《投资学原理 Investments》课程教学资源（阅读资料）On the Impossibility of Informationally Efficient Markets

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On the Impossibility of Informationally Efficient Markets By Sanford J. GRoSSMan AND JoSEPH E STiglitz* If competitive equilibrium is defined as a jectures concerning certain properties of the situation in which prices are such that all equilibrium. The remaining analytic sections arbitrage profits are eliminated, is it possible of the paper are devoted to analyzing in hat a competitive economy always be in detail an important example of our general equilibrium? Clearly not, for then those who model, in which our conjectures concerning arbitrage make no(private) return from the nature of the equilibrium can be shown their (privately) costly activity. Hence the to be correct. We conclude with a discussion assumptions that all markets, including that of the implicat pur app for information, are always in equilibrium results, with particular emphasis on the rela and always perfectly arbitraged are incon- tionship of our results to the literature on sistent when arbitrage is costly. efficient capital markets We propose here a model in which there is an equilibrium degree of disequilibrium L. The model prices reflect the information of informed ndividuals(arbitrageurs) but only partially, Our model can be viewed as an extension so that those who expend resources to ob- of the noisy rational expectations model in tain information do receive compensation. troduced by Robert Lucas and applied to How informative the price system is de- the study of information flows between pends on the number of individuals who are traders by Jerry Green (1973) ossman informed; but the number of individuals (1975, 1976, 1978); and Richard Kihlstrom who are informed is itself an endogenous and Leonard mirman There are two assets variable in the model a safe asset yielding a return R, and a risky The model is the simplest one in which asset, the return to which, u, varies ran prices perform a well-articulated role in con- domly from period to period The variable u veying information from the informed to the consists of two parts, serve information that the return to a secur- (1) ity is going to be high, they bid its price up, and conversely when they observe informa- where 0 is observable at a cost c, and e is ion that the return is going to be low. Thus unobservable. Both 0 and e are random the price system makes publicly available varables. There are two types of individu the information obtained by informed indi- als, those who observe 0(informed traders), viduals to the uniformed. In general, how- and those who observe only price (unin- ever, it does this imperfectly; this is perhaps formed traders). In our simple model, all lucky, for were it to do it perfectly, an individuals are, ex ante, identical; whether equilibrium would not exist they are informed or uninformed just de In the introduction, we shall discuss the pends on whether they have spent c to ob. general methodology and present some con- tain information. Informed traders'de 以0 y asset P. Uninformed traders'demands 'An alternative interpretation is that g is a"me This is a revised er presen he Econometric alternative interpretation differ slightly, but the Society inter 1975. at Dallas Texas are identical 393 I 1 Sep 2013 OR Terms and Conditions

On the Impossibility of Informationally Efficient Markets By SANFORD J. GROSSMAN AND JOSEPH E. STIGLITZ* If competitive equilibrium is defined as a situation in which prices are such that all arbitrage profits are eliminated, is it possible that a competitive economy always be in equilibrium? Clearly not, for then those who arbitrage make no (private) return from their (privately) costly activity. Hence the assumptions that all markets, including that for information, are always in equilibrium and always perfectly arbitraged are inconsistent when arbitrage is costly. We propose here a model in which there is an equilibrium degree of disequilibrium: prices reflect the information of informed individuals (arbitrageurs) but only partially, so that those who expend resources to obtain information do receive compensation. How informative the price system is depends on the number of individuals who are informed; but the number of individuals who are informed is itself an endogenous variable in the model. The model is the simplest one in which prices perform a well-articulated role in conveying information from the informed to the uninformed. When informed individuals observe information that the return to a security is going to be high, they bid its price up, and conversely when they observe information that the return is going to be low. Thus the price system makes publicly available the information obtained by informed individuals to the uniformed. In general, however, it does this imperfectly; this is perhaps lucky, for were it to do it perfectly, an equilibrium would not exist. In the introduction, we shall discuss the general methodology and present some conjectures concerning certain properties of the equilibrium. The remaining analytic sections of the paper are devoted to analyzing in detail an important example of our general model, in which our conjectures concerning the nature of the equilibrium can be shown to be correct. We conclude with a discussion of the implications of our approach and results, with particular emphasis on the relationship of our results to the literature on "efficient capital markets." I. The Model Our model can be viewed as an extension of the noisy rational expectations model introduced by Robert Lucas and applied to the study of information flows between traders by Jerry Green (1973); Grossman (1975, 1976, 1978); and Richard Kihlstrom and Leonard Mirman. There are two assets: a safe asset yielding a return R, and a risky asset, the return to which, u, varies randomly from period to period. The variable u consists of two parts, (1) = +e where 9 is observable at a cost c, and e is unobservable.' Both 9 and E are random variables. There are two types of individuals, those who observe 9 (informed traders), and those who observe only price (uninformed traders). In our simple model, all individuals are, ex ante, identical; whether they are informed or uninformed just depends on whether they have spent c to obtain information. Informed traders' demands will depend on 9 and the price of the risky asset P. Uninformed traders' demands *University of Pennsylvania and Princeton University, respectively. Research support under National Science Foundation grants SOC76-18771 and SOC77- 15980 is gratefully acknowledged. This is a revised version of a paper presented at the Econometric Society meetings, Winter 1975, at Dallas, Texas. 'An alternative interpretation is that 0 is a "measurement" of u with error. The mathematics of this alternative interpretation differ slightly, but the results are identical. 393 This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:12:49 AM All use subject to JSTOR Terms and Conditions

