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Equilibrium in Economy Where Trades Have

Differential Information

August 2005

This paper describes alternative models for a speculative market economyin which investors have differential information regarding returns of risky

assets. It explains the crucial role played by equilibrium prices to aggre-

gate and transmit information, and how such markets might become over-

informationally efficient. The paper goes on to describe a more complex and

realistic model of a multi-asset economy, and a model where information is

costly to acquire.

1 Introduction

Information relating to prices and expected payoffs helps investors accurately

value the financial assets being traded in the market based, on which theydecide whether to buy, sell or hold on to their investments. But informa-tion is not homogeneous among investors. The implication and importanceof diversity in information among investors is clearly demonstrated by theextreme case of insider trading. Insiders have access to special informationwhich others dont, and hence have an unfair advantage. Fund managersalso claim to outperform others based on their superior information.

It is now well known that in a speculative economy where traders havediverse information, the equilibrium price acts as an aggregator and trans-mitter of information. In the next section we will see how our model confirmsthis result, but to get some intuition and see how the models incorporatethis idea, consider an economy which repeats itself. Then, over many cyclesof trading, investors will learn the joint distribution of the price of the assetand the information of individual investors. In the future when the investorsobserve the market price of an asset, they will back-out and extrapolate theaggregate information held by all investors in the market. When investorslearn from prices as demonstrated above, they are said to have rationalexpectations and any equilibrium where agents behavior is influenced bythe market price of assets along with their own beliefs is then a RationalExpectations Equilibrium.

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Fully Revealing Rational Expectations Equilibrium, equilibrium where

prices symmetrisize information by revealing all the information of the in-formed to the uninformed, runs into conceptual difficulties. When pricesreveal all the information in the market, individuals realize that informationobtained from these prices is superior to their own. This makes their per-sonal information redundant and removes any economic incentive to collectcostly information. If all investors think similarly, there is no justification forany individual to collect information. If nobody is informed, it is profitableto be informed. Hence fully revealing Rational Expectations Equilibrium isnot stable.

The only way equilibrium will be stable is when there is noise in the equi-librium price system so that prices cannot reveal all the information of the

informed to the uninformed. In such a Noisy Rational Expectations Equilib-rium, the investors benefit from information collected by others, which theyestimate from prices, but do not find their personal information redundant.There is no tension in the optimal use of price and the equilibrium can bemaintained in the market.

The following section describes various models which make our argu-ments formal and rigorous. We start with a simple model with only onerisky asset and demonstrates how fully revealing rational expectation equi-librium is not a stable equilibrium. The main problem with such a model,the absence of any randomness to prevent prices from being fully reveal-ing, is dealt in subsequent models by treating aggregate supply of assets as

random. Finally we consider more complex models dealing with multi-asseteconomy and costly information. For the multi-asset model, it is shown bymeans of an example, that results from single-asset models dont carry intothe multi-asset setting. The paper ends with a list of further readings anda few potential research directions.

2 The Model

We start by considering a two period economy with T traders and two assets,one risky and one riskless. Traders invest in the first period and consumein the second period. Each investor is assumed small so that the tradingactivities of any individual does not influence the equilibrium prices in theeconomy.

The unit cost of the risky asset is P at time 0 and the time 1 payoff is arandom variable denoted by F. The riskless asset is treated as the numeraireand its time 0 and time 1 price is normalized to 1. Before trading begins,each investor receives a piece of private information about the time 1 payoffof the asset. Noise in his information prevents exact knowledge of the payoff.The precision of the signal received by all the investors is assumed identical.The signal is given by

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Yi = F + i, i = 1...T, (1)

where F is N(0,2f) and i, i = 1,...,T are i.i.d with distribution N(0,2 ). F

and the i are independent of each other.

W0i is the endowment of investor i, whose utility has an exponentialform given as Ui(x) = e

aix, where ai is the coefficient for risk aversion fortrader i. Therefore investors have constant absolute risk aversion and theirdemands are independent of initial endowment.

If i is the demand for shares of the risky asset, the time 1 wealth ofinvestor i is

W1i = i(F P) + W0i.

Each investor maximizes expected utility of wealth conditional on the signalreceived i.e. individual solves

max iE[U(W1i) | Yi = yi]. (2)

The conditional distribution ofW1i given Yi is normal. This gives us thedemand for investor i for the risky asset,

i =E[F | Yi] P

aivar(F | Yi). (3)

The assumption that densities are normal allows for closed-form solutions

and ensures that joint and conditional distributions are normal and thedemand function is linear in Yi and P.

