foi di= n Let So(i) and B (i) represent piecewise continuous functions giving the initial stock and bond endowments of uninformed traders Then their total endowments are so(if(o di= So and B6()f(1)d=B The functions So(i) and Bo (i are common knowledge in the econ The functions S(i and B(i are the respective demand densities for the stock and bond by type i uninformed traders. The aggregate demand is represented by the variables S and B. Thus, s(ifci di= S and B(f(1)d=B. Notice that the informed has an informational advantage because she knows her endowment (w) and two of the factors generating the return of the risky asset (e and up); whereas the uninformed only know up. Hence, o2 is a measure of the small investors" informational disadvantage. A similar metric for "informational disadvantage' appears throughout most of the noisy rational expectations literature. 2. Characterization of Equilibrium The Bayesian-Nash equilibrium must satisfy two conditions. First, each of the uninformed investors must submit a demand schedule that maximizes his expected utility, subject to his budget constraint and the available information, including P. Second, knowing this demand schedule, the informed investor has to submit a demand function so as to maximize her utility subject to her wealth and infor mational endowments, which are exogenous. The problem is con siderably simplified by noting that there is just one non-price-taker and that supplies add to unity. Therefore, one can reduce the insider's 259
problem to that of picking a price P on the aggregate demand sched ule of the uninformed traders, S(P) The next fact is used extensively throughout the article Fact 1. If X, and X, bave a bivariate normal distribution, wbere ui 42, 0m, 02, and p are the unconditional means, standard deviations, and correlation of the two random variables, then the condition al distribution of x, given that X,= x2 is a normal distribution whose mean is E(X,I x2)=u,+ po, (x2-u2/02, and variance is Var(X1|x2)=(1-p2) It is now possible to analyze the problem of the informed monop alist 2.1 Problem of the informed monopolist Using the well-known properties of exponential utility functions and normal distributions, the monopolist wishes to maximize over P and B the following value function maxv=(1-S+u)(H+e)+(1-B) 0.50(1-S+u)2o (1) subject to her budget constraint [(1-S)-(1-S)]P+[(1-B)-(1-B0)]=0.(2) Since the insider is effectively selecting a price on the aggregate demand function generated by the uniforme optimization problem becomes very simple (2), one elimi- from (1), and then derives the first-o condition for a maximum of (1) with respect to Pas S(P-kp-e)+(S-S)+62S(1-S+t)=0.(3) A more convenient and informative representation of the above equa tion Is P=μp+a(P)+r, a(P)=(S0-5)/S′-62(1-S) r=∈-602 (5) Equation(4) brings into sharp focus a number of insights devel
Insiders Outsiders, and Market breakdowns oped in the noisy rational expectations literature. First, given P, the riable T is fully revealed. This tells us that although there may be a nonlinear term in the price, the price is still a linear function of the normally distributed information variable of the insider, T, and it reveals this. Importantly, this revelation would not occur if the aggre gate demand function submitted by the outsiders was a correspon dence. In the Appendix, we prove that it is not optimal for the out. siders to submit such a correspondence. Second, even though the outsiders know T, they cannot fully identify e, since w also enters the equation. The general public is therefore left uncertain as to whether the primary motivation for trading by the informed is"hedging"or nformational. Third, if e is the only variable that the informed has n informational advantage in(o,=0), the market clearing price P fully reveals e. Fourth, although the uninformed cannot disentangle , they can learn something about Pi from the offer price P. We now proceed to analyze how this learning takes place 2.2 Problem of the uninformed competitors Equation (5) tells us that t is a linear function of two normally dis- tributed random