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Competition and InformationProduction in Market Maker

Models

A LAN D. MORRISON*

Abstract: The microstructure literature models the mechanisms throughwhich fundamental information is incorporated into market prices. Thispaper extends previous models by endogenising information productionand analysing incentives for costly information production. In contrast tothe existing literature, increasing the number of informed traders can resultin reduced price informativeness. When prices have an allocative role this haswelfare consequences: the regulatory implications of a dichotomy betweenprivate and public incentives for information gathering are discussed.

Keywords: market maker model, information accquisition, competition,liquidity, price informativeness

1. INTRODUCTION

Starting from the seminal work of Kyle (1985), a substantialliterature has arisen which examines market maker interme-

diated markets. These models analyse in a noisy rational expec-tations framework the strategic behaviour of informed traderswho wish to maximise the return which they earn on theirinformation in anonymous trading with uninformed ‘liquidity’

* The author is from Saıd Business School and Merton College, University of Oxford.He is grateful to the anonymous referee whose careful reading of the paper identifiedsome problems in the first version. He would also like to thank his colleagues at Oxfordand in particular Jos van Bommel, Alexander Gumbel and William J. Wilhelm, Jr. for

useful discussions and for comments upon earlier drafts of this paper. (Paper received January 2003, revised and accepted May 2003)

Address for correspondence: Alan Morrison, Merton College, Oxford OX1 4JD, UK.e-mail: [email protected]

Journal of Business Finance & Accounting, 31(7) & (8), September/October 2004, 0306-686X

# Blackwell Publishing Ltd. 2004, 9600 Garsington Road, Oxford OX4 2DQ, UKand 350 Main Street, Malden, MA 02148, USA. 1171

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traders. This literature provides an information-theoreticfoundation for price formation in traded markets and hasdeepened our understanding of the mechanisms by whichinformation is reflected in prices, and of the relationship

between adverse selection and market liquidity. In this paper Iextend the existing literature by modelling the costly pro-duction of information in the Kyle framework. Conclusions inthe existing literature about price informativeness are shownnot to be robust. The model of this paper has implications forthe design of exchanges and it suggests avenues for furtherresearch into the welfare economics of financial markets.

In standard microstructure models a risk neutral and com-

petitive market-maker interacts with informed and uninformedtraders. The informed traders strategically anticipate the effectsof their trading upon the market and so are not competitiveprice-takers. The trading motives of the uninformed tradersand the source and quality of the informed traders’ informationare typically unmodelled. Much of the literature in this fieldexamines the consequences of relaxing some of these assump-tions. For example, Spiegel and Subrahmanyam (1992) and

Dow and Rahi (2000, 2001) examine models in whichthe uninformed traders are strategic agents with a hedgingmotive for trade. Admanti and Pfleiderer (1988) use the Kyleformalism to examine intraday trading patterns. Chowdhry andNanda (1991), Pagano (1989) and Bhushan (1991) examine thedistribution of trade between different marketplaces. SeeO’Hara (1995) for a review of the applied theoretical literaturein this field.

One area which has been underinvestigated is the relation-ship between the cost of information and competition levels between informed traders. The related incentives determinethe level of information which is generated and therefore theinformativeness of prices in financial markets. Some work doesconsider the acquisition of information. Verrecchia (1982)examines information acquisition by risk averse competitive(price-taking) traders and derives some comparative statics forprice informativeness. A classic paper by Grossman and Stiglitz

(1980) examines the equilibrium level of informed trade andits effect upon price informativeness when information has afixed cost, access to information production technologies is

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# Blackwell Publishing Ltd 2004

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unrestricted, and traders are risk-averse competitive price-takers. Their paper demonstrates that perfectly informativefinancial markets are impossible and shows that the informa-tiveness of prices is inversely related to the cost of information.This work is extended to the imperfectly competitive case byKyle (1989), who considers free entry by risk averse strategictraders with access to a costly signal of given quality. Incentivesfor information production and revelation have been examinedin a different context in models of Initial Public Offerings.1

Several papers examine the relation2 between public revelationand private foreknowledge of information,2 but none examinesthe effect of competition upon the intensity of information

production and hence upon the informativeness of prices.In this paper I approach the problem of information gener-

ation from a different angle. Rather than fixing the cost and thequality of information and endogenising informed trader entryas the above authors do, I follow the market microstructureliterature and fix the number of informed traders exogenouslyand I endow them with a technology which allows them todecide the amount which they will spend upon information

production and hence the quality of the information which isavailable to them. This allows me to quantify incentives forinformation gathering.

