscretionaryliquiditytraderstomoveforward,andhencereducethemarketeÆciency
andtheliquiditytraders'welfare. Thoughmanypastpapershavementionedalotofbene ts
broughtbylong-termcommitment,thispaperpointsout several possible aws and provides
persuasive explanations. Proof of Lemma 1: Since (S 2 )= + (1 )(1 ) +(1 )(1 ) ; (S 1 )= (1 ) 1 ; subtract (S 1 )from (S 2 ),we obtain that (S 2 ) (S 1 )= (1 )(1 )(1 )(1 ) (1 )[ +(1 )(1 )] >0:
Supposethat 2 > 1 ,andE 1 andE 2
are thecorresponding equilibriumswhen =
1 and
=
2
. Wenow discuss this case by case.
(a) BothE
1
and E
2
are separating ones.
Since @ @ s ( )= (1 )( +1 ) (1 )(1 ) [ +(1 )(1 )] 2 <0; we obtain that ( 2 )<( 1 ). (b) BothE 1 and E 2
are pooling ones.
Since @ @ s ( )= (2 )<0,( 2 )<( 1 ). (c) (E 1 ,E 2 )=(S,P). Suppose 3
isthecriticalvalueofwhichmakes
+(1 )(1 )
(1 )(1 )
=2,bytheabovediscussion,
( 2 )( 3 )( 1 ). Note that(E 1 , E 2
)willneverbe(S,P) because
2 >
1
, the lemmaholds forallpossible
conditions.
Proof of Proposition 3:
This proposition isa combinationof Lemmas1 and 2.
Proof of Lemma 3:
Since a pooling equilibriumwillexist when 2
+(1 )(1 ) (1 )(1 ) and @ @ [ +(1 )(1 ) (1 )(1 ) ]= (1 )(1 )+(1 )(1 + ) [(1 )(1 )] 2 >0;
Suppose that a separating equilibrium exists in the rst period. If the long-term manager
submitsaone-unitorder,histotalexpectedpro tis1 1
2
+( (S 1
));ifinsteadatwo-unit
orderissubmitted,histotalpro tis2
1=2(1 )(1 )
+(1 )(1 )
+( (S 2
)). Thenecessary condition
for a separating equilibriumtosurvive is
2 1=2(1 )(1 ) +(1 )(1 ) +( (S 2 ))>1 1 2 +( (S 1 ))
Rewrite it, weobtain
2> +(1 )(1 ) (1 )(1 ) + 1=2(1 )(1 ) +(1 )(1 ) [( (S 1 )) ( (S 2 ))]: (1) ByLemma1, (S 2 )> (S 1
),andLemma2ensures that( (S 2
))<( (S 1
)). Sincethe
second termof the right-handside inEq. (1)isstrictly positive,if forsome the condition
holds, it shall satisfy 2 >
+(1 )(1 )
(1 )(1 )
. Thus the range that a separating equilibrium
will sustain under the long-term contractual relationship is narrower than that under the
short-term one, i.e., the pooling equilibriumwillcome up more likely.
Proof of Lemma 4:
It is easyto verify that
p (S 1 )= + (1 ) +(1 ) ; s (S 1 )= (1 ) 1 :
By subtractingthem, we obtain that
p (S 1 ) s (S 1 )= ( 1) (1 )[ +(1 )] <0; and henceforth p (S 1 )> s (S 1 ). Similarly, (S 2 )= (1 )+ (1 )(1 ) ; (S 2 )= + (1 )(1 ) :
s (S 2 ) p (S 2 )= (1 )(1 ) [ +(1 )(1 )][ (1 )+(1 )(1 )] >0: Thus p (S 2 )< s (S 2
). The other twoclaims can beshawn analogously.
Proof of Proposition 5:
By Lemmas3 and 4, the proposition holds.
Proof of Proposition 6:
The expected rst-periodpro t of I
L is 2 1=2(1 )(1 ) (1 )+(1 )(1 ) ; and that of I 1 is 2 1=2(1 )(1 ) +(1 )(1 ) :
Due to the largerdenominator, the rst-period pro t of I
L
ishigher.
