Abstract
In this article, we show that in models where location is endogenous, maximum welfare losses arising from non-optimal locations or from the lack of market coverage may be substantial. In contrast, maximum welfare losses arising from non-optimal quality choices are more modest, but they might vary discontinuously with the dispersion in consumer tastes. Very often, welfare losses can be inferred from data.
Appendix
ALemma 1
There is a unique local monopolistic equilibrium and
for a linear horizontal market
with
.
There is a unique touching equilibrium and
for a linear horizontal market
with
.
There is a unique touching equilibrium and
for a linear horizontal market
with
.
There is a unique oligopolistic equilibrium and
for a linear horizontal market
with
.
Proof:
Local monopolistic equilibrium. In this equilibrium, some consumers lying between two firms do not purchase the differentiated good, so the market is not fully covered. Each firm charges monopoly price . A consumer with preferred brand
is indifferent between purchasing from firm 1 and not purchasing the differentiated good if
. Thus, firm 1’s demand is
. In the second stage, firm 1’s profit maximization with respect to
yields

Plugging into firm 1’s profit yields

After tedious calculations, one finds that , so firms have incentives to relocate toward the market center still maintaining local monopoly power. Firms will move to the market center until consumers at the edges of the market are just indifferent between buying the differentiated good and not. One can check that in this case the firms’ marginal relocation tendency becomes zero.
Consumers at the edges of the market are indifferent between buying the good and not buying it, which amounts to . Consumers at the market center do not buy the differentiated good, which amounts to
. These two conditions yield a unique local monopolistic equilibrium: firm 1 chooses
, and firm 2 chooses
.13 Both firms charge the same price
. This equilibrium exists for
. Therefore, there is a unique local monopolistic equilibrium
and
for a linear horizontal market
with
.
Touching equilibrium. In a touching equilibrium, markets just “touch”. A consumer with preferred brand specification is indifferent between purchasing from firm 1 or from firm 2 at price
and not purchasing the differentiated good if
. Thus,
. At the same time, firms still enjoy local monopolistic power, therefore
, which yields a unique touching equilibrium:

In the touching equilibrium, firms behave as local monopolists but maintain full market coverage so or
. The consumers at the edges of the market purchase the differentiated good so
, which simplifies to
. Thus, there is a unique touching equilibrium
and
for a linear horizontal market
with
.
Note that for , there is a unique touching equilibrium with
and
, which is an intermediate case between the touching equilibrium described above and the oligopolistic equilibrium, which is analyzed below.
Oligopolistic equilibrium. A consumer with preferred brand specification is indifferent between purchasing brand
and purchasing brand
, if

so the demands and
faced by firms 1 and 2, respectively, read


Firm i’s profit maximization with respect to yields

and corresponding profits become

The “marginal relocation tendency of firms” reads and
. Thus, the firms have incentives to relocate marginally away from each other. The unique equilibrium has two firms locating at the two extremes of the product space (maximal differentiation)
> and
and charging the same price
. In the oligopolistic equilibrium, the entire market is covered, so
, which amounts to
. Thus, there is a unique oligopolistic equilibrium
and
for a linear horizontal market
with
. ■
B Decomposition of social welfare in the Hotelling model
The social welfare in case non-optimal locations of the firms were the only source of distortion is equal to

The PWL in case non-optimal locations of the firms were the only source of distortion is given by

C Proof of Proposition 1
First, consider the case where the market is uncovered, i.e. and
. Let
and
. We easily see that linear horizontal market
yields a local monopolistic equilibrium where
and
. Therefore, given an observation
such that
and
, there is a linear horizontal market
such that this observation is consistent with a local monopolistic equilibrium for this market. When
in the optimum the market is uncovered and