THE AMERICAN ECONOMIC REVIEW JUNE 1980 will depend only on P, but we shall assume which the informed can gain relative to the that they have rational expectations; they uninformed-is reduced learn the relationship between the distribu (b)Even if the above effect did not tion of return and the price, and use this in occur, the increase in the ratio of informed r dema If to uninformed means that the relative gains x denotes the supply of the risky asset, an of the informed, on a per capita basis, in equilibrium when a given percentage, A, of trading with the uninformed will be smaller. ders are informed, is thus a We summarize the above characterization P(8, x)such that, when demands are for- of the equilibrium of the economy in the mulated in the way described, demand following two conjectures equals supply. We assume that uninformed Conjecture 1: The more individuals who traders do not observe x. Uninformed are informed, the more informative is the traders are prevented from learning 6 via price system observations of Pa(0, x) because they can Conjecture 2: The more individuals who not distinguish variations in price due to are informed, the lower the ratio of expected changes in the informed trader's informa- utility of the informed to the uninformed tion from variations in price due to changes Conjecture I obviously requires a defini- in aggregate supply. Clearly, P(e, x) reveals tion of"more informative"; this is given in some of the informed trader's information the next section and in fn. 7.) to the uninformed trade The equilibrium number of informed We can calculate the expected utility of uninformed individuals in the economy will the informed and the expected utility of the depend on a number of critical parameters uninformed. If the former is greater than the the cost of information, how informative the latter (taking account of the cost of infor- price system is(how much noise there is to mation), some individuals switch from being interfere with the information conveyed by uninformed to being informed(and con- the price system), and how informative versely). An overall equilibrium requires the information obtained by an informed indi two to have the same expected utility. As vidual is more individuals become informed. the ex- Conjecture 3: The higher the cost of pected utility of the informed falls relative information, the smaller will be the equi- to the uninformed for two reasons librium percentage of individuals who are (a) The price system becomes more in- informed formative because variations in have a Conjecture 4: If the quality of the greater effect on aggregate demand and thus formed trader's information increases, the on price when more traders observe 8. Thus, more their demands will vary with their more of the information of the informed is information and thus the more prices will available to the uninformed. Moreover, the vary with 0. Hence, the price system be informed gain more from trade with the comes more informative. The equilibrium uninformed than do the uninformed. th proportion of informed to uninformed may informed, on average, buy securities when be either increased or decreased, because they are"underpriced"and sell them when even though the value of being informed has they are overpriced"(relative to what increased due to the increased quality of 0, they would have been if information were the value of being uninformed has also in equalized). As the price system becomes creased because the price system becomes more informative, the difference in their in- more informative formation-and hence the magnitude by Conjecture 5: The greater the magn tude of noise the less informative will the price system be, and hence the lower the 2The framework described herein does not explicitly expected utility of uninformed individuals of variations of futures markets and Hence, in equilibrium the greater the magni- ativeness of the price tude of noise, the larger the proportion of formed individuals OR Terms and Conditions

394 THE AMERICAN ECONOMIC REVIEW JUNE 1980 will depend only on P, but we shall assume that they have rational expectations; they learn the relationship between the distribution of return and the price, and use this in deriving their demand for the risky assets. If x denotes the supply of the risky asset, an equilibrium when a given percentage, X, of traders are informed, is thus a price function PA(O,x) such that, when demands are formulated in the way described, demand equals supply. We assume that uninformed traders do not observe x. Uninformed traders are prevented from learning 9 via observations of PA(O,x) because they cannot distinguish variations in price due to changes in the informed trader's information from variations in price due to changes in aggregate supply. Clearly, PA(O,x) reveals some of the informed trader's information to the uninformed traders. We can calculate the expected utility of the informed and the expected utility of the uninformed. If the former is greater than the latter (taking account of the cost of information), some individuals switch from being uninformed to being informed (and conversely). An overall equilibrium requires the two to have the same expected utility. As more individuals become informed, the expected utility of the informed falls relative to the uninformed for two reasons: (a) The price system becomes more informative because variations in 9 have a greater effect on aggregate demand and thus on price when more traders observe 9. Thus, more of the information of the informed is available to the uninformed. Moreover, the informed gain more from trade with the uninformed than do the uninformed. The informed, on average, buy securities when they are "underpriced" and sell them when they are "overpriced" (relative to what they would have been if information were equalized).2 As the price system becomes more informative, the difference in their information-and hence the magnitude by which the informed can gain relative to the uninformed-is reduced. (b) Even if the above effect did not occur, the increase in the ratio of informed to uninformed means that the relative gains of the informed, on a per capita basis, in trading with the uninformed will be smaller. We summarize the above characterization of the equilibrium of the economy in the following two conjectures: Conjecture 1: The more individuals who are informed, the more informative is the price system. Conjecture 2: The more individuals who are informed, the lower the ratio of expected utility of the informed to the uninformed. (Conjecture 1 obviously requires a definition of "more informative"; this is given in the next section and in fn. 7.) The equilibrium number of informed and uninformed individuals in the economy will depend on a number of critical parameters: the cost of information, how informative the price system is (how much noise there is to interfere with the information conveyed by the price system), and how informative the information obtained by an informed individual is. Conjecture 3: The higher the cost of information, the smaller will be the equilibrium percentage of individuals who are informed. Conjecture 4: If the quality of the informed trader's information increases, the more their demands will vary with their information and thus the more prices will vary with 9. Hence, the price system becomes more informative. The equilibrium proportion of informed to uninformed may be either increased or decreased, because even though the value of being informed has increased due to the increased quality of 9, the value of being uninformed has also increased because the price system becomes more informative. Conjecture 5: The greater the magnitude of noise, the less informative will the price system be, and hence the lower the expected utility of uninformed individuals. Hence, in equilibrium the greater the magnitude of noise, the larger the proportion of informed individuals. 2The framework described herein does not explicitly model the effect of variations in supply, i.e., x on commodity storage. The effect of futures markets and storage capabilities on the informativeness of the price system was studied by Grossman (1975, 1977). This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:12:49 AM All use subject to JSTOR Terms and Conditions

