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Convex analysis and financial equilibrium

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Abstract

Convexity has long had an important role in economic theory, but some recent developments have featured it all the more in problems of equilibrium. Here the tools of convex analysis are applied to a basic model of incomplete financial markets in which assets are traded and money can be lent or borrowed between the present and future. The existence of an equilibrium is established with techniques that include bounds derived from the duals to problems of utility maximization. Composite variational inequalities furnish the modeling platform. Models with and without short-selling are handled, moreover in the absence of any requirement that agents must initially have a positive amount of every asset, as is typical in equilibrium work in economics.

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Notes

  1. Even with borrowing and lending, however, the portfolios may fall short of being able to produce every pattern of payments in the future states, and in that sense the market would be incomplete. In the version of the Arrow-Debreu model with financial-like markets in “contingent” goods, incompletness can block the existence of equilibrium.

  2. For related model with explicit consumption, see our paper [10], which likewise works with variational inequalities.

  3. Differentiability could be dropped, with gradients replaced by subgradients, but for that we would need to work in Sect. 3 with a more complicated type of variational inequality, as we did in [10]. Upper semicontinuity corresponds to the upper level sets of \(u_i\) being closed, as is usual in preference theory.

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Authors and Affiliations

  1. Center for Mathematical Modeling & DIM, University of Chile, Santiago, Chile

    A. Jofré

  2. University of Washington, Seattle, WA, 98195-4350, USA

    R. T. Rockafellar

  3. University of California, Davis, CA, 95616, USA

    R. J.-B. Wets

Authors
  1. A. Jofré
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  2. R. T. Rockafellar
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  3. R. J.-B. Wets
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Correspondence to A. Jofré.

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Jofré, A., Rockafellar, R.T. & Wets, R.JB. Convex analysis and financial equilibrium. Math. Program. 148, 223–239 (2014). https://doi.org/10.1007/s10107-014-0747-3

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  • DOI: https://doi.org/10.1007/s10107-014-0747-3

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