IDEAS home Printed from https://ideas.repec.org/p/hal/wpaper/hal-02882403.html
   My bibliography  Save this paper

Linear demand systems for differentiated goods: Overview and user's guide

Author

Listed:
  • Philippe Choné

    (CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - CNRS - Centre National de la Recherche Scientifique)

  • Laurent Linnemer

    (CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - CNRS - Centre National de la Recherche Scientifique)

Abstract

Linear demand systems and quasi-linear quadratic utility models are widely used in industrial economics. We clarify the link between the two settings and explain their exact origin as it seems to be little known by practitioners. We offer practical recommendations to achieve consistency, tractability and a reasonable degree of generality when using the linear demand framework. We show that all tractable versions of the model used in practice are (almost) identical and have a mean-variance structure. We provide concise, ready-to-use formulae for the symmetric model. Finally, we revisit and extend the asymmetric model of Shubik and Levitan.

Suggested Citation

  • Philippe Choné & Laurent Linnemer, 2020. "Linear demand systems for differentiated goods: Overview and user's guide," Working Papers hal-02882403, HAL.
    • Handle: RePEc:hal:wpaper:hal-02882403
    Note: View the original document on HAL open archive server: https://hal.science/hal-02882403
    as

    Download full text from publisher

    File URL: https://hal.science/hal-02882403/document
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Michael Spence, 1976. "Product Selection, Fixed Costs, and Monopolistic Competition," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 43(2), pages 217-235.
    2. Allain, Marie-Laure & Chambolle, Claire, 2011. "Anti-competitive effects of resale-below-cost laws," International Journal of Industrial Organization, Elsevier, vol. 29(4), pages 373-385, July.
    3. Kyle Bagwell & Garey Ramey, 1991. "Oligopoly Limit Pricing," RAND Journal of Economics, The RAND Corporation, vol. 22(2), pages 155-172, Summer.
    4. Upender Subramanian & Jagmohan S. Raju & Sanjay K. Dhar & Yusong Wang, 2010. "Competitive Consequences of Using a Category Captain," Management Science, INFORMS, vol. 56(10), pages 1739-1765, October.
    5. Richard E. Levitan & Martin Shubik, 1967. "The Cournot Equilibrium in a Nonsymmetric Oligopolistic Market (A Business Game for Teaching and Research Purposes: Part VI)," Cowles Foundation Discussion Papers 224, Cowles Foundation for Research in Economics, Yale University.
    6. Varian, Hal R, 1980. "A Model of Sales," American Economic Review, American Economic Association, vol. 70(4), pages 651-659, September.
    7. Curtis Eaton, B. & Lipsey, Richard G., 1989. "Product differentiation," Handbook of Industrial Organization, in: R. Schmalensee & R. Willig (ed.), Handbook of Industrial Organization, edition 1, volume 1, chapter 12, pages 723-768, Elsevier.
    8. Ming Chung Chang & Hsiao-Ping Peng, 2012. "Cournot And Bertrand Equilibria Compared: A Critical Review And An Extension From The Output-Structure Viewpoint," The Japanese Economic Review, Japanese Economic Association, vol. 63(4), pages 467-496, December.
    9. Zhao, Jingang, 2001. "A characterization for the negative welfare effects of cost reduction in Cournot oligopoly," International Journal of Industrial Organization, Elsevier, vol. 19(3-4), pages 455-469, March.
    10. Avner Shaked & John Sutton, 1990. "Multiproduct Firms and Market Structure," RAND Journal of Economics, The RAND Corporation, vol. 21(1), pages 45-62, Spring.
