Imitation in location choice

a, however, firms experience actual costs from two separate categories. First is the cost
of deviating from a, which should be interpreted as reduced sales, extra transportation
costs, or higher taxes. Second is the cost of private information, which is acquired
specifically to reduce the expected deviation of the firms actual location from a. With
quadratic costs of deviating from a, the profit function takes the form π0 - (yi - a)² -
C (θi), i =1, 2. Because a is uncertain, the ex ante objective functions are stated in
expected form:

∏ ι( y 1 ,θ 1)  =  П о - E[( y 1 - a )2 ∖x ι; θ i] — C ( θ i),                       (1)

^π2⁽y2^,θ2)  =  ^πо — ^E[(y2 ^- a)2∖yi⁽θi)^,x2;θ2] ^- C⁽θ2).                ⁽2)

Note that Firm 1 and Firm 2’s expected profit functions differ only in the information
upon which expectations are conditioned. The notation makes clear that Firm 1’s
expectation of expressions involving a is conditioned by its private information xi,
which depends on its choice of θi. Firm 2’s expectation of expressions involving
a is conditioned by Firm 1’s location yi(θi) and Firm 2’s privately acquired signal
x2. Firm 2’s expectations depend on its choice of θ2. The notation in equation (2)
expresses yi as a function of θi to make the dependence of Firm 2’s information
acquisition on Firm 1’s choice of information explicit.

Expected deviations from a, which appear in each firm’s profit, can be decomposed
as follows:

E[(yi - a)²∖xi; θi] = (yi - E[a∖xi; θi])² + var(a∖xi; θi),                  (3)

E[(y2 - a)²∖yi(θi), x2; θ2] = (y2 - E[a∖yi(θi), x2; θ2])² + var(a∖yi(θi), xi; θ2). (4)

Because the first terms on the right hand side of (3) and (4) have unique minima
at zero, and because yi appears nowhere else in Firm i’s objective function, optimal
location choice rules are given by:

y↑ = E[a∖xi; θ 1] and y*₂ = E[a∖yi(θi),x2; θ2].                     (5)

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