American Institute of Mathematical Sciences

Advanced
April  2016, 3(2): 121-142. doi: 10.3934/jdg.2016006

Localization and prices in the quadratic Hotelling model with uncertainty

Alberto A. Pinto 1, and  Telmo Parreira 2,

1. 

LIAAD - INESC TEC and Department of Mathematics, Faculty of Science, University of Porto, Rua do Campo Alegre, 687, 4169-007
2. 

Department of Mathematics and Applications, University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal

Received  October 2015 Revised  February 2016 Published  April 2016

For the quadratic Hotelling model, we study the optimal localization and price strategies under incomplete information on the production costs of the firms. We compute explicitly the pure Bayesian-Nash price duopoly equilibrium and we prove that it does not depend upon the distributions of the production costs of the firms, except on their first moments. We find when the maximal differentiation is a local optimum for the localization strategy of both firms.
Keywords: uncertainty., product differentiation, Hotelling model, duopoly.
Mathematics Subject Classification: Primary: 91A40; Secondary: 91B24, 91B3.
Citation: Alberto A. Pinto, Telmo Parreira. Localization and prices in the quadratic Hotelling model with uncertainty. Journal of Dynamics and Games, 2016, 3 (2) : 121-142. doi: 10.3934/jdg.2016006
References:
[1]

C. D'Aspremont, J. Gabszewicz and J.-F. Thisse, On Hotelling's "Stability in Competition'', Econometrica, 47 (1979), 1145-1150. doi: 10.2307/1911955.

[2]

R. Biscaia and P. Sarmento, Spatial Competition and Firms' Location Decisions under Cost Uncertainty, in FEP Working Papers no. 445, 2012.

[3]

M. Boyer, J. Laffont, P. Mahenc and M. Moreaux, Location distortions under incomplete information, Regional Science and Urban Economics, 24 (1994), 409-440. doi: 10.1016/0166-0462(93)02048-8.

[4]

M. Boyer, P. Mahenc and M. Moreaux, Asymmetric information and product differentiation, Regional Science and Urban Economics, 33 (2003), 93-113. doi: 10.1016/S0166-0462(01)00108-9.

[5]

F. Ferreira, F. A. Ferreira, M. Ferreira and A. A. Pinto, Flexibility in a Stackelberg leadership with differentiated goods, Optimization: A Journal of Mathematical Programming and Operations Research, 64 (2015), 877-893. doi: 10.1080/02331934.2013.836649.

[6]

F. Ferreira, F. A. Ferreira and A. A. Pinto, Price-setting dynamical duopoly with incomplete information, in Nonlinear Science and Complexity, J. A. Machado, M. F. Silva, R. S. Barbosa, and L. B. Figueiredo, editors, Springer Netherlands, 2011, 397-403. doi: 10.1007/978-90-481-9884-9_46.

[7]

F. Ferreira, F. A. Ferreira and A. A. Pinto, Flexibility in stackelberg leadership, in Intelligent Engineering Systems and Computational Cybernetics, J. A. Machado, B. Patkai, I. J. Rudas, editors, Springer Netherlands, 2009, 399-405. doi: 10.1007/978-1-4020-8678-6_34.

[8]

F. Ferreira, F. A. Ferreira and A. A. Pinto, Bayesian price leadership, in Mathematical Methods in Engineering, K. Tas, et al., editors, Springer Netherlands, 2007, 371-379. doi: 10.1007/978-1-4020-5678-9_32.

[9]

F. Ferreira, F. A. Ferreira and A. A. Pinto, Unknown costs in a duopoly with differentiated products, in Mathematical Methods in Engineering, K. Tas, et al., editors, Springer Netherlands, 2007, 359-369. doi: 10.1007/978-1-4020-5678-9_31.

[10]

M. Ferreira, I. P. Figueiredo, B. M. P. M. Oliveira and A. A. Pinto, Strategic optimization in R&D Investment, Optimization: A Journal of Mathematical Programming and Operations Research, 61 (2012), 1013-1023. doi: 10.1080/02331934.2011.653357.

[11]

F. A. Ferreira and A. A. Pinto, Uncertainty on a Bertrand duopoly with product differentiation, in Nonlinear Science and Complexity, J.A. Machado, M.F. Silva, R.S. Barbosa, L.B. Figueiredo, editors, Springer Netherlands, 2011, 389-395. doi: 10.1007/978-90-481-9884-9_45.