    11. Bernstein, Fernando & Federgruen, Awi, 2004. "Comparative statics, strategic complements and substitutes in oligopolies," Journal of Mathematical Economics, Elsevier, vol. 40(6), pages 713-746, September.
    12. Valletti, Tommaso M., 2003. "Input price discrimination with downstream Cournot competitors," International Journal of Industrial Organization, Elsevier, vol. 21(7), pages 969-988, September.
    13. Dixit, Avinash K & Stiglitz, Joseph E, 1977. "Monopolistic Competition and Optimum Product Diversity," American Economic Review, American Economic Association, vol. 67(3), pages 297-308, June.
    14. Fevrier, Philippe & Linnemer, Laurent, 2004. "Idiosyncratic shocks in an asymmetric Cournot oligopoly," International Journal of Industrial Organization, Elsevier, vol. 22(6), pages 835-848, June.
    15. Marc Bourreau & Johan Hombert & Jerome Pouyet & Nicolas Schutz, 2011. "Upstream Competition between Vertically Integrated Firms," Journal of Industrial Economics, Wiley Blackwell, vol. 59(4), pages 677-713, December.
    16. Schinkel, Maarten Pieter & Spiegel, Yossi, 2017. "Can collusion promote sustainable consumption and production?," International Journal of Industrial Organization, Elsevier, vol. 53(C), pages 371-398.
    17. Lars-Hendrik Röller & Mihkel M. Tombak, 1993. "Competition and Investment in Flexible Technologies," Management Science, INFORMS, vol. 39(1), pages 107-114, January.
    18. Hans Jarle Kind & Tore Nilssen & Lars Sørgard, 2009. "Business Models for Media Firms: Does Competition Matter for How They Raise Revenue?," Marketing Science, INFORMS, vol. 28(6), pages 1112-1128, 11-12.
    19. Serdar Sayman & Stephen J. Hoch & Jagmohan S. Raju, 2002. "Positioning of Store Brands," Marketing Science, INFORMS, vol. 21(4), pages 378-397, June.
    20. Marie-Laure Allain & Emeric Henry & Margaret Kyle, 2016. "Competition and the Efficiency of Markets for Technology," Management Science, INFORMS, vol. 62(4), pages 1000-1019, April.
    21. Abito, Jose Miguel & Wright, Julian, 2008. "Exclusive dealing with imperfect downstream competition," International Journal of Industrial Organization, Elsevier, vol. 26(1), pages 227-246, January.
    22. Vibhanshu Abhishek & Kinshuk Jerath & Z. John Zhang, 2016. "Agency Selling or Reselling? Channel Structures in Electronic Retailing," Management Science, INFORMS, vol. 62(8), pages 2259-2280, August.
    23. Choné, Philippe & Linnemer, Laurent, 2008. "Assessing horizontal mergers under uncertain efficiency gains," International Journal of Industrial Organization, Elsevier, vol. 26(4), pages 913-929, July.
    24. Dobson, Paul W & Waterson, Michael, 1997. "Countervailing Power and Consumer Prices," Economic Journal, Royal Economic Society, vol. 107(441), pages 418-430, March.
    25. Herweg, Fabian & Müller, Daniel, 2016. "Discriminatory nonlinear pricing, fixed costs, and welfare in intermediate-goods markets," International Journal of Industrial Organization, Elsevier, vol. 46(C), pages 107-136.
    26. Chiara Fumagalli & Massimo Motta, 2008. "Buyers’ Miscoordination, Entry and Downstream Competition," Economic Journal, Royal Economic Society, vol. 118(531), pages 1196-1222, August.
    27. Paul W. Dobson & Michael Waterson, 1996. "Product Range and Interfirm Competition," Journal of Economics & Management Strategy, Wiley Blackwell, vol. 5(3), pages 317-341, September.
    28. Amr Farahat & Georgia Perakis, 2011. "TECHNICAL NOTE---A Comparison of Bertrand and Cournot Profits in Oligopolies with Differentiated Products," Operations Research, INFORMS, vol. 59(2), pages 507-513, April.
    29. Levitan, Richard & Shubik, Martin, 1972. "Price Duopoly and Capacity Constraints," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 13(1), pages 111-122, February.