From the market clearing condition,

Z =Ti=1

i =Ti=1

E[F | Yi, P] P

aivar(F | Yi, P), (4)

where Z is the total supply of the risky asset.Grossman (1976) finds a close form solution for the price function for

the model developed above. The price conjecture used is

P = 0 + 1Y , (5)

where

Y =

Ti=1 Yi

T.

Solving the model gives,

0 =2fZ

(2 + T 2f)T

i=11ai

.

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1 =T 2f

(2 + T 2f).

The initial endowments do not enter the expression, a consequence ofusing the exponential utility function.

On closer inspection, 1 > 0 in the above equations would enable anytrader to calculate Y by observing P. This means that price is aggregat-ing investor information and transmitting it to everyone. Grossman (1976)shows estimate of F made by using both Y and Yi is independent of Yi i.e.Y is a sufficient statistic for Yi. Hence after observing P the investor findsthat the information obtained from prices is of better quality and his owninformation is redundant. This eliminates any motivation for individuals tocollect costly information. But when nobody collects any information, thereis nothing for prices to aggregate. Therefore the equilibrium is not stableand breaks down.

To overcome the limitation of the above model, Grossman (1976) andGrossman and Stiglitz (1980) suggest that there is an equilibrium degreeof disequilibrium in the market. They argue that there has to be noisein the equilibrium prices so that they do not reveal all the information. Intheir paper noise is introduced by taking the aggregate supply of assets asrandom. The equilibrium price is now affected by both, changes in supplyof the asset or changes in the quality of information, but investors cannotdistinguish between the two and hence price is not fully revealing. There isthen reward from investing in information and the equilibrium thus obtaineddoes not suffer from the earlier shortcoming of individuals lacking motivationto collect information.

To develop such a model, let the endowment of investor i be vi whichis N(0, 2v) and independent of F and the i, i = 1...T. The total supply ofasset in the market is now random denoted by

Z =Ti=1

vi.

Z is then N(0, T 2v). The investors are still expected utility maximizers but

the only information they have about the supply of risky asset in the marketcomes from their endowment of the asset. The form of the price function istaken as

P = 0 + 1Y 2Z, 2 = 0. (6)

The reason 2 has to be non-zero is to prevent equilibrium prices frombeing fully revealing. The closed form solution has been found in [5], Huangand Litzenberger. From the price function above, it is clear that price isaffected by both Y and Z. It is not possible to isolate the effect of either ofthe two individually and correspondingly price does not reveal all the infor-mation. A similar but generalized model has been developed by Diamond

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and Verracchia (1980). They consider the equilibrium price as a function of

investor information only through their individual demands.A natural extension is to consider more assets and Admati (1985) uses

similar framework to extend the model to multiple assets and a continuumof investors. In the paper she is able to find closed-form expressions forprices and, more interestingly, develop number of examples to show theexistance of counter-intuitive results in the multi-asset environment. Theseresults cannot be obtained from a generalization of the two asset models.An example demonstrating how such results can be obtained is delineatedlater but the driving force behind such results is the correlation among thepayoffs and among the supply of different assets i.e. the structure of thevariance-covariance matrices.

For the model with N risky and one riskless asset with T traders, theexpression for time 1 wealth is given by

W1i =

i(F P) + W0i,

and the price function is taken as

P = 0 +Ti=1

1iYi 2Z.

The notation now represent matrices rather than scalars. Lastly, the signalsof individuals are allowed to be asymmetric by having different variances.

Admati develops her model in the case of infinite investors. These con-tinuum of investors are indexed by a [0, 1] and the economy is definedas a function (, S1) : [0, 1] + X

nXn , where (a, S1a ) is the value

of the function at a. Each agent observes the signal Ya = F + a where(a)a[0,1] are i.i.d normal with mean vector zero and covariance Sa. Theprice function used by Admati in the continuum of agent case is

P = 0 + 1X 2Z. (7)

Here price is a function of the actual future payoff of the asset and notthe aggregate information becuase the collective information of all infinite

agents in the economy averages out the error terms and the market as awhole has perfect information about the cash flows. 1 The paper describes

1If (a)a[0,1] is a stochastic process, then the Lebegue integral10ada might or might

not be well defined because of measurability constraint on the realization of our process(as a function of a). But ifE(a) = 0, a and Var(a) are uniformly bounded, then forevery sequence {an} of distinct indices from [0,1], the strong law of large numbers applied

to the sequence ( an) yields that (1/N)N

n=1an 0 almost surely. Admati thus claims

it is reasonable to define10ada = 0. She first assumes the integral she is defining is linear

and writes it as10ada =

10

(a E(a))da +10E(a)da. The first term on the right

goes to zero by the argument above and hence the result follows. She continues with the

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how a closed form solution for the model is obtained and what are the factors

affecting the constants in the price function. The counter-intuitive resultsobtained for a multi-asset can be best seen using a simple example takenfrom the paper.