The incentives which informed agents have to gatherinformation are of interest for two reasons. Firstly, they will affectthe profitability of their trades and hence the expected costswhich liquidity traders must meet. These issues are the primaryfocus of the existing literature and this paper does not address

them in detail. Instead it concentrates upon a second concern:the informativeness, or efficiency, of prices. When stock priceshave an allocative role in guiding productive investment, priceinformativeness has welfare consequences (Grossman, 1995; andSubrahmanyam and Titman, 2001). The relationship betweenthe social and private incentives for information production byspeculators was examined in Hirshleifer’s (1971) paper. More

1 See Benveniste and Spindt (1989), Benveniste and Wilhelm (1990), Sherman andTitman (2002) and references therein.2 See for example Foster and Viswanathan (1993), Kim and Verrecchia (1991),Hirshleifer, Subrahmanyam and Titman (1994), Leland (1992), Zhang (2001) andHolden and Subrahmanyam (2002).

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recently, Dow and Rahi (2001) provide a welfare analysis of theeffects documented by Hirshleifer, but without endogenising theinformation acquisition process.

In providing an explicit model for information production,this paper identifies an effect which has been ignored in theexisting literature. When several informed traders compete,each will earn a lower rent upon his informational assets.When information is costly, there will be a reduced incentivefor information production and prices will tend to be lessinformative. I show that when a plausible concave relationshipis assumed between investment in information and its precision,this effect will dominate the usual positive efficiency effects of

increased competition. In other words, and contrary to previouswork in this field, increasing the number of informed traders ina market place can reduce information-gathering incentives andwith them the informativeness of prices.

Similarly, when information is given exogenously an increasein fundamental uncertainty reduces the informativeness of ex ante prices and hence increases the rents which informedtraders earn from their information. In this paper, increased

informational rents provide incentives for increased informa-tion production. I show that this effect can outweigh the effectidentified in the existing literature: in other words, an increasein fundamental uncertainty can, by increasing information-gathering incentives, increase the informativeness of prices.

Finally, I examine the role of uninformed traders. In theexisting literature, uninformed traders hide the activities of speculators so that an increase in the number of liquidity traders

increases speculator trading volume and with it informationrents. The informativeness of prices is unaffected. In contrast,I show that with endogenous information production an increasein the number of liquidity traders increases price informativeness

by raising information rents and hence information-gatheringincentives. As I discuss in the conclusion to this paper, my resultson price informativeness have implications for the optimal designof financial markets.

Section 2 presents the model and defines the equilibrium

concept which is used in the remainder of the paper. Section 3shows that a unique equilibrium exists. Section 4 discussesthe comparative statics of the equilibrium and examines the

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informativeness of prices. The final section concludes the paper,notes some of the policy implications of the work and suggestsavenues for further investigation. One of the proofs appears inthe Appendix.

2. THE MODEL

I consider a market for a single risky security, whose value S is arandom variable:

~SS ¼ SS þ ~;

where the expected value SS of the security is common knowledgeand ~ is a randomly distributed noise term with mean 0 andvariance V

.3 There are three types of agents in the model: a

risk neutral competitive market maker, m liquidity traders, and n informed traders.

As Spiegel and Subrahmanyam (1992) note, the trading motivesof the liquidity traders can be endogenised by assuming that theywish to hedge endowment shocks. In this paper, I instead followthe standard microstructure line and assume that the trade sizes~ y y1, ~ y y2,. . .,~ y y m of the liquidity traders are determined exogenously asindependent draws from a normal distribution with mean 0 andvariance V L V

. For every i, ~ y y i is independent of ~.

The informed traders have access to a costly information-gathering technology which gives them an imperfect signal of ~. I assume that each of the informed traders can access at a costthe same information set. Each trader can observe a signal

~ þ ~"" ffiffiffiffiffiffiffiffiffi

V ðcÞ

s ;

where ~"" is a draw from N (0,1) which is the same for everytrader, and c is the trader’s expenditure on information. ~"" isindependent of ~ and of every ~ y y i. The precision ðcÞ

V of the signal

is an increasing concave function of its cost c: in other words,0 (.) > 0 and 00 (.) < 0. Conditional upon their signals, theinformed traders will select trade sizes ~ x x1, ~ x x2,. . ., ~ x x n.

