400 THE AMERICAN ECONOMIC REVIEW JUNE 1980 other so that the equilibrium informative- improvement in the precision of informed ness of the price system is unchanged.This traders'information,with the cost of the can be illustrated diagrammatically if we information fixed,increases the benefit of note from(l6a)that for a given入，anin- being informed.However,some of the im- crease in o raises m which from(18)lowers proved information is transmitted,via a (Var(u*0))/Var(u*wa).Thus from (14)a more informative price system,to the unin- rise in o:leads to a vertical downward shift formed;this increases the benefits of being of the y(A)curve in Figure 1,and thus a uninformed.If n is small,both the price higher value ofλe. system m is not very informative and the 5)Similarly an increase in o2 for marginal value of information to informed a constant n (equivalent to an increase in traders is high.Thus the relative benefits of the variance of u since n is constant)leads being informed rises when n rises;implying to an increased proportion of individuals that the equilibrium A rises.Conversely becoming informed-and indeed again just when n is large the price system is very enough to offset the increased variance,so informative and the marginal value of infor- that the degree of informativeness of the mation is low to informed traders so the price system remains unchanged.This can relative benefits of being uninformed rises. also be seen from Figure I if (16)is used to 7)From (14)it is clear that an increase note that an increase in o2 with n held in the cost of information c shifts the y(A) constant by raising og leads to an increase in curve up and thus decreases the percentage m for a given A.From (18)and (14)this of informed traders. leads to a vertical downward shift of the The above results are summarized in the Y(A)curve and thus a higher value of A. following theorem. 6)It is more difficult to determine what happens if,say og increases,keeping o con- THEOREM 4:For equilibrium A such that stant (implying a fall in o2),that is,the 0<λ<1： information obtained is more informative. A.The equilibrium informativeness of the This leads to an increase in n,which from price system,pa,rises if n rises,c falls,or a (19b)implies that the equilibrium infor- falls. mativeness of the price system rises.From B.The equilibrium informativeness of the (16)it is clear that m and nm both fall when price system is unchanged if of changes,or if 0g rises (keeping o=og+o2 constant).This o changes with n fixed. implies that the y(A)curve may shift up or C.The equilibrium percentage of informed down depending on the precise values of c, traders will rise if o rises,rises for a fixed a,and n.s This ambiguity arises because an n,or c falls. D.If n satisfies (e2ac-1)/(n-(e2ac-1))= (<) 8From (14)and (18)it is clear that A rises if and only n/(n+1),then nn implies that A falls if Var(u)+Var(uw)falls due to the rise in og for (rises)due to an increase in n. a given A.This occurs if and only if nm/(1+m)rises. Using (16)to differentiate nm/(1+m)with respect to PROOF: a2 subject to the constraint that doz=0(i.e.,dog= -do),we find that the sign of Parts A-C are proved in the above re- marks.Part D is proved in footnote 8. 品（平）8(）] I.Price Cannot Fully Reflect Costly Information =g（之，） We now consider certain limiting cases, where y=e24-1 and the last equality follows from fory(o)≤1≤y(l),and show that equi- equation (19a).Thus for n very large the derivative is librium does not exist if c>0 and price is negative so that A falls due to an increase in the fully informative. precision of the informed trader's information.Simi- larly if n is sufficiently small,the derivative is positive 1)As the cost of information goes to and thus入rises. zero,the price system becomes more infor-