This study investigates spatial price discrimination with two types of market competition – price competition and quantity competition – and two kinds of cross-relations between goods – substitutes and complements – with endogenous location choices in a barbell model. The results herein present that the maximum differentiation (end point agglomeration) is the unique location equilibrium with substitutes (complements), irrespective of what type of competition. We demonstrate that if the unit transportation rate is sufficiently high, then consumer surplus, profits, and social welfare are higher under price competition than under quantity competition for both substitutes and complements. This means that introducing a spatial barrier to competition generated through transportation costs may solve the problem of inconsistency from the conflict interests between consumers, firms, and a social planner.
Proof of Proposition 3
The derivations of the case of complements are omitted, because the results are directly implied by the non-spatial Singh and Vives (1984) model, in which each firm’s constant marginal cost is assumed to be the same. In the case of substitutes, direct calculations yield:
where the function is easily shown to be concave in t, for . Solving for yields the following two roots:
It follows that for all . With a straightforward derivation, we yield:
Here, the function can be shown as a convex function with respect to t, for . Solving for obtains the following two roots:
Thus, for and for .
Regarding the social welfare and consumer surplus rankings, we solve the equilibrium prices and quantities as:
Here, and ( and ) are respectively firm i’s equilibrium price and quantity in market j under Cournot competition (under Bertrand competition), and and respectively denote firm i’s price and quantity in the limit-pricing equilibrium, for and .9 After substituting eqs , , and  into the utility (total surplus) function, , and the consumer surplus function, , straightforward calculations yield the following social welfare difference and consumer surplus difference in each market j, for :
The four functions in eq.  are all quadratic functions with respect to t. A similar logic to the above analysis shows that the signs of the four functions are all negative for all the admissible values of the model’s parameters. Q.E.D.
Anderson, S. P., and D. J.Neven. 1991. “Cournot Competition Yields Spatial Agglomeration.” International Economic Review32:793–808. Search in Google Scholar
Arevalo-Tome, R., and J. M.Chamorro-Rivas. 2006. “Location as an Instrument for Social Welfare Improvement in a Spatial Model of Cournot Competition.” Investigaciones Economicas30:117–36. Search in Google Scholar
Boone, J. 2001. “The Intensity of Competition and the Incentive to Innovate.” International Journal of Industrial Organization19:705–26. Search in Google Scholar
Bulow, J. L., J. D.Geanakoplos, and P. D.Klemperer. 1985. “Multimarket Oligopoly: Strategic Substitutes and Complements.” Journal of Political Economy91:488–511. Search in Google Scholar
Colombo, S. 2011. “Taxation and Predatory Prices in a Spatial Model.” Papers in Regional Science90:603–12. Search in Google Scholar
Delbono, F., and V.Denicolo. 1990. “R&D Investment in a Symmetric and Homogeneous Oligopoly.” International Journal of Industrial Organization8:297–313. Search in Google Scholar
Gross, J., and W. L.Holahan. 2003. “Credible Collusion in Spatially Separated Markets.” International Economic Review44:299–312. Search in Google Scholar
Hackner, J. 2000. “A Note on Price and Quality Competition in Differentiated Oligopolies.” Journal of Economic Theory93:233–39. Search in Google Scholar
Hamilton, J. H., J. F.Thisse, and A.Weskamp. 1989. “Spatial Discrimination: Bertrand vs. Cournot in a Model of Location Choice.” Regional Science and Urban Economics19:87–102. Search in Google Scholar
Hinloopen, J., and C.Van Marrewijk. 1999. “On the Limits and Possibilities of the Principle of Minimum Differentiation.” International Journal of Industrial Organization17:735–50. Search in Google Scholar