3 Throughout the paper, random variables are indicated by a tilde ( ~ X X ): realisations of the variables are shown unadorned ( X ).



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Denote by

~!! X

n

i

¼1

~ x x i þX

m

j

¼1

~ y y j

the total order flow which the market maker observes. Shecannot distinguish between informed and uninformed tradersand this therefore comprises her entire information set.

The market functions in the following fashion. Firstly, theinformed traders decide how much to spend upon informationproduction. Then informed and uninformed traders pass theirorders to the market maker as limit orders. The market maker

is competitive and sets a price at which her expected profit isequal to zero. Amongst the multiplicity of possible equilibria for this model,

I will examine those which are linear and symmetric.

Definition A symmetric linear Nash equilibrium for this game is onein which:

1. The market maker’s pricing strategy is:

(a) Competitive: Ef ~ P Pj~!!g ¼ 0;(b) Linear: ~ P P ¼ SS þ ~!!.

2. The informed traders’ strategies are:

(a) Symmetric: all spend c*;(b) Linear: informed trader order size has a linear depen-

dence upon the received signal:

x ¼ ~ þ ~""

ffiffiffiffiffiffiffiffiffiV

ðcÞ

s !:

3. Informed traders’ actions maximise their expectedincome, given the actions of other informed traders.

In searching for linear Nash equilibria I am following theexisting literature. There are a multiplicity of possible Nash

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strategies for information production. Restriction to symmetricequilibria is necessary to ensure model tractability and seemsplausible when all traders have the same information productiontechnology. Throughout the paper, I will further follow the exist-ing literature by assuming that the respective numbers m and n of uninformed and informed traders are common knowledge.

3. EXISTENCE AND UNIQUENESS OF EQUILIBRIA

Proposition 1 proves the existence of equilibria for the model,and gives the equilibrium parameter values. Its proof appears

in the Appendix.

Proposition 1: A unique symmetric linear Nash equilibriumalways exists. The equilibrium parameter values * and c*solve the following simultaneous equations:

¼ 1

n þ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðcÞ

mð1 þ ðcÞÞV L

s ; ð1Þ

1 ¼ V 0ðcÞð2ðcÞ þ n þ 1Þ2ð1 þ ðcÞÞ2ð n þ 1Þ2 : ð2Þ

The equilibrium value for the informed order size parameter is as follows:

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m

n

ðcÃÞ1 þ ðcÃÞ V L

s : ð3Þ

Equation (2) is the first order condition which each informedtrader solves when selecting the information production level cin a symmetric equilibrium. It states that the marginal cost 1 of information production must be equal to the marginal incomeon the right hand side of the equation which new informationgenerates. Note that:

d

dc

0ðcÞð2ðcÞ þ n þ 1Þ

ð1 þ ðcÞÞ2 ! ¼ 00ðcÞ ð2ðcÞ þ n þ 1Þ

ð1 þ ðcÞÞ2

À 20ðcÞ2 1

ð1 þ ðcÞÞ2þ n À 1

ð1 þ ðcÞÞ3

( )< 0;



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since the precision (.) is assumed concave and n ! 1.Equation (2) therefore yields a unique optimal level of informa-tion production. The cost and income curves are illustrated inFigure 1.

Note that the income curve is shifted upwards by a higher V

and by a higher market depth 1, so that c* is increasing in these

quantities. Increasing the number n of informed traders lowersthe income curve and hence results in a reduced level of equilib-rium information production. This is because an increase in nintensifies the competition between the market makers and sodiminishes their expected information rents. It thereforereduces their expenditure on information production.

Equation (1) is derived in the Appendix using standardarguments which rest upon the projection theorem. It showsthat the market liquidity 1

is increasing in both informed anduninformed trader numbers, in the expenditure c uponinformation, and in the variance of the liquidity traders’order size. The two reaction curves are plotted together onFigure 2. They have a unique crossing point, from which theexistence and uniqueness of the symmetric linear equilibrium

follows.

