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Core equivalences for equilibria supported by non-linear prices

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Abstract

The goal of this paper is to provide some new cooperative characterizations and optimality properties of competitive equilibria supported by non-linear prices. The general framework is that of economies whose commodity space is an ordered topological vector space which need not be a vector lattice. The central notion of equilibrium is the one of personalized equilibrium introduced by Aliprantis et al. (J Econ Theory 100:22–72, 2001). Following Herves-Beloso and Moreno-Garcia (J Math Econ 44:697–706, 2008), the veto power of the grand coalition is exploited in the original economy and in a suitable family of economies associated to the original one. The use of Aubin coalitions allows us to connect results with the arbitrage free condition due to non-linear supporting prices. The use of rational allocations allows us to dispense with Lyapunov convexity theorem. Applications are provided in connection with strategic market games and economies with asymmetric information.

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Notes

  1. It is a larger, with respect to the dual cone, lattice ordered cone containing non-linear prices.

  2. The Lyapunov convexity theorem, through an application of the Vind’s theorem on the measure of blocking coalitions (see [32]), is a central tool in [20] for finite dimensional economies.

  3. Remind that the positive orthant \(L_+\) of an ordered vector space is a proper cone i.e. a convex cone with vertex \(0\) and such that \(L_+\cap (- L_+)=\{0\}\). If a topological vector space is also an ordered vector space, then we say that it is an ordered topological vector space if its positive cone is closed.

  4. Of course the generating property is automatic in a vector lattice as well as the boundedness of order intervals in case \(L\) is a topological vector lattice.

  5. If \(L\) is a vector lattice, then it would be enough to assume that the positive cone of \(L\) is \(\tau \)-closed and that \(L^{^{\prime }}\) is a vector sublattice of \(L^{\sim }\).

  6. Convex-valued is required if we are under the assumption (A1, 2.) of convexity of preferences.

  7. Same as f-core or fuzzy core.

  8. In [27] an example is provided of an economy with three commodities in which there is no Walrasian equilibrium and the second welfare theorem fails for linear prices.

  9. \(\psi _p \cdot e\ge \psi _p\cdot (\sum _{t\in T}\, x_t) \ge \sum _{t\in T}\, \psi _p\cdot x_t \). So when (iii) holds, then necessarily the inequalities become equalities. We shall discuss the condition in details in Sect. 3.3. Note also that from (iii)\(_b\) it follows that \(\psi _p\cdot e_t\le \psi _p\cdot x_t\) for any agent \(t\).

  10. The interpretation of condition (iii)\(_b\) proposed in [9] relies on coalitions of a continuum economy canonically associated to the finite one.

  11. The notion of Aubin core is due to [11] for finite economies and has been extended in [29] to arbitrary measure spaces of agents, in [12, 16, 18] it has been studied for economies with infinitely many goods and allowing for production.

  12. Notice that \(x_ t^*=0\) for a \(t\in T\) would imply, by monotonicity, that \(x^ *\) can be blocked by \({t}\) and this contradicts the fact that \(x^*\) belongs to \(\mathcal{C}^\mathrm{A}\).

  13. \(q_ t(\omega )\) is defined by \(E[p_t|\mathcal{F}_t](\omega )=\frac{1}{\sharp A_ t(\omega )}\sum _{\omega ^{^{\prime }}\in A_ t(\omega )}p(\omega ^{^{\prime }})\), where \(A_ t(\omega )\) is the unique element of \(\mathcal{F}_ t\) containing \(\omega \).

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Authors and Affiliations

  1. Dipartimento di Matematica e Statistica and CSEF, Universitá di Napoli Federico II, Naples, Italy

    Achille Basile & Maria Gabriella Graziano

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  1. Achille Basile
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  2. Maria Gabriella Graziano
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Correspondence to Achille Basile.

Additional information

The authors thank Y. Polyrakis and seminar participants in the Conference on Ordered Spaces and Applications (National Technical University of Athens, November 25–27, 2011) for helpful comments. The kind hospitality of organizers of Manchester Workshop in Economic Theory: Ambiguity and Equilibrium (University of Manchester, May 9, 2012) and XXI European Workshop on General Equilibrium Theory (University of Exeter, May 31–June 2, 2012), where the paper has been also presented, is acknowledged.

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Basile, A., Graziano, M.G. Core equivalences for equilibria supported by non-linear prices. Positivity 17, 621–653 (2013). https://doi.org/10.1007/s11117-012-0195-3

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