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assuming that the closest center captures the demand. Later, Marianov et al. ( 2004 )
extend these results to facilities with waiting lines.
14.4.7
UD1b, Linear Market, Nash Equilibria
Consider now models that employ the customer choice rule UD1b, i.e., models
in which customers patronize the least expensive facility. Hotelling's original
model belongs into that group, which, with its linear transportation costs, does
not exhibit an equilibrium. This was pointed out by d'Aspremont et al. ( 1979 )
who also demonstrated that as soon as quadratic transportation costs are used, an
equilibrium does exist with maximum differentiation, i.e., the two facilities locate
at opposite ends of the market. Anderson ( 1988 ) provided further insight into the
case: he demonstrated that in case of linear-quadratic transportation cost functions,
i.e., cost functions that have a quadratic and a linear component, equilibria only
exist, if there is no linear component and the cost function is purely quadratic.
Hamoudi and Moral ( 2005 ) extend the analysis and investigate linear-quadratic
transportation cost functions with different parameters, which result in convex and
concave transportation cost functions, respectively. The authors then define profit
functions for the two cases. Because a price equilibrium does not exist for all pairs
of locations, the authors delineate pairs of locations, for which such an equilibrium
does exist. It turns out that the region, in which price equilibria exist in the concave
case is complete enclosed in the region, in which equilibria exist in the convex case.
Tabuchi and Thisse ( 1995 ) analyze Hotelling's model with a quadratic transport
cost function and triangular customer density. Again, a subgame-perfect equilibrium
is sought. It turns out that no symmetr ic location equilibriu m exists. Instead,
asymmetric equilibria exist at 0;
p 2 p 33C2 ;1 , .e ,
(0, 0.3736) and (0.2527, 1), given that we restrict facility locations to the inside
of the market. Cremer et al. ( 1991 ) locate n facilities on a linear market. Given
quadratic transportation costs and the usual Hotelling assumptions (including the
“first simultaneous choice of location, then simultaneous choice of mill prices”),
the model includes m public and n-m private firms. While private firms maximize
their individual profits, public firms maximize the social surplus, which, with the
assumption of inelastic demand, reduces to the minimization of transportation
costs. For n D 2, one public and one private firm perform best. The two facilities
will locate at the social optimum of 1 = 4 and 3 = 4 , respectively. For n D 3 and one
public facility, profits of the private firms are higher and general welfare is lower
than in the all-private case. With two public facilities, the social optimum is reached.
Some additional combinations of public and private facilities are also investigated.
An important strand of research considers the original Hotelling model, but
allows mixed strategies on prices and pure strategies for the location subgame.
Among the earlier attempts is the contribution by Osborne and Pitchik ( 1987 ), who
determine that facilities will locate at about 0.27 away from the ends of the market
and 1 p 3 3 3
p 33 3
p 2 p 33C2
2
 
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