Thus, with the cost function (20), Firm 1 acquires a positive quantity of private
information (provided that the marginal cost of the first unit, c, is not prohibitively
high) and Firm 2 absolutely imitates Firm 1 without acquiring any private information
of its own. Without the benefit of observing Firm 1’s location, Firm 2 would have
acquired the same positive quantity of private information that Firm 1 did. But
because the observation of Firm 1’s location reduces Firm 2’s marginal benefit of
private information so much that it lies uniformly below marginal cost, Firm 2 never
acquires private information after observing Firm 1’s location. Firm 2 locates exactly
where Firm 1 locates, y⅛ = y↑, illustrating the case of absolute imitation described in
Result 1.
2.3 Aggregate efficiency and absolute imitation
Firm 1 cannot capture the positive informational externality its choice of location
provides to Firm 2. To measure the aggregate inefficiency resulting from this infor-
mational externality, it is useful to compare aggregate profits between two cases: the
decentralized case in which both firms make information and location choices on their
own, versus the centralized case in which a central planner simultaneously chooses
θ1, θ2, y1 and y2 to maximize the aggregate profit function π1(y1 , θ1) + π2(y2, θ2). In
the centralized case where y1 and y2 are chosen by the central planner to equal each
firm’s respective conditional expectation of a, the information acquisition variables
θ1 and θ2 must be chosen simultaneously to maximize:
2π0 — σa (1 — θI) — σa [1 — (θI + θ2 — 2θIθ2) /(1 — θIθ2)] + c log(1 — θ 1) + c log(1 — θ2) ∙ (22)
The central planner’s first-order condition for θ2 is the same as that faced by Firm 2.
Therefore, the central planner also chooses θ2 =0andy2 = y1. The central planner,
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