Journal of Financial Economics 9(1981)221-235.North-Holland Publishing Company INFORMATION AGGREGATION IN A NOISY RATIONAL EXPECTATIONS ECONOMY Douglas W.DIAMOND and Robert E.VERRECCHIA* University of Chicago,Chicago,IL 60637,USA Received July 1980,final version received December 1980 This paper analyzes a general equilibrium model of a competitive security market in which traders possess independent pieces of information about the return of a risky asset.Each trader conditions his estimate of the return both on his own private source of information and price, which in equilibrium serves as a 'noisy'aggregator of the total information obscrved by all traders.A closed-form characterization of the rational expectations equilibrium is presented.A counter-example to the existence of 'fully revealing'equilibrium is developed. 1.Introduction This paper analyzes a model of a competitive security market in which the partial aggregation of diverse sources of information results from a rational expectations equilibrium.The economy analyzed yields a unique,closed-form equilibrium in which participants are able to obtain supplementary information from market prices without rendering their own information redundant.The analysis presented here is of interest for at least two reasons. First,it provides a reasonable characterization of the economic concept of an informationally efficient market.Secondly,it introduces a definition of equilibrium which restricts prices to depend on traders'information only through their demand correspondences. The idea that equilibrium prices in competitive security markets largely reflect the information possessed by various traders is both widely accepted and often analyzed by financial economists.Empirically,evidence presented in Fama (1970)and elsewhere shows that predictions conditioned on market prices are not dominated by predictions using many other sources of information.Several theoretical papers have also addressed this issue. Lintner (1969)analyzes an economy in which beliefs are exogenous.This leads to a characterization of equilibrium prices as the weighted average of these beliefs [see also Rubinstein (1975)and Verrecchia (1980)].Green *The authors are grateful to Philip Dybvig,Gur Huherman,Jon Ingersoll,Michael Jensen, Richard Leftwich,Tom Turnbull and the referee,Alan Kraus,for helpful comments on an earlier draft. 0304-405X/81/0000-0000/S02.50©North-Holland
Journal of Financial Economics 9 (1981) 221-235. North-Holland Publishing Company INFORMATION AGGREGATION IN A NOISY RATIONAL EXPECTATIONS ECONOMY Chicago, Chicago, IL 60637, USA Received July 1980, final version received December 1980 This paper analyzes a general equilibrium model of a competitive security market in whtch traders possess independent pieces of information about the return of a risky asset. Each trader conditions his estimate of the return both on his own private source of information and price, which in equilibrium serves as a ‘noisy’ aggregator of the total information observed by all traders. A closed-form characterizatton of the rational expectations equilibrium is presented. A counter-example to the existence of ‘fully revealing’ equilibrium is developed. 1. Introduction This paper analyzes a model of a competitive security market in which the partial aggregation of diverse sources of information results from a rational expectations equilibrium. The economy analyzed yields a unique, closed-form equilibrium in which participants are able to obtain supplementary information from market prices without rendering their own information redundant. The analysis presented here is of interest for at least two reasons. First, it provides a reasonable characterization of the economic concept of an informationally efficient market. Secondly, it introduces a definition of equilibrium which restricts prices to depend on traders’ information only through their demand correspondences. The idea that equilibrium prices in competitive security markets largely reflect the information possessed by various traders is both widely accepted and often analyzed by financial economists. Empirically, evidence presented in Fama (1970) and elsewhere shows that predictions conditioned on market prices are not dominated by predictions using many other sources of information. Several theoretical papers have also addressed this issue. Lintner (1969) analyzes an economy in which beliefs are exogenous. This leads to a characterization of equilibrium prices as the weighted average of these beliefs [see also Rubinstein (1975) and Verrecchia (1980)]. Green *The authors are grateful to Philip Dybvig, Cur Huberman, Jon Ingersoll, Michael Jensen, Richard Lcftwich, Tom Turnbull and the referee, Alan Kraus, for helpful comments on an earlier draft. 0304405X/81/000&0000/$02.50 0 North-Holland