Hinloopen, J., and J.Vandekerckhove. 2011. “Product Market Competition and Investments in Cooperative R&D.” The B.E. Journal of Economic Analysis & Policy11:Article 55. Search in Google Scholar
Hotelling, H. 1929. “Stability in Competition.” Economic Journal39:41–57. Search in Google Scholar
Hwang, H., Y. S.Lin, and C. C.Mai. 2007. “Spatial Pricing, Optimal Location and Social Welfare with Consumer Arbitrage.” Annals of Regional Science41:619–38. Search in Google Scholar
Hwang, H., and C. C.Mai. 1990. “Effects of Spatial Price Discrimination on Output, Welfare, and Location.” American Economic Review80:567–75. Search in Google Scholar
Liang, W. J., H.Hwang, and C. C.Mai. 2006. “Spatial Discrimination: Bertrand vs. Cournot with Asymmetric Demands.” Regional Science and Urban Economics36:790–802. Search in Google Scholar
Pal, D. 1998. “Does Cournot Competition Yield Spatial Agglomeration?” Economics Letters60:49–53. Search in Google Scholar
Porter, M. E. 2000. “Location, Competition and Economic Development: Local Clusters in the Global Economy.” Economic Development Quarterly14:15–31. Search in Google Scholar
Qiu, L. D. 1997. “On the Dynamic Efficiency of Bertrand and Cournot Equilibria.” Journal of Economic Theory75:213–29. Search in Google Scholar
Salop, S. C. 1979. “Monopolistic Competition with Outside Goods.” Bell Journal of Economics10:141–56. Search in Google Scholar
Shimizu, D. 2002. “Product Differentiation in Spatial Cournot Markets.” Economics Letters76:317–22. Search in Google Scholar
Singh, N., and X.Vives. 1984. “Price and Quantity Competition in a Differentiated Duopoly.” Rand Journal of Economics15:546–54. Search in Google Scholar
Sun, C. H. 2009. “A Note on Arevalo-Tom and Chamorro-Rivas: Location as an Instrument for Social Welfare Improvement in a Spatial Model of Cournot Competition.” Investigaciones Economicas33:131–41. Search in Google Scholar
Vives, X. 1991. “Banking Competition and European Integration.” In European Financial Integration, edited by A.Giovannini and C.Mayer, 9–31. Cambridge: Cambridge University Press. Search in Google Scholar
Yu, C. M. 2007. “Price and Quantity Competition Yield the Same Location Equilibrium in a Circular Market.” Papers in Regional Science86:643–55. Search in Google Scholar
Zanchettin, P. 2006. “Differentiated Duopoly with Asymmetric Costs.” Journal of Economics and Management Strategy15:999–1015. Search in Google Scholar
The discrete linear-city model can be extended to consumers located at the nodes in a finite transportation network.
Consider a firm that produces coffee. If its competitor produces coffee (cream, respectively), then the firm definitely wants its competitor to charge a high price for coffee (the firm definitely wants the quantity of cream to be high, respectively).
More specifically, a limit-pricing strategy occurs in a situation of strategic entry deterrence, by setting the price at the highest level that is consistent with keeping the potential entrant out, and a firm works as a monopolist, but bears competitive pressure from its rival. Without loss of generality, it is assumed that the low transportation cost firm 1 (firm 2) obtains the whole share of market A (market B) in equilibrium.
The detailed derivation of the sign of the function in eq.  is omitted to save space. It can be checked that if , then the function in eq.  is concave in the transportation rate t and has two roots, with the large root greater than and the small root smaller than . If , then the function in eq.  is convex in the transportation rate t and has two roots, with the large root smaller than . In either case, the sign of the function in eq.  is positive for all .
Empirical evidence proposed by Porter (2000) supports a similar idea that Manhattan hoteliers tend to locate new hotels sufficiently close to established hotels that are similar on one product dimension in order to benefit from agglomeration economies, but different on another product dimension in order to create complementary differences.
It is well known that a spatial model also has an explanation for product locations.
Note that firm 2’s quantity supplied is zero in firm 1’s home market in the limit-pricing equilibrium.
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