A nonnegative extension of the affine demand function and equilibrium analysis for multiproduct price competition

We derive a nonnegative extension of the affine demand function for differentiated substitute products from the optimization problem facing a representative consumer whose utility function is quadratic. We show that the extended demand function reduces to a linear program. The linear program has a simple intuitive interpretation in terms of a shifted price vector. We prove the existence and uniqueness of the Bertrand equilibrium in oligopolies consisting of multiproduct firms under the proposed demand function. The equilibrium, available in closed form, coincides with that obtained when allowing negative demands.

[1]  M. Shubik,et al.  Market structure and behavior , 1982 .

[2]  Fernando Bernstein,et al.  Comparative statics, strategic complements and substitutes in oligopolies , 2004 .

[3]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[4]  Amr Farahat,et al.  Profit loss in differentiated oligopolies , 2009, Oper. Res. Lett..

[5]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[6]  X. Vives,et al.  Price and quantity competition in a differentiated duopoly , 1984 .

[7]  J. Harrington.  Experimentation and Learning in a Differentiated-Products Duopoly , 1995 .

[8]  Sudheer Gupta,et al.  Process Innovation, Product Differentiation, and Channel Structure: Strategic Incentives in a Duopoly , 1998 .

[9]  Hongtao Zhang.  VERTICAL INFORMATION EXCHANGE IN A SUPPLY CHAIN WITH DUOPOLY RETAILERS , 2002 .

[10]  P. Klemperer,et al.  Price competition vs. quantity competition: the role of uncertainty , 1986 .

[11]  André de Palma,et al.  Discrete Choice Theory of Product Differentiation , 1995 .

[12]  Manu Goyal,et al.  Strategic Technology Choice and Capacity Investment Under Demand Uncertainty , 2007, Manag. Sci..

[13]  Awi Federgruen,et al.  Competition in Service Industries , 2005, Oper. Res..

[14]  S. C. Choi,et al.  Price Competition in a Channel Structure with a Common Retailer , 1991 .

[15]  Richard Staelin,et al.  An Industry Equilibrium Analysis of Downstream Vertical Integration , 2008, Mark. Sci..

[16]  J. Miguel Villas-Boas,et al.  Bertrand Supertraps , 2005, Manag. Sci..

[17]  Puneet Manchanda,et al.  Differences in Dynamic Brand Competition Across Markets: An Empirical Analysis , 2005 .

[18]  Olvi L. Mangasarian,et al.  Linear complementarity problems solvable by A single linear program , 1976, Math. Program..

[19]  Gongyun Zhao,et al.  Complementarity Demand Functions and Pricing Models for Multi-product Markets , 2009 .

[20]  Pradeep K. Chintagunta,et al.  Investigating Dynamic Multifirm Market Interactions in Price and Advertising , 1999 .

[21]  V. Kerry Smith,et al.  A GENERALIZED CONSUMER SURPLUS MEASURE: THE MINERAL KING CONTROVERSY , 1977 .