Competitive Location Models - Location Science - page 351

Fig. 14.2 Profit functions in Hotelling's model with linear transportation costs

undercut him slightly and thus capture the entire market. This results in facilities
moving apart so as to position themselves in a region with less competitive pressure.
The obvious question is whether or not there is a locational arrangement and a
price structure, which represents a stable solution, i.e., an equilibrium. Temporarily
holding the location of both and the price of one of the competitors, say, B , constant,
Fig. 14.2a, b shows competitor A 's profit function in the case of firms A and B
locating close to each other (Fig. 14.2a ) or a significant distance apart (Fig. 14.2b ).
First consider Fig. 14.2a . From left to right, A 's profit function is linearly
increasing for low prices p A (as firm B is cut out and A 's profit increases proportional
to the price); then, as p A increases, at some point, B is no longer cut out, there is a
marginal customer in the competitive region, and A 's profit function is an inverted
ellipse. As p A increases further, there is a point, at which it is sufficiently high so that
firm B cuts out firm A , and thus A 's profit drops to zero. Notice that there are two
local maxima, one at the first breakpoint from the left, and the second in the domain
of the quadratic piece of the function. In Fig. 14.2b , the linearly increasing part is
valid only for negative prices, which are nonsensical in this application. Other than
that, the function is similar to that in Fig. 14.2a , but with a single maximum.
d'Aspremont et al. ( 1979 ) first demonstrated that Hotelling's model does not
possess an equilibrium in pure strategies, i.e., as long as each player chooses exactly
one strategy, rather than randomize. They then demonstrated that an equilibrium
was restored in the model if we were to use a quadratic, rather than a linear,
transportation cost function. Later, Gabszewicz et al. ( 1986 ) pointed out that the
lack of the existence of equilibria in Hotelling's model is due to the lack of
quasiconcavity of the profit functions of the duopolists (see again Fig. 14.2a ).
Figure 14.3a, b shows again competitor A 's profit , given a quadratic, rather than
linear transportation cost function: Fig. 14.3a for competitors' locations that are
close to each other, and Fig. 14.3b for locations far apart. Note that the functions are
both quasiconcave.
In general, many competitive location models have shown major signs of
instability: Hotelling's original model with variable prices and linear cost func-
tions has no equilibrium, the same model with quadratic transportation costs has
one—with firms located at opposite ends of the market. Hotelling's model with a