Marginal income

from information

production

Marginal cost

of information

c c *

Figure 1

Optimal Expenditure on Information Production is at c*: Changing , n, or V

will Shift the Income Curve and Hence Change c*

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4. PROPERTIES OF THE EQUILIBRIUM

(i) Comparative Statics

Proposition 2: The comparative statics of the equilibrium liquidityand information production parameters * and c* are as follows:

1:@ cÃ

@ V > 0;

@Ã

@ V > 0:

2:

@ cÃ

@ V L > 0;

@Ã

@ V L < 0:

3:@ cÃ

@ m> 0;

@Ã

@ m< 0:

4:@ cÃ

@ n< 0;

@Ã

@ n< 0:

Proof : The bold lines in Figure 3 are the reaction curveswhich are discussed in Section 3 and illustrated in Figure 2.

We can determine the comparative statics of the equilibrium byconsidering the curve movements which are induced bychanges in the exogenous parameters of the model.

c

c*

λ λ *

m

n1

σ L(n+1)

λ (c,n,m,V L)

c(λ ,n,V α )

Figure 2

The Unique Symmetric Linear Nash Equilibrium Occurs at (*, c*),where the Reaction Curves of the Market Maker and the Informed

Traders Cross



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Recall from the discussion in Section 3 that increases in m andin V L will shift the reaction curve to the left without alteringthe c reaction curve. The equilibrium point will therefore beshifted to a point A, as illustrated in the figure. This proves parts2 and 3 of the proposition. As discussed in Section 3, increasesin V

will shift the c reaction curve upwards without altering the

curve, so that the equilibrium moves to a point B as in thefigure: this proves part 1 of the proposition.

Finally, recall from Section 3 that increases in n will lower the and the c reaction curves so that the equilibrium point

becomes C, as indicated in the figure. The new value of c* iscertainly lower. Equation (1) implies that:

@Ã

@ n¼ @

@ nc¼cÃþ@

@ cÃc¼cÃ

@ cÃ

@ n

¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ã

ð1 þ ÃÞV L

s 1

2ffiffiffi

np ð n þ 1Þ À

ffiffiffi np

ð n þ 1Þ2

( )þ @ cÃ

@ n

1

2ð n þ 1Þ ffiffiffiffiffiffiffiffiffi

n

mV L

r 0ðcÃÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ãð1 þ ÃÞp ¼ 1 À n

2ð n þ 1Þ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÃ

nð1 þ ÃÞV L

s þ @ cÃ

@ n

1

2ð n þ 1Þ ffiffiffiffiffiffiffiffiffi

n

mV L

r 0ðcÃÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ãð1 þ ÃÞp < 0:

This concludes the proof. &

The intuition for these results is as follows.When the ex ante variability V

of the security’s value increases,

the value of the informed traders’ information increases and

c

λ

c(λ ,n,V α )

λ (c,n,m,V L)

•

• •A B

C

Figure 3

Comparative Statics of the Equilibrium

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hence expenditure on information production will alsoincrease. When there is more fundamental uncertainty tradingis riskier and trading volumes will be lower. Informed tradingwill therefore be more visible and as a result market liquiditywill drop: equivalently, * will increase.

An increase in the noise level V L will enable informed traders tohide their activities more effectively and so will raise the equili-

brium return on information production, which will thereforeincrease. A greater degree of noise will render informed tradingless visible and market liquidity will therefore increase. Anidentical argument explains part 3 of the proposition.

As discussed in Section 3, an increase in the number of

informed traders results in higher competition levels, reducedinformational rent and hence a reduced level of informationproduction, so that @ cÃ

@ n < 0. The increase in n has two effectsupon . Firstly, an increased n with no change in informationproduction renders the order flow observed by the market makermore informative and hence decreases market liquidity (increases). However, the increased competition between informed tra-ders and the reduction in information production has an offset-

ting effect: although more informed traders are passing orders tothe market maker, the information content of each order isreduced. This will tend to increase market liquidity (reduce ).It transpires that the second effect dominates.

(ii) Price Informativeness

In this section I examine the consequences of endogenous

information accquisition for price informativeness. Priceinformativeness is inversely related to the following expression:

varfV j Pg ¼ varf~j SS þ ~!!g ¼ V ð1 À 2Þ;

where ¼ covð~;~!!ÞV

12V

12!

is the correlation coefficient between ~ andSS þ ~!!.

Proposition 3 shows how price informativeness is related to n,V

, V L and m.