[12]

R. Gibbons, A Primer in Game Theory, Prentice Hall, 1992.

[13]

D. Graitson, Spatial competition á la Hotelling: A selective survey, The Journal of Industrial Economics, 31 (1982), 11-25. doi: 10.2307/2098001.

[14]

H. Hotelling, Stability in competition, The Collected Economics Articles of Harold Hotelling, (1990), 50-63. doi: 10.1007/978-1-4613-8905-7_4.

[15]

P. Lederer and A. Hurter, Competition of Firms: Discriminatory pricing and location, Econometrica, 54 (1986), 623-640. doi: 10.2307/1911311.

[16]

M. J. Osborne and C. Pitchick, Equilibrium in Hotelling's model of spatial competition, Econometrica, 55 (1987), 911-922. doi: 10.2307/1911035.

[17]

A. A. Pinto, F. A. Ferreira, M. Ferreira and B. M. P. M. Oliveira, Cournot Duopoly with Competition in the R&D Expenditures, in Proceedings of Symposia in Pure Mathematics,, Wiley-VCH Verlag, Weinheim, 7 (2007), 1060311-1060312. doi: 10.1002/pamm.200701031.

[18]

A. A. Pinto, B. M. P. M. Oliveira, F. A. Ferreira and F. Ferreira, Stochasticity favoring the effects of the R&D strategies of the firms, in Intelligent Engineering Systems and Computational Cybernetics, J. A. Machado, B. Patkai, I. J. Rudas, editors, Springer Netherlands, 2009, 415-423. doi: 10.1007/978-1-4020-8678-6_36.

[19]

A. A. Pinto and T. Parreira, Price competition in the Hotelling model with uncertainty on costs, Optimization: A Journal of Mathematical Programming and Operations Research, 64 (2015), 2477-2493. doi: 10.1080/02331934.2014.917304.

[20]

A. A. Pinto and T. Parreira, A Hotelling-type Network, in Dynamics, Games and Science I, M. Peixoto, A. A. Pinto, and D. Rand, editors, Springer Proccedings in Mathematics Series, 1 (2011), 709-720. doi: 10.1007/978-3-642-11456-4_45.

[21]

S. Salop, Monopolistic competition with outside goods, Bell Journal of Economics, 10 (1979), 141-156. doi: 10.2307/3003323.

[22]

T. Tabuchi and J. F. Thisse, Asymmetric equilibria in spatial competition, International Journal of Economic Theory, 13 (1995), 213-227. doi: 10.1016/0167-7187(94)00449-C.

[23]

S. Ziss, Entry deterrence, cost advantage and horizontal product differentiation, Regional Science and Urban Economics, 23 (1993), 523-543. doi: 10.1016/0166-0462(93)90045-G.

show all references

References:
[1]

C. D'Aspremont, J. Gabszewicz and J.-F. Thisse, On Hotelling's "Stability in Competition'', Econometrica, 47 (1979), 1145-1150. doi: 10.2307/1911955.

[2]

R. Biscaia and P. Sarmento, Spatial Competition and Firms' Location Decisions under Cost Uncertainty, in FEP Working Papers no. 445, 2012.

[3]

M. Boyer, J. Laffont, P. Mahenc and M. Moreaux, Location distortions under incomplete information, Regional Science and Urban Economics, 24 (1994), 409-440. doi: 10.1016/0166-0462(93)02048-8.

[4]

M. Boyer, P. Mahenc and M. Moreaux, Asymmetric information and product differentiation, Regional Science and Urban Economics, 33 (2003), 93-113. doi: 10.1016/S0166-0462(01)00108-9.

[5]

F. Ferreira, F. A. Ferreira, M. Ferreira and A. A. Pinto, Flexibility in a Stackelberg leadership with differentiated goods, Optimization: A Journal of Mathematical Programming and Operations Research, 64 (2015), 877-893. doi: 10.1080/02331934.2013.836649.

[6]

F. Ferreira, F. A. Ferreira and A. A. Pinto, Price-setting dynamical duopoly with incomplete information, in Nonlinear Science and Complexity, J. A. Machado, M. F. Silva, R. S. Barbosa, and L. B. Figueiredo, editors, Springer Netherlands, 2011, 397-403. doi: 10.1007/978-90-481-9884-9_46.

