On the Efficiency of Competitive Stock Markets Where Trades Have Diverse Information TOR Sanford grossman Journal of finance, Volume 31, Issue 2, Papers and Proceedings of the Thirty-Fourth Annual Meeting of the American Finance Association Dallas, Texas December 28-30 1975(May,1976),573-585 Your use of the jStor database indicates your acceptance of jSTOR's Terms and Conditions of Use. A copy of UsTor'sTermsandConditionsofUseisavailableathttp://www.jstor.ac.uk/about/terms.htmlbycontacting JSTOR at jstor@mimas. ac uk, or by calling JSTOR at 0161 2757919 or(FAX)0161 275 6040. No part of a JSTOR transmission may be copied, downloaded, stored, further transmitted, transferred, distributed, altered, or therwise used, in any form or by any means, except: (1)one stored electronic and one paper copy of any article solely for your personal, non-commercial use, or(2)with prior written permission of jSTOR and the publisher of the article or other text Each copy of any part of a STOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission Journal of finance is published by American Finance Association. Please contact the publisher for further permissions regarding the use of this work. Publisher contact information may be obtained at http://www.jstor.ac.uk/journals/afina. nal of finance 76 American Finance Association jSTOR and the jStor logo are trademarks of JSTOR, and are registered in the u.s. Patent and Trademark Office For more information on JSTOR contact jstor@mimas. ac uk @2001 JSTOR http:// on feb2620:11:402001
THE JOURNAL OF FINANCE VOL XXXI. NO. 2 MAY 1976 ON THE EFFICIENCY OF COMPETITIVE STOCK MARKETS WHERE TRADES HAVE DIVERSE INFORMATION SANFORD GROSSMAL NTRODUCTION I HAVE SHOWN elsewhere that competitive markets can be"over-informationally" efficient. (See Grossman [1975] for this and a review of other work in this area )If competitive prices reveal too much information, traders may not be able to earn a return on their investment in information. This was demonstrated for a market with two types of traders, "informed"and"uninformed. " Informed"traders learn the true underlying probability distribution which generates a future price, and they take a position in the market based on this information. when all informed traders do this, current prices are affected. Uninformed"traders invest no resources in collecting information, but they know that current prices reflect the nformation of informed traders. uninformed traders form their beliefs about a future price from the information of informed traders which they learn from observing current price In the above framework, prices transmit information. However, it is often claimed that prices aggregate information. In this paper we analyze a market where there are n-types of informed traders. Each gets a"piece of information. In a simple model we study the operation of the price system as an aggregator of the different pieces of information We consider a market where there are two assets: a risk free asset and a risk asset. Each unit of the risky asset yields a return of PI dollars. PI will also be referred to as the price of the risky asset in period 1. In period 0(the current period), each trader gets information about Pi and then decides how much of risky which will depend on the information received by all traders. We assume that f and non-risky assets to hold. This determines a current price of the risky asset, P( ith trader observes yi, where yi=PI+E. There is a noise term, e,, which prevents any trader from learning the true value of P. The current equilibrium price is a function of (y ,,n): write The main result of this paper is that when there are n-types of traders(n>D), Po reveals information to each trader which is of "higher quality"than his own information. That is, the competitive system aggregates all the market's informa- tion in such a way that the equilibrium price summarizes all the information in the Graduate School of Business, Stanford University. I Michael Rothschild, Joseph pants of the Summer Seminar 1975 at the Institute for Mathematical Studies in the Social Sciences, Stanford University for their helpful comments. This work was supported by National Science Foundation Grant SOC74-11446 at the Institute for Mather Studies in the Social Sciences, Stanford University, and the Dean Witter Foundation. Due to space limitations, an Appendix on the subject of the"Uniqueness of Equilibrium"is not included in the article and is available from the author upon request. 573