VOL, 70 NO. 3 GROSSMAN AND STIGLITZ. EFFICIENT MARKETS Conjecture 6: In the limit, when there is the conjectures provided above can be veri no noise, prices convey all information, and fied. The next sections are devoted to solv there is no incentive to purchase informa- ing for the equilibrium in this particular tion. Hence, the only possible equilibrium is example. information. But if everyo uninformed, it clearly pays some individual II. Constant Absolute Risk-Aversion Model to become informed. 3 Thus, there does not exist a competitive equilibrium. 4 The securities Trade among individuals occurs either be- cause tastes (risk aversions)differ, endot The ith trader is assumed to be endowed ments differ, or beliefs differ. This paper with 'o types of securities focuses on the last of these three. An inter- the riskless asset, and x, a risky asset. Let P esting feature of the equilibrium is that be- be the current price of risky assets and set liefs may be precisely identical in either one the price of risk free assets equal to unity of two situations: when all individuals are The ith traders budget constraint is informed or when all individuals are unin- formed. This gives rise to PX2+M1=Wo≡M+PX Conjecture 7: That, othe equal, markets will be thinner Each unit of the risk free asset pays conditions in which the percent indi-“ dollars” at the end of the period, while viduals who are informed ()is either near each unit of the risky asset pays u dollars. If zero or near unity. For example, markets at the end of the period, the ith trader holds will be thin when there is very little noise in a portfolio(M, X,), his wealth will be the system(so A is near zero), or when costs of information are very low(so A is near (3) WERM+uX In the last few paragraphs, we have pro- B. Individuals Utility Maximization ided a number of conjectures describing the nature of the equilibrium when prices Each individual has a utility function convey information. Unfortunately, we have v(wl). For simplicity, we assume all indi not been able to obtain a general proof of viduals have the same utility function and any of these propositions. What we have so drop the subscripts i. Moreover,we been able to do is to analyze in detail an assume the utility function is exponential, interesting example, entailing constant ab-1.e solute risk-aversion utility functions and a>0 normally distributed random this example, the equilibrium price distribu- where a is the coefficient of absolute risk tion can actually be calculated, and all aversion, Each trader desires to maximize expected utility, using whatever information is available to him. and to decide on what That is. with no one informed an individual nformation to acquire on the basis of the nly get informa consequences to his expected utility. formation is revealed by the pri Assume that in equation(1)8 and e have than the market when it is optimal to hold the risky a multivariate normal distribution, with c dollars an individual will be able oposed to the risk-free asset. Thus his e Ee=0 gross of information costs. Thus for c sufficiently lot See Grossman(1975, 1977)for a formal example of (6) Var(u*0)=VarE*=02>0 atures markets. See Stiglitz(1971 1974) for a general discussion of information and the SThe possibility of nonexistence of equilibrium in capital not, in general, exist. See Green(1977). Of course, for the utility function we choose equilibrium does exis I 1 Sep 2013 OR Terms and Conditions

VOL. 70 NO. 3 GROSSMAN AND STIGLITZ: EFFICIENT MARKETS 395 Conjecture 6: In the limit, when there is no noise, prices convey all information, and there is no incentive to purchase information. Hence, the only possible equilibrium is one with no information. But if everyone is uninformed, it clearly pays some individual to become informed.3 Thus, there does not exist a competitive equilibrium.4 Trade among individuals occurs either because tastes (risk aversions) differ, endowments differ, or beliefs differ. This paper focuses on the last of these three. An interesting feature of the equilibrium is that beliefs may be precisely identical in either one of two situations: when all individuals are informed or when all individuals are uninformed. This gives rise to: Conjecture 7: That, other things being equal, markets will be thinner under those conditions in which the percentage of individuals who are informed (X) is either near zero or near unity. For example, markets will be thin when there is very little noise in the system (so X is near zero), or when costs of information are very low (so X is near unity). In the last few paragraphs, we have provided a number of conjectures describing the nature of the equilibrium when prices convey information. Unfortunately, we have not been able to obtain a general proof of any of these propositions. What we have been able to do is to analyze in detail an interesting example, entailing constant absolute risk-aversion utility functions anid normally distributed random variables. In this example, the equilibrium price distribution can actually be calculated, and all of the conjectures provided above can be verified. The next sections are devoted to solving for the equilibrium in this particular example.5 II. Constant Absolute Risk-Aversion Model A. The Securities The ith trader is assumed to be endowed with stocks of two types of securities: Mi, the riskless asset, and Xi, a risky asset. Let P be the current price of risky assets and set the price of risk free assets equal to unity. The ith trader's budget constraint is (2) PXI+ Ml=Woi0Mi+ PXi Each unit of the risk free asset pays R "dollars" at the end of the period, while each unit of the risky asset pays u dollars. If at the end of the period, the ith trader holds a portfolio (Mi,X), his wealth will be (3) Wli = RM, + uX, B. Individual's Utility Maximization Each individual has a utility function Vi(Wli). For simplicity, we assume all individuals have the same utility function and so drop the subscripts i. Moreover, we assume the utility function is exponential, i.e., V(Wli)= e-awl a>O where a is the coefficient of absolute risk aversion. Each trader desires to maximize expected utility, using whatever information is available to him, and to decide on what information to acquire on the basis of the consequences to his expected utility. Assume that in equation (1) 9 and e have a multivariate normal distribution, with (4) Ee = 0 (5) EOe = O (6) Var(u*19)= Vare*=_a~2>O 3That is, with no one informed, an individual can only get information by paying c dollars, since no information is revealed by the price system. By paying c dollars an individual will be able to predict better than the market when it is optimal to hold the risky asset as opposed to the risk-free asset. Thus his expected utility will be higher than an uninformed person gross of information costs. Thus for c sufficiently low all uninformed people will desire to be informed. 4See Grossman (1975, 1977) for a formal example of this phenomenon in futures markets. See Stiglitz (1971, 1974) for a general discussion of information and the possibility of nonexistence of equilibrium in capital markets. 5The informational equilibria discussed here may not, in general, exist. See Green (1977). Of course, for the utility function we choose equilibrium does exist. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:12:49 AM All use subject to JSTOR Terms and Conditions

