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Evolutionary Variational Formulation for Oligopolistic Market Equilibrium Problems with Production Excesses

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Abstract

The paper is devoted to generalize a previous model of the dynamic oligopolistic market equilibrium problem allowing the presence of production excesses and assuming, in a more reasonable way that the total amounts of commodity shipments from a firm to all the demand markets be upper bounded. First, we give equilibrium conditions in terms of the well-known dynamic Cournot–Nash equilibrium principle. Then we show that such conditions can be expressed in terms of Lagrange multipliers; namely, by means of an utility function, prove that both equilibrium conditions can be equivalently expressed by a variational inequality. The variational formulation allows us to provide existence theorems and qualitative properties for equilibrium solutions. At last, a numerical example illustrates the results obtained.

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Notes

  1. In the Hilbert space L 2([0,T],ℝk), let us recall that

    $$\langle\! \langle \phi, y \rangle\! \rangle := \int _0^T \bigl\langle \phi(t), y(t) \bigr\rangle \,dt,$$

    is its duality mapping, where ϕ∈(L 2([0,T],ℝk))∗=L 2([0,T],ℝk) and y∈L 2([0,T],ℝk).

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Authors and Affiliations

  1. Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, via Cintia, 80126, Naples, Italy

    Annamaria Barbagallo

  2. Department of Mathematics and Computer Science, University of Catania, viale Andrea Doria n. 6, 95125, Catania, Italy

    Paolo Mauro

Authors
  1. Annamaria Barbagallo
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  2. Paolo Mauro
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Correspondence to Annamaria Barbagallo.

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Communicated by Antonino Maugeri.

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Barbagallo, A., Mauro, P. Evolutionary Variational Formulation for Oligopolistic Market Equilibrium Problems with Production Excesses. J Optim Theory Appl 155, 288–314 (2012). https://doi.org/10.1007/s10957-012-0056-z

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  • DOI: https://doi.org/10.1007/s10957-012-0056-z

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