57 The Journal of finance market. Po(r 2,.,yn) is a sufficient statistic for the unknown value of Pr. A trader who invests nothing in information and observes the market price can trader who purchases y and then observes Po(y)(where y=(,2,,,n)), fidf p achieve a utility as high as traders who pay for the information y. Similarly, that yi is redundant; Po(y) contains all the information he requires. That is informationally efficient price systems aggregate diverse information perfectly, but in doing this the price system eliminates the private incentive for collecting the lt is demonstrated in the context of a simple mean-variance model.The result that the price system perfectly aggregates information is not robust. This is shown in the context of the above model when "noise""is added One example of"noise "is an uncertain total stock of the risky asset. However, the paradoxical nature of"perfect markets, "which the model illustrates, is robust. When a price system is a perfect aggregator of information it removes private incentives to collect information. If information is costly, there must be noise in the price system so that traders can earn a return on information gathering. If there is no noise and information collection is costly, then a perfect competitive market will break down because no equilibrium exists where information collectors earn a return on their information, and no equilibrium exists where no one collects information. The latter part follows from the fact that if no one collects informa tion then there is an incentive for a given individual to collect costly information because he does not affect the equilibrium price. When many individuals attempt to earn a return on information collection, the equilibrium price is affected and it perfectly aggregates their information. This provides an incentive for individuals to stop collecting information. In Grossman [1975] there is a more detailed analysis of the breakdown of markets when price systems reveal too much information On the other hand, when there is noise so that the price system does not aggregate information perfectly, the allocative efficiency properties of a competi tive equilibrium may break down. Hayek [1945] argues that the essence of a competitive price system is that when a commodity becomes scarce its price rises and this induces people to consume less of the commodity and to invest more in the production of the commodity. Individuals need not know why the price ha risen, the fact that there is a higher price induces them to counteract the scarcity in n efficient way. This argument breaks down when the price system is noisy. We will show that in such cases each individual wants to know why the price has risen (i.e, what exogenous factors make the price unusually high), and that an optimal allocation of resources involves knowing why the price has risen (i. e, knowledge of e states of nature determining current prices is required) 2. THE MODEL Assume that trader"i"has an initial wealth Wo Using Woi, he can purchase two assets; a risk free asset and a risky asset. His wealth in period l, wI is given by W1=(1+r)X+P1X where Xe is the value of risk free assets purchased in period 0, X, is the number of
Efficiency of Competitive Stock Markets Where Trades have Diverse Information575 nits of risky assets purchased in period 0, r>0 is the exogenous rate of return on the risk free asset, and P, is the(unknown) exogenous payoff per unit on the risky asset(also called the period I price of the risky asset). The budget constraint is Wor= Xa+ Pox here Po is the current price of the risky asset. Substituting(2)into(I)to eliminate Xo yields H1=(1+n)Ha+[F1-(1+)Pl]x At time zero, P, is unknown. The ith trader observes y, where P1+ and P, is a realization of the random variable PI. Thus, a fixed, but unknown, realization of P, mixes with noise, e, to produce the observed yr. Later, we shall argue that traders also get information from Po. For the present, let 1, denote the information available to the ith trader. assume that the ith trader has a utility function where a is the coefficient of absolute risk aversion Each trader is assumed to maximize the expected value of U, (W,)conditional on I;. If WI is normally distributed conditional on then where Var[ WIilL,] is the conditional variance of wu given I. It follows that to maximize E[U(W,) 1] is equivalent to maximizing E[Wn4-za[n1小 since the expression in (7) in a monotone increasing transformation of the expres sion in (6). All we have shown is that mean-variance analysis in the Normal case can be derived from the utility function in(5) E[m小]=(1+)+{E[1-(1+)Px
