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
THEAMERICAN ECONOMIC REVIEW JUNE 980 E. Equilibrium in the Information Market F. Existence of Overall equilibrium What we have characterized so far is the Theorem 2 is useful, both in equilibrium price distribution for given A. uniqueness of overall equilibrium and in We now define an overall equilibrium to be analyzing comparative statics. Overall equi- a pair ( P*)such that the expected utility librium, it will be recalled, requires that for of the informed is equal to that of the unin- 0<A<l, EV(WA/EV(WO)=1. But from formed if0<λ<1;λ=0 if the expected utility of the informed is less than that of utility of the informed is greater than the (14) EV(WA) the uninformed at Po; A=l if the expected Ey(WU) uninformed at P. Let (12a)WA=R(Woi-c) var(u*a) +[u-RPa(,x)X, (P(e, x), 0) par(*1w3)=() (12b)Wt1≡RWo Hence overall equilibrium simply requires or0<λ<1 +[u-RP(e, x)JXu(P(e,x); P*) (15) y(入)=1 where c is the cost of observing a realization More precisely, we now prove of 8*. Equation(12a) gives the end of period wealth of a trader if he decides to become THEOREM 3 If0<A<l, y()=l, and normed, while(12b) gives his wealth if he is given by(A10) in Appendix B, then(, Px) decides to be uninformed. Note that end of is an overall equilibrium. If y(1)<l, then period wealth is random due to the random- (1, P*) is an overall equilibrium. If y(o)>I ness of Wo: u.e. and x then(0, Po)is an overall equilibrium. For all In evaluating the expected utility of wi, price equilibria we do not assume that a trader knows which tions of wa, there exists a unique overall if he pays c dollars. A trader pays c dollars and then gets to observe some realization of 8*. PROOF The overall expected utility of wi averages The first three sentences follow im over all possible 0*,e*, x*, and Wor. The mediately from the definition of overall variable Woi is random for two reasons. equilibrium given above equation(12), and First from(2) it depends on P(e, x), which Theorems 1 and 2. Uniqueness follows from is random as(0, x) is random. Secondly, in the monotonicity of y()which follows from what follows we will assume that X, is ran-(All)and(14). The last two sentences in the statement of the theorem follow im- We will show below that EV(WA)/ mediately. Ev(wA) is independent of i, but is a func tion of A, a, c, and a. More precisely In the process of prov Th Appendix B we prove THEOREM 2: Under the assumptions of COROLLARY 1: y()is a strictly mono- Theorem 1, and if x is independent of tone increasing function of n (u*, 8*, x*)then This looks paradoxical Ev(N (13) Var(u*0) utility to be a decreasing function of A. But, Ev(WA) we have defined utility as negative. Therefore OR Terms and Conditions
398 THE AMERICAN ECONOMIC REVIEW JUNE 1980 E. Equilibrium in the Information Market What we have characterized so far is the equilibrium price distribution for given X. We now define an overall equilibrium to be a pair (X, PA*) such that the expected utility of the informed is equal to that of the uninformed if 0 <X < 1; X =0 if the expected utility of the informed is less than that of the uninformed at Po*; X= 1 if the expected utility of the informed is greater than the uninformed at P*. Let (12a) WI'S=R(Woj-c) + I u-RP,(9, x) ] X, (P.(9, x), 9) ( 12b) WuA-=R W0j + [ U- RPA(9, x) ] Xu(Px(0, x); PA*) where c is the cost of observing a realization of 9*. Equation (12a) gives the end of period wealth of a trader if he decides to become informed, while (12b) gives his wealth if he decides to be uninformed. Note that end of period wealth is random due to the randomness of W0i, u, 9, and x. In evaluating the expected utility of W,i, we do not assume that a trader knows which realization of 9* he gets to observe if he pays c dollars. A trader pays c dollars and then gets to observe some realization of 9*. The overall expected utility of W1?, averages over all possible 9*, E*, x*, and W0i. The variable W0i is random for two reasons. First from (2) it depends on P,(9,x), which is random as (9,x) is random. Secondly, in what follows we will assume that Xi is random. We will show below that EV( W,'')/ E V( Wu) is independent of i, but is a function of X, a, c, and a2. More precisely, in Appendix B we prove THEOREM 2: Under the assumptions of Theorem 1, and if Xi is independent of (u*, 9*, x*) then (13) EV( W'') =e ac r(u*10) EV( Wui) Var(u*Iwx) F. Existence of Overall Equilibrium Theorem 2 is useful, both in proving the uniqueness of overall equilibrium and in analyzing comparative statics. Overall equilibrium, it will be recalled, requires that for 0<X<1, EV(WI')/EV(Wu")=1. But from (13) (14) E V( Wjx) EV( Wui) =eac (U -y ) Vr(u* Iwx) Hence overall equilibrium simply requires, for 0<X< 1, (15) y(X)=I More precisely, we now prove THEOREM 3: If 0< X< 1, y(X) = 1, and P* is given by (A 10) in Appendix B, then (X, P*) is an overall equilibrium. If y(1) < 1, then (1,P*) is an overall equilibrium. If y(O)> 1, then (0, P*) is an overall equilibrium. For all price equilibria Px which are monotone functions of wx, there exists a unique overall equilibrium (X, Px*). PROOF: The first three sentences follow immediately from the definition of overall equilibrium given above equation (12), and Theorems 1 and 2. Uniqueness follows from the monotonicity of y(-) which follows from (Al 1) and (14). The last two sentences in the statement of the theorem follow immediately. In the process of proving Theorem 3, we have noted COROLLARY 1: y(X) is a strictly monotone increasing function of A. This looks paradoxical; we expect the ratio of informed to uninformed expected utility to be a decreasing function of X. But, we have defined utility as negative. Therefore 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