The name is absent

A Appendix 1: Proofs

Proof of Proposition 1. Let us denote by σ (hj, h_k, h_ε), 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

|mk ^- mj |2 ʌ   hε W = , Z *Λ

2σ² (hj ,hk ,h_e)) hj + hε j ^{ai) ,}

which depends on ∆ ≡ |m1 - m2 | but not on m1 and m2 individually.
Applying the implicit function theorem yields

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 a_j^* . Applying the implicit function
theorem to (11) and rearranging terms yields

exp (-_i⅜⅛ ) W,⅛
√2πσ (hj, hk, hε) c (a**)
`---------------------V---------------------'

>0

It is easy to see from (4) that ^dσ(hj,_hh ,h < 0. This implies that whenever (m_k
σ² (h_j,h_k,h_ε), then dhɪ < 0. However, the sign of dj is ambiguous if (m_k

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 lim_h →₀ da = ∞. Moreover, since lim_h →∞ ^dP(^αj,^α = 0,
j  d'j                     j     ∂aj

we have lim_hj→∞ (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 h_j∙ bexond a certain
threshold, j’s profit maximization problem has the corner solution a* = 0.

40

<<   <   43   44   45   46   47   48   49   >   >>