THE AMERICAN ECONOMIC REVIEW JUNE 1980 since 0 and e are uncorrelated. Throughout Then, we can write for the uninformed his paper we will put a* above a symbol to dividual emphasize that it is a random variable. Since Wu is a linear function of e, for a (7)E((W)IP")=-exp-aELWtlP* given portfolio allocation, and a linear func- tion of a normally distributed random vari able is normally distributed, it follows that m Var[ wilP* Wu is normal conditional on 6. Then, using (2)and (3)the expected utility of the in formed trader with information 6 can be =-exp -a RWor +Xu(E[ulP*]-RP (7)E(V(W)θ Xi var[uP* The demands of the uninformed will thus be a function of the price function P* and the actual price P. exp(-a RWo, + X,( E(u10)-RP)(8)Xu(P: P") E[uP(8, x)=P-RP gx? var(.10) C. Equilibrium Price Distribution exp(-a「RW+x,(6-RP If A is some particular fraction of traders who decide to become informed, then define Xo an equilibrium price system as a function of (0, x), P(0, x), such that for all (0, x)per capita demands for the risky assets equal where X is an informed individual,'s de- supplies mand for the risky security. Maximizing(7) with respect to X, yields a demand function (9) AX, (P(e, x), 0) for risky assets 8-RP +(1-入)XU(PA(0,x);P)=x XCP 8) The function P(0, x)is a statistical The right-hand side of (8)shows the familiar equilibrium in the following sense. If over result that with constant absolute risk aver- time uninformed traders observe many re- sion, a trader's demand does not depend on alizations of(u*, PA), then they wealth; hence the subscript i is not on the joint distribution of (u*, P*). After all learn left-hand side of (8) ing about the joint distribution of (u*P* We now derive the demand function for ceases all traders will make allocations and the uninformed. Let us assume the only form expectations such that this joint dis- source of"noise"is the per capita supply of tribution persists over time. This follows the risky sec from( 8),(8), and (9), where the market Let P*() be some particular price func- clearing price that comes about is the one tion of (0, x)such that u* and P* are jointly which takes into account the fact that unin- normally distributed. (We will prove that formed traders have learned that it contains this exists below.) information OR Terms and Conditions

396 THE AMERICAN ECONOMIC REVIEW JUNE 1980 since 9 and e are uncorrelated. Throughout this paper we will put a * above a symbol to emphasize that it is a random variable. Since Wli is a linear function of e, for a given portfolio allocation, and a linear function of a normally distributed random variable is normally distributed, it follows that W11 is normal conditional on 0. Then, using (2) and (3) the expected utility of the informed trader with information 9 can be written (7) E( V( Wl*i)10)= -exp(-a E[ Wl*10] _ a Var[ Wl*1'] ) =-exp( -a[ RWOi + X1{ E(u*I9)-RP} -2 X2 Var(u*I9)]) =-exp(-a[RJWoiV+X1( -RP) -2 Xl a, x2 ] ) where X, is an informed individual's demand for the risky security. Maximizing (7) with respect to X, yields a demand function for risky assets: (8) X,(P, 9) = aa2 The right-hand side of (8) shows the familiar result that with constant absolute risk aversion, a trader's demand does not depend on wealth; hence the subscript i is not on the left-hand side of (8). We now derive the demand function for the uninformed. Let us assume the only source of "noise" is the per capita supply of the risky security x. Let P*(.) be some particular price function of (9,x) such that u* and P* are jointly normally distributed. (We will prove that this exists below.) Then, we can write for the uninformed individual (7') E( V( W*i)P*) =-exp -a ttE[ W*I P*] a - Var[ W*IP*1]) 2 LlJJ =-exp[-a RWoi+Xu(E[u*IP*]1RP) -2X Var[u*IP*]}] The demands of the uninformed will thus be a function of the price function P* and the actual price P. (8') Xu(P; P*) E[ u*I P*(9,x) = P] -RP a Var[ u*I P*(9, x) =P] C. Equilibrium Price Distribution If X is some particular fraction of traders who decide to become informed, then define an equilibrium price system as a function of (9, x), PA(O, x), such that for all (9, x) per capita demands for the risky assets equal supplies; (9) XXI(PA(9,x),0) + (1- X)XU(PA(9, x); PA ) = x The function PA(O, x) is a statistical equilibrium in the following sense. If over time uninformed traders observe many realizations of (u*,Px*), then they learn the joint distribution of (u*, P*). After all learning about the joint distribution of (u*,P,*) ceases, all traders will make allocations and form expectations such that this joint distribution persists over time. This follows from (8), (8'), and (9), where the marketclearing price that comes about is the one which takes into account the fact that uninformed traders have learned that it contains information. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:12:49 AM All use subject to JSTOR Terms and Conditions

