GROSSMAN AND STIGLITZ: EFFICIENT MARKETS mative, but at a positive value of c, say c, all while if a>0, by(18) traders are informed. From(14) and(15)c (W合) -e ar(u*0) Ev(WA) Var(u*wu But if o =0 or o=0, then m=0, nm=0 for λ>0, and hence 2)From(19a)as the precision of the formed traders information n goes to in- Id fixed. (21) E(W合) the price system becomes perfectly informa A→0E(W) tive. Moreover the percentage of informed traders goes to zero! This can be seen from It immediately follows that (18)and(15). That is, as of-0, nm/(1+m) ust stay constant for equilibrium to THEOREM 5:(a)If there is no noise (of= maintained. But from(19b) and(17), m O), an overall equilibrium does not exist jf falls as 0? goes to zero. Therefore nm must (and only ife<V1+n(b)If information fall,but nm must not go to zero or else nm/ is perfect(o=0, n=oo), there never exists an (1+ m)would not be constant. From (16 nm=(a/A)o o 2, and thus A must go to zero to prevent nm from going to zero as 20. PROOF 3)From(16a)and (19a)it is clear that (a)If e ac<vi+n, then by(20)and(21), informed traders goes to zero. Further, since equilibrium since by(20)r(O)<1; 1>0? s noise ox goes to zero, the percentage of y()is discontinuous at A =0; A =0 is not 19a)implies that m does not change as not an equilibrium since by (21)Y()>I hanges, the informativeness of the price (b)If g2=0 and o=o so that informa- system is unchanged as 0 x0 tion is perfect, then for A>0, nm=0 by(16) Assume that c is small enough so that it is and hence ya)>l by (21). From(20)Y(O) worthwhile for a trader to become informed 0<l when no other trader is informed. Then if 02=0 or 02=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 ra 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 x>0 is not an is less than unity so that A=0 cannot be an equilibrium. On the other hand, if ne equilibrium; but when A>0, it is greater traders are informed, then each uninformed than unity. That is, if o=0 or 0=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 ofλatλ=0. formed (if e<(1+n)/2). Similarly if the This follows immediately from observing informed traders get perfect information, hat at x=0, var(u*wo)=varu*, and thus then their demands are very sensitive to by(14) their information so that the market -clear (20)2(wa ing price becomes very sensitive to their information and thus reveals 0 to the unin Ey(wO) 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 1+n each trader desires to be informed OR Terms and ConditionsVOL. 70 NO. 3 GROSSMAN AND STIGLITZ: EFFICIENT MARKETS 401 mative, but at a positive value of c, say c, all traders are informed. From (14) and (15) e satisfies eac Vr(u*1 Var(u*lwl) 2) From (19a) as the precision of the informed trader's information n goes to in￾finity, i.e., a2-*O and a92 -u, a2 held fixed, the price system becomes perfectly informa￾tive. Moreover the percentage of informed. traders goes to zero! This can be seen from (18) and (15). That is, as q20, nm/(I + m) must stay constant for equilibrium to be maintained. But from (19b) and (17), m falls as 2 goes to zero. Therefore nm must fall, but nm must not go to zero or else nm/ (1+ m) would not be constant. From (16) nm = (a/X)2q2a.2, and thus X must go to zero to prevent nm from going to zero as a2 --O. 3) From (16a) and (19a) it is clear that as noise a2 goes to zero, the percentage of informed traders goes to zero. Further, since (19a) implies that m does not change as changes, the informativeness of the price system is unchanged as U2O. Assume that c is small enough so that it is worthwhile for a trader to become informed when no other trader is informed. Then if a2=0 or a 2=0, there exists no competitive equilibrium. To see this, note that equi￾librium requires either that the ratio of ex￾pected utility of the informed to the unin￾formed be equal to unity, or that if the ratio is larger than unity, no one be informed. We shall show that when no one is informed, it is less than uipity so that X =0 cannot be an equilibrium; but when X > 0, it is greater than unity. That is, if a2 =0 or a2=0, the ratio of expected utilities is not a continuous function of X at X = 0. This follows immediately from observing that at X = 0, Var(u *wo) = Var u *, and thus by (14) (20) EV(W,) W+_ eac EV(Wug)2 eac ____ - 1+ n while if X>0, by (18) EV(Wjs) _eac 1 EV(Wui) /l+n m+ But if q2=0 or a,2=0, then m=O, nm=O for X > 0, and hence (21) lim EV( W) =_ eac A-0EV( Wul) It immediately follows that THEOREM 5: (a) If there is no noise (a.2= 0), an overall equilibrium does not exist if (and only if) eac < 1 + . (b) If information is perfect (a,2 = 0, n = x), there never exists an equilibrium. PROOF: (a) If eac < 1+ n , then by (20) and (21), y(X) is discontinuous at X = 0; X = 0 is not an equilibrium since by (20) y(O) < 1; X >0 is not an equilibrium since by (21) -y(X) > 1. (b) If a,2=0 and a 2= a2 so that informa￾tion is perfect, then for X >0, nm = 0 by (16) and hence -y(Q)> 1 by (21). From (20) y(O)= 0<1. If there is no noise and some traders be￾come informed, then all their information is transmitted to the uninformed by the price system. Hence each informed trader acting as a price taker thinks the informativeness of the price system will be unchanged if he becomes uninformed, so X> 0 is not an equilibrium. On the other hand, if no traders are informed, then each uninformed trader learns nothing from the price system, and thus he has a desire to become in￾formed (if eac <(1 + n)"'2). Similarly if the informed traders get perfect information, then their demands are very sensitive to their information, so that the market-clear￾ing price becomes very sensitive to their information and thus reveals 0 to the unin￾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 each trader desires to be infermed. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:12:49 AM All use subject to JSTOR Terms and Conditions