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On Contractual Approach in Competitive Economies with Constrained Coalitional Structures

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Mathematical Optimization Theory and Operations Research (MOTOR 2020)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1275))

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Abstract

We establish a theorem that equilibria in an exchange economy can be described as allocations that are stable under the possibilities: (i) agents can partially and asymmetrically break current contracts, after that (ii) a new mutually beneficial contract can be concluded in a coalition of a size not more than 1 plus the maximum number of products that are not indifferent to the coalition members.

The presented result generalizes previous ones on a Pareto improvement in an exchange economy with l commodities that requires the active participation of no more than \(l+1\) traders. This concerned with Pareto optimal allocations, but we also describe equilibria. Thus according to the contractual approach to arrive at equilibrium only coalitions of constrained size can be applied that essentially raise the confidence of contractual modeling.

The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. 0314-2019-0018).

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Notes

  1. 1.

    Here “solid” is equivalent to “having nonempty interior.”

  2. 2.

    The symbol \(\overline{A}\) denotes the closure of A and \(\setminus \) is set for the set-theoretical difference.

  3. 3.

    Otherwise, it would occur that an allocation realized via breaking contracts is not feasible.

  4. 4.

    This is not a contract, because its key property \( \sum _\mathcal{I}t_iv_i = 0 \) is violated.

  5. 5.

    A linear segment with ends \(a, b\in L\) is the set \( [ a,b]=\mathrm{conv}\{ a,b\}=\{\lambda a+(1-\lambda )b\mid 0\le \lambda \le 1\}\).

  6. 6.

    Conditions, providing this fact are well known in the literature, e.g. it can be irreducibility.

  7. 7.

    For example, an ordinary consumer on the market is not interested in all kinds of spare parts, parts and structural elements (bolts, nuts, gears, transistors ...).

  8. 8.

    Here we indirectly assume that all bundles we need belong to consumption set, i.e., \(((y_i)_{-j},\mathbf{e}_i^j)\), \(((x_i)_{-j},\mathbf{e}_i^j)\in X_i\); it is a specific constraint for \(X_i\), \(i\in \mathcal{I}\).

  9. 9.

    Under classical assumptions they are equivalent, but it is not so in general case.

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Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Russian Academy of Sciences, 4 Acad. Koptyug Avenue, Novosibirsk, 630090, Russia

    Valeriy Marakulin

  2. Novosibirsk State University, 2 Pirogova Street, Novosibirsk, 630090, Russia

    Valeriy Marakulin

Authors
  1. Valeriy Marakulin
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Correspondence to Valeriy Marakulin .

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Editors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Yury Kochetov

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Igor Bykadorov

  3. Matrosov Institute for System Dynamics and Control Theory, Irkutsk, Russia

    Tatiana Gruzdeva

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Marakulin, V. (2020). On Contractual Approach in Competitive Economies with Constrained Coalitional Structures. In: Kochetov, Y., Bykadorov, I., Gruzdeva, T. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2020. Communications in Computer and Information Science, vol 1275. Springer, Cham. https://doi.org/10.1007/978-3-030-58657-7_21

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  • DOI: https://doi.org/10.1007/978-3-030-58657-7_21

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-58656-0

  • Online ISBN: 978-3-030-58657-7

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