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 marketsuch 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, andwhich 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
where 
and either 
and 
or 
and 
, there is a linear vertical market![[5]](/document/doi/10.1515/bejeap-2012-0045/asset/graphic/bejeap-2012-0045_eq5.png)
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
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
where 
, 
and 
, there is a linear vertical market 
such that this observation is consistent with an uncovered market equilibrium for this market, and
![[6]](/document/doi/10.1515/bejeap-2012-0045/asset/graphic/bejeap-2012-0045_eq6.png)
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.
References
Aghion, P., and M. Schankerman. 2004. “On the Welfare Effects and Political Economy of Competition-Enhancing Policies.” Economic Journal 114:800–24.10.1111/j.1468-0297.2004.00244.xSearch in Google Scholar
Armstrong, M., and D. Sappington. 2007. “Recent Developments in the Theory of Regulation.” In Handbook of Industrial Organization, edited by M. Armstrong and R. Porter, Vol. 3, 1557–700. Netherlands: Elsevier.10.1016/S1573-448X(06)03027-5Search in Google Scholar
Becker, G., M. DeGroot, and J. Marschak. 1964. “Measuring Utility by a Single-response Sequential Method.” Behavioral Science 9:226–36.10.1002/bs.3830090304Search in Google Scholar
Chen, Y., and M. H. Riordan. (2007). “Price and Variety in the Spokes Model.” Economic Journal 117:897–921.Search in Google Scholar
Corchón, L. 2008. “Welfare Losses under Cournot Competition.” International Journal of Industrial Organization 26:1120–31.10.1016/j.ijindorg.2007.10.003Search in Google Scholar
Corchón, L., and G. Zudenkova. 2009. “Computing Welfare Losses from Data under Imperfect Competition with Heterogeneous Goods.” International Journal of Industrial Organization 27:646–54.10.1016/j.ijindorg.2009.04.002Search in Google Scholar
Cowling, K., and D. C. Mueller. 1978. “The Social Costs of Monopoly Power.” Economic Journal 88:727–48.10.2307/2231975Search in Google Scholar
Crawford, G. 2012. “Endogenous Product Choice: A Progress Report.” International Journal of Industrial Organization 30, 3:315–20.10.1016/j.ijindorg.2011.12.006Search in Google Scholar
Crawford, G., A. Shcherbakov, and M. Shum. 2011. “The Welfare Effects of Endogenous Quality Choice: Evidence from Cable Television Markets.” Mimeo, University of Warwick.Search in Google Scholar
Draganska, M., M. Mazzeo, and K. Sei. 2009. “Beyond Plain Vanilla: Modeling Joint Product Assortment and Pricing Decisions.” Quantitative Marketing and Economics 7:105–46.10.1007/s11129-008-9047-7Search in Google Scholar
D’Aspremont, C., J. J. Gabszewicz, and J.-F. Thisse. 1979. “On Hotelling’s Stability in Competition.” Econometrica 47:1145–50.10.2307/1911955Search in Google Scholar
Eaton, B. C., and R. G. Lipsey. 1978. “Freedom of Entry and the Existence of Pure Profit.” Economic Journal 88:455–69.10.2307/2232046Search in Google Scholar
Eaton, B. C., and R. G. Lipsey. 1981. “Capital, Commitment, and Entry Equilibrium.” Bell Journal of Economics 12:593–604.10.2307/3003574Search in Google Scholar
Eaton, B. C., and R. G. Lipsey. 1982. “An Economic Theory of Central Places.” Economic Journal 92:56–72.10.2307/2232256Search in Google Scholar
Economides, N. 1984. “The Principle of Minimum Differentiation Revisited.” European Economic Review 24:345–68.10.1016/0014-2921(84)90061-8Search in Google Scholar
Economides, N. 1986. “Minimal and Maximal Product Differentiation in Hotelling’s Duopoly.” Economics Letters 21:67–71.10.1016/0165-1765(86)90124-2Search in Google Scholar
Economides, N. 1989. “Symmetric Equilibrium Existence and Optimality in Differentiated Product Markets.” Journal of Economic Theory 47:178–94.10.1016/0022-0531(89)90108-7Search in Google Scholar
Eizenberg, A. 2011. “Upstream Innovation and Product Variety in the U.S. Home PC Market.” Working Paper, Hebrew University of Jerusalem.10.2139/ssrn.1760828Search in Google Scholar
Fan, Y. 2012. “Ownership Consolidation and Product Characteristics: A Study of the U.S. Daily Newspaper Market.” American Economic Review forthcoming.10.1257/aer.103.5.1598Search in Google Scholar
Gabszewicz, J. J., and J.-F. Thisse. 1979. “Price Competition, Quality and Income Disparities.” Journal of Economic Theory 20:340–59.10.1016/0022-0531(79)90041-3Search in Google Scholar
Gabszewicz, J. J., and J.-F. Thisse. 1980. “Entry (and Exit) in a Differentiated Industry.” Journal of Economic Theory 22:327–38.10.1016/0022-0531(80)90046-0Search in Google Scholar
Gandhi, A., L. Froeb, S. Tschantz, and G. J. Werden. 2008. “Post-Merger Product Repositioning.” Journal of Industrial Economics 56:49–67.10.1111/j.1467-6451.2008.00332.xSearch in Google Scholar
Harberger, A. C. 1954. “Monopoly and Resource Allocation.” American Economic Review: Papers and Proceedings 44:77–87.Search in Google Scholar
Horowitz, J. 2006. “The Becker-DeGroot-Marschak Mechanism Is Not Necessarily Incentive Compatible, Even for Non-Random Goods.” Economics Letters 93:6–11.10.1016/j.econlet.2006.03.033Search in Google Scholar
Hotelling, H. 1929. “Stability in Competition.” Economic Journal 39:41–57.10.2307/2224214Search in Google Scholar
Leibenstein, H. 1966. “Allocative Efficiency versus X-Efficiency.” American Economic Review 56:392–425.Search in Google Scholar
Salop, S. C. 1979. “Monopolistic Competition with Outside Goods.” Bell Journal of Economics 10:141–56.10.2307/3003323Search in Google Scholar
Shaked, A., and J. Sutton. 1982. “Relaxing Price Competition through Product Differentiation.” Review of Economic Studies 49:3–13.10.2307/2297136Search in Google Scholar
Shaked, A., and J. Sutton. 1983. “Natural Oligopolies.” Econometrica 51:1469–83.10.2307/1912285Search in Google Scholar
Tirole, J. 1988. The Theory of Industrial Organization. Cambridge, Massachusetts: MIT Press.Search in Google Scholar
Tullock, G. 1967. “The Welfare Costs of Tariffs, Monopolies, and Theft.” Western Economic Journal 5:224–32.10.1111/j.1465-7295.1967.tb01923.xSearch in Google Scholar
Wauthy, X. 1996. “Quality Choice in Models of Vertical Differentiation.” Journal of Industrial Economics 44:345–53.10.2307/2950501Search in Google Scholar
- 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.
with 
. 
.©2013 by Walter de Gruyter Berlin / Boston