Top

Schmalenbach Business Review

Published in:

17-12-2018 | Original Article

Location in a Disk City with Consumer Concentration Around the Center

Author: Manfred Stadler

Published in: Schmalenbach Business Review | Issue 1/2019

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The paper studies a two-stage location-price duopoly game in a disk city with consumer concentration around the city center. When consumers are uniformly distributed over the plane, unconstrained firms locate outside of the city. Consumer concentration, however, induces firms to locate nearer to each other and, when the degree of concentration is sufficiently high, inside of the city. Prices and firm profits decrease in the degree of consumer concentration. We explicitly solve the model for classes of cone-shaped, dome-shaped, and bell-shaped consumer densities. In all cases we identify a loss of welfare due to the strategic effect which causes the firms’ spatial differentiation being too large.

previous article Discussion of “Consequences of Unfair Job Promotions in Organizations”

next article Discussion of “Location in a Disk City with Consumer Concentration Around the Center”

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

The disk area on the right side of the vertical line \(\tilde{x}\) is \(2\int_{\tilde{x}}^{1}r\arccos(\tilde{x}/r)dr\). Multiply at each point in this area by the local consumer density \(f(r)\) to obtain the demand function \(D(\tilde{x})\).

This is a two-dimensional extension inter alia of the triangular density function as used by Tabuchi and Thisse (1995). A corresponding specification can be found in Mazalov and Sakaguchi (2003).

If the city border is binding, firms locate on the disk’s perimeter at maximal distance. Substituting \(x_{1}=-x_{2}=1\), \(D(\tilde{x}=0)=1/2\), and \(D^{\prime}(\tilde{x}=0)=-2/\pi\) into the price equations (4) shows that the firms charge the prices \(p=\pi<p^{*}\) and hence realize the profits \(\Pi=(1/2)\pi<\Pi^{*}\). This solution is the constrained optimum as it was derived by Mazalov and Sakaguchi (2003) and Feldin (2012).

This is a two-dimensional counterpart to the one-dimensional beta density function as used by Stadler (2012).

Note that \(\Gamma(1/2)=\sqrt{\pi}\), \(\Gamma(1)=1\), and \(\Gamma(n+1)=n\Gamma(n)\) for \(n=\{1/2,1\}\).

Note that the tiny area differential \(dA\) for the polar coordinates (\(r,\phi\)) is \(dA=rdrd\phi\).

Additionally, the population of the city – an alternative measure of the “city size” – does not have to be restricted to the mass of 1. An increasing mass would leave the location and price decisions of the firms unchanged but would proportionally increase their profits.

In case of finite support, a rectangular (or a quadratic) product space would certainly be preferable to a disk space. A quadratic product space, however, is not tractable for an analysis of demand concentration since the property of polar-symmetry is not satisfied. In the case of infinite support, the geometrical difference between a disk and a square diminishes such that this problem is resolved.

As pointed out e.g. by Ansari et al. (1994), the distribution of consumer preferences in many markets is unlikely to be uniform.

Note that \(\arccos(0)=\pi/2\) and \(2\pi\int_{0}^{1}rf(r)dr=1\) for all feasible polar-symmetric consumer densities.

Anderson, S.P., J.K. Goeree, and R. Ramer. 1997. Location, Location, Location. Journal of Economic Theory 77:102–127.CrossRef

Ansari, A., N. Economides, and A. Ghosh. 1994. Competitive Positioning in Markets with Nonuniform Preferences. Marketing Science 13:248–273.CrossRef

Ansari, A., N. Economides, and J. Steckel. 1998. The Max-Min-Min Principle of Product Differentiation. Journal of Regional Science 38:207–230.CrossRef

Böckem, S. 1994. A Generalized Model of Horizontal Product Differentiation. Journal of Industrial Economics 17:287–298.CrossRef

Brenner, S. 2005. Hotelling Games with Three, Four, and More Players. Journal of Regional Science 45:851–864.CrossRef

Casado-Izaga, F.J. 2000. Location Decisions: The Role of Uncertainty about Consumer Tastes. Journal of Economics 71:31–46.CrossRef

D’Aspremont, C., J.J. Gabszewicz, and J.-F. Thisse. 1979. On Hotelling’s Stability in Competition. Econometrica 47:1145–1150.CrossRef

Economides, N. 1986. Minimal and Maximal Product Differentiation in Hotelling’s Duopoly. Economics Letters 21:67–71.CrossRef

Feldin, A. 2012. Three Firms on a Unit Disk Market: Intermediate Product Differentiation. Economic and Business Review 14:321–345.

Hotelling, H. 1929. Stability in Competition. Economic Journal 39:41–57.CrossRef

Irmen, A., and J.-F. Thisse. 1998. Competition in Multi-Characteristics Spaces: Hotelling Was Almost Right. Journal of Economic Theory 78:76–102.CrossRef

Lambertini, L. 1994. Equilibrium Locations in the Unconstrained Hotelling Game. Economic Notes 24:438–446.

Mazalov, V., and M. Sakaguchi. 2003. Location Game on the Plane. International Game Theory Review 5:13–25.CrossRef

Meagher, K.J., and K.G. Zauner. 2004. Product Differentiation and Location Decisions under Demand Uncertainty. Journal of Economic Theory 117:201–216.CrossRef

Meagher, K.J., and K.G. Zauner. 2005. Location-then-Price Competition with Uncertain Consumer Tastes. Economic Theory 25:799–818.CrossRef

Meagher, K.J., E.G.S. Teo, and W. Wang. 2008. A Duopoly Location Toolkit: Consumer Densities which Yield Unique Spatial Duopoly. Equilibria. B.E. Journal of Theoretical Economics 8:1–21.

Stadler, M. 2012. Standortwahl und Produktdifferenzierung. In Zur Zukunft des Wettbewerbs. Marburg. Metropolis Verlag, ed. H. Enke, A. Wagner, 323–338.

Tabuchi, T. 1994. Two-Stage Two-Dimensional Spatial Competition between Two Firms. Regional Science and Urban Economics 24:207–227.

Tabuchi, T. 2012. Multiproduct Firms in Hotelling’s Spatial Competition. Journal of Economics and Management Strategy 21:445–467.CrossRef

Tabuchi, T., and J.-F. Thisse. 1995. Asymmetric Equilibria in Spatial Competition. International Journal of Industrial Organization 13:213–227.CrossRef

Veendorp, E.C.H., and A. Majeed. 1995. Differentiation in a Two-Dimensional Market. Regional Science and Urban Economics 25:75–83.

Woeckener, B. 2002. Spatial Competition with an Outside Good and Distributed Reservation Prices. Journal of Economics 71:31–46.

Title: Location in a Disk City with Consumer Concentration Around the Center
Author: Manfred Stadler
Publication date: 17-12-2018
Publisher: Springer International Publishing
Published in: Schmalenbach Business Review / Issue 1/2019
Print ISSN: 1439-2917
Electronic ISSN: 2194-072X
DOI: https://doi.org/10.1007/s41464-018-0064-0

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Springer Professional "Wirtschaft+Technik"

Other articles of this Issue 1/2019

Discussion of Martin Windl: Net Stable Funding Ratio and Liquidity Hoarding

Consequences of Unfair Job Promotions in Organizations

Special Section: Economic Analyses in Business Administration

Is the Time Ripe? How the Value of Waiting and Incentives Affect Users’ Switching Behaviors for Smart Home Devices

Discussion of “Location in a Disk City with Consumer Concentration Around the Center”

Discussion of “Consequences of Unfair Job Promotions in Organizations”