# 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

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

and

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,

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.

**Published Online:**2013-07-30

©2013 by Walter de Gruyter Berlin / Boston