    30. Armstrong, Mark & Vickers, John, 2015. "Which demand systems can be generated by discrete choice?," Journal of Economic Theory, Elsevier, vol. 158(PA), pages 293-307.
    31. Deneckere, R., 1983. "Duopoly supergames with product differentiation," Economics Letters, Elsevier, vol. 11(1-2), pages 37-42.
    32. Jean Tirole, 1988. "The Theory of Industrial Organization," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262200716, April.
    33. Giacomo Calzolari & Vincenzo Denicolò, 2015. "Exclusive Contracts and Market Dominance," American Economic Review, American Economic Association, vol. 105(11), pages 3321-3351, November.
    34. Inderst, Roman & Valletti, Tommaso, 2011. "Incentives for input foreclosure," European Economic Review, Elsevier, vol. 55(6), pages 820-831, August.
    35. Rabah Amir & Jim Y. Jin & Gerald Pech & Michael Tröge, 2016. "Prices and Deadweight Loss in Multiproduct Monopoly," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 18(3), pages 346-362, June.
    36. Joris Pinkse & Margaret E. Slade & Craig Brett, 2002. "Spatial Price Competition: A Semiparametric Approach," Econometrica, Econometric Society, vol. 70(3), pages 1111-1153, May.
    37. Symeonidis, George, 2003. "Comparing Cournot and Bertrand equilibria in a differentiated duopoly with product R&D," International Journal of Industrial Organization, Elsevier, vol. 21(1), pages 39-55, January.
    38. Jaffe Sonia & Weyl E. Glen, 2010. "Linear Demand Systems are Inconsistent with Discrete Choice," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 10(1), pages 1-8, December.
    39. Symeonidis, George, 1999. "Cartel stability in advertising-intensive and R&D-intensive industries," Economics Letters, Elsevier, vol. 62(1), pages 121-129, January.
    40. Sutton, John, 1996. "Technology and market structure," European Economic Review, Elsevier, vol. 40(3-5), pages 511-530, April.
    41. Avinash Dixit, 1979. "A Model of Duopoly Suggesting a Theory of Entry Barriers," Bell Journal of Economics, The RAND Corporation, vol. 10(1), pages 20-32, Spring.
    42. Chen, Yongmin & Gayle, Philip G., 2007. "Vertical contracting between airlines: An equilibrium analysis of codeshare alliances," International Journal of Industrial Organization, Elsevier, vol. 25(5), pages 1046-1060, October.
    43. Greg Shaffer, 1991. "Slotting Allowances and Resale Price Maintenance: A Comparison of Facilitating Practices," RAND Journal of Economics, The RAND Corporation, vol. 22(1), pages 120-135, Spring.
    44. Lloyd Shapley & Martin Shubik, 1969. "Price Strategy Oligopoly With Product Variation," Kyklos, Wiley Blackwell, vol. 22(1), pages 30-44, February.
    45. Nocke, Volker & Schutz, Nicolas, 2017. "Quasi-linear integrability," Journal of Economic Theory, Elsevier, vol. 169(C), pages 603-628.
    46. Jonathan Kluberg & Georgia Perakis, 2012. "Generalized Quantity Competition for Multiple Products and Loss of Efficiency," Operations Research, INFORMS, vol. 60(2), pages 335-350, April.
    47. Spence, Michael, 1976. "Product Differentiation and Welfare," American Economic Review, American Economic Association, vol. 66(2), pages 407-414, May.
    48. Levitan, Richard & Shubik, Martin, 1971. "Price variation duopoly with differentiated products and random demand," Journal of Economic Theory, Elsevier, vol. 3(1), pages 23-39, March.
    49. Martin Shubik, 1961. "A Business Game for Teaching and Research Purposes," Cowles Foundation Discussion Papers 115R1, Cowles Foundation for Research in Economics, Yale University.
    50. Xavier Vives, 2001. "Oligopoly Pricing: Old Ideas and New Tools," MIT Press Books, The MIT Press, edition 1, volume 1, number 026272040x, April.
