57 The Journal of finance market. Po(r 2,.,yn) is a sufficient statistic for the unknown value of Pr. A trader who invests nothing in information and observes the market price can trader who purchases y and then observes Po(y)(where y=(,2,,,n)), fidf p achieve a utility as high as traders who pay for the information y. Similarly, that yi is redundant; Po(y) contains all the information he requires. That is informationally efficient price systems aggregate diverse information perfectly, but in doing this the price system eliminates the private incentive for collecting the lt is demonstrated in the context of a simple mean-variance model.The result that the price system perfectly aggregates information is not robust. This is shown in the context of the above model when "noise""is added One example of"noise "is an uncertain total stock of the risky asset. However, the paradoxical nature of"perfect markets, "which the model illustrates, is robust. When a price system is a perfect aggregator of information it removes private incentives to collect information. If information is costly, there must be noise in the price system so that traders can earn a return on information gathering. If there is no noise and information collection is costly, then a perfect competitive market will break down because no equilibrium exists where information collectors earn a return on their information, and no equilibrium exists where no one collects information. The latter part follows from the fact that if no one collects informa tion then there is an incentive for a given individual to collect costly information because he does not affect the equilibrium price. When many individuals attempt to earn a return on information collection, the equilibrium price is affected and it perfectly aggregates their information. This provides an incentive for individuals to stop collecting information. In Grossman [1975] there is a more detailed analysis of the breakdown of markets when price systems reveal too much information On the other hand, when there is noise so that the price system does not aggregate information perfectly, the allocative efficiency properties of a competi tive equilibrium may break down. Hayek [1945] argues that the essence of a competitive price system is that when a commodity becomes scarce its price rises and this induces people to consume less of the commodity and to invest more in the production of the commodity. Individuals need not know why the price ha risen, the fact that there is a higher price induces them to counteract the scarcity in n efficient way. This argument breaks down when the price system is noisy. We will show that in such cases each individual wants to know why the price has risen (i.e, what exogenous factors make the price unusually high), and that an optimal allocation of resources involves knowing why the price has risen (i. e, knowledge of e states of nature determining current prices is required) 2. THE MODEL Assume that trader"i"has an initial wealth Wo Using Woi, he can purchase two assets; a risk free asset and a risky asset. His wealth in period l, wI is given by W1=(1+r)X+P1X where Xe is the value of risk free assets purchased in period 0, X, is the number of