The Journal of finance he spot price, as he could if he also purchased a y Note that perfect competition is assumed among the traders, so that the information of all type"i"traders taken together affects Po. However each individual trader of type"i "assumes that his trading activity has no affect on Po. Thus, when one type"i"trader stops getting information via yo, Po is not affected, and y can still be deduced from Po If it costs C>0 dollars to become informed then equilibrium will not exist. Each trader of type"i"stops collecting information because the information in Po superior to y, and free. Is there another equilibrium with fewer types of informed traders? No. Let there be m types of informed traders, then the price will be a linear function of 2m,y, and thus transmits all information to uninformed traders dollars and l given informed trader of type m; he feels that he could stop paying c Consider an though he would no longer get the information ym rice system reveals the superior information, 2m-1yp. Hence, it is not an equilibrium to have m pes of informed traders. If no traders are informed, then(for sufficiently small cost of becoming informed) each trader would want to become informed because he gets no information for free via the price system. Hence, with information costs positive equilibrium does not exist. The key to the argument is that no matter how many types of informed traders there are, the price system perfectly aggregates their information and removes the incentive from a trader of a particular type to become informed. This is because traders are price takers and assume the price system is not affected by their actions The result that there exists a price which is a sufficient statistic is not robust. For example if the dimensionality of the price system(i.e, the number of commodities less one)is smaller than the dimensionality of the sufficient statistic, then the price function cannot reveal the sufficient statistic. However, Grossman [1975] shows that when prices do not symmetrize people's information, there is a private incentive to open new markets and thus increase the dimensionality of the price system. The rest of this section is devoted to the uniqueness of equilibrium and the notion of "noise. The following section discusses the welfare, aspects of equi- librium We now show that if there are two equilibria then they must contain different information. If Po() and Po* ()are equilibria and they contain the same information, then there exists a strictly monotone function H( such that Po() H(P(). Below we show that either H( is the identity mapping (i.e, Po*(y) =P() or one of them is not an equilibrium THEOREM 2. If Po()is an equilibrium, and Po*()=H(Po()), where H()is strictly monotone function which is not the identity mapping, then Po() is not an Another paradoxical aspect of markets where prices are sufficient statistics is that eac demand function is a function only of the price and not his own information. If all traders nformation how does the information Dint is strongly related to th the demand function in(13)is not an ordinary demand function. It gives the demands of equilibrium. In models where the price conveys information, there is no longer the classical prices. Classically, demand functions can independently of the distribution of equilibrium prices. Here this is no longer possible. See glitz [1976] for an elaboration of this point