A Appendix 1: Proofs
Proof of Proposition 1. Let us denote by σ (hj, hk, hε), equal to the square root of σ2
defined in (4), the standard deviation of the distributions φ1 (∙) and φ2 (∙). Making use
of the normality of φj (∙), the first-order condition defining a* can then be rewritten as
____________exp
2πσ (hj, hk, hε)
which is equivalent to
(mk
2
mj)
2σ2 (hj, hk, hε)
)h⅛ Wj=c j
(11)
.—----------exp
2πσ (hj, hk, hε)
|mk - mj |2 ʌ hε W = , Z *Λ
(12)
2σ2 (hj ,hk ,he)) hj + hε j ai) ,
which depends on ∆ ≡ |m1 - m2 | but not on m1 and m2 individually.
Applying the implicit function theorem yields
d∆=W(0)(
-2∆
2σ2 (hj, hk, hε)
(13)
Since c" > 0, (13) implies that dj < 0 if ∆ > 0, and dj = 0 if ∆ = 0.
Next, we consider the impact of a change in hj on aj* . Applying the implicit function
theorem to (11) and rearranging terms yields
aj*
dhj
exp (-i⅜⅛ ) W,⅛
√2πσ (hj, hk, hε) c (a**)
`---------------------V---------------------'
(14)
>0
∂σ(hj ,hk ,hε)
dhj
(mk
2
mj)
σ (hj, hk, hε)
`-----------V-----------'
>0
σ2 (hj, hk, hε)
V -L-
hj + hε
Δ
<0
(15)
It is easy to see from (4) that dσ(hj,hh ,h < 0. This implies that whenever (mk
σ2 (hj,hk,hε), then dhɪ < 0. However, the sign of dj is ambiguous if (mk
mj)2 ≥
mj)2 <
σ2 (hj, hk, hε). This inequality always holds ifmk = mj; moreover, since lim'j →0 σ (hj, hk, hε) =
∞, it also always holds for hj close enough to 0, even if the difference between mk and
mj is large.
Lemma A1 below tells us that limh →0 da = ∞. Moreover, since limh →∞ dP(αj,α = 0,
j d'j j ∂aj
we have limhj→∞ (a = 0.37 Given continuity, this implies that (a < 0 for some hj. It
d d d ■ da d dα*
remains to show that there exists a unique hj, such that > 0 if and only if hj < hj.
37 An interior solution exists for arbitrarily large hj if c,(0) = 0. Otherwise, for all hj∙ bexond a certain
threshold, j’s profit maximization problem has the corner solution a* = 0.
40
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