576 he Journal of finance var[W1小]=x2ar[1小 In deriving( 8)and(9)we have used the fact that Woi, r, and Po are known to the firm in period 0. Thus, from(7)9), the consumer's problem is to maximize (1+r)Wo+E[P1]-(1+r)Po)x-2X Var[P1,] by choosing X. Using the calculus, an optimal Xi, Xi, satisfies E[P]-(1+ Thus, the demand for the risky asset depends on its expected price appreciation and on its variance. Let X be the total stock of the risky asset. An equilibrium price in period 0 must cause >i,Xd=X. From(11), the ith trader's demand for the risky asset depends on the information he receives. This depends on the observa- tion he gets, y:. Thus, since the total demand for the risky asset depends on yi2,,,,,,n, it is natural to think of the market clearing price as depending on the yi,i=1, 2,..., n. Let y=(,y2,.,yn), then the equilibrium price is some function of y, Po(y). That is, different information about the return on an asset leads to a different equilibrium price of There are many different functions of y. For a particular function, Po(y) to be equilibrium we require that: for all y, /E[m-(1+n)0=x a, var[ Pily, Po(] (12)states that the total demand for the risky asset must equal the total supply for each y.( Throughout we put no non-negativity constraint on prices. By proper choice of parameters the probability of a negative price can be made arbitrarily small. The ith trader's demand function under the price system Po()is X[P:, E{P1VP(y)]-(1+) a, var[ P1 v, Po(y)] The ith trader's information I; is y, and Po(). He is able to observe his own sample yi and Po(). Po()gives the ith trader some information about the sample
Efficiency of Competitive Stock Markets Where Trades have Diuerse Information 577 of other traders. The next section shows that Po()reveals"all"the information of the traders Po(y) can be interpreted as a stationary point of the following process. Suppose traders initially begin in a naive way, thinking of Po as a number and conditioning nly Let an auctioneer call out prices until the market clears. Call thi olution Po(). That is Po() solves 受-0+)-x a,Var[Ply] (13a) Each period traders come to the market with another realization of y, and another Po()is found where the auction stops. After many repetitions traders can tabulate the empirical distribution of (Po, P1 pairs. From this they get a good estimate of the joint distribution of Po and Pr. After this joint distribution is learned, traders will have an incentive to change their bids just as the market is about to clear. Thi allows from the fact that if everyone observes that the market is about to clear at PoC), they can condition their beliefs on Po() and learn something more about PI. This changes their demands and thus the market will not clear at Po(). Suppose instead that the market has been clearing for a long time with prices traders come to the market with some y, if the market is about to clear at Po(), and traders then realize that Po() is the equilibrium, they will not change their bids due to the new information they get about P from Po(y). Po() is a self fulfilling expectations equilibrium: when all traders think prices are generated by Po(), they will act in such a way that the market clears at Po() 3. THERE IS AN EQUILIBRIUM PRICE WHICH IS A SUFFICIENT STA Assume that in (4), e is a random variable which is normally distributed, with mean0 and variance 1. Thus, each trader"i"observes y,=P+ej, and given PI,yi is Normal with mean P, and variance 1. Each trader gets information of equal precision in that Vare=l for each trader"i. Further assume that ep,E2,,., E, is jointly normally distributed and covariance(e, 6 )=0 if i+j. Thus, we assume that the joint density of y given P, say f( P1), is multivariate Normal with mean vector(P,PI, PI,,, P) and covariance matrix which is the identity matrix. PI is assumed unknown at time zero, however, traders believe that P is distributed dependently of E1, E2,..., n, and PI is Normal (P1,0). This marginal distribution of PI has two interpretations. Under a Bayesian interpretation, next periods price is some fixed number, and people represent their uncertainty about the value of that number with a prior distribution which is Normal(P1,0). A non-Bayesian interpretation is that nature draws the true price next period from an urn with distribution Normal(P, 02). Nature makes the drawing before period 0. After a
