be Review of Financial Studies/v 4n 1991 K,+ Ks+ Ks<o (15) and (K3+K2)(K3P*+K)一K4K1+KKS*≤0 (16) Proof See the Appendix Graphically, the conditions can be related to Figure 1. The com plete set of feasible aggregate demand curves lies in the shaded region of Figure 1. Equality(13)is line 2, whereas equality(14)is line 1. Note that the shaded region is bounded by the familiar linear demand curve(line 1)and the no-trade demand curve(quantity demanded=S, aggregate endowment of outsiders), and all the feasible nonlinear demand curves lie in between 2.4 Equilibria Because an aggregate demand curve must lie entirely within the shaded region of Figure 1, the set of possible equilibria, for a given T, turns out to be closed compact, and located along a line segment The characterization is contained in the next theorem Theorem 2. For a given value of T, the set of all equilibrium pairs (S, P) lies in a bounded segment of the line + 1+K3 1+K3 (17) The bounds are SE(A, A,/, wbere A1=S0 (18) A2=⑥6So3(1+K)[(1+K3)+n+(n(K3-1)/S) +6S:K3(1+K3)-Tn(K3-1)K3} X{K(2+n)(1+K3)+2K0]}-1 (19) graphically, the set of all equilibrium pairs is depicted by line 3 in Figure 2. If the insider buys the risky stock, the top line 3 represents this set; if tbe insider sells the risky stock, tbe bottom line 3 represents this set. Note that the equilibrium price-quantity pair chosen(C), if anonlinear demand curve is played, lies between tbe no-trade price- quantity pair (A, ) and the price-quantity pair that occurs if the linear demand curve is played(A2) Proof. Given the aggregate demand curve S(P), the informed investor chooses a price P. Theorem 1 derives the set of feasible aggregate 264
Insiders, Outsiders, and Market Breakdowns Price Line 2 Line 1 S Quantity The aggregate quantities of shares of the risky stock dem es lie in the shaded region. Note nonlinear demand curves lie in between. The curved lines represent nonlinear solutions to (1 demand curves, which can then be substituted into(4) to give us (17), the offer price P of the informed. Now, equilibrium price quantity pairs will lie on the points of intersection of the price rule of the informed(17)-given by line 3 in Figure 2-and the feasible aggregate demand curves(12). This means that they will lie between A, and Az. At Au, S=So. The point A, is the intersection of (17), line 3, and the linear demand curve(20), line 1(given in Figures 1 and 2). Solving(20)and(17) simultaneously, we obtain(19). Q.E.D Theorem 2 determines whether the insider will purchase or sell securities. When A2> So, the insider is a net seller, while a value of A2<So implies that the insider is a net buyer of stock, Since the aggregate supply of the stock is fixed, the uninformed are necessarily forced to take the opposite position Many researchers tacitly assume that nonlinear demand functions implausible under the mean-variance framework with Gaussian assumptions, but Theorem 1 shows that this is not necessarily so This may appear to be discouraging news, since Gale and Hellwig (1988) show that the study of markets as a communication system