Consider two risky assets a and b with the covariance matrix for F andZ respectively being

V =

vaa vabvba vbb

U =

uaa uabuba ubb

Let vbb >> vaa > 0 and uaa > ubb > 0 for our example. Consider an increase

in the price of asset a, Pa with the price of asset b, Pb held constant. Arational investor will take this as a signal of higher future payoff for asset a,Fa. The high degree of positive correlation between the payoffs of the twoassets would then suggest that the payoff of asset b, Fb, would also increase,and by a greater amount. The only way Pb is expected to remain constantis if its supply, Zb, increases. The positive correlation between the supplyof the assets would again mean that supply of a, Za, also increases, andby more than Zb. This will tend to decrease Pa beyond the initial increase.So an increase in Fa resulting from an increase in Pa with Pb held constantproduces inconsistent result. A decrease in Fa is more consistent: When Fadecreases, Fb also reduces and to keep prices constant, there is an drop in Zb.

This is accompanied by decrease in Za, which raises Pa and reinforces theinitial increase. Hence we see that increase in the price of the asset mightlead to decrease in future payoff. This is one example which goes against theintuition developed in earlier models and shows how the variance-covariancestucture can produce suprising results when multiple assets are considered.

Another impressive feature of a multi-asset economy which comes out ofthe Admati paper is that the supply of each asset in the economy does nothave to be unknown for the equilibrium to be stable. Noise in the supplyin any asset affects the prices of remaining assets (the exact degree dependson the correlation structure) and prevents the prices of assets with knownsupplies from becoming fully revealing. Secondly, even if the payoff of an

asset is known perfectly, the investor can obtain information about otherassets through signal relating to this first asset. This is again due to thecorrelation between different assets in the economy. So the conditions forthe equilibrium to exist in this setting are not as strict as those in the earliermodels.

argument that if (a)a[0,1] are as above and (a)a[0,1] is almost surely integrable, then10

(a + a)da =10

ada. Now if we return to our model, the error terms {a}a[0,1] is

similar as the (a )a[0,1] above and so we can write10Yada =

10

(X+ a)da = X almostsurely.

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An alternative direction of reseach relates to models where investors

have to pay to acquire information. In such models, they have to make thedecision to buy the information or not, depending on how that effects theirexpected utility. Grossman and Stiglitz (1980) explore this in a Two-AssetTwo-Period setting.

The model is constructed as follows. There is one piece of informationand all investors who decide to buy get the same information. The payoffF of the risky asset is random and given by F = + . Investors can knowthe value of at a price k. The informed and the uninformed investorsare identical apriori and the classification just depends on whether theypurchase information or not. When traders buy information, they obtainknowledge of the payoff accurately to a precision of , and their demand for

the risky asset then depends on and P, the price in the market. Demandof uninformed individuals just depends on P. The supply of the asset israndom denoted by Z. If denotes the percentage of investors who decideto become informed, then the price function is of the form P(z, ). Theuninformed dont know the value of z and hence cannot extrapolate fromprice.

The amount of information the investors can extract from the equilib-rium price depends on how noisy the price is. The investors weigh thisagainst the benefit if they buy the information. At the margin, the ex-pected benefit from buying information is zero. Thus the equilibrium canbe obtained by equating the expected utility of the informed and the unin-

formed. As long as these two are different, investors will find it profitableto switch sides. Grossman and Stiglitz come up with a number of generalconjectures which should be true for a general model of this form. Some ofthem are discussed below.

As the number of informed investors increase, the price system be-comes more informative. This reduces the utility of being informedand the ratio of utility of informed to the uninformed decreases. Thisdrives the ratio of utilities to one and the market towards equilibrium.

A rise in the price of information reduces the expected benefit of theinformation and consequently the fraction of investors who decide tobecome informed in equilibrium is a decreasing function of the priceof information.

