Imitation in location choice

The key feature of this cost function is that the first unit of information has zero
marginal cost, implying that both firms always acquire positive quantities of private
information. Solving (13) leads to:

θ*_γ = σ²_a /c, for σ²_α ≤ c, and 1 otherwise,                   (25)

or θ[ = min{σa/c, 1}.

With quadratic costs, Firm 2’s objective function can be written as:

= ^πо — ⁽У2 ^- E[^α∖y 1⁽θ 1)^, x2; θ2^{])2 -} ⅛ ^{- (}θ 1 + θ2 ^{- 2}θ 1 θ2)/^{(1 -} θ 1 θ2^{)] -} cθ²/²∙

The first-order condition for θ2 is:

^σa(^{1 - θ} 1⁾²/^{(1 - θ} 1^θ2^{)2 - cθ}2 = ⁰∙                         ⁽²⁶⁾

Assuming σa ≤ c, one divides (26) through by c and substitutes θ ₁ = σa/c, which
gives rise to a cubic in θ2 that turns out to have a unique solution on the unit interval.
Following these steps, it is straightforward to re-express (26) using the characteristic
equation h(θ2):

h(θ2) ≡ θ₁²θ₂³ - 2θ1θ₂² + θ2 - θ1(1 - θ1)² =0∙                  (27)

Because h(0) = -θ1(1 - θ1)² ≤ 0, and h(1) = (1 - θ1)³ ≥ 0, there exists at least one
solution on the interval 0 ≤ θ2 ≤ 1∙ To rule out the possibility of multiple solutions on
the closed unit interval, one may examine possible nonmonotonicities of h(θ2), which
must occur at zeros of the equation:

— = = = ^3θ2θ2 ^{- 4}θ 1 θ2 + 1 = ⁽1 ^- θ 1 θ2⁾⁽1 ^{- 3}θ 1 θ2) = ⁰∙           (²8)

∂θ2

There are two points at which the sign of the curve’s slope can change: θ₂ = 1 /(3θ ₁ )
and θ₂ = 1 /θ ₁. The second of these is necessarily to the right of θ₂ = 1, implying
that (27) has exactly one solution on the unit interval.

Result 3 (Partial Imitation): Given jointly normal a, x1 and x2, and non-
decreasing cost function C(θ) such that C'(0) < σ2(1 — θ↑), Firm 2’s demand for

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