Geoscience Reference
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of unit length. Matsumura and Matsushima (
2009
) use heterogeneity in the form
of different production costs, and if those result in pure strategy equilibria not to
exist, then mixed strategy equilibria are used. Location equilibria with minimal and
maximal differentiation appear each with probability of
1
=
2
.
Anderson (
1987
) showed that in the “first location, then price” two-stage game
if facility
A
were to lead in the first-stage location game, then it would be best
for its opponent
B
to be a leader in the second-stage pricing game. As a result,
firm
A
would locate at the center at the market, while firm
B
will locate at 0.131
(or, symmetrically, at 0.869). Anderson and Neven (
1989
) use the usual Hotelling
assumptions, including duopolists on a linear market, mill pricing and “first location,
then price” competition, but allow customers to purchase goods from both firms
according to some loss function and the use of a quadratic transportation cost
function. The result is maximal differentiation with the duopolists locating at the
two ends of the market. In another contribution, the same authors (Anderson and
Neven 1991) employ spatial price discrimination in a two stage “first location, than
quantity” procedure. The result is an equilibrium with minimum differentiation. The
authors also demonstrate that for more than two firms, given linear transportation
costs and a regularity condition, all firms will locate at the center of the market. Such
agglomeration is often observed in practice, see, e.g., Marianov and Eiselt (
2014
).
Hamilton et al. (
1989
) describe a Hotelling model with spatial price discrimination
and a linear price-quantity relation. The authors compare the results of Cournot
(i.e., quantity) and Bertrand (i.e., price) competition. Throughout, Cournot prices are
higher than those in Betrand competition, and aggregate welfare (i.e., total surplus-
total transport costs) is higher under Bertrand than under Cournot.
Anderson et al. (
1997
) drop the assumption of uniform demand and consider
logconcave demand functions, coupled with quadratic transportation costs. It turns
out that if customers are more spread out, prices are higher, and that symmetric
demand densities lead to symmetric locations of firms. Bester et al. (
1996
)reex-
amine d'Aspremont et al.'s (
1979
) Hotelling game without coordination (firm
A
isassumedtolocatetotheleftoffirm
B
) and allow mixed strategies. An infinite
number of mixed-strategy Nash equilibria exists, and without coordination, the
result of maximum differentiation is invalidated. Eaton (
1972
) follows Smithies
(
1941
) by considering a model, which includes a linearly sloping price-demand
function. The author also uses a modified zero conjectural variation assumption,
according to which a firm will react unless undercut. In case of a short market, the
result will be agglomeration of the firms, as the length of the market grows, duopoly
locations approach the social optimum. Behavior in case of a triopoly is similar:
as the length of the market grows, agglomeration forces get weaker. The paper by
Kohlberg and Novshek (
1982
) examines a similar model.
There are a few contributions that examine spaces similar to a line: Eaton's
(
1976
) model allows free entry on a circle, Kats's (
1995
) model locates duopolists
on a circular market, whereas Tsai and Lai (
2005
) investigate the case of a market,
in which customers are distributed along the sides of a triangle, and Braid (
1989
,
2013
) looks at the case of intersecting roadways, i.e., intersecting lines.
