V0L.70N0.3 GROSSMAN AND STIGLITZ:EFFICIENT MARKETS 401 mative,but at a positive value of c,say c,all while ifλ>0，by(l8) traders are informed.From (14)and (15)6 satisfies EV(WA) Var(u*0) EV(W) 1+n =I m+1 Var(u*w) But if o2=0 or o2=0,then m=0,nm=0 for λ>0，and hence 2)From (19a)as the precision of the informed trader's information n goes to in- EV(WA) finity,i.e,a→0 and o→o,o2 held fixed, (21) lim =eac the price system becomes perfectly informa- EV(W) tive.Moreover the percentage of informed traders goes to zero!This can be seen from It immediately follows that (l8)and(15).That is,asoe→0，nm/(1+m) must stay constant for equilibrium to be THEOREM 5:(a)If there is no noise (2= maintained.But from (19b)and (17),m 0),an overall equilibrium does not exist if falls as o2 goes to zero.Therefore nm must (and only if)eae<V1+n.(b)If information fall,but nm must not go to zero or else nm/ is perfect (2=0,n=oo),there never exists an (1+m)would not be constant.From (16) equilibrium. nm=(a/入)2a,2a2,and thus入must go to zero to prevent nm from going to zero as o20. PROOF: 3)From (16a)and (19a)it is clear that (a)If eac<V1+n,then by (20)and (21), as noise o goes to zero,the percentage of y()is discontinuous at入=0；入=0 is not an informed traders goes to zero.Further,since equilibrium since by (20)y(0)<1;A>0 is (19a)implies that m does not change as o2 not an equilibrium since by (21)Y(A)>1. changes,the informativeness of the price (b)If o2=0 and og=o so that informa- system is unchanged as o20. tion is perfect,.then forλ>0，nm=0by(l6) Assume that c is small enough so that it is and hence y(A)>I by (21).From(20)Y(0)= worthwhile for a trader to become informed 0<1. when no other trader is informed.Then if =0 or o2=0,there exists no competitive If there is no noise and some traders be- equilibrium.To see this,note that equi- come informed,then all their information is librium requires either that the ratio of ex- transmitted to the uninformed by the price pected utility of the informed to the unin- system.Hence each informed trader acting formed be equal to unity,or that if the ratio as a price taker thinks the informativeness is larger than unity,no one be informed.We of the price system will be unchanged if he shall show that when no one is informed,it becomes uninformed,so A>0 is not an is less than unity so that A=0 cannot be an equilibrium.On the other hand,if no equilibrium;but when A>0,it is greater traders are informed,then each uninformed than unity.That is,if o2=0 or o2=0,the trader learns nothing from the price system, ratio of expected utilities is not a continuous and thus he has a desire to become in- function of入at入=0. formed (if eac<(1+n)2).Similarly if the This follows immediately from observing informed traders get perfect information, that at入=0，Var(u|wo)=Varu*,and thus then their demands are very sensitive to by(14) their information,so that the market-clear- (20) EV(W) ing price becomes very sensitive to their information and thus reveals a to the unin- EV(WO) 02+0 formed.Hence all traders desire to be un- informed.But if all traders are uninformed, each trader can eliminate the risk of his portfolio by the purchase of information,so +n each trader desires to be informed