[7]

F. Ferreira, F. A. Ferreira and A. A. Pinto, Flexibility in stackelberg leadership, in Intelligent Engineering Systems and Computational Cybernetics, J. A. Machado, B. Patkai, I. J. Rudas, editors, Springer Netherlands, 2009, 399-405. doi: 10.1007/978-1-4020-8678-6_34.

[8]

F. Ferreira, F. A. Ferreira and A. A. Pinto, Bayesian price leadership, in Mathematical Methods in Engineering, K. Tas, et al., editors, Springer Netherlands, 2007, 371-379. doi: 10.1007/978-1-4020-5678-9_32.

[9]

F. Ferreira, F. A. Ferreira and A. A. Pinto, Unknown costs in a duopoly with differentiated products, in Mathematical Methods in Engineering, K. Tas, et al., editors, Springer Netherlands, 2007, 359-369. doi: 10.1007/978-1-4020-5678-9_31.

[10]

M. Ferreira, I. P. Figueiredo, B. M. P. M. Oliveira and A. A. Pinto, Strategic optimization in R&D Investment, Optimization: A Journal of Mathematical Programming and Operations Research, 61 (2012), 1013-1023. doi: 10.1080/02331934.2011.653357.

[11]

F. A. Ferreira and A. A. Pinto, Uncertainty on a Bertrand duopoly with product differentiation, in Nonlinear Science and Complexity, J.A. Machado, M.F. Silva, R.S. Barbosa, L.B. Figueiredo, editors, Springer Netherlands, 2011, 389-395. doi: 10.1007/978-90-481-9884-9_45.

[12]

R. Gibbons, A Primer in Game Theory, Prentice Hall, 1992.

[13]

D. Graitson, Spatial competition á la Hotelling: A selective survey, The Journal of Industrial Economics, 31 (1982), 11-25. doi: 10.2307/2098001.

[14]

H. Hotelling, Stability in competition, The Collected Economics Articles of Harold Hotelling, (1990), 50-63. doi: 10.1007/978-1-4613-8905-7_4.

[15]

P. Lederer and A. Hurter, Competition of Firms: Discriminatory pricing and location, Econometrica, 54 (1986), 623-640. doi: 10.2307/1911311.

[16]

M. J. Osborne and C. Pitchick, Equilibrium in Hotelling's model of spatial competition, Econometrica, 55 (1987), 911-922. doi: 10.2307/1911035.

[17]

A. A. Pinto, F. A. Ferreira, M. Ferreira and B. M. P. M. Oliveira, Cournot Duopoly with Competition in the R&D Expenditures, in Proceedings of Symposia in Pure Mathematics,, Wiley-VCH Verlag, Weinheim, 7 (2007), 1060311-1060312. doi: 10.1002/pamm.200701031.

[18]

A. A. Pinto, B. M. P. M. Oliveira, F. A. Ferreira and F. Ferreira, Stochasticity favoring the effects of the R&D strategies of the firms, in Intelligent Engineering Systems and Computational Cybernetics, J. A. Machado, B. Patkai, I. J. Rudas, editors, Springer Netherlands, 2009, 415-423. doi: 10.1007/978-1-4020-8678-6_36.

[19]

A. A. Pinto and T. Parreira, Price competition in the Hotelling model with uncertainty on costs, Optimization: A Journal of Mathematical Programming and Operations Research, 64 (2015), 2477-2493. doi: 10.1080/02331934.2014.917304.

[20]

A. A. Pinto and T. Parreira, A Hotelling-type Network, in Dynamics, Games and Science I, M. Peixoto, A. A. Pinto, and D. Rand, editors, Springer Proccedings in Mathematics Series, 1 (2011), 709-720. doi: 10.1007/978-3-642-11456-4_45.

[21]

S. Salop, Monopolistic competition with outside goods, Bell Journal of Economics, 10 (1979), 141-156. doi: 10.2307/3003323.

[22]

T. Tabuchi and J. F. Thisse, Asymmetric equilibria in spatial competition, International Journal of Economic Theory, 13 (1995), 213-227. doi: 10.1016/0167-7187(94)00449-C.

[23]

S. Ziss, Entry deterrence, cost advantage and horizontal product differentiation, Regional Science and Urban Economics, 23 (1993), 523-543. doi: 10.1016/0166-0462(93)90045-G.

