THEAMERICAN ECONOMIC REVIEW JUNE 980 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 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 ( P*)such that the expected utility librium, it will be recalled, requires that for of the informed is equal to that of the unin- 0<A<l, EV(WA/EV(WO)=1. But from formed if0<λ<1;λ=0 if the expected utility of the informed is less than that of utility of the informed is greater than the (14) EV(WA) the uninformed at Po; A=l if the expected Ey(WU) uninformed at P. Let (12a)WA=R(Woi-c) var(u*a) +[u-RPa(,x)X, (P(e, x), 0) par(*1w3)=() (12b)Wt1≡RWo Hence overall equilibrium simply requires or0<λ<1 +[u-RP(e, x)JXu(P(e,x); P*) (15) y(入)=1 where c is the cost of observing a realization More precisely, we now prove of 8*. Equation(12a) gives the end of period wealth of a trader if he decides to become THEOREM 3 If0<A<l, y()=l, and normed, while(12b) gives his wealth if he is given by(A10) in Appendix B, then(, Px) decides to be uninformed. Note that end of is an overall equilibrium. If y(1)<l, then period wealth is random due to the random- (1, P*) is an overall equilibrium. If y(o)>I ness of Wo: u.e. and x then(0, Po)is an overall equilibrium. For all In evaluating the expected utility of wi, price equilibria we do not assume that a trader knows which tions of wa, there exists a unique overall if he pays c dollars. A trader pays c dollars and then gets to observe some realization of 8*. PROOF The overall expected utility of wi averages The first three sentences follow im over all possible 0*,e*, x*, and Wor. The mediately from the definition of overall variable Woi is random for two reasons. equilibrium given above equation(12), and First from(2) it depends on P(e, x), which Theorems 1 and 2. Uniqueness follows from is random as(0, 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 the statement of the theorem follow im- We will show below that EV(WA)/ mediately. Ev(wA) is independent of i, but is a func tion of A, a, c, and a. More precisely In the process of prov Th Appendix B we prove THEOREM 2: Under the assumptions of COROLLARY 1: y()is a strictly mono- Theorem 1, and if x is independent of tone increasing function of n (u*, 8*, x*)then This looks paradoxical Ev(N (13) Var(u*0) utility to be a decreasing function of A. But, Ev(WA) we have defined utility as negative. Therefore OR Terms and Conditions398 THE AMERICAN ECONOMIC REVIEW JUNE 1980 E. Equilibrium in the Information Market What we have characterized so far is the equilibrium price distribution for given X. We now define an overall equilibrium to be a pair (X, PA*) such that the expected utility of the informed is equal to that of the unin￾formed if 0 <X < 1; X =0 if the expected utility of the informed is less than that of the uninformed at Po*; X= 1 if the expected utility of the informed is greater than the uninformed at P*. Let (12a) WI'S=R(Woj-c) + I u-RP,(9, x) ] X, (P.(9, x), 9) ( 12b) WuA-=R W0j + [ U- RPA(9, x) ] Xu(Px(0, x); PA*) where c is the cost of observing a realization of 9*. Equation (12a) gives the end of period wealth of a trader if he decides to become informed, while (12b) gives his wealth if he decides to be uninformed. Note that end of period wealth is random due to the random￾ness of W0i, u, 9, and x. In evaluating the expected utility of W,i, we do not assume that a trader knows which realization of 9* he gets to observe if he pays c dollars. A trader pays c dollars and then gets to observe some realization of 9*. The overall expected utility of W1?, averages over all possible 9*, E*, x*, and W0i. The variable W0i is random for two reasons. First from (2) it depends on P,(9,x), which is random as (9,x) is random. Secondly, in what follows we will assume that Xi is ran￾dom. We will show below that EV( W,'')/ E V( Wu) is independent of i, but is a func￾tion of X, a, c, and a2. More precisely, in Appendix B we prove THEOREM 2: Under the assumptions of Theorem 1, and if Xi is independent of (u*, 9*, x*) then (13) EV( W'') =e ac r(u*10) EV( Wui) Var(u*Iwx) F. Existence of Overall Equilibrium Theorem 2 is useful, both in proving the uniqueness of overall equilibrium and in analyzing comparative statics. Overall equi￾librium, it will be recalled, requires that for 0<X<1, EV(WI')/EV(Wu")=1. But from (13) (14) E V( Wjx) EV( Wui) =eac (U -y ) Vr(u* Iwx) Hence overall equilibrium simply requires, for 0<X< 1, (15) y(X)=I More precisely, we now prove THEOREM 3: If 0< X< 1, y(X) = 1, and P* is given by (A 10) in Appendix B, then (X, P*) is an overall equilibrium. If y(1) < 1, then (1,P*) is an overall equilibrium. If y(O)> 1, then (0, P*) is an overall equilibrium. For all price equilibria Px which are monotone func￾tions of wx, there exists a unique overall equilibrium (X, Px*). PROOF: The first three sentences follow im￾mediately from the definition of overall equilibrium given above equation (12), and Theorems 1 and 2. Uniqueness follows from the monotonicity of y(-) which follows from (Al 1) and (14). The last two sentences in the statement of the theorem follow im￾mediately. In the process of proving Theorem 3, we have noted COROLLARY 1: y(X) is a strictly mono￾tone increasing function of A. This looks paradoxical; we expect the ratio of informed to uninformed expected utility to be a decreasing function of X. But, we have defined utility as negative. Therefore 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