578 The Journal of Finance particular P, is drawn, traders do their research and the ith type trader is able learn the true value of P, to within Gj, where E is distributed as Normal(0, 1) Under either interpretation, the following is true TE HEOREM Under the above assumption about the joint distribution of y and P, if Po() is given by y (15) (1+m2)(l (1+no2)(1+r) (17) hen Po()is an equilibrium. That is, it is a solution to(12) Before proving the theorem we present some comments on its significance. First y is the sample mean of the y;. The equilibrium price depends on the information y only through y. Second, any trader by observing the value of Po() can learn y from(14), since by(17), a,>0. y is a more precise estimate of PI, than is y. Thus the market price aggregates all the information collected by the traders in an optimal"way. y is a sufficient statistic for the family of densities f( P).The market aggregation is optimal to the extent that it produces a sufficient statistic. The following lemma is used to prove the theorem: there are functions g ( )and g2( )such that, for all y, and y=ZiiMie n P, then LEMMA l. If h, (i, y P) is the joint density of y and yi conditional h(y2列P1)=g1(y)g2(元,P1 That is, y is a suficient statistic for h, (,FIP) Proof. Conditional on PI, y is Normal (PI, 1)and y is Normal(Pl,1/n) Conditional on PI, covariance (, D)=1/n. Thus conditional on PI, ( D)is normall 11/ P1/(1/n1/
Eficiency of Competitive Stock Markets Where Trades have Diverse Information 579 h(yy|P1)=(2丌) /n1/n PI 1/n)|y-P1 -1八(1n1n)(p (27) [=[-P)(-P(y-P (P1-y)p-P)+n(-P1)] 2n-1 g2(y, Pi=exp( 2n [2P1-P2+n(-P)2 (19) Then h(y,列|P1)=g1(y,)g2(元,P1.QED We use Lemma I to prove Lemma 2 below. Lemma 2 states that if a trader is given y, then yi provides no additional information about P, over that provided LEMMA 2. Let m(PI ly) be the density of P, conditional on y. Let m(Pi,yi)be the density of PI conditional on y and y, Then m(P,lD)=m(Pily, y)and hence E[P1列=E[P1 P B (P)=一8(P)(FP g(Pih, o P) where g(Pi) is the marginal density of P 病(P1)=8( Pig yi,y)g(PD g(P1)g2(,P1) g(P)g1(y,y)g2(,P1) g(P1)(, P)dP
580 he Journal of finance The density of y given PI, f( P)), satisfies f(列P)广M(列P减-8(%)(P g(,P1)g;(y列) The second equality in(22)follows from(18). By Bayes rule mP1)=。8(PD(yP) g(P1)g2(,P1)g1(y) g(PDf( Pdp g(P)32(,P)∫g1(y,)d where the second equality follows from(22),/oo g (ya, y)dy can be cancelled from the numerator and denominator in(23), hence m(P1)=-8(PB,B g(Pg2(, pidP Comparing(24)and (21), we see that m(PID)=m(Pily, yi). QED An immediate consequence of Lemma 2 is that if a trader is given y and ye, then inferences about P will be made independently of y. That is y, is extraneous information if y is known We now prove the main theorem. The proof uses the fact that if ao and a1>0 are known constants, then the conditional distribution of P, given ao +av, is the same as the conditional distribution of Pi given y Proof. We show that if Po()=ao+a,y, where ao and a, are given in(16)and (17), then Po()satisfies(12)for all y From Lemma 2, E[P,D F]=E[PDy] and Var[P,l, ]=Var[P,I]. Under the distribution assumpti ning of this section it can be shown that the conditional distribution of P, given y is normal with moments given by E[F|=万 5 十no [F1 I+no
Efficiency of Competitive Stock Markets Where Trades have Diverse Information 581 (see Degroot 1970], p. 167). That is, the posterior mean of P, is a weighted average of the prior mean PI and the sample mean y. Note that E[Pvyi, ao+aiy] =ElP,l]=E[PI D] if aI>0. Hence E[PIly, P0()]=I+na Similarly, var[ Ply, Po()]=Var[Ply] Using(13),(14,(26),and(27) (1+ny) X3=2〈+m (1+)(ao+a1 Using the definitions of ao and a, given in(16)and (17),(28)becomes ∑ⅹ[P:,y yI (F+no分) (1+m2) (1+r) (1+m3)+)21(1+m0301+) (29) The right hand side of (29)reduces to X. Thus for all y d[ Po, y]=X. QED Thus, in equilibrium the current price summarizes all the information in the market. Each trader finds his own yi, redundant. This creates strong disincentives for investment in information, since each trader could do as well by observing only