VOL. 70 NO. 3 GROSSMAN AND STIGLITZ. EFFICIENT MARKETS We shall now prove that there exists an For each replication of the economy, e is equilibrium price distribution such that P* the information that uninformed traders and u* are jointly normal. moreover, we would like to know. But the noise shall be able to characterize the price dis- prevents w* from revealing 6. How well tribution. We define formed uninformed traders can become from observing Pa(equivalently w*) is (0)(0,x)=0-a82(x-Ex) measured by Var[w*e]. When Var[w*8] is zero, w* and 8 are perfectly correlated Hence when uninformed firms observe wX for A>0, and define wo(0, x) as the numb this is equivalent to observing 8. On the other hand, when Var[* 0] is very large, (10b) wo(0, x)=x for all(0, x) ere are“many” realizations of w, that are sSociated with a given 0. In this case the where wa is just the random variable 8, plus observation of a particular wa tells very noise. The magnitude of the noise is in- little about the actual 0 which generated it. versely proportional to the proportion of From equation (11) it is clear that large informed traders, but is proportional to the noise(high Varx")leads to an imprecise variance of e. We shall prove that the price system. The other factor which de- equilibrium price is just a linear function of termines the precision of the price system w.Thus, if A>0, the price system conveys (a00:/2)is more subtle. When a is small information about 0, but it does so imper-(the individual is not very risk averse)or o is small(the information is very precise),an informed trader will have a demand for D. Existence of equilibrium and risky assets which is very responsive to changes in 0. Further, the larger A is, the more responsive is the total demand of in THEOREM 1: If(8*e* x*)has a nonde- formed traders. Thus small(a0:/A)means generate joint normal distribution such that that the aggregate demand of informed 8*, e, and x* are mutually independent, then traders is very responsive to 8. For a fixed there exists a solution to(9)which has the amount of noise (i. e, fixed Var x*)the form P(e, x)=a+awa(8, x), where a and larger are the movements in aggregate de a, are real numbers which may depend on A, mand which are due to movements in 8, the such that a2>0.(If x=0, the price contains more will price movements be due to move- no information about 0. The exact form ments in 6. That is, x* becomes less im- Pa(e, x) is given in equation(A10) in Appen- portant relative to 8 in determining price ix B. The proof of this theorem is also in movements. Therefore, for small(a20:/22 Appendix B uninformed traders are able to confident know that price is, for example, unusually The importance of Theorem I rests in the high due to 0 being high. In this way infor simple characterization of the information mation from informed traders is transferred in the equilibrium price system: PX is infor- to uninformed traders. nationally equivalent to w*. From(10)wx is a"mean-preserving spread"of 8;i.e E[w米|]=6and (1)mn"1)= 'Formally, wo is 6If y'=y+Z, and E[Zl]=0, then y' is just y plus the experiment; see Grossman, (p.539) OR Terms and Conditions

VOL. 70 NO. 3 GROSSMAN AND STIGLITZ: EFFICIENT MA RKETS 397 We shall now prove that there exists an equilibrium price distribution such that P* and u* are jointly normal. Moreover, we shall be able to characterize the price distribution. We define a2 (lOa) w,(9,x)=9 0- a (x-Ex*) for X> 0, and define wo(9,x) as the number: (lOb) wo(9,x)=x for all (9,x) where wX is just the random variable 9, plus noise.6 The magnitude of the noise is inversely proportional to the proportion of informed traders, but is proportional to the variance of E. We shall prove that the equilibrium price is just a linear function of wx. Thus, if X>0, the price system conveys information about 9, but it does so imperfectly. D. Existence of Equilibrium and a Characterization Theorem THEOREM 1: If (0*,?*,x*) has a nondegenerate joint normal distribution such that 9*, E*, and x* are mutually independent, then there exists a solution to (9) which has the form PX(9,x)=a1+a2wX(9, x), where a1 and a2 are real numbers which may depend on A, such that a2 >0- (If X = 0, the price contains no information about 9.) The exact form of PX(9,x) is given in equation (A 10) in Appendix B. The proof of this theorem is also in Appendix B. The importance of Theorem 1 rests in the simple characterization of the information in the equilibrium price system: Px* is informationally equivalent to w*. From (10) w* is a "mean-preserving spread" of 9; i.e., E[w*10]=9 and (1 1) Var[ IwxI Varx* For each replication of the economy, 9 is the information that uninformed traders would like to know. But the noise x * prevents w* from revealing 9. How wellinformed uninformed traders can become from observing Px* (equivalently wx*) is measured by Var[w*10]. When Var[w*10] is zero, w,* and 9 are perfectly correlated. Hence when uninformed firms observe w*, this is equivalent to observing 9. On the other hand, when Var[w* 10] is very large, there are "many" realizations of w,* that are associated with a given 9. In this case the observation of a particular w,* tells very little about the actual 9 which generated it.7 From equation (11) it is clear that large noise (high Varx*) leads to an imprecise price system. The other factor which determines the precision of the price system (a2a4'/X2) is more subtle. When a is small (the individual is not very risk averse) or a,2 is small (the information is very precise), an informed trader will have a demand for risky assets which is very responsive to changes in 9. Further, the larger X is, the more responsive is the total demand of informed traders. Thus small (a2a'4/X2) means that the aggregate demand of informed traders is very responsive to 9. For a fixed amount of noise (i.e., fixed Var x*) the larger are the movements in aggregate demand which are due to movements in 9, the more will price movements be due to movements in 9. That is, x* becomes less important relative to 9 in determining price movements. Therefore, for small (a2a,'/X2) uninformed traders are able to confidently know that price is, for example, unusually high due to 9 being high. In this way information from informed traders is transferred to uninformed traders. 61f y'= y + Z, and E[Z Iy] = O, then y' is just y plus noise. 7Formally, wA is an experiment in the sense of Blackwell which gives information about 9. It is easy to show that, ceteris paribus, the smaller Var(wxI1) the more "informative" (or sufficient) in the sense of Blackwell, is the experiment; see Grossman, Kihlstrom, and Mirman (p. 539). This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:12:49 AM All use subject to JSTOR Terms and Conditions

