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On the microeconomic foundations of linear demand for differentiated products

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  • Amir, Rabah
  • Erickson, Philip
  • Jin, Jim

Abstract

This paper provides a thorough exploration of the microeconomic foundations for the multi-variate linear demand function for differentiated products, which is widely used in industrial organization. The setting is the standard representative consumer with a quasi-linear utility function. A key finding is that strict concavity of the quadratic utility function is critical for the demand system to be well defined. Otherwise, the true demand function may be quite complex: Multi-valued, non-linear and income-dependent. We uncover failures of duality relationships between substitute products and complementary products, as well as the incompatibility between high levels of complementarity and concavity. The two-good case emerges as a special case with strong but non-robust properties. A key implication is that all conclusions derived in applied economic models via the use of linear demand that does not satisfy the Law of Demand ought to be regarded with some suspicion.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jetheo:v:169:y:2017:i:c:p:641-665
    DOI: 10.1016/j.jet.2017.03.005
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    References listed on IDEAS

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    1. Rabah Amir, 2005. "Supermodularity and Complementarity in Economics: An Elementary Survey," Southern Economic Journal, Southern Economic Association, vol. 71(3), pages 636-660, January.
    2. 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.
    3. 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.
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    5. Okuguchi, Koji, 1987. "Equilibrium prices in the Bertrand and Cournot oligopolies," Journal of Economic Theory, Elsevier, vol. 42(1), pages 128-139, June.
    6. 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.
    7. 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.
    8. Francis Bloch, 1995. "Endogenous Structures of Association in Oligopolies," RAND Journal of Economics, The RAND Corporation, vol. 26(3), pages 537-556, Autumn.
    9. Xavier Vives, 1987. "Small Income Effects: A Marshallian Theory of Consumer Surplus and Downward Sloping Demand," Review of Economic Studies, Oxford University Press, vol. 54(1), pages 87-103.
    10. Deneckere, R., 1983. "Duopoly supergames with product differentiation," Economics Letters, Elsevier, vol. 11(1-2), pages 37-42.
    11. 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.
    12. 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.
    13. repec:eee:jetheo:v:169:y:2017:i:c:p:603-628 is not listed on IDEAS
    14. Amir, Rabah & Halmenschlager, Christine & Jin, Jim, 2011. "R&D-induced industry polarization and shake-outs," International Journal of Industrial Organization, Elsevier, vol. 29(4), pages 386-398, July.
    15. 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.
    16. 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.
    17. LaFrance, Jeffrey T., 1985. "Linear demand functions in theory and practice," Journal of Economic Theory, Elsevier, vol. 37(1), pages 147-166, October.
    18. Nocke, Volker & Schutz, Nicolas, 2017. "Quasi-linear integrability," Journal of Economic Theory, Elsevier, vol. 169(C), pages 603-628.
    19. 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.
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    1. repec:eee:gamebe:v:109:y:2018:i:c:p:99-103 is not listed on IDEAS
    2. repec:eee:jetheo:v:175:y:2018:i:c:p:58-87 is not listed on IDEAS
    3. repec:eee:energy:v:150:y:2018:i:c:p:1006-1017 is not listed on IDEAS
    4. Camille Cornand & Rodolphe Dos Santos Ferreira, 2017. "The social value of information and the competition motive: Price vs. quantity games," Working Papers halshs-01614815, HAL.
    5. repec:eee:jetheo:v:169:y:2017:i:c:p:603-628 is not listed on IDEAS
    6. Maria Rosa, Battaggion & Vittoria, Cerasi, 2018. "Endogenous interlocking directorates," Working Papers 380, University of Milano-Bicocca, Department of Economics, revised 01 May 2018.
    7. Nocke, Volker & Schutz, Nicolas, 2017. "Quasi-linear integrability," Journal of Economic Theory, Elsevier, vol. 169(C), pages 603-628.

    More about this item

    Keywords

    Linear demand; Gross substitutes; Gross complements; Edgeworth complements; Representative consumer; Law of Demand;

    JEL classification:

    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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