When in the optimum the entire market should be covered. Plugging
and
into the second line of eq. [2] yields

which is strictly decreasing for .
Next, consider the case where the entire market is covered, i.e. , and the firms do not locate at the edges of the market, i.e.
. Let
and
. It is straightforward to check that linear horizontal market
yields a touching equilibrium where
and
. Therefore, given an observation
such that
and
, there is a linear horizontal market
such that this observation is consistent with a touching equilibrium for this market. Plugging
and
into the third line of eq. [2] yields

which is strictly decreasing for .
Finally, let us consider the case where the entire market is covered, , and the firms locate at the edges of the market,
. Here, with available observables, there is no way to distinguish between a touching equilibrium with the firms located at the market edges and an oligopolistic equilibrium.
In case of a touching equilibrium with the firms located at the market edges, let us fix (for condition
to hold) and let
. It is straightforward to check that linear horizontal market
with
yields a touching equilibrium where
and
. Therefore, given an observation
such that
and
, there is a linear horizontal market
with
such that this observation is consistent with a touching equilibrium for this market. From the fourth line of eq. [2], we get PWL in the touching equilibrium, denoted as
, as a function of observables and
:

which is increasing in and achieves its maximal value of
at
and its minimal value of
at
. This and the continuity of
with respect to
imply that
.
In case of an oligopolistic equilibrium, let us fix (for condition
to hold) and let
. It is easy to show that linear horizontal market
with
yields an oligopolistic equilibrium where
and
. Therefore, given an observation
such that
and
, there is a linear horizontal market
with
such that this observation is consistent with an oligopolistic equilibrium for this market. From the fourth line of eq. [2], we get PWL in the oligopolistic equilibrium, denoted as
, as a function of observables and
:

which is decreasing in and achieves its maximal value of
at
and goes to 0 as
goes to infinity. This and the continuity of
with respect to
imply that
. Thus, for an observation with the covered market and the firms located at the market edges,
and
,

D Lemma 2
There is a unique local monopolistic equilibrium and
,
, for a circular market
with
.
There is a unique touching equilibrium and
,
, for a circular market
with
.
There is a unique oligopolistic equilibrium and
,
, for a circular market
with
.
Proof:
Local monopolistic equilibrium. In the local monopolistic equilibrium, some consumers lying between two neighboring firms do not purchase the differentiated commodity, so the market is not covered. Each firm charges monopoly price . A consumer with preferred brand specification located at the distance
from firm i’s brand specification is indifferent between purchasing from firm i and not purchasing the differentiated commodity if
. Thus, firm i’s demand is

Firm i’s profit maximization yields . In local monopolistic equilibrium, the market is uncovered, which amounts to
. Therefore, there is a unique local monopolistic equilibrium
and
,
, for a circular market
with
.
Touching equilibrium. In a touching equilibrium, markets just “touch”. A consumer with preferred brand specification located at the distance from a firm’s brand specification is indifferent between purchasing from a firm or from its closest neighbor at price
and not purchasing the differentiated commodity if
. Thus,
. In touching equilibrium, the entire market is covered,
, but there is no tangency of demand. These conditions amount to
. Thus, there is a unique touching equilibrium
and
,
, for a circular market
with
.
Oligopolistic equilibrium. Firms are located equidistant from one another and compete in prices given these locations. Since they are located equidistant from one another, they will charge the same price in the equilibrium. Firm i has two potential competitors, namely firms
and
. Suppose that it chooses price
. A consumer with preferred brand specification located at the distance
from firm i’s brand specification, is indifferent between purchasing from firm i and from i’s closest neighbor if
. Thus, i’s demand reads

Firm i’s profit maximization yields (in equilibrium )
. In oligopolistic equilibrium all consumers receive positive net surplus, so the entire market is covered, which amounts to
. So there is a unique oligopolistic equilibrium
and
,
, for a circular market
with
E Proof of Proposition 2
Consider the case of uncovered market, . Let
and
. We easily see that circular market
yields a local monopolistic equilibrium where
and
. Therefore, given an observation
such that
, there is a circular market
such that this observation is consistent with a local monopolistic equilibrium for this market. When
in the optimum, the market is uncovered and