222 D.W.Diamond and R.E.Verrecchia,Information aggregation and rational expectations (1973),Grossman (1976,1978),Grossman-Stiglitz (1980)and others have used the concept of rational expectations equilibrium,which makes beliefs endogenous,to study the utilization of prices themseves as sources of information. The analysis of the information content of prices in economies with exogenous beliefs [e.g.,Lintner (1969)]is subject to the objection that when prices do contain information that a particular trader does not possess he ought to make use of it;this,however,is not generally consistent with assuming that beliefs are exogenous.Grossman (1976,1978)analyzes an economy in which traders have diverse pieces of information about the rcturn of risky asscts,and he claims that the rational cxpectations equilibrium price reveals to all traders all of the information of the traders taken together;that is,it reveals a sufficient statistic of that information.A major implication of this result is that when traders take prices as given,they have no economic incentive to acquire information.One problem with fully revealing equilibrium is that it does not exist under the definition presented below,which may be a more plausible definition of equilibrium than that of Grossman. Analyses assuming the exogeneity of people's beliefs produce results which are not consistent with the notion that expectations are formed rationally, and analyses suggesting the full revelation of aggregate information produce results which are too strong on empirical grounds.Therefore,an alternative is to study markets in which an equilibrium results in some aggregation of individuals'information without revealing all of it.This type of partial aggregation environment implies the following.When,in addition to his privately collected information,an individual trader uses observable endogenous variables,such as prices,as pieces of information,he benefits from the information collected by others without bclicving that his own is superfluous.This is consistent with a private incentive to collect information, and an equilibrium where information affects price through supply and demand. The only existing studies of partially revealing prices do not analyze the role of prices as aggregators of information.The models of Green (1973) and Grossman-Stiglitz (1980)analyze economies where there is only a single piece of information which anyone can observe,such as a private weather forecast.In their analyses there are two factors which may be unknown to market participants,and which affect the determination of prices.If a single piece of information,e.g.,the weather forecast,is the only factor which is unknown,and each weather forecast implies a different equilibrium price, then the market price can fully reveal the content of the forecast.The 'After completion of this manuscript,it was brought to the authors'attenton that Hellwig (1980)analyzes a somewhat different model of the aggregation of infirmation
222 D.W Diamond and R.E. Verrecchia, Information aggregation and rational expectations (1973), Grossman (1976, 1978), Grossman-Stiglitz (1980) and others have used the concept of rational expectations equilibrium, which makes beliefs endogenous, to study the utilization of prices themseves as sources of information. The analysis of the information content of prices in economies with exogenous beliefs [e.g., Lintner (1969)] is subject to the objection that when prices do contain information that a particular trader does not possess he ought to make use of it; this, however, is not generally consistent with assuming that beliefs are exogenous. Grossman (1976, 1978) analyzes an economy in which traders have diverse pieces of information about the return of risky assets, and he claims that the rational expectations equilibrium price reveals to all traders all of the information of the traders taken together; that is, it reveals a sufficient statistic of that information. A major implication of this result is that when traders take prices as given, they have no economic incentive to acquire information. One problem with fully revealing equilibrium is that it does not exist under the definition presented below, which may be a more plausible definition of equilibrium than that of Grossman. Analyses assuming the exogeneity of people’s beliefs produce results which are not consistent with the notion that expectations are formed rationally, and analyses suggesting the full revelation of aggregate information produce results which are too strong on empirical grounds. Therefore, an alternative is to study markets in which an equilibrium results in some aggregation of individuals’ information without revealing all of it. This type of partial aggregation environment implies the following. When, in addition to his privately collected information, an individual trader uses observable endogenous variables, such as prices, as pieces of information, he benefits from the information collected by others without believing that his own is superfluous. This is consistent with a private incentive to collect information, and an equilibrium where information affects price through supply and demand. The only existing studies of partially revealing prices do not analyze the role of prices as aggregators of information.’ The models of Green (1973) and Grossman-Stiglitz (1980) analyze economies where there is only a single piece of information which anyone can observe, such as a private weather forecast. In their analyses there are two factors which may be unknown to market participants, and which affect the determination of prices. If a single piece of information, e.g., the weather forecast, is the only factor which is unknown, and each weather forecast implies a different equilibrium price, then the market price can fully reveal the content of the forecast. The ‘After completion of this manuscript, It was brought to the authors’ attention that Hellwig (1980) analyzes a somewhat different model of the aggregation of infkmation
