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(μa,σθ), x 1 = a + e 1, and x2 = a + e2, (9)
where ei is normal, with mean -μa (so that, without loss of generality, E[xi] = 0),
and independent from a,fori =1, 2. Thus, each firm’s acquisition of information can
be expressed as:
θi ≡ [corr(xi, a)]2 = (σ2)2/(σ2σ2 = σ2/σ2, (10)
where σi2 ≡ 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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