VOL. 70 NO. 3 GROSSMAN AND STIGLITZ: EFFICIENT MARKETS as A rises, the expected utility of informed Ev(w traders does go down relative to uninformed EV(w y() Note that the function y(0)=e(var(u* g)/Varu*/2. Figure 1 illustrates the de termination of the equilibrium A. The figure assumes that y(O)<1<y(1). G. Characterization of equilibrium We wish to provide some further char- acterization of the equilibrium. Let us define FIGURE I (16a) Note that(19)holds for y(O)<I<y(1), since hese conditions insure that the equilibrium A is between zero and one. Equation (19b) (16b) shows that the equilibrium informativeness of th by the cost of information c, the quality of Note that m is inversely related to the the informed traders information n, and the normativeness of the price system since the degree of risk aversion quare correlation coefficient between PX andθ*, pg is given by H. Comparative static p0-1+m From equation(19b), we immediately ob. tain some basic comparative statics results Similarly, n is directly related to the quality 1)An increase in the quality of infor f the informed trader's information be- mation(n)increases the informativeness of cause n/(1+n) is the squared correlation the price system coefficient between 6* and u* 2)A decrease in the cost of information Equations(14)and(15)show that the increases the informativeness of the pric cost of information c, determines the equi- system librium ratio of information quality be 3)a decrease in risk aversion leads tween informed and uninformed traders informed individuals to take larger posi- (var(u* e))/Var(u*wA). From(1),(All)of tions, and this increases the informativeness Appendix A, and(16), this can be written as of the price system Further, all other changes in parameters, (18) such that n, a, and c remain constant, Var(u"w2)I+m+ mm(1+ nm-1 formativeness of the price system;other changes lead only to particular changes in X of a magnitude to exactly offset them. For Substituting(18)into(14) and using(15) example we obtain, for 0<x<l, in equilibrium 4)An increase in noise(o+ incre the proportion of informed traders. At any 1+ increases the returns to information and leads more individuals to become informed (19b)1-m2≈c2a-1 lishes that the two effects exactly offset each OR Terms and Conditions

VOL. 70 NO. 3 GROSSMAN AND STIGLITZ: EFFICIENT MARKETS 399 as X rises, the expected utility of informed traders does go down relative to uninformed traders. Note that the function y (0) = eac(Var(u* I 9)/Var u*)l/2. Figure 1 illustrates the determination of the equilibrium X. The figure assumes that y(O) < 1 <y(l). G. Characterization of Equilibrium We wish to provide some further characterization of the equilibrium. Let us define /22 2 (16a) m=( aa= X) (16b) n=aO2 ae2 Note that m is inversely related to the informativeness of the price system since the squared correlation coefficient between Px* and 9*, p92 is given by (17) po2 = I Similarly, n is directly related to the quality of the informed trader's information because n/(l + n) is the squared correlation coefficient between 9* and uO. Equations (14) and (15) show that the cost of information c, determines the equilibrium ratio of information quality between informed and uninformed traders (Var(u*I9))/ Var(u*Iwx). From (1), (A1) of Appendix A, and (16), this can be written as (18) Var(u*10) 1 + m = + nm Var(u*lw,) l+m+nm I l m Substituting (18) into (14) and using (15) we obtain, for 0< X < 1, in equilibrium e2ac I (19a) e- 1 + n - e2ac or e2ac_ (19b\ I 2_ e0 EV (W) EV(W)) riX e ac |v ar (ut1l) 9 0 A e 1 FIGURE 1 Note that (19) holds for y(O) < 1 <y(l), since these conditions insure that the equilibrium X is between zero and one. Equation (19b) shows that the equilibrium informativeness of the price system is determined completely by the cost of information c, the quality of the informed trader's information n, and the degree of risk aversion a. H. Comparative Statics From equation (19b), we immediately obtain some basic comparative statics results: 1) An increase in the quality of information (n) increases the informativeness of the price system. 2) A decrease in the cost of information increases the informativeness of the price system. 3) A decrease in risk aversion leads informed individuals to take larger positions, and this increases the informativeness of the price system. Further, all other changes in parameters, such that n, a, and c remain constant, do not change the equilibrium degree of informativeness of the price system; other changes lead only to particular changes in X of a magnitude to exactly offset them. For example: 4) An increase in noise (a2) increases the proportion of informed traders. At any given X, an increase in noise reduces the informativeness of the price system; but it increases the returns to information and leads more individuals to become informed; the remarkable result obtained above establishes that the two effects exactly offset each This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:12:49 AM All use subject to JSTOR Terms and Conditions

