Imitation in location choice

in own private information acquisition (by convexity of Firm 2’s conditional variance),
and marginal cost is increasing (by convexity of the cost function). Therefore, the first
inequality rules out Firm 2 acquiring private information, and the second inequality
ensures that Firm 1 acquires some.³

2.1 Joint normality

To examine the interdependence of the firms’ levels of information acquisition using
tractable functional forms, attention now turns to the consequences of assuming joint
normality:

a ~ N(μ_a,σθ), x 1 = a + e 1, and x2 = a + e2,                (9)

where e_i is normal, with mean -μ_a (so that, without loss of generality, E[x_i] = 0),
and independent from a,fori =1, 2. Thus, each firm’s acquisition of information can
be expressed as:

θi ≡ [c^orr(xi^, a^)]2 = ⁽σ2)2/^(σ2^σ2 = ^σ2/^σ2^,                   (1⁰)

where σ_i² ≡ var(xi), i =1, 2.

Joint normality allows conditional expectations to be expressed explicitly. Recall-
ing that x1 has mean zero by definition, Firm 1’s conditional expectation of a [and
by equation (5), its expected-profit-maximizing location] is:

y*₁ = μa + [cov( x 1 ,a )/σ2] x 1 = μa + θ 1 x 1.                   (11)

Conditional variance of a given Firm 1’s observed signal is given by the convenient
formula:

var[ a∣x ₁] = σa — [cov( x ₁ ,a )]2/σj = σa(1 — θ ₁).                 (12)

The condition under which Firm 1 acquires a positive quantity of information [the
second inequality in (8), which requires that marginal cost of the first unit of infor-

³ For more on the value of information and its interactions with risk aversion, not considered further here, see
Willinger (1989), Hilton (1981), and Eeckhoudt, Godfroid and Gollier (2001).

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