Increase in the quality of information without any increase in its priceproduces ambigious results. On one side, this will encourage investorsto buy information, since it is more informative. But better qualityinformation will mean that investors who buy information are makingmore informed decisions, and so the price system is more revealing.This increases the expected benefit for the uninformed, discouragingthem from buying information.

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The noise in the system in the form stochastic supply of assets is the

reason prices are not fully revealing. It is clear that more the noise inthe system, less informative is the price.

Grossman and Stiglitz are not able to verify the conjectures for a generalmodel, but they are able to prove their arguments for a special case where in-vestors have constant risk aversion and all variables are normally distributed.Their closed form solution shows how the ratio of utilities change with thefraction of informed investors. They analyze the case where prices reveal allinformation as limiting cases of their model by letting the variance of noisego to zero. The results show that as the variance of noise goes to zero, theproportion of informed traders also go to zero. Hence we have fully revealing

prices and the equilibrium breaks down.In the final section of their paper, Grossman and Stiglitz argue whyprices cannot be fully revealing. They start by pointing out that differencesin preference is not the only reason driving trade, and differences in endow-ment and belief are also important reasons why people trade. Disregardingdifferences in preferences, if all traders have identical endowments and be-liefs then a competitive equilibrium would leave them with exactly the sameshare as their endowments, and nobody would trade if there is some costinvolved. Grossman and Stiglitz then go on to show there there is continuityin net trade with respect to differences in beliefs. Based on this argument,as the noise in prices go to zero, traders become identically informed having

identical beliefs, and trading thins down to zero. If operating the market iscostly it will close down before the equilibrium ceases to exist.

3 Future Reading and Research

Some areas for further reading and consideration for research are highlightedbelow.

1. Noise, in the form of stochastic supply of assets, prevent prices frombeing fully revealing and makes the above models stable. Diamondand Verracchia (1980) write that introducing noise in such a manner

may appear somewhat artificial. They propose other plausible sourcesof noise such as individually stochastic life cycle motives for trade,individually stochastic taxes and wonder whether these alternativesources might give more insight into the nature and role of noise inthe rational expectations equilibrium.

I argue that treating supply of assets as random in the models is notunreasonable and should be interprated as a mixture of stochastic lifecycle of individuals trade, which Diamond and Verracchia suggest,and Limited Participation in trading. For the former I reason thatmost investors have some fixed random time they would like stick

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with their investment before they engage in buying or selling the asset

in question again. During this time, their supply is absent from themarket when the price is being determined. The Limited ParticipationI mention is meant to capture the idea that Not all investors arewatching prices of all the assets all the time. This might be speciallytrue for assets which are not traded often or assets which make a verysmall percentage of portfolios. This makes it impossible to give anaccurate figure for the total supply of the asset.

2. All the above models are for a competitive equilibrium where individ-ual investors are price takers. Kyles (1989) work addresses the casewhere the market is imperfect and individual traders can manipulate

prices. A Bayesian-Nash Equilibrium framework is used to explorethe implications and develop the model. The justification for explor-ing such a market is that the best informed traders are usually thevery large traders who have the capacity of moving the market.

3. Except for the last model, investors are costlessly endowed with infor-mation. Treating information gathering as costly would enhance ourunderstanding of the market equilibrium. A more complicated modelcan be constructed by assuming investors have a choice to buy privateheterogeneous information corresponding to payoffs of risky assets.There could be multiple sources of information with non-uniform cost.The resulting model could be much closer to reality but tractability

of such a model is a serious concern. The other aspect to consider ishow much would it really add to our understanding of the equilibriumprocess.

We can start by looking at Grossman and Stiglitz (1980) and analyzewhat happens if we increase from one to two, at non-uniform cost,the pieces of information investors can buy. Do we just get a morecomplicated model, with no enhancement of our understanding, or dowe see some interesting results? What if these pieces of informationare not independent? There is scope in this direction and reasonableassumptions might provide mathematical tractability.

4. A prominent work for future reference is by Campbell and Kyle(1993).

5. An idea suggested by Prof David Brown was to use the Multi-AssetAdmati model but have teh cash flows take on a factor structure. Un-fortunately, the analysis became intractable because the simple struc-ture exploited by Admati for her paper vanished. A different approachwill have to be used if a solution is to be obtained. Hughes, Liu andLiu (2005) have tried to accomplish a similar goal in their paper andit should be interesting going through their analysis.

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