    51. Judy Hsu & X. Wang, 2005. "On Welfare under Cournot and Bertrand Competition in Differentiated Oligopolies," Review of Industrial Organization, Springer;The Industrial Organization Society, vol. 27(2), pages 185-191, September.
    52. Jin, Jim Y., 1997. "Comparing Cournot and Bertrand equilibria revisited," Discussion Papers, Research Unit: Market Dynamics FS IV 97-4, WZB Berlin Social Science Center.
    53. Lu, Yuanzhu & Wright, Julian, 2010. "Tacit collusion with price-matching punishments," International Journal of Industrial Organization, Elsevier, vol. 28(3), pages 298-306, May.
    54. LaFrance, Jeffrey T., 1985. "Linear demand functions in theory and practice," Journal of Economic Theory, Elsevier, vol. 37(1), pages 147-166, October.
    55. Patrick Rey & Thibaud Vergé, 2010. "Resale Price Maintenance And Interlocking Relationships," Journal of Industrial Economics, Wiley Blackwell, vol. 58(4), pages 928-961, December.
    56. Marx, Leslie M. & Shaffer, Greg, 2004. "Opportunism and menus of two-part tariffs," International Journal of Industrial Organization, Elsevier, vol. 22(10), pages 1399-1414, December.
    57. Michael Kopel & Anna Ressi & Luca Lambertini, 2017. "Capturing Direct and Cross Price Effects in a Differentiated Products Duopoly Model," Manchester School, University of Manchester, vol. 85(3), pages 282-294, June.
    58. Xavier Vives, 2008. "Innovation And Competitive Pressure," Journal of Industrial Economics, Wiley Blackwell, vol. 56(3), pages 419-469, December.
    59. Wang, X. Henry & Zhao, Jingang, 2007. "Welfare reductions from small cost reductions in differentiated oligopoly," International Journal of Industrial Organization, Elsevier, vol. 25(1), pages 173-185, February.
    60. Majerus, David W., 1988. "Price vs. quantity competition in oligopoly supergames," Economics Letters, Elsevier, vol. 27(3), pages 293-297.
    61. Piercarlo Zanchettin, 2006. "Differentiated Duopoly with Asymmetric Costs," Journal of Economics & Management Strategy, Wiley Blackwell, vol. 15(4), pages 999-1015, December.
    62. Amir, Rabah & Jin, Jim Y., 2001. "Cournot and Bertrand equilibria compared: substitutability, complementarity and concavity," International Journal of Industrial Organization, Elsevier, vol. 19(3-4), pages 303-317, March.
    63. Richard E. Levitan & Martin Shubik, 1967. "Part IV: Mathematical Structure and Analysis of the Nonsymmetric Game," Cowles Foundation Discussion Papers 219, Cowles Foundation for Research in Economics, Yale University.
    64. Andrew F. Daughety & Jennifer F. Reinganum, 2008. "Imperfect competition and quality signalling," RAND Journal of Economics, RAND Corporation, vol. 39(1), pages 163-183, March.
    65. Benjamin Edelman & Julian Wright, 2015. "Price Coherence and Excessive Intermediation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 130(3), pages 1283-1328.
    66. Ulsaker, Simen A., 2020. "Competing buyers, rent extraction and inefficient exclusion," International Journal of Industrial Organization, Elsevier, vol. 68(C).
    67. Vernon L. Smith, 1992. "Game Theory and Experimental Economics: Beginnings and Early Influences," History of Political Economy, Duke University Press, vol. 24(5), pages 241-282, Supplemen.
    68. Laurent Linnemer, 2003. "Backward Integration by a Dominant Firm," Journal of Economics & Management Strategy, Wiley Blackwell, vol. 12(2), pages 231-259, June.
    69. Bernd Theilen, 2012. "Product differentiation and competitive pressure," Journal of Economics, Springer, vol. 107(3), pages 257-266, November.
    70. Bos Iwan & Vermeulen Dries, 2022. "On the Microfoundation of Linear Oligopoly Demand," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 22(1), pages 1-15, January.
    71. Raith, Michael, 1996. "A General Model of Information Sharing in Oligopoly," Journal of Economic Theory, Elsevier, vol. 71(1), pages 260-288, October.
