398 THE AMERICAN ECONOMIC REVIEW JUNE 1980 E.Equilibrium in the Information Market F.Existence of Overall Equilibrium What we have characterized so far is the Theorem 2 is useful,both in proving the equilibrium price distribution for given A. uniqueness of overall equilibrium and in We now define an overall equilibrium to be analyzing comparative statics.Overall equi- a pair (A,P*)such that the expected utility librium,it will be recalled,requires that for of the informed is equal to that of the unin- O<入<1，EV(W)/EV(Wt)=1.But from formed if0<λ<l;入=0 if the expected (13) utility of the informed is less than that of the uninformed at P;A=I if the expected (14) EV(WA) utility of the informed is greater than the EV(W) uninformed at P*.Let (12a)W≡R(Wo-c) Var(u*0) ≡Y(A) +[4-RPx(8,x)]X,(P(0,x),8) Var(u*wa) (12b)W1三RWo Hence overall equilibrium simply requires, for0<λ<1， +[u-RPx(8,x)]X(P(0,x):P*) (15) y()=1 where c is the cost of observing a realization More precisely,we now prove of *Equation(12a)gives the end of period wealth of a trader if he decides to become THEOREM3:If0≤入≤1，Y()=1,andP* informed,while (12b)gives his wealth if he is given by (A10)in Appendix B,then (A,P*) decides to be uninformed.Note that end of is an overall equilibrium.If y(1)<1,then period wealth is random due to the random- (1,P)is an overall equilibrium.If y(0)>1, ness of Woi,u,6,and x. then (0,P)is an overall equilibrium.For all In evaluating the expected utility ofW, price equilibria P,which are monotone func- we do not assume that a trader knows which tions of wx,there exists a unique overall realization of he gets to observe if he equilibrium(入，P). pays c dollars.A trader pays c dollars and then gets to observe some realization of * PROOF: The overall expected utility of W averages The first three sentences follow im- over all possible 0*,e*,x*,and Woi.The mediately from the definition of overall variable Wo:is random for two reasons. equilibrium given above equation (12),and First from(2)it depends on P(,x),which Theorems I and 2.Uniqueness follows from is random as (,x)is random.Secondly,in the monotonicity of y()which follows from what follows we will assume that X,is ran- (All)and (14).The last two sentences in dom. the statement of the theorem follow im- We will show below that EV(W)/ mediately. Ev(W)is independent of i,but is a func- tion of入，a,c,anda.More precisely,.in In the process of proving Theorem 3,we Appendix B we prove have noted THEOREM 2:Under the assumptions of COROLLARY 1:Y(A)is a strictly mono- Theorem 1,and if X:is independent of tone increasing function of A. (u*,0*,x*)then This looks paradoxical;we expect the (13 EV(WA) Var(u*0) ratio of informed to uninformed expected utility to be a decreasing function of A.But, EV(W) Var(u*wa) we have defined utility as negative.Therefore