Proposition 3:

1. Price informativeness is decreasing in n if and only if:



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À @ cÃ

@ n0ðcÃÞ >

ðcÃÞð1 þ ðcÃÞÞ nð n þ 1Þ ;

2. Price informativeness is increasing in V

if and only if:

@ cÃ

@ V 0ðcÃÞ >

ð n þ 1 þ ðcÃÞÞð1 þ ðcÃÞÞ nV

;

3. Price informativeness is increasing in V L and m.

Increases in n have two opposing effects upon the informa-tiveness of prices. Firstly, a higher n results in more competition

between market makers and so increases the information

content of prices. Secondly, the competition which comes froma higher n reduces the rent which informed traders earn oninformation and so reduces information production and with itthe informativeness of prices. Proposition 3 indicates that when

À @ cÃ

@ n0ðcÃÞ >

ðcÃÞð1 þ ðcÃÞÞ nð n þ 1Þ ;

the second effect is stronger than the first. This will be the case

when either n is sufficiently high, so that the effects of competi-tion upon information rents are particularly severe, or when theprecision of information is sufficiently cost-sensitive (0 (c*) issufficiently high), so that a slight reduction in expenditurein response to increased competition has a large effect uponinformation production.

Similar intuition applies to increases in fundamentaluncertainty, V

. For a given precision (c), a higher V

reduces

the informativeness of prices and hence increases the informa-tion rent which accrues to informed traders. As shown inProposition 2, this in turn increases information gatheringincentives and hence the informativeness of prices. Part 2 of Proposition 3 indicates that the second effect will outweigh thefirst precisely when

@ cÃ

@ V 0ðcÃÞ >

ð n þ 1 þ ðcÃÞÞð1 þ ðcÃÞÞ nV

:

Increases in V L and m allow informed traders better to hidetheir activities. In models in which their signal precision is fixedthey respond by increasing their trade size: they earn higher

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information rents, and price informativeness is unaffected.However, as demonstrated in Section 4( i), with endogneousinformation production increased rents will lead to greaterinvestment in information. Proposition 3 demonstrates thatthe increased level of information production will increaseprice informativeness.

The results of this proposition are at variance with micro-structure models in which information is acquired exogenously.Since speculators cannot in those models reduce their expendi-ture on information production increased competition betweenthem results unambiguously in increased price informativeness,and increased fundamental uncertainty in reduced price

informativeness. Proposition 3 shows that endogenisinginformation production can in some circumstances reversethese conclusions. The results for the number of informedtraders and the variance of their order size are even morestriking: endogenising information production renders priceinformativeness an increasing function of these parametersand hence always alters the conclusions of previous models.

These results may have implications for market efficiency

and hence for social welfare. I return to this theme in theconclusion.

5. CONCLUSION

The literature on market microstructure examines thestrategic interplay between informed and uninformed traders

who meet in an anonymous marketplace, typically inter-mediated by a risk neutral competitive market maker. Itmodels the process by which information is impounded inprices and shows how the informativeness of prices is affected

by exogenous parameters, such as the number of informedand uninformed traders. The relevant models have however,typically ignored the process by which information isacquired. This is important, as price informativeness dependsnot only upon the numbers of traders and the uncertainties

to which they are exposed, but also upon the ex anteincentives for gathering information. This paper extends theexisting literature by endogenising information production.



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It is thus able to comment upon the relationship between theprivate incentives for information acquisition and the publicgood which price informativeness provides.

When prices have an allocative role, price informativeness hasimportant welfare consequences. These effects are typicallyignored by the agents who generate information and in conse-quence there may be either under- or over- production of information (Hirshleifer, 1971). I show in this paper thatinformation production incentives depend critically upon com-petition levels between market makers and upon the nature of the liquidity traders with whom they trade. When competitionlevels increase information rents are reduced and incentives for

information production are diminished. I show in Proposition 3that this may reduce the informativeness of prices and thusreduce the allocative efficiency of market prices. Conversely,an increase in the number of uninformed traders will increaseinformation production and with it the informativeness of prices.

These effects are ignored by the existing literature, in whichincreased competition between informed traders leads

inevitably to more informative prices and hence to welfaregains. They suggest that access to the information gatheringtechnology should be limited. This might be achieved by char-ging traders a Pigouvian tax for access to information technol-ogy. Although trader entry is unmodelled in this paper, it seemslikely that such a tax would in equilibrium reduce the numberof informed traders to the level where their expected profits justequalled the tax. One could think of the high cost of an

exchange seat as such a tax. Restrictions upon participation inthe initial stage of an Initial Public Offering could be explainedin a similar way (see Sherman and Titman, 2001).