D.W.Diamond and R.E.Verrecchia,Information aggregation and rational expectattons 223 introduction of unobserved variation of another factor (generally referred to as 'noise'and represented by aggregate supply)prevents identifying the aggregate demand curve by simply observing the equilibrium price.The result is an equilibrium in which the price contains information but does not fully reveal the forecast.The fraction of the traders who directly observe the forecast(as contrasted with the fraction which infers something about it from prices)determines how fully the forecast will be revealed.This analysis leads to a theory of the private incentives for acquiring costly information in which an equilibrium fraction of traders expends resources to predict the weather. However,for the expenditure of his resources,each informed trader receives an identical prediction duplicating the effort of many others.With only a single piece of information considered,there is no diversity of information for the market to aggregate. The model presented here concerns the partial aggregation of many diverse sources of information.Such a model has not yet been analyzed previously because the analytical methods used to determine fully revealing equilibrium with diverse information [e.g.,Grossman (1976,1978)]proceed by demonstrating that the equilibrium price is a one-to-one function of the aggregate information.The presence of noise in rational expectations equilibrium requires explicit analysis of the statistical decision problem in which conjectures made by competitive traders about the amount of information revealed by prices are self-fulfilling.By making some strong assumptions about the preferences of traders and imposing a simple information structure with one risky asset [assumptions similar to those made in Grossman (1976)]this analysis characterizes a noisy,partially aggregating equilibrium.The results provide a description of how information is impounded into prices when prices are partially revealing. Fully revealing equilibria,and equilibria which are identical to exogenous beliefs equilibria are interpreted as limiting results.The endogenous production of costly information by traders is not analyzed,but the equilibrium which is presented is consistent with such production.This is because with partial aggregation competitive traders perceive that their private information is valuable. 2.An equilibrium model of exchange with diverse information A simple exchange model of a two-asset economy is used to analyze incomplete aggregation of traders'information by competitive markets.The model assumes that all traders have identical preferences and identical prior beliefs,but that each trader costlessly observes information about the return of one of the assets.Each trader is endowed with information of the same precision,but the information is diverse.Conditional upon the true realized
D.W Diamond and R.E. Verrecchia, Information aggregation and rational expectations 223 introduction of unobserved variation of another factor (generally referred to as ‘noise’ and represented by aggregate supply) prevents identifying the aggregate demand curve by simply observing the equilibrium price. The result is an equilibrium in which the price contains information but does not fully reveal the forecast. The fraction of the traders who directly observe the forecast (as contrasted with the fraction which infers something about it from prices) determines how fully the forecast will be revealed. This analysis leads to a theory of the private incentives for acquiring costly information in which an equilibrium fraction of traders expends resources to predict the weather. However, for the expenditure of his resources, each informed trader receives an identical prediction duplicating the effort of many others. With only a single piece of information considered, there is no diversity of information for the market to aggregate. The model presented here concerns the partial aggregation of many diverse sources of information. Such a model has not yet been analyzed previously because the analytical methods used to determine fully revealing equilibrium with diverse information [e.g., Grossman (1976,1978)] proceed by demonstrating that the equilibrium price is a one-to-one function of the aggregate information. The presence of noise in rational expectations equilibrium requires explicit analysis of the statistical decision problem in which conjectures made by competitive traders about the amount of information revealed by prices are self-fulfilling. By making some strong assumptions about the preferences of traders and imposing a simple information structure with one risky asset [assumptions similar to those made in Grossman (1976)] this analysis characterizes a noisy, partially aggregating equilibrium. The results provide a description of how information is impounded into prices when prices are partially revealing. Fully revealing equilibria, and equilibria which are identical to exogenous beliefs equilibria are interpreted as limiting results. The endogenous production of costly information by traders is not analyzed, but the equilibrium which is presented is consistent with such production. This is because with partial aggregation competitive traders perceive that their private information is valuable. 2. An equilibrium model of exchange with diverse information A simple exchange model of a two-asset economy is used to analyze incomplete aggregation of traders’ information by competitive markets. The model assumes that all traders have identical preferences and identical prior beliefs, but that each trader costlessly observes information about the return of one of the assets. Each trader is endowed with information of the same precision, but the information is diverse. Conditional upon the true realized
