Geoscience Reference
In-Depth Information
Prescott and Visscher (
1977
) that popularized the methodology and the results.
In one of their examples, the authors look at a duopoly on a linear market—the
simplest possible case—and determine that the leader will locate at the center of the
market, while the follower will locate next to the leader, thus resulting in central
agglomeration. The authors then extend their analysis to the case of three firms.
After considering many cases and subcases (see, e.g., Younies and Eiselt
2011
),
it is determined that one of the outcomes (arguable the most likely one) is that
the three facilities locate at
1
=
4
,
3
=
4
,and
1
=
2
of the market, capturing
3
=
8
,
3
=
8
and
1
=
4
of the market, respectively. The fact that the first two facilities to locate earn
50 % more than the last entrant into the market is, however, troublesome: having
established that it takes capability and incentive to be a leader (see, e.g., Younies
and Eiselt
2011
), we can consider the second and third firms to enter the market
as followers. However, why would any follower accept being the third rather than
the second entrant, if the latter course of action is much more profitable? A similar
result had already been obtained by Teitz (
1968
), who considered duopolists, so that
the location leader would locate two facilities, while the location follower would
locate a single facility. He suggested “conservative optimization,” i.e., a minimax
strategy. While the leader locates his two facilities at
1
=
4
and
3
=
4
of the market, the
follower will locate his single facility anywhere between the leader's facilities.
An interesting extension is provided by Thisse and Wildasin (
1995
), who locate
private facilities alongside a centrally located public facility. Households have
incomes, which they spend on trips to the facilities and paying land rent. In the first
stage of the game, all firms locate first, followed by stage two, in which customers
locate. The result is that high travel costs yield maximal differentiation, while low
travel costs result in minimal differentiation. Bhadury (
1996
) considers a Hotelling
model on the line with fixed and equal mill prices, in which the leader does not have
perfect information regarding the follower's variable costs. For a general demand
distribution, the author shows that market failure is possible (i.e., the leader may not
wish to locate any facilities) and that a greedy strategy is not bad (optimal for an
atomistic leader, i.e., one who wishes to locate only a small number of facilities).
Osborne and Pitchik (1986) allow the demand distribution to be not necessarily
uniform. Allowing mixed strategies, the result for a three-firm problem has all three
firms randomize over the central half of the market. Dasci and Laporte (
2005
) allow
facilities to have different cost functions. The paper is novel in that it does not deal
with exact facility locations, but with the density of retails branches that are located.
14.4.3
UD1a, Plane, Nash Equilibrium
In two-dimensional space, Okabe and Aoyagi (
1991
) attempt to prove a conjecture
by Eaton and Lipsey (
1975
) in the two-dimensional plane. With fixed demand
and equal mill prices, customers patronize the closest facility. In the infinite two-
dimensional plane with Euclidean distances and an infinite number of independent