    72. Nirvikar Singh & Xavier Vives, 1984. "Price and Quantity Competition in a Differentiated Duopoly," RAND Journal of Economics, The RAND Corporation, vol. 15(4), pages 546-554, Winter.
    73. John Sutton, 1997. "One Smart Agent," RAND Journal of Economics, The RAND Corporation, vol. 28(4), pages 605-628, Winter.
    74. Symeonidis, George, 2003. "Quality heterogeneity and welfare," Economics Letters, Elsevier, vol. 78(1), pages 1-7, January.
    75. James W. Friedman, 1983. "Advertising and Oligopolistic Equilibrium," Bell Journal of Economics, The RAND Corporation, vol. 14(2), pages 464-473, Autumn.
    76. Rosenthal, Robert W, 1980. "A Model in Which an Increase in the Number of Sellers Leads to a Higher Price," Econometrica, Econometric Society, vol. 48(6), pages 1575-1579, September.
    77. Øystein Foros & Kåre P. Hagen & Hans Jarle Kind, 2009. "Price-Dependent Profit Sharing as a Channel Coordination Device," Management Science, INFORMS, vol. 55(8), pages 1280-1291, August.
    78. Pinkse, Joris & Slade, Margaret E., 2004. "Mergers, brand competition, and the price of a pint," European Economic Review, Elsevier, vol. 48(3), pages 617-643, June.
    79. Jagmohan S. Raju & Raj Sethuraman & Sanjay K. Dhar, 1995. "The Introduction and Performance of Store Brands," Management Science, INFORMS, vol. 41(6), pages 957-978, June.
    80. Yongmin Chen & Michael H. Riordan, 2007. "Price and Variety in the Spokes Model," Economic Journal, Royal Economic Society, vol. 117(522), pages 897-921, July.
    81. Martin Shubik, 1964. "A Business Game for Teaching and Research Purposes. Part III. Discussion and Manual for Users," Cowles Foundation Discussion Papers 180, Cowles Foundation for Research in Economics, Yale University.
    82. Marie-Laure Allain & Emeric Henry & Margaret Kyle, 2016. "Competition and the Efficiency of Markets for Technology," Management Science, INFORMS, vol. 62(4), pages 1000-1019, April.
    83. Hackner, Jonas, 2000. "A Note on Price and Quantity Competition in Differentiated Oligopolies," Journal of Economic Theory, Elsevier, vol. 93(2), pages 233-239, August.
    84. Dobson, Paul W. & Waterson, Michael, 2007. "The competition effects of industry-wide vertical price fixing in bilateral oligopoly," International Journal of Industrial Organization, Elsevier, vol. 25(5), pages 935-962, October.
    85. David Besanko & Martin K. Perry, 1993. "Equilibrium Incentives for Exclusive Dealing in a Differentiated Products Oligopoly," RAND Journal of Economics, The RAND Corporation, vol. 24(4), pages 646-668, Winter.
    86. Bresnahan, Timothy F, 1987. "Competition and Collusion in the American Automobile Industry: The 1955 Price War," Journal of Industrial Economics, Wiley Blackwell, vol. 35(4), pages 457-482, June.
    87. Vives, Xavier, 1985. "On the efficiency of Bertrand and Cournot equilibria with product differentation," Journal of Economic Theory, Elsevier, vol. 36(1), pages 166-175, June.
    88. Amir, Rabah & Erickson, Philip & Jin, Jim, 2017. "On the microeconomic foundations of linear demand for differentiated products," Journal of Economic Theory, Elsevier, vol. 169(C), pages 641-665.
    89. Raymond Deneckere & Carl Davidson, 1985. "Incentives to Form Coalitions with Bertrand Competition," RAND Journal of Economics, The RAND Corporation, vol. 16(4), pages 473-486, Winter.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Philippe Choné & Laurent Linnemer, 2019. "The quasilinear quadratic utility model: An overview," Working Papers hal-02318633, HAL.
    2. Amir, Rabah & Erickson, Philip & Jin, Jim, 2017. "On the microeconomic foundations of linear demand for differentiated products," Journal of Economic Theory, Elsevier, vol. 169(C), pages 641-665.