224 D.W.Diamond and R.E.Verrecchia,Information aggregation and rational expectations return per unit of the risky asset,the information of each is stochastically independent of that of all other traders.The objective is to understand the relationship between the exogenous parameters and the total amount of information which traders obtain in market equilibrium. The economy has two assets,a riskless bond and a risky asset,both of which pay off in the single consumption good in the economy.There are T consumers in the economy,called traders.There are two time periods.In the first period traders are endowed with assets and trade them in a competitive market,but do not consume.In the second period there is no trading,and each trader consumes the return realized from his portfolio. Trader t is endowed with B,riskless bonds.Endowments of risky asscts are stochastic with each trader knowing only his own endowment.These are discussed later. The numeraire in the economy is the price of a bond which returns R at the end of the period.We normalize R=1,as time preference is not essential to this analysis.All traders know the return of this asset. In the initial period the return of the risky asset is not known to traders:it is a random variable denoted by i.Let u denote the realized return per unit of the risky asset.Each trader has identical prior beliefs about specifically, each believes that u has a Normal distribution with mean yo and precision ho (where precision is defined to be the inverse of variance).To analyze the ability of markets to aggregate information,it is assumed that each trader is costlessly endowed with information about what the realization of will be in the second period.The information observed by trader t is the observation of a random variable.Trader t makes this observation before the market opens.The random variables i and y.are jointly distributed with a Bivariate Normal distribution with mean (yo,yo)and covariance matrix, ho ho ho h1+s-1 Each trader observes an independent realization,=y,.The random variable y.can be thought of as the sample mean of s independent observations from a Normal distribution whose mean and variance,conditional on =u,is u and 1,respectively.Let$be the vector=(1....,r).Each trader uses the observation y=y:,in conjunction with his prior beliefs and the information implicit in prices,to form posterior judgments about 4. In addition to the information received by traders,there is another stochastic factor influencing the security market.This factor introduces noise into the relationship between traders'demands and equilibrium prices,and results in prices revealing only part of the aggregate information.The noise is represented by uncertainty about the aggregate supply of risky assets
224 D.W Diamond and R.E. Verrecchia, Information aggregation and rational expectations return per unit of the risky asset, the information of each is stochastically independent of that of all other traders. The objective is to understand the relationship between the exogenous parameters and the total amount of information which traders obtain in market equilibrium. The economy has two assets, a riskless bond and a risky asset, both of which pay off in the single consumption good in the economy. There are T consumers in the economy, called traders. There are two time periods. In the first period traders are endowed with assets and trade them in a competitive market, but do not consume. In the second period there is no trading, and each trader consumes the return realized from his portfolio. Trader t is endowed with B, riskless bonds. Endowments of risky assets are stochastic with each trader knowing only his own endowment. These are discussed later. The numeraire in the economy is the price of a bond which returns R at the end of the period. We normalize R = 1, as time preference is not essential to this analysis. All traders know the return of this asset. In the initial period the return of the risky asset is not known to traders: it is a random variable denoted by 6. Let u denote the realized return per unit of the risky asset. Each trader has identical prior beliefs about u”; specifically, each believes that C has a Normal distribution with mean y, and precision h, (where precision is defined to be the inverse of variance). To analyze the ability of markets to aggregate information, it is assumed that each trader is costlessly endowed with information about what the realization of u’ will be in the second period. The information observed by trader t is the observation of a random variable jj,. Trader t makes this observation before the market opens. The random variables 6 and jjt are jointly distributed with a Bivariate Normal distribution with mean (y,,, yO) and covariance matrix, h,’ h,’ h, l 1 h,'+s-' Each trader observes an independent realization jj, = y,. The random variable jj, can be thought of as the sample mean of s independent observations from a Normal distribution whose mean and variance, conditional on C=U, is u and 1, respectively. Let J be the vector jj = (PI,. ., Each uses observation in with prior and information in to posterior about In to information by there another factor the market. factor noise the between demands equilibrium and in revealing part the information. noise represented uncertainty the supply risky