When in the optimum, the whole market should be covered. Plugging
and
into the second line of eq. [3] yields

which is decreasing in the market coverage for
F Lemma 3
There is a unique equilibrium with uncovered market

for a linear vertical market when either
or
and
.
There is a unique equilibrium with corner solution

for a linear vertical market when
and
.
There is a unique equilibrium with covered market

for a linear vertical market when
and
.
Proof:
Uncovered market. When , some consumers with low taste for quality purchase neither good. Firms’ profit maximization yields

Then, profits read

For , the condition
is satisfied for all
. For
and
, the condition
amounts to
.
Covered market. When , the market is covered and the consumer with the lowest taste parameter strictly prefers to purchase product
- 1.
Firms’ profit maximization yields
r

For and
, the conditions
and
amount to
. For
and
, these conditions amount to
.
Corner solution. When , the market is covered with firm 1 quoting the price which is just sufficient to cover the market:
. Firm 2’s profit maximization yields

The profits are
This case arises whenever ,
, and
or
,
, and
.
In the first stage, each firm i maximizes over
. The “marginal relocation tendency” of firm 2 reads
in each of the three configurations of the price game, therefore firm 2 chooses the maximal quality level
. Next, we consider firm 1’s profit maximization.
Uncovered market. Firm 1’s profit maximization yields . Firm 1’s profit becomes

This uncovered market equilibrium outcome arises either when or when
and
.
Covered market. The “marginal relocation tendency” of firm 1 reads . Therefore, maximal differentiation holds, subject to the restriction that the market is covered. It implies that
when
and
, and
when
and
. For both firms to compete for consumers the condition
should hold. (Otherwise, the market is preempted by firm 2.) It is straightforward to check that
is satisfied only when
,
, and
. Firm 1’s profit becomes

Corner solution. Firm 1’s profit maximization yields . Firm 1’s profit is equal to

The corner solution arises when and
.
Plugging and
into the formulas for
and
yields equilibrium prices.
Having considered quality choices for each market configuration separately, we now compare these choices. Direct computations show that whenever
. Since
corresponds to the upper bound for the low quality to define a corner solution, it is always beaten by
, whenever the latter is defined, i.e. whenever
. (Indeed,
for
). Finally, we show that
whenever
and
whenever
, with
G Decomposition of social welfare in the vertical differentiation model
The social welfare in case non-optimal market coverage was the only source of distortion is equal to

The PWL in case non-optimal market coverage was the only source of distortion is given by

H Proposition 3 (positive marginal cost)
- i)
Given an observation
where
and either
and
or
and
, there is a linear vertical market
such that this observation is consistent with a covered market equilibrium for this market, and
[5]which for the case of zero marginal cost simplifies to
- ii)
Given an observation
where
and either
and
, or
and
, or
and
, or
and
, with
, there is a linear vertical market
such that this observation is consistent with a corner solution for this market, and
which for the case of zero marginal cost simplifies to
- iii)
Given an observation
where
and either
and
or
and
, there are two linear vertical markets
and
such that this observation is consistent with two corner solutions (one for each of the markets), and
which for the case of zero marginal cost simplifies to
- iv)
Given an observation
where
,
and
, there is a linear vertical market
such that this observation is consistent with an uncovered market equilibrium for this market, and
[6] - v)
Given an observation
where
,
and
, there is a linear vertical market
such that this observation is consistent with an uncovered market equilibrium for this market, and
Proof:
Consider the case with covered market, , which might correspond either to covered market equilibrium or to corner solution. First, assume that it corresponds to the covered market equilibrium. Let

which yields
![[7]](/document/doi/10.1515/bejeap-2012-0045/asset/graphic/bejeap-2012-0045_eq7.png)
![[8]](/document/doi/10.1515/bejeap-2012-0045/asset/graphic/bejeap-2012-0045_eq8.png)
Conditions and
amount either to
and
or to
and
. Then by construction the market

yields an equilibrium where . Plugging eqs [7] and [8] into the first line of eq. [4] we get [5] for PWL as a function of an observation
for which the above conditions hold.
Assume now that an observation with covered market, , corresponds to the corner solution. Let

which yields 2 roots and
such that

![[9]](/document/doi/10.1515/bejeap-2012-0045/asset/graphic/bejeap-2012-0045_eq9.png)
and

