578 The Journal of Finance particular P, is drawn, traders do their research and the ith type trader is able learn the true value of P, to within Gj, where E is distributed as Normal(0, 1) Under either interpretation, the following is true TE HEOREM Under the above assumption about the joint distribution of y and P, if Po() is given by y (15) (1+m2)(l (1+no2)(1+r) (17) hen Po()is an equilibrium. That is, it is a solution to(12) Before proving the theorem we present some comments on its significance. First y is the sample mean of the y;. The equilibrium price depends on the information y only through y. Second, any trader by observing the value of Po() can learn y from(14), since by(17), a,>0. y is a more precise estimate of PI, than is y. Thus the market price aggregates all the information collected by the traders in an optimal"way. y is a sufficient statistic for the family of densities f( P).The market aggregation is optimal to the extent that it produces a sufficient statistic. The following lemma is used to prove the theorem: there are functions g ( )and g2( )such that, for all y, and y=ZiiMie n P, then LEMMA l. If h, (i, y P) is the joint density of y and yi conditional h(y2列P1)=g1(y)g2(元,P1 That is, y is a suficient statistic for h, (,FIP) Proof. Conditional on PI, y is Normal (PI, 1)and y is Normal(Pl,1/n) Conditional on PI, covariance (, D)=1/n. Thus conditional on PI, ( D)is normall 11/ P1/(1/n1/