D.W.Diamond and R.E.Verrecchia,Information aggregation and rational expectations 225 Each trader's endowment of risky assets represents an independent draw from a Normal population.Let =x,denote the realization of a random variable;x,is the initial endowment of the risky asset to trader t.Let x be the vector(1,...,).The are mutually independent Normal random variables with a mean of zero and a variance v and are independent of ii. (The results are not influenced by the mean of zero.)Aggregate supply of risky assets is the random variable Every trader knows his own cndowment,but not the endowment of others.Because the arc independent,and T is finite,a trader's own endowment provides some information about the aggregate endowmentX.Analysis is presented for >0,which introduces noise into the relationship between equilibrium prices and traders'information.The random variables i,X,y and are all real- valued,and are all defined on the same probability space. Every trader has a negative exponential utility function for wealth w;that is,for each trader t, U,(w)=-exp(-w), where U,(w)represents his utility for wealth w. The use of exponential utility is important to the analysis in this paper because of the well-known property that it implies demand correspondences for the risky asset which depend upon a trader's beliefs,but not directly on his wealth.Although it is straightforward to construct parametric examples of partially aggregating equilibrium for other preferences,a general method for proving that such cquilibria exist is not currently known.Without assuming exponential utility,the equilibrium distribution of prices would be difficult to characterize,and it is highly unlikely that a closed form expression could be obtained.The assumption that the risk tolerance of each trader is identical can be relaxed because demand correspondences would continue to have their simple form.This would produce a result similar to Lintner (1969)or Verrecchia (1980)where a trader's risk tolerance affects the weighting of his posterior belief in the price,but the added complication would add little to the economic understanding obtained from the model. Rational expectations equilibrium Traders in the economy formulate their beliefs conditional on the information observable to them.Because of the diverse information in the economy,traders will find that the price of risky assets will be a useful piece of information about i.This is because the private information of each trader will influence that trader's demand,and prices reflect supply and demand.Prices are endogenous variables.Because prices are used as a source of information,beliefs become endogenous.This use of price,in turn
D.W Diamond and R.E. Verrecchia, Information aggregation and rational expectations 225 Each trader’s endowment of risky assets represents an independent draw from a Normal population. Let 2, =x, denote the realization of a random variable 2,; x, is the initial endowment of the risky asset to trader t. Let x’ be the vector 2 = (%r , . . ., 5&). The z?~ are mutually independent Normal random variables with a mean of zero and a variance K and are independent of C. (The results are not influenced by the mean of zero.) Aggregate supply of risky assets is the random variable x =c,T= 1 2,. Every trader knows his own endowment, but not the endowment of others. Because the z?~ are independent, and T is finite, a trader’s own endowment provides some information about the aggregate endowment 8. Analysis is presented for V > 0, which introduces noise into the relationship between equilibrium prices and traders’ information. The random variables ti, _%?, j and 2 are all realvalued, and are all defined on the same probability space. Every trader has a negative exponential utility function for wealth w; that is, for each trader t, u,(w)= -exp( -w), where U,(w) represents his utility for wealth w. The use of exponential utility is important to the analysis in this paper because of the well-known property that it implies demand correspondences for the risky asset which depend upon a trader’s beliefs, but not directly on his wealth. Although it is straightforward to construct parametric examples of partially aggregating equilibrium for other preferences, a general method for proving that such equilibria exist is not currently known. Without assuming exponential utility, the equilibrium distribution of prices would be difficult to characterize, and it is highly unlikely that a closed form expression could be obtained. The assumption that the risk tolerance of each trader is identical can be relaxed because demand correspondences would continue to have their simple form. This would produce a result similar to Lintner (1969) or Verrecchia (1980) where a trader’s risk tolerance affects the weighting of his posterior belief in the price, but the added complication would add little to the economic understanding obtained from the model. Rational expectations equilibrium Traders in the economy formulate their beliefs conditional on the information observable to them. Because of the diverse information in the economy, traders will find that the price of risky assets will be a useful piece of information about 12. This is because the private information jYt of each trader will influence that trader’s demand, and prices reflect supply and demand. Prices are endogenous variables. Because prices are used as a source of information, beliefs become endogenous. This use of price, in turn