The relevance of uninformed traders to informationproduction is also critical. The profits which informed tradersmake come at the expense of uninformed traders, who ignorethe social benefits which price informativeness brings. If uninformed traders can trade without participating in theprice formation process, they will do so. Prices will be less

informative, but the expected losses sustained by uninformedtraders will also be reduced. Trading off-exchange at exchangeprices is one way in which uninformed traders can accomplish

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this. This type of trade will reduce the rent earned by informedtraders on their rent and so will diminish the informativeness of prices and possibly also social welfare. This argument provides astrong rationale for limiting access to exchange prices by delay-ing price reporting to non-exchange members. It also suggeststhat the recent explosion in trading via electronic crossing net-works may be socially undesirable.

The observations in the above paragraphs suggest a possibleresearch programme. Categorical policy statements can bemade only in the context of a general equilibrium model inwhich all of the players are acting strategically and where thesocial benefits of improved allocative efficiency are captured

through the utility functions of shareholders. This work would bridge the finance and welfare economics literatures and mightgenerate insights into the importance of market institutions.This paper takes a first step towards an understanding of therelevant issues.

APPENDIX

Proof of Proposition 1Suppose that every informed trader except the kth followsstrategies given by parts 2(a) and 2(b) of Definition 1. Todetermine the kth trader’s optimal information expenditure c k,I determine for a given c k the optimal x k conditional upon a

given signal þ "

ffiffiffiffiffiffiffiffiV

ðc kÞq . This yields a maximum expected profit

conditional upon þ " ffiffiffiffiffiffiffiffi

V ðc kÞ

q . Taking expectations of this with

respect to and " yields the expected profit from an infor-mation expenditure of c k. c k is selected to maximise thisexpression. To ensure a symmetric equilibrium, c* and are finally selected so that the optimal c k is equal to c* andso that the kth trader’s decision rule is given by part 2(b) of Definition 1.

Write k for (c k) and * for (c*). Then given expenditure c k

and a signal þ " ffiffiffiffi

V k

q , the expected profit which the kth trader

earns from a position size x k is



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E x k~SS À ~ P Pð~!!ÞÀ ÁÀ c k þ "

ffiffiffiffiffiffiV k

r

& '

¼ E x k ~ À x kX n

i¼1

~ x x i þX m

j¼1

~ y y j

0@ 1AÀ c k þ " ffiffiffiffiffiffi

V k

r 8<: 9=;¼ x k

V V þ V = k

& ' þ "


r

À E x2 k þ x k

X n

i¼1 i6¼ k

~ þ ~""

ffiffiffiffiffiffiV Ã

r ! þ "


r

8<

:

9=

;À E

X m

j¼1

~ y y j

8<

:

9=

;À c k

¼ x k k

1 þ k þ "


r À x2

k À x k n À 1ð Þ k þ ffiffiffiffi

k

Ãp

1 þ k þ "


r À c k: (4)

The kth informed trader will select x k so as to maximise thisexpression. Differentiation immediately yields

x k ¼ þ "

ffiffiffiffiV k

q 1

þ k

k

2À

n À 1

2 k þ ffiffiffiffiffi k

Ãr & ': ð5Þ

Inserting equation (5) into the objective function equation (4)yields the following:

1 þ kð Þ2 þ "


r 2 k

2À

n À 1

2

k þ

ffiffiffiffiffi k

Ã

r & '2

Àc k: ð6Þ

To determine the optimal information production for the kth

informed trader, I firstly take expectations of equation (6)to determine the trader’s expected profit from informationexpenditure of c k:

V

1 þ kð Þ2

1 þ k

k

k

2À

n À 1

2

k þ

ffiffiffiffiffi k

Ã

r & '2

Àc k

¼ V k

1 þ k

1

2À

n À 1

2

1 þ 1

ffiffiffiffiffiffiffiffiffi ffi kÃp

& '2

Àc k:

The first order condition at the optimal level c k of informationproduction is as follows:

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V 0ðc kÞ B B þ k ð n À 1Þ2 k

ffiffiffiffiffiffiffiffiffiffiÃ k

p & '