    3. Ming Chang & Yan-Ching Ho, 2014. "Comparing Cournot and Bertrand equilibria in an asymmetric duopoly with product R&D," Journal of Economics, Springer, vol. 113(2), pages 133-174, October.
    4. Belleflamme,Paul & Peitz,Martin, 2015. "Industrial Organization," Cambridge Books, Cambridge University Press, number 9781107687899, September.
    5. Amir, Rabah & Jin, Jim Y. & Lasselle, Laurence, 2023. "Uniform, efficient and independent Ramsey taxes across markets," Games and Economic Behavior, Elsevier, vol. 138(C), pages 373-386.
    6. Wang, X. Henry & Zhao, Jingang, 2007. "Welfare reductions from small cost reductions in differentiated oligopoly," International Journal of Industrial Organization, Elsevier, vol. 25(1), pages 173-185, February.
    7. Jurgan, Jens, 2009. "Conjectural cost variations in a differentiated good oligopoly," W.E.P. - Würzburg Economic Papers 80, University of Würzburg, Department of Economics.
    8. Jens Jurgan, 2009. "Cost Variations in a Differentiated Good Oligopoly," Working Papers 069, Bavarian Graduate Program in Economics (BGPE).
    9. Ming Chang, 2010. "An Asymmetric Oligopolist can Improve Welfare by Raising Price," Review of Industrial Organization, Springer;The Industrial Organization Society, vol. 36(1), pages 75-96, February.
    10. Häckner, Jonas & Herzing, Mathias, 2016. "Welfare effects of taxation in oligopolistic markets," Journal of Economic Theory, Elsevier, vol. 163(C), pages 141-166.
    11. Joachim Heinzel & Simon Hoof, 2020. "Oligopolistic Upstream Competition with Differentiated Inputs," Working Papers CIE 129, Paderborn University, CIE Center for International Economics.
    12. Neelanjan Sen & Uday Bhanu Sinha, 2023. "When to merge with a lower quality producer?," Journal of Economics, Springer, vol. 138(2), pages 165-188, March.
    13. Xavier Vives, 2008. "Innovation And Competitive Pressure," Journal of Industrial Economics, Wiley Blackwell, vol. 56(3), pages 419-469, December.
    14. Ming Chung Chang & Hsiao‐Ping Peng, 2009. "Structure Regulation, Price Structure, Cross‐Subsidization And Marginal Cost Of Public Funds," Manchester School, University of Manchester, vol. 77(6), pages 675-698, December.
    15. Xing Gao & Weijun Zhong, 2016. "Economic incentives in security information sharing: the effects of market structures," Information Technology and Management, Springer, vol. 17(4), pages 361-377, December.
    16. Luciano Fanti & Nicola Meccheri, 2015. "On the Cournot–Bertrand Profit Differential and the Structure of Unionisation in a Managerial Duopoly," Australian Economic Papers, Wiley Blackwell, vol. 54(4), pages 266-287, December.
    17. Chang, Ming Chung & Lin, Yun-Chieh & Lin, Chiu Fen, 2012. "Comparing Cournot Duopoly And Monopoly With Asymmetric Differentiated Goods," Hitotsubashi Journal of Economics, Hitotsubashi University, vol. 53(2), pages 177-200, December.
    18. Hattori Keisuke & Lin Ming-Hsin, 2011. "Alliance Partner Choice in Markets with Vertical and Horizontal Externalities," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 11(1), pages 1-27, June.
    19. Aymeric Lardon, 2019. "On the coalitional stability of monopoly power in differentiated Bertrand and Cournot oligopolies," Theory and Decision, Springer, vol. 87(4), pages 421-449, November.
    20. Corchón, Luis C. & Zudenkova, Galina, 2009. "Computing welfare losses from data under imperfect competition with heterogeneous goods," International Journal of Industrial Organization, Elsevier, vol. 27(6), pages 646-654, November.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:wpaper:hal-02882403. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.