226 D.W.Diamond and R.E.Verrecchia,Information aggregation and rational expectations influences the joint distribution of price and Prices are also directly influenced by aggregate supply X and traders can use their own endowments as information about X.A rational expectations equilibrium occurs when the conjecture traders make about the joint distribution of ,and prices is self-fulfilling.This concept of equilibrium captures the idea that individuals make use of statistical relationships between endogenous and exogenous variables;it was developed by Lucas (1972),Green (1973), Grossman(1976),and Kreps(1977). Let D(x,y,P)and B.(x,y,P)represent the real-valued demand functions of trader t for the risky and riskless assets,respectively,given ,=x,月=y,and庐=P. A rational expectations competitive equilibrium is defined here to be circumstance in which markets clear,traders'beliefs about the distribution of all observable random variables are fulfilled,and prices depend on information through supply and demand.Formally,a rational expectations competitive equilibrium price is a random variable p whose realization is P, where and P()is a real-valued function,such that if for t=1,...,T,the realizations of demand, D(x,y,P)and B(x,y,P), maximize equilibrium conditional expected utility,i.e.,they solve max E[U(D.+B)=x=y,P=P], (1) {D·B) subject to the budget constraint PD+B=Px:+B, (2) then for every y and x,markets clear,or 三%x川-三xX 三BkP-三豆 (3)
226 D.W Diamond and R.E. Verrecchia, Information aggregation and rational expectations influences the joint distribution of price and u”. Prices are also directly influenced by aggregate supply _$? and traders can use their own endowments 2, as information about x. A rational expectations equilibrium occurs when the conjecture traders make about the joint distribution of 6, r?, Jt, &, and prices is self-fulfilling. This concept of equilibrium captures the idea that individuals make use of statistical relationships between endogenous and exogenous variables; it was developed by Lucas (1972), Green (1973), Grossman (1976) and Kreps (1977). Let D,(x,,y,,P) and B,(x,,y,,P) represent the real-valued demand functions of trader t for the risky and riskless assets, respectively, given Zt=xt,_iif=yl, and p=P. A rational expectations competitive equilibrium is defined here to be circumstance in which markets clear, traders’ beliefs about the distribution of all observable random variables are fulfilled, and prices depend on information through supply and demand. Formally, a rational expectations competitive equilibrium price is a random variable P” whose realization is P, where P=P D, ,..., D,,B, ,..., Br,X, i Bt , I=1 > and P( . ) is a real-valued function, such that if for t = 1,. . ., T, the realizations of demand, PI and 4(x,, yt, PI, maximize equilibrium conditional expected utility, i.e., they solve maxE[U,(u”.D,+B,)I~,=x,,~~=y,,P=P], W’,, BJ subject to the budget constraint PD,+B,=Px,+B,, then for every y and x, markets clear, or (1) (2) (3)
D.W.Diamond and R.E.Verrecchia,Information aggregation and rational expectations 227 This definition is distinct from that used in Grossman (1978).The definition given there allows P()to depend directly on the vector y.The definition used here requires the dependence of price on traders'information only through their demand functions.Requiring prices to depend on information through demand only is a further restriction.It rules out schemes such as the one discussed by Kreps (1977).in which a Walrasian auctioneer,instead of simply finding a market clearing price,observes y directly,and uses it to set prices.The restriction captures the idea that the competitive market process aggregates information despite the fact that traders observe only their own private information and the price which clears the market.This does not imply that traders are unsophisticated.In equilibrium traders know the true joint distribution of every variable in the model.It only rules out situations which imply that some trader must observe more than what he actually does.On these grounds the definition offered above is economically plausible. 3.Determining an equilibrium In this section an equilibrium price random variable p is characterized which clears the market,and fulfills each trader's conjecture about its behavior.Broadly stated,this is achieved in four steps.Let and o2 represent the mean and variance,respectively,of ii conditional on tradert observing=x,=y,and P=P.First,imagine that traders conjecture some behavior for P;the Bayesian statistical decision problem traders face is analyzed to determine expressions for and o conditional on-y, and P=P.Second,the portfolio decision which traders face is analyzed to determine an expression for the market clearing price of the risky asset,P, conditional on u,and o2.Third,the expressions for and o2 derived in the first step are substituted into the expressions for p derived in the second step;this determines a price random variable p which clears the market. Finally,by equating this expression for P with the initial conjecture about P, and solving for endogenous parameters,an expression for the equilibrium price random variable results. Expressions for and o2 are determined as follows.Suppose that each trader conjectures that the equilibrium price random variable is an expression of the form P=yo+(B/T)∑元-(/T)x, (4) where P-(+B)yo has a Normal distribution with mean zero and precision H,where H is given by H≡{var[P}-1={B2ho1+β2(Ts)1+y2(V/T)}-1. (5)
