Eficiency of Competitive Stock Markets Where Trades have Diuerse Information 583 Proof. E[Pily Po()]=E[Pilya, P*() and Var[P,lyi,Po()]=Var [P,Vs Po*(] since Po and Po* contain the same information (i.e, they generate the same a-field). Let X,[Po, y] be as defined in(13). Then if y'is some realization of y, x∑X=4n=(+)P a, Var[ Ply, Po] E[P"]-(+)P( [Pelvo] (30) Assume, without loss of generality, that Po()>Po*()>0. Then sfPP”]-(1+)(y) E[,P}-(+)P3().( The right hand side of (31)is just 2,xi[Po*,y]. Thus by(30), 2X/[Po",y>X (This clearly also holds in a non-degenerate neighborhood of yi, as the prices are continuous functions of y Thus Po* is not an ec Thus there cannot be two equilibria with the same information content. The appendix shows that equilibrium is unique in the class of all linear functions of We do not know whether there are prices which are non-linear functions of y and are also equilibria The result that there is an equilibrium which is a sufficient statistic is not robust It will not hold if there is noise in the price system. (See Grossman [1975] for an elaboration of the notion of"noise " )Suppose the total stock of the risky asset, x is unknown to all traders. Suppose that they have a common prior distribution on a,such that i is independent of P, and E, E2,..., n, When x is random all traders will know that the price which clears the market depends not only on y but also on the realization of i. Define an equilibrium as a mapping Po(,x), such that for all sE「Dy8)1-(+)P2)- a, Var Pili, Po(, x) Clearly an equilibrium Po(, x)cannot be a constant function of x(i. e, a function of y alone). This is because if Po(y, x) is a function only of y, then the left hand side of (32)does not depend on x, while the right hand side of(32)does depend on x.As x and y are independent this is impossible. Thus it will not be possible to infer y from Po(, x)unless x has a degenerate distribution