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