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. 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.
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. Luo Economics Letters 70 2001 59 –68
4. Conclusion