The Nash equilibrium Directory UMM :Data Elmu:jurnal:E:Economics Letters:Vol70.Issue1.Jan2001:

62 S . Luo Economics Letters 70 2001 59 –68 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ E[v 2 P2x 2 u ] 5 2 E[E[v 2 Px 1 u us ,x 1 u]] c ˜ ˜ ˜ ˜ ˜ ˜ 5 2 E[E[v us ,x 1 u] 2 Px 1 u ] c 5 0.

3. The Nash equilibrium

We only consider linear Nash equilibria. The following result is a generalization of Kyle 1985. Taking t → `, we recover Kyle’s result. Proposition 1. The unique linear Nash equilibrium is given by ˜ ˜ ˜ x 5 as 1 bs , 1 c i ˜ ˜ ˜ P 5 gs 1 lx 1 u , 2 c with ]]]]]]]] 2 s u ]]]]]]]] a 5 2 , 3 2 2 t s 1 1 1 ts 1 1 t œ v i ]]]]] 2 1 1 t s u ]]]]] b 5 , 4 2 2 t s 1 1 1 ts œ v i 1 ]] g 5 , 5 1 1 t 2 t s v ]]]]]]]]] l 5 . 6 ]]]]]]]] 2 2 2 2 t s 1 1 1 ts 1 1 ts œ v i u Proof. See Appendix A. From Eqs. 3 and 4, we see that the insider put different weights on the public information and the private information in formulating his trading strategy. The relative weight is a b 5 2 1 1 1 t. Since g 5 2 a b , the market makers put a weight which is exactly the negative of the insider’s relative weight on the public information in forming their pricing rule. When t → ` so that there is actually now public information, we have ]]] 2 2 s s u v ]]] ]]]]] a → 0, b → , g → 0, l → . ]]]] 2 2 2 2 2 s 1 s œ 2 s 1 s s v i œ v i u This is exactly Kyle’s result 1985. An important quantity describing the informativeness of the price is the conditional variance ˜ var[v uP], which measures the residual variance after information public and private is incorporated ˜ ˜ into the price. Thus It ; var[v] 2 var[vuP] measures how much information has been incorporated into the equilibrium price. S . Luo Economics Letters 70 2001 59 –68 63 Proposition 2. We have 2 2 2 t 1 2t s 1 21 1 ts v i 2 ]]]]]]]] It 5 ? s . 2 2 2 2 v 2t 1 2t s 1 21 1 t s v i In particular , It is a decreasing function of t. Consequently, the price is more informative when the public information is more precise . Proof. See Appendix A. Two extreme cases are of particular interest. First, when t 5 0, the public information is perfect, and 2 I0 5 s , thus the equilibrium price incorporates all the information about the future payoff, this is v ˜ evident since by Proposition 1, in this situation, P 5 v. Second, when t → `, there is actually no public information, and 2 s 1 v 2 2 ]]]] ] I` 5 ? s s . 2 2 v v 2 2 s 1 s v i Thus the equilibrium price can at most reveal half of the future payoff information. ˜ ˜ Proposition 3. In the equilibrium, the ex ante expected profit of the insider is Pt ; E[v 2 Px] 5 2 ls , where l is given by Eq. 6. Consequently, Pt is an increasing function of t, and thus a u decreasing function of the precision of the public information. In particular, ]]] 2 2 s s v u ] ]]] Pt , P` 5 . 2 2 2 s 1 s œ v i Proof. By Eqs. 1 and 2, and Proposition 1, the ex ante expected profit of the insider is ˜ ˜ Pt ; E[v 2 Px] ˜ ˜ ˜ ˜ ˜ ˜ ˜ 5 E[v 2 gs 2 las 1 bs 1 u as 1 bs ] c c i c i 2 2 2 2 2 2 2 2 2 5 a 1 b s 2 ag1 1 ts 2 bgs 2 la 1 b s 1 a ts 1 b s v v v v v i 2 5 ls . u The last equality follows from observing that 2 2 2 s l 2 s 1 u u ]] ]] ]] g 5 , a 5 2 , b 5 1 1 t l, 2 2 1 1 t t s t s v v which are implied by Eqs. 3–6. 64 S . Luo Economics Letters 70 2001 59 –68

4. Conclusion