Imitation in location choice

mation is less than its marginal benefit] simplifies to C'(0) < σ22. When this condition
is satisfied, the interior solution θ↑ solves:

σ²_a = C ' ( θ ι).

After making the substitutions σ1 = σ2/θ 1, σ2 = σ2/θ₂, and var(y 1) = θ²(σ2/θ 1) =
θ1σ_a², Firm 2’s conditional expectations, E[a|y1, x2] and var(a|y1, x2), can be expressed
in terms of θ1 and θ2:

-1

= ^σ2[^{1 — (θ} 1 + ^θ2 ^{— 2θ} 1 ^θ2)/^{(1 — θ} 1^θ2^)].              ⁽¹⁷⁾

Equation (15) implies that the more information Firm 1 acquires, the less weight
Firm 2 places on its own private signal (i.e., ∂[θ₂(1 — θ 1)/(1 — θ 1 θ₂)] = — θ₂(1 —
θ₂)/(1 — θ 1 θ₂)² ≤ 0). As long as θ ₁ > θ₂, equation (15) also shows that Firm 2 will
weight Firm 1’s information more than its own, and if θ2 =0, locate exactly where
Firm 1 does: y2 = y↑ = μ₂ + θ ₁ x ₁.

The following expression, which is positive, measures the marginal benefit to Firm
2 of its private information:

_ ^{d var}( a^ly,(^θ 1) ,^{χ 2}; θ 2) = σ²(1 — θ 1)2 / ₍₁ — θ 1 θ                    (18)

∂θ2

12

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