D.W Diamond and RX. Verrecchia, Information aggregation and rational expectations 221 This definition is distinct from that used in Grossman (1978). The definition given there allows P(. ) to depend directly on the vector jj. The definition used here requires the dependence of price on traders’ information only through their demand functions. Requiring prices to depend on information through demand only is a further restriction. It rules out schemes such as the one discussed by Kreps (1977) in which a Walrasian auctioneer, instead of simply finding a market clearing price, observes jj directly, and uses it to set prices. The restriction captures the idea that the competitive market process aggregates information despite the fact that traders observe only their own private information and the price which clears the market. This does not imply that traders are unsophisticated. In equilibrium traders know the true joint distribution of every variable in the model. It only rules out situations which imply that some trader must observe more than what he actually does. On these grounds the definition offered above is economically plausible. 3. Determining an equilibrium In this section an equilibrium price random variable P is characterized which clears the market, and fulfills each trader’s conjecture about its behavior. Broadly stated, this is achieved in four steps. Let pt and C$ represent the mean and variance, respectively, of u” conditional on trader t observing 2, = xt, pt = yt, and P”=P. First, imagine that traders conjecture some behavior for P; the Bayesian statistical decision problem traders face is analyzed to determine expressions for 1~~ and C$ conditional on 2, =x,, it = y,, and P’=P. Second, the portfolio decision which traders face is analyzed to determine an expression for the market clearing price of the risky asset, P, conditional on pl, and c$. Third, the expressions for pt and rr: derived in the first step are substituted into the expressions for P derived in the second step; this determines a price random variable p which clears the market. Finally, by equating this expression for P with the initial conjecture about 6, and solving for endogenous parameters, an expression for the equilibrium price random variable results. Expressions for pt and c$ are determined as follows. Suppose that each trader conjectures that the equilibrium price random variable is an expression of the form where p-- (a+/?)~, has a Normal distribution with mean zero and precision H, where H is given by
228 D.W.Diamond and R.E.Verrecchia,Information aggregation and rational expectations A price random variable of this form implies that the covariances between p and the random variables i,X,,and are given by c1≡cov[P,闪=ho', c2≡cov[庐，]=-yV, c3≡cov[产，x]=-(y/T)V, c4=COv[P,]=B(ho1+(Ts)1). (6) Thus,the vectorZ=(,,-[+]yo)has a joint five-variate Normal distribution with mean m=(yo,0,0,yo,0)and covariance matrix ho1 00 h6' Ci 0 TV V 0 C2 M= 0 VV 0 C3 00 ho1+s-1 C2 C3 Ca H-1 Partition the vectors Z,m,and the matrix M as follows: m /M11M12 m= M= m M21M22 where Z=(,X).m,=(yo,0)and M,:is a two-by-two matrix,and the sizes of the other matrices and vectors are determined.Let F,=F(a,X|=x,立=y,庐=P) represent trade t's distribution function for the joint random variables i and conditional on observing=x=y:and p=P.LetZ=(x y,P).It is a well-known result [see,e.g.,Mood-Graybill (1963,p.213)]that the conditional distribution of ii and &given Z,is a Bivariate Normal with mean m=m,+M2M22 (Z-m2),and convariance matrix M1= M11-M12M22 M21.In particular,this implies that the mean and variance of i conditional on observing==y,and P=P is given by =yo+(1/K)[(cic3(ho1+s-1)-ho 'c4c3)x +(e1c4+ho'{c3-VH-1})(y.-o) +(ho1e4-V{ho1+s-1}c1)P-{a+}yo门， (7)
228 D.W Diamond and R.E. Verrecchia, Information aggregation and rational expectations A price random variable of this form implies that the covariances between P and the random variables C, 8, &, and A are given by c,~cov[P,izJ=/3h~‘, c,ECOV[B,X]= -yV, c3 =cov [P, Z,] = - (y/T)V, C~~COV[&yJ=/Y{h,‘+(Ts)-‘}. (6) Thus, the vector Z = (t?, x, gt, j$ P - [a + /I] yo) has a joint five-variate Normal distribution with mean m = (y,,, O,O, y,, 0) and covariance matrix ;i && J. Partition the vectors Z, m, and the matrix M as follows: where Z: = (G,x), m1 = (y,,O) and Ml1 is a two-by-two matrix, and the sizes of the other matrices and vectors are determined. Let represent trade t’s distribution function for the joint random variables u” and w conditional on observing 2, = xt, j$ = y,, and P = P. Let Z,* = (xt, y,, P). It is a well-known result [see, e.g., Mood-Graybill (1963, p. 213)] that the conditional distribution of u’ and 1, given Zy, is a Bivariate Normal with mean rn: = m, + M,,M;,’ (Zz -m,), and convariance matrix Mf, = MI, -M,,M;,‘Mz,. In particular, this implies that the mean and variance of C conditional on observing zt = x,, yt = y,, and P = P is given by ~,=y,+(1/K)[(c,c,{h,‘+s-‘}-h,‘c,c,)x, + (Vc,c,+h,‘{c:- VK’})(y,-yo) +(h~‘~lc,-I/{h~‘+s~‘}cl)(P-{a+~)yo)], (7)
D.W.Diamond and R.E.Verrecchia,Information aggregation and rational expectations 229 o=h61+(1K)[V(h1+s-1)c-2ho'c1c4V +ho2VH-1-ho 2c3], (8) where K=e经+c3(ho1+s-1)-VH-1(ho1+s-1) Note that traders'assessments of the mean of a are heterogeneous since each is determined in part by the private observation of=x and =y. However,precision is identical across traders since each trader has the same prior belief about is endowed with private information of the same precision,and conditions his belief on a commonly observed source of public information,price. Recall that conditional on i=u,trader t has final period wealth uD.+B.. From (2).B.=B,+P.{x,-D,}and thus his utility for his final period wealth U({u-P}·D,+B,+P·x)=-exp({P-u}·D.-B,-P·x) Using the conjectured conditional distribution function for and X,each trader's conditional expected utility is given by the expression -exp(P-~D-B-Px)dFaX=x,成=庐=P) Integrating the above expression over and X,and ignoring terms that are irrelevant because they have no effect on a choice of D.,yields -exp(PD-uD+D2:), (9) where and o?represent the mean and variance,respectively,of conditional on=x =y,and P=P.(The expressions for u and which result in equilibrium are derived below.)To maximize eq.(9), differentiate that expression with respect to D.and set the resulting expression equal to zero.This yields as a maximum D,=(4-P)/a, (10) where it can be shown that the second-order condition for a maximum is clearly satisfied.This expression for the demand for the risky asset is the difference between the mean of trader t's posterior distribution of conditional on observing=x=y,and p=P,and the price of the risky asset,weighted by the precision of his posterior distribution (i.e.,the inverse