THE AMERICAN ECONOMIC REVIEW JUNE 1980 other so that the equilibrium informative- improvement in the precision of informed ness of the price system is unchanged. This traders'information, with the cost of the can be illustrated diagrammatically if we information fixed, increases the benefit of note from(16a) that for a given A,an being informed. However, some of the im- crease in o raises m which from(18)lowers proved information is transmitted, via a (Var(uo))/Var(u*wx). Thus from(14)a more informative price system, to the unin rise in a leads to a vertical downward shift formed; this increases the benefits of being of the y()curve in Figure 1, and thus a uninformed. If n is small, both the price higher value ofλ° system m is not very informative and the 5)Similarly an increase in o2 for marginal value of information to informed a constant n (equivalent to an increase in traders is high. Thus the relative benefits of the variance of u since n is constant) leads being g informed rises when n rises; implying enough to offset the increased va ega/als that the equilibrium A rises.Conversely to an increased proportion of individu becoming informed-and indeed again just when n is large the price system is very price system remains unchanged. This can relative benefits of being uninformed rise that the degree of informativeness of the mation is low to informed traders so also be seen from Figure 1 if (16)is used to 7)From(14)it is clear that an increase note that Increase with n held n the cost of information c shifts the ya) constant by raising ob leads to an increase in curve up and thus decreases the percentage m for a given A. From(18)and(14) this of informed traders r()curve and thus a higher value of e the ads to a vertical downward shift of the The above results are summarized in the following theorem. 6)It is more difficult :o determine what happens if, say o increases, keeping o2 con- THEOREM 4: For equilibrium A such that stant (implying a fall in o,), that is, the 0n implies that A falls if Var(u 10)+Var(u wA) falls due to the rise in od for (rises)due to an increase in n given A. This occurs if and only if nm/(1+m)rises PROOF Parts A-C are proved in the above re- arks. Part d is proved in footnote 8 I. Price Cannot Fully Reflect Costly Information We now consider certain limiting cases where y=ezac-1 and the last equality follows from for y(O)0 and price is egative so that a falls due to fully informative. precision of the informed trader's information. Simi larly if n is sufficiently small, the derivative is positive 1)As the cost of information goes to and thusλ rises. zero, the price system becomes more info I 1 Sep 2013 OR Terms and Conditions

400 THE AMERICAN ECONOMIC REVIEW JUNE 1980 other so that the equilibrium informativeness of the price system is unchanged. This can be illustrated diagrammatically if we note from (16a) that for a given X, an increase in a2 raises m which from (18) lowers (Var(u*J0))/Var(u*Iw,). Thus from (14) a rise in a2 leads to a vertical downward shift of the y(X) curve in Figure 1, and thus a higher value of Xe. 5) Similarly an increase in a2 for a constant n (equivalent to an increase in the variance of u since n is constant) leads to an increased proportion of individuals becoming informed-and indeed again just enough to offset the increased variance, so that the degree of informativeness of the price system remains unchanged. This can also be seen from Figure 1 if (16) is used to note that an increase in a2 with n held constant by raising ad, leads to an increase in m for a given X. From (18) and (14) this leads to a vertical downward shift of the yy(A) curve and thus a higher value of Me. 6) It is more difficult -o determine what happens if, say a9 increases, keeping a,2 constant (implying a fall in a2), that is, the information obtained is more informative. This leads to an increase in n, which from (19b) implies that the equilibrium informativeness of the price system rises. From (16) it is clear that m and nm both fall when a,9 rises (keeping au2= a,9 + a2 constant). This implies that the y(X) curve may shift up or down depending on the precise values of c, a, and n.8 This ambiguity arises because an improvement in the precision of informed traders' information, with the cost of the information fixed, increases the benefit of being informed. However, some of the improved information is transmitted, via a more informative price system, to the uninformed; this increases the benefits of being uninformed. If n is small, both the price system m is not very informative and the marginal value of information to informed traders is high. Thus the relative benefits of being informed rises when n rises; implying that the equilibrium X rises. Conversely when n is large the price system is very informative and the marginal value of information is low to informed traders so the relative benefits of being uninformed rises. 7) From (14) it is clear that an increase in the cost of information c shifts the y(X) curve up and thus decreases the percentage of informed traders. The above results are summarized in the following theorem. THEOREM 4: For equilibrium X such that 0 ii implies that X falls (rises) due to an increase in n. PROOF: Parts A - C are proved in the above remarks. Part D is proved in footnote 8. I. Price Cannot Fully Reflect Costly Information We now consider certain limiting cases, for -y(O) 0 and price is fully informative. 1) As the cost of information goes to zero, the price system becomes more infor- 8From (14) and (18) it is clear that X rises if and only if Var(u*1j)4 Var(u*lwx) falls due to the rise in a3 for a given X. This occurs if and only if nm/(l + m) rises. Using (16) to differentiate nm/(l + m) with respect to a2 subject to the constraint that dao2=0 (i.e., da3= - da2), we find that the sign of d nm )sgn[m n+I _1 dc~~ [(2 n- n )( where y _=-e2_1 and the last equality follows from equation (19a). Thus for n very large the derivative is negative so that X falls due to an increase in the precision of the informed trader's information. Similarly if n is sufficiently small, the derivative is positive and thus X rises. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:12:49 AM All use subject to JSTOR Terms and Conditions

