Efficiency of Competitive Stock Markets Where Trades have Diuerse Information 577 of other traders. The next section shows that Po()reveals"all"the information of the traders Po(y) can be interpreted as a stationary point of the following process. Suppose traders initially begin in a naive way, thinking of Po as a number and conditioning nly Let an auctioneer call out prices until the market clears. Call thi olution Po(). That is Po() solves 受-0+)-x a,Var[Ply] (13a) Each period traders come to the market with another realization of y, and another Po()is found where the auction stops. After many repetitions traders can tabulate the empirical distribution of (Po, P1 pairs. From this they get a good estimate of the joint distribution of Po and Pr. After this joint distribution is learned, traders will have an incentive to change their bids just as the market is about to clear. Thi allows from the fact that if everyone observes that the market is about to clear at PoC), they can condition their beliefs on Po() and learn something more about PI. This changes their demands and thus the market will not clear at Po(). Suppose instead that the market has been clearing for a long time with prices traders come to the market with some y, if the market is about to clear at Po(), and traders then realize that Po() is the equilibrium, they will not change their bids due to the new information they get about P from Po(y). Po() is a self fulfilling expectations equilibrium: when all traders think prices are generated by Po(), they will act in such a way that the market clears at Po() 3. THERE IS AN EQUILIBRIUM PRICE WHICH IS A SUFFICIENT STA Assume that in (4), e is a random variable which is normally distributed, with mean0 and variance 1. Thus, each trader"i"observes y,=P+ej, and given PI,yi is Normal with mean P, and variance 1. Each trader gets information of equal precision in that Vare=l for each trader"i. Further assume that ep,E2,,., E, is jointly normally distributed and covariance(e, 6 )=0 if i+j. Thus, we assume that the joint density of y given P, say f( P1), is multivariate Normal with mean vector(P,PI, PI,,, P) and covariance matrix which is the identity matrix. PI is assumed unknown at time zero, however, traders believe that P is distributed dependently of E1, E2,..., n, and PI is Normal (P1,0). This marginal distribution of PI has two interpretations. Under a Bayesian interpretation, next periods price is some fixed number, and people represent their uncertainty about the value of that number with a prior distribution which is Normal(P1,0). A non-Bayesian interpretation is that nature draws the true price next period from an urn with distribution Normal(P, 02). Nature makes the drawing before period 0. After a