These results mean that equilibrium in the second

These results mean that equilibrium in the second stage pricing game exists if firms are
located at the same place, or if they are separated by some minimum distance. Indeed,
d’Aspremont et al . (1979) show that there are, in fact four different regimes:
a. Firms that are located in the same place are in a Bertrand equilibrium with
zero profits;
b. Firms that are ‘too close’ can never establish an equilibrium because they
continuously undercut each other;
c. More differentiated firms establish an equilibrium in which they share the
market;
4
‘The competition for votes between the Republican and Democrat parties does not lead to a clear drawing of
issues, an adoption of two strongly contrasting positions between which the voter may chose. Instead, each
party strives to make its platform as much like the other’s as possible.’ [Hotelling (1929), p. 482]. And:
‘Methodists and Presbyterian churches are too much alike; cider is too homogenous.’ [Hotelling (1929), p. 484].
a b L + =
0
A B
p p = =
a b L + <
( )
2
4 2
3 3
L a b
a b
a b L
+
-
æ ö
> Þ + ³
ç ÷
è ø
( )
2
4 2
3 3
L a b
b a
a b L
+
-
æ ö
< Þ + ³
ç ÷
è ø

Industrial Economics & Policy (ECO3026) Topic Five: Product Differentiation (ii) - Horizontal
d. And finally, if locations are so far apart that there exists a consumer who buys
from neither firm, then the firms operate as local monopolies.
4. Salop’s Circular City Model
An alternative to Hotelling’s linear model is developed by Salop (1979) who assumes a
circular city of circumference 1. As with Hotelling, this location model can be given an
interpretation of describing differentiated products that differ from the physical-location
interpretation. Consider, for example, airline, bus and train firms that can provide a round-
the-clock service. If we treat the circle as twenty-four hours, each brand can be interpreted as
the time when such a firm schedules as departure.
Firms
The model does not explicitly model how firms choose where to locate. It assumes, instead, a
monopolistic-competition market structure in which the number of firms, N , is endogenously
determined. All of the (potentially infinite) firms have the same technology. Let F denote the
fixed cost of production, c the marginal cost of production, and and the output and
profit level of the firm producing brand i . We assume:
(36)
Consumers
Consumers are uniformly distributed on the unit circle. We again denote by t the consumer’s
transportation cost per unit of distance, and we assume that each consumer purchase one unit
of he brand that minimises the sum of the price and transportation cost.
Assuming that each of the N firms is located at an equal distance from one another
implies that the distance between any two firms is . Consider Firm 1 setting in Figure
5. Firm 1 faces Firm 2 to its right and Firm N to its left, and can attract potential customers of
each by setting its price accordingly. If we assume that Firms 2 and N set a uniform price, p
then the consumer who is indifferent between buying from Firms 1 and 2 is located at point
on the circle, where is defined implicitly through:
(37)
And since Firm 1 has potential customers on its left and right, its demand function is given
by:
(37)
,
9