Imitation in location choice

Locations are indexed on the unbounded real line. Because the focus is information
acquisition rather than strategic considerations or other interesting problems such as
multiple equilibria, Knightian uncertainty, and complex dynamics, the model takes
a shortcut by assuming the existence of a unique profit-maximizing location a ∈ fà,
referred to as the ideal location. This unique profit-maximizing location, a, is assumed
to be the same for both firms.

Information acquisition is important in the model because a is unknown to both
firms. Firms must therefore condition predictions of a on private signals they acquire,
denoted x1 and x2 for Firms 1 and 2, respectively. In addition to x2 , Firm 2 also
conditions its expectations of a on the observed location of Firm 1, which is denoted
y1. Firm 2’s choice of location is denoted y2.

Quantities of private information are denoted θ1 and θ2,0≤ θi ≤ 1, i =1, 2.
The “quantity of information” θi represents R² in a univariate regression of a on
the privately acquired signal xi . Larger θi means that Firm i chooses more private
information or, equivalently, lower conditional variance of a given xi .

Privately acquired signals come from a variety of sources, including public data
sets and private vendors, both of which incur time, processing and sometimes explicit
financial costs. The model captures the costs of acquiring private information with a
continuously differentiable and weakly increasing cost function C (θ), with C(0) = 0
and C'(θ) ≥ 0. Firm 1’s location is a crucial piece of information for Firm 2. Under
the assumption that this information is easily observable, it makes sense to keep it
distinct from the privately acquired signal x2 and without any effect on Firm 2’s total
cost of information C (θ2).

If either firm (or both) knew where the best location was (i.e., knew a), profit
would be given by the exogenous parameter π0, interpreted as maximized profit in
the ideal case of full knowledge with zero information costs. Given uncertainty about

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