variables, implying that it is also a normally distrib- uted random variable. Simple calculations can then be used to show that E(T)=0 and var()=02=02+6如2AsP1=μp+∈+nby construction, one also knows that Pi is a normally distributed random variable with E(P)=μ and var(P1)=σ2+ p(P,T)=σ/2(02+σ2)]/2 Hence, by observing the equilibrium price P, and thereby T, the uninformed update their priors on P,. The posterior distribution of Pi given P, using fact 1, is now normally distributed with E(P1|P=E(P1|) =p+o{r-0]1σ2=μp+σ{P-(p+a(P)/a2 var(P1|P)=Ⅴar(P1|r)=(2+a2)-0:/σ2. (7) Equations (6)and(7) make precise the " learning procedure""of the uninformed. Equation(6) gives them the posterior mean of P1 as a function of P, while Equation (7) is used to update the variance Notice that the posterior precision on P is higher than the prior precision, and this improvement does not depend on P(assuming P exists) The above arguments show that the final period 1 wealth of the ith 261
Tbe Review of Financial Studies/v4n21991 uninformed trader, S(i)P,+ B(i), is normally distributed. Therefore one can use the assumption that their utility functions are negative exponential and rewrite their objective in a mean-variance frame vork as maxv=S()E(P|P)+B()-0.50s()var(f|P),(8) subject to the budget constraint [S()-S0()P+[B()-B()=0. The first-order condition of (8), subject to(9), is then used to deter mine the individual demand schedule of each uninformed investor The demand, it should be noted is the same for all the uninformed 2.3 Feasible aggregate demand curves Aggregating all the individual demands we get S()f( nE(P I P)-PI 8 Var(P I P) (10) Substituting for E(P I P)from(6) and for Var(P I P) from(7) in(10), we obtain a differential equation in S(P) K t k2st KSPt KS'st KsS=0 (11) where K1=S,K2=-1 G2-0282o2 K={1+1+K3+k32}≥ K5=-(2K3+6o2) The solution of(11) gives us all the possible aggregate demand curves that satisfy the first-order condition of the informed investor Her second-order condition [obtained by differentiating (3) with respect to P] further restricts this set, and provides the complete set of feasible aggregate demand curves. In order to find the set of equi libria, we analyze the problem in three steps. First, Proposition 1 establishes the set of solutions satisfying Equation(11). Second, Prop osition 2 uses the second-order condition to find restrictions that any 262
Insiders, Outsiders, and Market Breakdowns solution to the problem must satisfy. Third, Theorem 1 uses Propo- tions 1 and 2 to determine the set of feasible equilibrium aggregate demand functions of the uninformed Proposition I. Tbere exists a complex number C such that tbe set of feasible aggregate demand curves that satisfy Equation(11) can be [(K,+ K2)(K,P+ Ks)-K,K,+ K, KSI(K,+ K2S)K3/K2]=C, for any initial condition(S*, P*). In the present application, one can simplify tbe above equation by substituting out K, and k2to obtain (K3-1)(KP+K)-K4S+KKSI(S-S)8]=C.(12) Proof Differentiate(12) with respect to P to get back (11). For a detailed exposition of the "method of integrating factors"that was utilized to solve (11), refer to the appendix Q ED While there always exists a curve passing though any (S, P) pair, not all of them satisfy every equilibrium condition imposed by the economics of the problem. As the next proposition shows, the insi der's second-order conditions impose the intuitive restriction that an equilibrium aggregate demand curve cannot have any upward sloping region Proposition 2. In equilibrium, it is necessary and suficient for an ggregate demand curve to bave a sufficiently negative slope for all S, and satisfy(12). The exact requirement is dP/ds s-on-t war(P|P< O for S∈(-∞,∞) in equilibrium Proof See the Appendix. Using Propositions 1 and 2, it is now possible to find a general characterization of the set of equilibrium demand curves Tbeorem 1. Any curve satisfying(12) is an equilibrium demand urve if and only if it passes througb a point(S", P*), wbere tbe following conditions are met. IfS*< So, then K3P*+K1S*+K5>0 (13) (K3+K2)(K3P*+K5)-KK1+K3KS*≥0 (14) Conversely, if S*>So, then 263