mation is less than its marginal benefit] simplifies to C'(0) < σ22. When this condition
is satisfied, the interior solution θ↑ solves:
σ2a = C ' ( θ ι).
(13)
After making the substitutions σ1 = σ2/θ 1, σ2 = σ2/θ2, and var(y 1) = θ2(σ2/θ 1) =
θ1σa2, Firm 2’s conditional expectations, E[a|y1, x2] and var(a|y1, x2), can be expressed
in terms of θ1 and θ2:
-1
E[a∖y 1(θ 1),x2; θ2] = μa + [cov(y 1 ,a) cov(x2,a)]
var(y1)
cov(y1, x2)
cov(y1 ,x2) var(x2)
y 1( θ 1) — μa
x2
and:
= μa+[θ 1 σa σa]
θ 1 σa θ 1 σa θ 1 x 1
θ 1 σa σa/θ2 x2
θ1(1 — θ2) θ2(1 — θ1)
= μa + 1 θlΓx 1 + 1 θlΓx 2
1 — θ1θ2 1 — θ1 θ2
(14)
(15)
var(a∖y 1(θ 1),x2; θ2)
σa2 — [θ1σa2 σa2]
θ 1 σ22 θ 1 σa θ 1 σi
θ 1 σ2 σ2/θ2 σ2
(16)
= σ2[1 — (θ 1 + θ2 — 2θ 1 θ2)/(1 — θ 1θ2)]. (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( aly,(θ 1) ,χ 2; θ 2) = σ2(1 — θ 1)2 / (1 — θ 1 θ (18)
∂θ2
12
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