Geoscience Reference
In-Depth Information
14.4.10
UD1, Linear Market, Nash Equilibria
Among the earliest papers to follow Hotelling's lead is the work by Lerner and
Singer (
1937
). The authors keep Hotelling's linear market and the assumption on
linear transportation costs, but introduced a finite reservation price, and assert that
each firm assumes that its competitor's location and price is fixed, and a firm only
reacts if undercut. In such a case, equilibria do exist. The authors also extend
their analysis to spatial price discrimination, which results in social optima. The
contribution by Economides (
1986
) is most interesting, as it includes Hotelling's
(
1929
) and d'Aspremont et al.'s (
1979
) results as special cases. The utility function
includes a budget and the utility inherent in the product. The transportation costs
are the facility—customer distance raised to some power '. The main result is
that for ' less than about 1.26 (which includes Hotelling's original case with
'
D
1), no subgame-perfect Nash equilibrium exists, whereas for ' greater than
about 1.26, it does exist (which includes d'Aspremont et al.'s case of '
D
2). More
specifically, for '
2
[1.26, 1.6667], the equilibrium locations are strictly interior,
while for '
1.6667, they are at the endpoints of the market.
Zhang (
1995
) discusses the case of a duopoly with quadratic transportation costs
and reservation prices, in which decision makers make their decisions in three
phases: locate first, then decide whether or not to adopt a price-matching policy, and
then determine the price. The paper shows that if both players use price matching,
high reservation prices lead to a unique Nash equilibrium “with tacit collusion on
prices.” Equilibrium locations for high reservation prices lie at the center of the
market (minimum differentiation). Not surprisingly, they find that price matching
reduces price competition. The paper of Smithies (
1941
), which has spawned many
followers, discusses a Hotelling model with elastic demand and reservation prices.
The author appears to have been the first to use “push” and “pull” forces (see also
Eiselt and Laporte
1995
). He also found that higher transportation costs lead to
less competition, and as unit transportation costs increase, firm will move farther
apart. Finally, the interesting contribution by Guo and Lai (
2014
) adds an online
dealer to the brick-and-mortar duopolists. While customers purchasing from the
latter, face the usual transportation costs, consumers who deal with the online firm
have waiting/inconvenience cost. The authors demonstrate that an equilibrium does
indeed exist given a relation between the unit transportation costs and the unit
inconvenience cost.
14.4.11
UD1, Linear Market, von Stackelberg Solution
Bonanno's (
1987
) model examines location, which an incumbent can use to deter
future entry of competitors. His model uses quadratic transportation costs, fixed
setup costs for new stores and finite reservation prices. The proposed three-stage
procedure has the incumbent decide how many stores to open, followed by the
