The name is absent



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 (hj,hh ,h0. 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 limh0 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 *

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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