![[10]](/document/doi/10.1515/bejeap-2012-0045/asset/graphic/bejeap-2012-0045_eq10.png)
Conditions and
amount either to
and
, or to
and
, or to
and
, or to
and
, with
. Then by construction, the market
yields an equilibrium where
. Plugging eq. [9] into the second line of eq. [4] we get PWL as a function of an observation
for which the above conditions hold.
Conditions and
amount either to
and
or to
and
. Then by construction, the market
yields an equilibrium where
. Plugging eq. [10] into the second line of eq. [4] we get PWL as a function of an observation
for which the aforementioned conditions hold.
For an observation such that
and
or
and
, both markets
and
are defined. For an observation
such that
and
, or
and
, or
and
, or
and
, just one market
is defined.
Consider now the case with uncovered market, . Let

which for yields

and, therefore,
![[11]](/document/doi/10.1515/bejeap-2012-0045/asset/graphic/bejeap-2012-0045_eq11.png)
Condition amounts to
and
. Conditions
and
amount to
and
(with
). Then by construction, the market

yields an equilibrium where .
for
and
. As for
and
, plugging eq. [11] into the third line of eq. [4] we get eq. [6] for PWL as a function of an observation
Acknowledgments
We thank Helmuth Cremer, Nicholas Economides, Ying Fan, Till Requate, Yosuke Yasuda, two anonymous referees, and seminar and conference participants at several institutions for useful comments and suggestions. The first author acknowledges financial support from Project grant ECO2011-25330 from the Spanish Ministry of Science and Innovation. The second author acknowledges financial support from Project grant ECO2011-25203 from the Spanish Ministry of Science and Innovation. Both authors acknowledge the hospitality of the department of Economics, Australian School of Business, during the revision of this article.
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- 1
See Cowling and Mueller (1978, 728) for a summary of the criticism of Harberger’s approach.
- 2
It can be argued that lack of market coverage cannot occur once entry is allowed in the model. At this stage, however, we want to concentrate the analysis on markets with an established oligopoly in the short run, leaving the issue of entry for future research.
- 3
See Economides (1986) for the general case of
with
. Economides (1986) showed existence of a Subgame Perfect Nash Equilibrium for
.
- 4
Following Economides (1984), we use term “touching” for an equilibrium in which the markets just touch and there is no tangency of demand.
- 5
Economides (1984) studied the case of a “not-too-high” reservation price where consumers at the edges of the market prefer not to purchase the differentiated good.
- 6
PWL could be calculated, if the reservation price was observed. The latter is usually thought to be private information but, in some cases, it can be elicited by the mechanism of Becker, DeGroot, and Marschak (1964). For the limitations of this mechanism, see Horowitz (2006) and the references there.
- 7
Knowledge of demand elasticity cannot be used to break the indeterminacy of PWL, since PWL is independent of demand elasticities (own and cross) and markups. This is explained by the fact that as demand is totally inelastic, a high price, unless it induces not buying the good, does not cause welfare losses. This makes a difference with models in which consumers may buy several goods where demand elasticities and markups can be used to find PWL, even though their impact is sometimes counterintuitive (see Corchón and Zudenkova 2009).
- 8
See Economides (1989) where this assumption emerges in equilibrium in a model which generalizes Salop model but in which transportation costs are quadratic. This is the case we consider here.
- 9
Results for linear transportation costs or positive fixed costs are available upon request.
- 10
These results generalize Wauthy’s (1996) findings for the case of zero costs.
- 11
Alternatively, only one firm may produce the whole output at the maximum quality.
- 12
Tirole (1988) made the observation that non-optimal locations or quality choices lead to distortions. Our contribution formally analyzes and quantifies these distortions.
- 13
It is important to stress that here, as well as in other equilibrium configurations in the models of horizontal and vertical differentiation, an equilibrium is unique up to a permutation of firms.
©2013 by Walter de Gruyter Berlin / Boston