Proof of Proposition 7:
AccordingtoLemma2,theexpectedsecond-periodpro toftheinformedtraderisdecreasing
in , and by Lemma 4, p (S 1 ); p (S 2 )< s (S 2
). The expected pro t of I
2 is ( s (S 2 ))= ( s (S 2 ))+(1 )( s (S 2 ))< ( p (S 1 ))+(1 )( p (S 2 ));
where the right-handside of the inequalityis the expectedpro t of I
L .
The expected two-periodpro t of the informed tradersis 2 ( 1 +E[ 2 j1 I ])+(1 ) 1 +(1 )E[ 2 j1 N ]; (2) where E[ 2 j1 I ];E[ 2 j1 N
] represent the expected second-period pro t given that the
rst-period order is submitted by, respectively, an informed trader and a liquidity trader. By
Propositions 5 and 6, the rst two terms of I
L
are higher than that of I
1 and I 2 . E[ 2 j1 N ]
can be explicitly expressed as ( (S 1
))+ (1 )( (S 2
)), and we have obtained that
( s (S 1 )) > ( p (S 1 )). Unless (1 )(1 )( (S 2
)) outweighs all the other terms in
Eq.(2), the expected pro t of I
L
willbe higher.
Proof of Lemma 5:
It is easyto verify that
p (S 2 ) p (S 1 )= (1 2 )(1 )(1 ) [ +(1 )][ (1 )+(1 )(1 )] ;
and the lemma follows directly.
Proof of Proposition 9:
First note that if 2<
+(1 (1 )
(1 (1 )
, the rst-period equilibrium under the short-term
con-tractualrelationshipwillbeapoolingone. AccordingtoProposition4,I
L
willalsorandomize
his orderin the rst period,and hisincentive compatibilitycondition is as follows:
2 1=2(1 )(1 ) (1 )+(1 )(1 ) +( p (S 2 ))>1 1=2(1 ) +(1 ) +( p (S 1 )):
Bythe continuityoftheaboveequation,thereexists asolution 0 :Since 0 < 1 2 ,( p (S 2 ))< ( p (S 1
))by Lemma5. One can easilyverify that 0
>
Since s (S 1 ) < s (S 2
), by Lemma 2 the price under
s (S
1
) is strictly preferred.
Discre-tionary liquidity traders will therefore show up in the second period more likely when the
order executed inthe rst periodis asmall-quantityone. Inother words, the second-period
pro t of informed trader is higher than that without the discretionary liquidity traders.
Moreover, the expected pro tdi erence (
s (S 1 )) ( s (S 2
))ishigher. ThusEq.(1) holds
withahigherprobability. ForI
L
the incentivetorandomize hisorderisstrengthened by the
discretionary liquidity traders if they do not share the trading opportunity signi cantly. If
thepopulationofthediscretionaryliquiditytraderislarge,leadingthemtothesecond-period
market willreduce the informedtrader's tradingopportunity.
Proof of Proposition 11: By Lemma 4, p (S 1 ); p (S 2 )< s (S 2
). Therefore the bid-ask spreads in the second period
under p (S 1 ) and p (S 2
) are strictly favorable to traders than that under
s (S
2
). Given
that the rst-period orderis submitted by the informedtrader,
2 = p (S 1 )or 2 = p (S 2 )
when the rst-period equilibrium is pooling and
2 =
s (S
2
) when it is separating. The
propositionfollows immediately.
Proof of Proposition 12:
Toverifythat adiscretionary liquiditytraderwithsmall-quantitydemand willparticipateis
easy, and therefore we consider that with two-unit demand. Suppose that the discretionary
liquiditytradersucceedstotradewiththemarketmakerinperiod2. Ifhesubmitsaone-unit
order,there isnoexpected lossduring thisround oftradeand he stillholdsthe opportunity
totradeinthenext period. Suchastrategyisstrictlypreferredtowaitingforthelastchance
It is easy to verify that the necessary condition of the existence of the second-period
sepa-rating equilibriumis still
2 1=2(1 2 )(1 ) 2 +(1 2 )(1 ) >1 1 2 ;
irrelevanttothepresenceofthediscretionaryliquiditytraders. However,ifthesecond-period
equilibriumispooling,both thesmall-quantityandlarge-quantityspreadsaredi erentfrom
those without the presence of them, and therefore the expected second-period pro t of the
informed trader willvary, onwhich the criterionof the rst-periodequilibriumdepends.
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