Imitation in location choice

Whenever y↑ is linear in x 1, then y% → y^ as θ2 → 0. The reason for this is that Firm
2 receives an undistorted version of Firm 1’s private signal while acquiring no private
information of its own, which implies that Firm 2 must have the same expectation of
a and consequently choose the same location as Firm 1.

After substituting (3) into (1), the first-order condition for θ1 is:

^dE[ax 1; θ 1]    ∂v^ar(ax 1; θ 1)   ∂C(θ 1)

2( y¹ - ^E[ a^lx 1; θ^1]) -_g^---∂e_l---∂θΓ = ⁰ ■        ⁽⁶⁾

but because the global solution for y↑ in (5) makes the first term in parentheses
on the left-hand side of (6) uniformly zero, this first-order condition simplifies to
—^dvarax^l;^{θ 1} ) = ^dCθ(-θɪ). This requires that Firm 1 choose θ 1 to set the marginal
reduction in conditional variance ofa    equal to the marginal cost of information. A

solution to the first-order condition for θ1 may not always exist and, even when it
does, it may not correspond to a global maximum. The boundary values, θ1 =0and
θ1 = 1, must also be checked. The choice θ1 = 0 maximizes expected profit when
information is too expensive to justify its acquisition at any level, in which case Firm 1
locates at the unconditional mean, y 1 = μ_a, and achieves expected profit π₀ — σ02. The
choice θ1 = 1 represents cases where Firm 1 acquires maximally precise information,
implying that var(α∣x 1; θ 1) ∣_{θ 1}=1 = 0, and profit π₀ — C(1), which is certain in this case
rather than expected.

Given the real-world policy problems associated with the hypothesis of imitation
causing inefficient spatial agglomerations that fail to utilize profitable opportunities
elsewhere, the most interesting case to consider is when Firm 1 acquires information
and Firm 2 does not: θ1 > 0andθ2 = 0, referred to as absolute imitation. One seeks to
identify the conditions under which imitation of this kind is consistent with expected-
profit maximization. Thus, it is assumed that the global maximizer θ1 lies on the strict
interior of the unit interval: that is, π 1(E[a∣x 1; θ↑],θ↑~) > max{π₀ — σa,π₀ — C(1)}.2
² A sufficient but not necessary condition for existence of an interior solution for θ1 is that the conditional variance

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