[1]

Jose S. Cánovas, Tönu Puu, Manuel Ruiz Marín. Detecting chaos in a duopoly model via symbolic dynamics. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 269-278. doi: 10.3934/dcdsb.2010.13.269
[2]

Serena Brianzoni, Giovanni Campisi. Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5807-5825. doi: 10.3934/dcdsb.2021116
[3]

Ming-hui Wang, Nan-jing Huang, Donal O'Regan. Optimal product release time for a new high-tech startup firm under technical uncertainty. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021186
[4]

Alberto A. Pinto, João P. Almeida, Telmo Parreira. Local market structure in a Hotelling town. Journal of Dynamics and Games, 2016, 3 (1) : 75-100. doi: 10.3934/jdg.2016004
[5]

Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd. Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (3) : 553-576. doi: 10.3934/mbe.2012.9.553
[6]

Francesca Biagini, Jacopo Mancin. Financial asset price bubbles under model uncertainty. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 14-. doi: 10.1186/s41546-017-0026-3
[7]

Yu Chen, Zixian Cui, Shihan Di, Peibiao Zhao. Capital asset pricing model under distribution uncertainty. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021113
[8]

Robert G. McLeod, John F. Brewster, Abba B. Gumel, Dean A. Slonowsky. Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs. Mathematical Biosciences & Engineering, 2006, 3 (3) : 527-544. doi: 10.3934/mbe.2006.3.527
[9]

Martin Strugarek, Nicolas Vauchelet, Jorge P. Zubelli. Quantifying the survival uncertainty of Wolbachia-infected mosquitoes in a spatial model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 961-991. doi: 10.3934/mbe.2018043
[10]

Ali Unver, Christian Ringhofer, Dieter Armbruster. A hyperbolic relaxation model for product flow in complex production networks. Conference Publications, 2009, 2009 (Special) : 790-799. doi: 10.3934/proc.2009.2009.790
[11]

Qing-you Yan, Juan-bo Li, Ju-liang Zhang. Licensing schemes in Stackelberg model under asymmetric information of product costs. Journal of Industrial and Management Optimization, 2007, 3 (4) : 763-774. doi: 10.3934/jimo.2007.3.763
[12]

Fengjun Wang, Qingling Zhang, Bin Li, Wanquan Liu. Optimal investment strategy on advertisement in duopoly. Journal of Industrial and Management Optimization, 2016, 12 (2) : 625-636. doi: 10.3934/jimo.2016.12.625
[13]

Rabah Amir, Igor V. Evstigneev. A new perspective on the classical Cournot duopoly. Journal of Dynamics and Games, 2017, 4 (4) : 361-367. doi: 10.3934/jdg.2017019
[14]

Ferenc A. Bartha, Hans Z. Munthe-Kaas. Computing of B-series by automatic differentiation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 903-914. doi: 10.3934/dcds.2014.34.903
[15]

Tom Maertens, Joris Walraevens, Herwig Bruneel. Controlling delay differentiation with priority jumps: Analytical study. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 657-673. doi: 10.3934/naco.2011.1.657
[16]

Jianzhong Wang. Wavelet approach to numerical differentiation of noisy functions. Communications on Pure and Applied Analysis, 2007, 6 (3) : 873-897. doi: 10.3934/cpaa.2007.6.873
[17]

Javad Taheri-Tolgari, Mohammad Mohammadi, Bahman Naderi, Alireza Arshadi-Khamseh, Abolfazl Mirzazadeh. An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1317-1344. doi: 10.3934/jimo.2018097
[18]

Colin Cotter, Dan Crisan, Darryl Holm, Wei Pan, Igor Shevchenko. Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model. Foundations of Data Science, 2020, 2 (2) : 173-205. doi: 10.3934/fods.2020010
[19]

Francesca Biagini, Katharina Oberpriller. Reduced-form setting under model uncertainty with non-linear affine intensities. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 159-188. doi: 10.3934/puqr.2021008
[20]

Georgios I. Papayiannis. Robust policy selection and harvest risk quantification for natural resources management under model uncertainty. Journal of Dynamics and Games, 2022, 9 (2) : 203-217. doi: 10.3934/jdg.2022004

 Impact Factor: 

Tools

Metrics

  • PDF downloads (241)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy