Imitation in location choice

Taking the second derivative with respect to θ2 reveals a negative sign, which shows
convexity of conditional variance [i.e., the second derivative of var(a∣y ₁(θ 1),χ2; θ2)
without the negative sign is positive], as required in Result 1. To investigate whether
the other part of Result 1’s absolute imitation condition holds, a specification of the
cost function is required, which subsequent sections provide. If both firms choose
interior quantities of information, then they will equate marginal benefit of private
information with marginal cost. For any strictly increasing cost function, Firm 2 will
acquire less private information than Firm 1 if and only if its marginal benefit is less
than Firm 1’s. This clearly holds in the case of joint normality:

^dva^r(a∖y 1⁽θ 1)^,χ2; θ2) _ _2_rn               2 2   2     2 _ ^dva^r(a^lx 1; θ 1)

--∂θ^---------- = ^σa^[(1 “ ^θ 1)/ ⁽¹ “ ^θ 1 ^θ2)] ≤ ^σa =--∂θ^₁------’    (¹⁹)

Result 2 (Firm 2 demands less private information than Firm 1): For
strictly increasing C(θ) and joint normality of a, x1 and x2, Firm 2 demands less
private information than Firm 1: θ2 ≤ θ^. This result follows from the fact that
Firm 1’s marginal benefit of private information is uniformly greater than Firm 2’s:
_ ^dv^ar(a^ly1(θ ₁),^χ2;θ2) ≤ _ ^dvar∂θ 1;^θɪ), as demonstrated in (19).

2.2 Example with exponential cost of information

Suppose the cost function has the following (inverse) exponential form:

C ( θ ) = _c log(1 _ θ ), c> 0.                        (20)

The key feature of this cost function is that the first unit of information has positive
marginal cost c and approaches infinity as θ approaches 1. Solving (13) leads to
θ*_x = 1 _ c∕σa for 0 ≤ c ≤ σ2, and 0 otherwise. Referring back to Result 1, one can
observe that, with c<σ², the condition for absolute imitation holds:
a

13

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