Imitation in location choice

Conditions are also sought under which Firm 2 acquires private information, but
strictly less information than Firm 1: 0 < θ2 < θ↑, referred to as partial imitation,
because y£ and y₁ are closer on average than they would be if location choices were
based solely on private information.

Turning to Firm 2’s condition for optimal information acquisition, the first-order
condition for θ2 is:

∂var(a∖yι(θ 1),x2; θ2) _ ∂C(θ2)
∂θ2        = ∂θ2

which makes clear the dependence of Firm 2’s choice of θ2 on θ1. Again, this first-
order condition may not have a solution, and its solution may be dominated by
choices at the boundaries. In cases where — ^dvar(ay^¹ )^,x2;^θ2) < ^dC_θ^θ²) for all θ₂, Firm
2’s marginal benefit of private information is never greater than its marginal cost.
The following condition describes when Firm 2 acquires no private information.

Result 1 (AbsoluteImitation): Providedthat C(θ) is convex, that var(a∖y ₁(θ ₁),x₂; θ₂)
is convex in θ2 (i.e., Firm 2’s marginal benefit of private information acquisition is
decreasing in θ2), and provided that the following inequality holds:

∂var(a∖yι(θ 1),x2; θ2),         ∂C(0)       ∂var(a∖x 1; θɪ)

∂θ2           ^lθ2 =⁰ <    ■■   <         ∂θ[      ^{lθ 1} =^{0,           (8)}

then Firm 2 absolutely imitates Firm 1. Absolute imitation means that Firm 2
acquires no information on its own, θ2 = 0, while Firm 1 acquires a strictly positive
quantity of information, θ*_γ > 0. If Firm 1’s location is linear in its private signal,
then this condition also implies that Firm 2 will choose the same location as Firm 1,

У 2 = VÏ.

Condition (8) relies on the fact that, if marginal benefit of private information is
less than marginal cost starting from an initial position of zero information, then there
is never any incentive to acquire information. Firm 2’s marginal benefit is decreasing
of a given x1 is weakly convex in θ1 , which holds, for example, in case a and x1 are jointly normal, as shown in later
sections.

10

<<   9   10   11   12   13   14   15   >>

More intriguing information