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