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2020 | OriginalPaper | Chapter

On Contractual Approach in Competitive Economies with Constrained Coalitional Structures

Author : Valeriy Marakulin

Published in: Mathematical Optimization Theory and Operations Research

Publisher: Springer International Publishing

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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.

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previous chapter Weak Berge Equilibrium
next chapter Pontryagin’s Maximum Principle for Non-cooperative Differential Games with Continuous Updating
Footnotes
1
Here “solid” is equivalent to “having nonempty interior.”
 
2
The symbol \(\overline{A}\) denotes the closure of A and \(\setminus \) is set for the set-theoretical difference.
 
3
Otherwise, it would occur that an allocation realized via breaking contracts is not feasible.
 
4
This is not a contract, because its key property \( \sum _\mathcal{I}t_iv_i = 0 \) is violated.
 
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
Conditions, providing this fact are well known in the literature, e.g. it can be irreducibility.
 
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
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
Under classical assumptions they are equivalent, but it is not so in general case.
 
Literature
1.
Marakulin, V.M.: Contracts and domination in competitive economies. J. New Econ. Assoc. 9, 10–32 (2011). (in Russian)
2.
Marakulin, V.M.: On the Edgeworth conjecture for production economies with public goods: a contract-based approach. J. Math. Econ. 49(3), 189–200 (2013)MathSciNetCrossRef
3.
Marakulin, V.M.: On contractual approach for Arrow-Debreu-McKenzie economies. Econ. Math. Methods 50(1), 61–79 (2014). (in Russian)
4.
5.
6.
Graham, D.A., Weintraub, E.R.: On convergence to Pareto allocations. Rev. Econ. Stud. 42, 469–472 (1975)CrossRef
7.
Madden, P.J.: Efficient sequences of non-monetary exchange. Rev. Econ. Stud. 42(4), 581–595 (1975)CrossRef
8.
Graham, D.A., Jennergren, L.P., Peterson, D.P., Weintraub, E.R.: Trader-commodity parity theorems. J. Econ. Theory 12, 443–454 (1976)MathSciNetCrossRef
9.
Polterovich, V.M.: Mathematical models of resources allocations. Central Economic and Mathematical Institute, Moscow, 107 p. (1970). (in Russian)
10.
Metadata
Title
On Contractual Approach in Competitive Economies with Constrained Coalitional Structures
Author
Valeriy Marakulin
Copyright Year
2020
Book
Mathematical Optimization Theory and Operations Research

Print ISBN: 978-3-030-58656-0

Electronic ISBN: 978-3-030-58657-7

Copyright Year: 2020

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

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