American Institute of Mathematical Sciences

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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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

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