¼ 1; ð7Þ

where

B 1

ð1 þ kÞ1

2À

n À 1

2

1 þ 1 ffiffiffiffiffiffiffiffiffi ffi

kÃp

:

For any and c*, the equilibrium information expenditure c kof the kth speculator must satisfy equation (7) and in response toa subsequent signal þ "

ffiffiffiffiV k

q he will select trade size x k given

by equation (5). A symmetric equilibrium will occur wheneverequations (7) and (5) are satisfied with c k

¼c* and in addition the

optimal x k satisfies x k ¼ þ " ffiffiffiffi

V k

q . The condition on x k is

equivalent to:

1

1 þ k

k

2À

n À 1

2

k þ

ffiffiffiffiffi k

Ã

r & '¼ : ð8Þ

Setting k¼ * and solving for we obtain

¼ Ã

ð n

þ1Þð

1þ

Ã

Þ: ð9Þ

Substituting equation (9) into equation (7) and setting k¼ *we obtain the following condition:

V 0ðcÃÞ 1

2À n À 1

2ð n þ 1Þ

1

1 þ Ã1

2À n À 1

2ð n þ 1Þ

þ n À 1

2ð n þ 1Þð1 þ ÃÞ2

( )¼ 1;

which reduces to equation (10), as required:

V 0ðcÃÞð2Ã þ n þ 1Þ2ð n þ 1Þ2ð1 þ ÃÞ2

¼ 1: ð10Þ

Part 1(a) of Definition 1 and the projection theorem implythat

Pð~!!Þ ¼ SS þ ~!!

¼ E ~SSj~!!È É ¼ SS þ covð

~SS; ~!!Þ

varð~!!Þ ~!!;



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whence

¼ cov ~SS; ~!!

À Ávar

ð~!!

Þ: ð11Þ

The equilibrium value for ~!! is n ~ þ ~"" ffiffiffiffi

V Ã

q þP m

j¼1 ~ y y j, so

cov ~SS; ~!!À Á ¼ covð~; ~!!Þ ¼ nV ; ð12Þ

varð~!!Þ ¼ n2 2ðV þ V =ÃÞ þ mV LV

¼ V Ã ð n2 2ð1 þ ÃÞ þ mÃV LÞ: ð13Þ

Substituting equations (12) and (13) into equation (11) andsolving for finally yields equation (14), which was to be proved:

¼ 1

n þ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nÃ

mð1 þ ÃÞV L

s : ð14Þ

The existence and uniqueness of the simultaneous solution toequations (10) and (14) is proved in the main body of the paper,using Figures 1 and 2. The expression for is derived by

substituting for in equation (9).

Proof of Proposition 3The correlation coefficient between ~ and SS þ ~!! is

¼ covð~; ~!!Þ

V 12V

12!

:

Using equation (11), we obtain

2 ¼ covð~; ~!!ÞV !

covð~; ~!!ÞV

¼ n ¼ n

n þ 1

Ã

1 þ Ã ;

so that varfV j Pg ¼ V ð1þ nþÃÞ

ð nþ1Þð1þÃÞ. It follows after some simplemanipulation that

d

dn varfV j Pg ¼@

@ n þ@ cÃ

@ n

@

@ cÃ varfV j Pg

¼ V

ð1 þ nÞ2ð1 þ Ã2Þ À @ cÃ

@ n0ðcÃÞ nð1 þ nÞ À Ãð1 þ ÃÞ

& '; ð15Þ

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from which part 1 of the proposition follows immediately.For the second part, note that

d

dV var

fV

j P

g ¼1

ð n þ 1Þð1 þ ÃÞ2 ð n

þ1

þÃ

Þð1

þÃ

ÞÀ nV

@ cÃ

@ V 0

ðcÃ

Þ& ':

Finally, note that

d

dV LvarfV j Pg ¼ ÀV

n

ð n þ 1Þð1 þ ÃÞ2

@ cÃ

@ V L0ðcÃÞ < 0;

d

dmvarfV j Pg ¼ ÀV

n

ð n þ 1Þð1 þ ÃÞ2

@ cÃ

@ m0ðcÃÞ < 0:

REFERENCES

Admati, A.R. and P. Pfleiderer (1988), ‘A Theory of Intraday Patterns:Volume and Price Variability’, Review of Financial Studies, Vol. 1, No. 1(Spring), pp. 3–40.

Benveniste, L.M. and P.A. Spindt (1989), ‘How Investment BankersDetermine the Offer Price and Allocation of New Issues’, Journal of

Financial Economics, Vol. 24, No. 2 (October), pp. 343–61.——— and W.J. Wilhelm (1990), ‘A Comparative Analysis of IPO Proceeds

under Alternative Regulatory Regimes’, Journal of Financial Economics,Vol. 28, No. 1,2 (November/December), pp. 173–207.

Bhushan, R. (1991), ‘Trading Costs, Liquidity, and Asset Holdings’, Review of Financial Studies, Vol. 4, No. 2, pp. 343–60.

Chowdhry, B. and V. Nanda (1991), ‘Multimarket Trading and MarketLiquidity’, Review of Financial Studies, Vol. 4, No. 3, pp. 483–511.

Dow, J. and R. Rahi (2000), ‘Should Speculators Be Taxed?’, Journal of Business, Vol. 73, No. 1 (January), pp. 89–107.

——— ——— (2003), ‘Informed Trading, Investment and Welfare’, Journal of Business, Vol. 76, No. 3 (July), pp. 439–54.

Foster, F.D. and S. Viswanathan (1993), ‘The Effect of Public Information andCompetition on Trading Volume and Price Volatility’, Review of FinancialStudies, Vol. 6, No. 1, pp. 23–56.

Grossman, S.J. (1995), ‘Dynamic Asset Allocation and the InformationalEfficiency of Markets’, Journal of Finance, Vol. 50, No. 3 (July), pp. 773–87.

——— and J.E. Stiglitz (1980), ‘On the Impossibility of InformationallyEfficient Markets’, American Economic Review, Vol. 70, No. 3 (June),pp. 393–403.

Hirshleifer, D., A. Subrahmanyam and S. Titman (1994), ‘Security Analysisand Trading Patterns When Some Investors Receive Information BeforeOthers’, Journal of Finance, Vol. 49, No. 5 (December), pp. 1665–98.

Hirshleifer, J. (1971), ‘The Private and Social Value of Information and theReward to Inventive Activity’, American Economic Review, Vol. 61, No. 4(September), pp. 561–74.



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Holden, C.W. and A. Subrahmanyam (2002), ‘News Events, Information Acquisition, and Serial Correlation’, Journal of Business, Vol. 75, No. 1(January), pp. 1–32.

Kim, O. and R.E. Verrecchia (1991), ‘Market Reactions to Anticipated Announcements’, Journal of Financial Economics, Vol. 30, No. 2 (December),pp. 273–310.

Kyle, A.S. (1985), ‘Continuous Auctions and Insider Trading’, Econometrica,Vol. 53, No. 6 (November), pp. 1315–35.

——— (1989), ‘Informed Speculation with Imperfect Competition’, Review of Economic Studies, Vol. 56, No. 3 (July), pp. 317–56.

Leland, H.E. (1992), ‘Insider Trading: Should It Be Prohibited?’, Journal of Political Economy, Vol. 100, No. 4 (August), pp. 859–87.

O’Hara, M. (1995), Market Microstructure Theory (Blackwell, Cambridge, MA).Pagano, M. (1989), ‘Trading Volume and Asset Liquidity’, Quarterly Journal of

Economics, Vol. 104, No. 2 (May), pp. 255–74.

Sherman, A.E. and S. Titman (2002), ‘Building the IPO Order Book: Under-pricing and Participation Limits with Costly Information’, Journal of Financial Economics, Vol. 65, No. 1 (July), pp. 3–29.

Spiegel, M. and A. Subrahmanyam (1992), ‘Informed Speculation andHedging in a Noncompetitive Securities Market’, Review of FinancialStudies, Vol. 5, No. 2, pp. 307–29.

Subrahmanyam, A. and S. Titman (2001), ‘Feedback from Stock Pricesto Cash Flows’, Journal of Finance, Vol. 56, No. 6 (December), pp.2389–413.

Verrecchia, R.E. (1982), ‘Information Acquisition in a Noisy RationalExpectations Economy’, Econometrica, Vol. 50, No. 6 (November), pp.

1415–30.Zhang, G. (2001), ‘Regulated Managerial Insider Trading as a Mechanism toFacilitate Shareholder Control’, Journal of Business Finance & Accounting,Vol. 28, Nos. 1 & 2 (January/March), pp. 35–62.

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