GROSSMAN AND STIGLITZ: EFFICIENT MARKETS mative, but at a positive value of c, say c, all while if a>0, by(18) traders are informed. From(14) and(15)c (W合) -e ar(u*0) Ev(WA) Var(u*wu But if o =0 or o=0, then m=0, nm=0 for λ>0, and hence 2)From(19a)as the precision of the formed traders information n goes to in- Id fixed. (21) E(W合) the price system becomes perfectly informa A→0E(W) tive. Moreover the percentage of informed traders goes to zero! This can be seen from It immediately follows that (18)and(15). That is, as of-0, nm/(1+m) ust stay constant for equilibrium to THEOREM 5:(a)If there is no noise (of= maintained. But from(19b) and(17), m O), an overall equilibrium does not exist jf falls as 0? goes to zero. Therefore nm must (and only ife0? s noise ox goes to zero, the percentage of y()is discontinuous at A =0; A =0 is not 19a)implies that m does not change as not an equilibrium since by (21)Y()>I hanges, the informativeness of the price (b)If g2=0 and o=o so that informa- system is unchanged as 0 x0 tion is perfect, then for A>0, nm=0 by(16) Assume that c is small enough so that it is and hence ya)>l by (21). From(20)Y(O) worthwhile for a trader to become informed 00 is not an is less than unity so that A=0 cannot be an equilibrium. On the other hand, if ne equilibrium; but when A>0, it is greater traders are informed, then each uninformed than unity. That is, if o=0 or 0=0, the trader learns nothing from the price system, ratio of expected utilities is not a continuous and thus he has a desire to become in function ofλatλ=0. formed (if e<(1+n)/2). Similarly if the This follows immediately from observing informed traders get perfect information, hat at x=0, var(u*wo)=varu*, and thus then their demands are very sensitive to by(14) their information so that the market -clear (20)2(wa ing price becomes very sensitive to their information and thus reveals 0 to the unin Ey(wO) formed. Hence all traders desire to be un informed. But if all traders are uninformed each trader can eliminate the risk of his portfolio by the purchase of information,so 1+n each trader desires to be informed OR Terms and Conditions

VOL. 70 NO. 3 GROSSMAN AND STIGLITZ: EFFICIENT MARKETS 401 mative, but at a positive value of c, say c, all traders are informed. From (14) and (15) e satisfies eac Vr(u*1 Var(u*lwl) 2) From (19a) as the precision of the informed trader's information n goes to infinity, i.e., a2-*O and a92 -u, a2 held fixed, the price system becomes perfectly informative. Moreover the percentage of informed. traders goes to zero! This can be seen from (18) and (15). That is, as q20, nm/(I + m) must stay constant for equilibrium to be maintained. But from (19b) and (17), m falls as 2 goes to zero. Therefore nm must fall, but nm must not go to zero or else nm/ (1+ m) would not be constant. From (16) nm = (a/X)2q2a.2, and thus X must go to zero to prevent nm from going to zero as a2 --O. 3) From (16a) and (19a) it is clear that as noise a2 goes to zero, the percentage of informed traders goes to zero. Further, since (19a) implies that m does not change as changes, the informativeness of the price system is unchanged as U2O. Assume that c is small enough so that it is worthwhile for a trader to become informed when no other trader is informed. Then if a2=0 or a 2=0, there exists no competitive equilibrium. To see this, note that equilibrium requires either that the ratio of expected utility of the informed to the uninformed be equal to unity, or that if the ratio is larger than unity, no one be informed. We shall show that when no one is informed, it is less than uipity so that X =0 cannot be an equilibrium; but when X > 0, it is greater than unity. That is, if a2 =0 or a2=0, the ratio of expected utilities is not a continuous function of X at X = 0. This follows immediately from observing that at X = 0, Var(u *wo) = Var u *, and thus by (14) (20) EV(W,) W+_ eac EV(Wug)2 eac ____ - 1+ n while if X>0, by (18) EV(Wjs) _eac 1 EV(Wui) /l+n m+ But if q2=0 or a,2=0, then m=O, nm=O for X > 0, and hence (21) lim EV( W) =_ eac A-0EV( Wul) It immediately follows that THEOREM 5: (a) If there is no noise (a.2= 0), an overall equilibrium does not exist if (and only if) eac 0 is not an equilibrium since by (21) -y(X) > 1. (b) If a,2=0 and a 2= a2 so that information is perfect, then for X >0, nm = 0 by (16) and hence -y(Q)> 1 by (21). From (20) y(O)= 0 0 is not an equilibrium. On the other hand, if no traders are informed, then each uninformed trader learns nothing from the price system, and thus he has a desire to become informed (if eac <(1 + n)"'2). Similarly if the informed traders get perfect information, then their demands are very sensitive to their information, so that the market-clearing price becomes very sensitive to their information and thus reveals 0 to the uninformed. Hence all traders desire to be uninformed. But if all traders are uninformed, each trader can eliminate the risk of his portfolio by the purchase of information, so each trader desires to be infermed. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:12:49 AM All use subject to JSTOR Terms and Conditions

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