D.W Diamond and R.E. Verrecchia, Information aggregation and rational expectations 229 a:=h,‘+(l/K)[l/(h,‘+s-‘)c:-2h,‘c,c,I/ where +h,WP--h;%~], (8) K=Vc:+c:(h,‘+s-‘)-~H-‘(h,‘+s-‘). Note that traders’ assessments of the mean of fi are heterogeneous since each is determined in part by the private observation of 3, =x, and ~,=JJ,. However, precision is identical across traders since each trader has the same prior belief about ti, is endowed with private information of the same precision, and conditions his belief on a commonly observed source of public information, price. Recall that conditional on ii=u, trader t has final period wealth u. D, +B,. From (2), B, = & + P. {x, - Dt> and thus his utility for his final period wealth is U({u-P} .D,+B,+P.x,)= -exp({P-u} .D,-B,--P.x,). Using the conjectured conditional distribution function for ti and 8, each trader’s conditional expected utility is given by the expression ~r-exp({P-~i.D,-8,-P.x,)d~(~,R(~,=x,,~,=y,,P=P). Integrating the above expression over t? and 8, and ignoring terms that are irrelevant because they have no effect on a -exp(PD,-pLrD,+fD:c$), where pr and 0: represent the mean choice of D,, yields (9) and variance, respectively, of zi conditional on z?~ =‘x$, ~7~ = y,, and P=P. (The expressions for u and 0: which result in equilibrium are derived below.) To maximize eq. (9), differentiate that expression with respect to D, and set the resulting expression equal to zero. This yields as a maximum D,=(P,-WJ,~, (10) where it can be shown that the second-order condition for a maximum is clearly satisfied. This expression for the demand for the risky asset is the difference between the mean of trader t’s posterior distribution of iZ, conditional on observing ZC = xt, jjC = y, and P” =P, and the price of the risky asset, weighted by the precision of his posterior distribution (i.e., the inverse
230 D.W.Diamond and R.E.Verrecchia,Information aggregation and rational expectations of the posterior variance).Because the equilibrium precision is identical across traders,let it be represented by h,where h=(o2)1.For a price to constitute an equilibrium,it must clear the market through eq.(3)and fulfill conjectures made about its behavior.Using the expression derived in eq.(10) to represent demand,eq.(3)requires x-ED-h(Eh-TP) or (11) P=(1/T)∑4-X/Th. A price random variable p which clears the market is found by substituting the expressions for u and h=(2)derived in eqs.(7)and (8), respectively,into eq.(11).By substituting the expressions for H,c,c2,c3 and ca derived in eqs.(5)and(6),respectively,P can be reduced to an expression whose only endogenous parameters are a,B and y.Finally,by equating this expression for p with the initial conjecture about p expressed in (4),ex- pressions for a,B and y can be derived in terms of exogenously specified parameters.This yields an equilibrium price random variable p which fulfills conjectures about its behavior. In particular,it can be shown that a (unique)closed-form expression for P which fulfills the conjecture expressed in eq.(4)is given by 户-{h(V+s) {nr可w+{sra可un2a Ts+V 一s(Ts+ho)+vh。+s) 1/T)x. (12) Furthermore,endogenously determined relationships for H,ci,c2,c3,and c4 that are implied by this equilibrium are given by H=tvar[}-'-Th。[五+ho]+Tho+s (Ts+V)2(s[Ts+ho]+hoV) s(Ts+V) C:=COv[P,ho ([Ts+ho]+VTho +5])' V(Ts+V) c2=cov[值，X=s(T5+ho)+Vho+)
230 D.W Diamond and R.E. Verrecchia, Information aggregation and rational expectations of the posterior variance). Because the equilibrium precision is identical across traders, let it be represented by h, where h 3 ($)-I. For a price to constitute an equilibrium, it must clear the market through eq. (3) and fulfill conjectures made about its behavior. Using the expression derived in eq. (10) to represent demand, eq. (3) requires or P=(l/T&,-X/Th. A price random variable p which clears the market is found by substituting the expressions for pLI and h= (a:)-’ derived in eqs. (7) and (8), respectively, into eq. (11). By substituting the expressions for H, ci, cl, c3 and cq derived in eqs. (5) and (6), respectively, p can be reduced to an expression whose only endogenous parameters are TV, /I and y. Finally, by equating this expression for p with the initial conjecture about P” expressed in (4), expressions for u, /I and y can be derived in terms of exogenously specified parameters. This yields an equilibrium price random variable p which fulfills conjectures about its behavior. In particular, it can be shown that a (unique) closed-form expression for P which fulfills the conjecture expressed in eq. (4) is given by p”= hoU’+s) s(Ts+ V) s(Ts+h,)+V(h,+s) s(Ts+h,)+V(h,+s) (llT)c3 f -i Ts+V s(Ts+h,)+V(h,+s) (l/T)x. (12) Furthermore, endogenously determined relationships for H, cl, c2, cg, and cq that are implied by this equilibrium are given by Th,(s[Ts + ho] + V[h, +s])’ H=‘varCP’)-‘= (Ts+V)Z(s[Ts+ho]+hoV) ’ cl~cov[&rJ= s(Ts+ V) ho(sCTs+h,l+V[ho+s])’ c2 = cov [P, 8-j = V(Ts + V) s(Ts+h,)+V(h,+s)’
