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《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

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