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Essential properties of Lp,q spaces (the amalgams) and the implicit function theorem for equilibrium analysis in continuous time

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  • Mertens, Jean-François
  • Rubinchik, Anna

Abstract

To extend the analysis of continuous-time general-equilibrium macro models we study 2 parameter variants Lp,q of the Lebesgue spaces, thus gaining separate control on the asymptotic behaviour (p) and the local behaviour (q): they behave w.r.t. p like the spaces ℓp and w.r.t. q like the spaces Lq on a probability space. Such spaces might naturally contain equilibrium variables (paths) as well as time-dependent policies of a macro model. Convolution behaves very well on those spaces, which can be used as a basis for the classical “comparative statics” (see e.g. Mertens and Rubinchik (2011)). Finally, we generalise the classical implicit function theorem (ift) for a family of Banach spaces, with the resulting implicit function having derivatives that are locally Lipschitz to very strong operator norms.

Suggested Citation

  • Mertens, Jean-François & Rubinchik, Anna, 2014. "Essential properties of Lp,q spaces (the amalgams) and the implicit function theorem for equilibrium analysis in continuous time," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 187-196.
  • Handle: RePEc:eee:mateco:v:50:y:2014:i:c:p:187-196
    DOI: 10.1016/j.jmateco.2013.06.002
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    References listed on IDEAS

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    1. Mertens, Jean-Francois & Rubinchik, Anna, "undated". "Regularity And Stability Of Equilibria In An Overlapping Generations Growth Model," Working Papers WP2014/2, University of Haifa, Department of Economics.
    2. Chris Shannon & William R. Zame, 2002. "Quadratic Concavity and Determinacy of Equilibrium," Econometrica, Econometric Society, vol. 70(2), pages 631-662, March.
    3. Jean-François Mertens & Anna Rubinchik, 2013. "Equilibria in an overlapping generations model with transfer policies and exogenous growth," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(3), pages 537-595, November.
    4. Debreu, Gerard, 1970. "Economies with a Finite Set of Equilibria," Econometrica, Econometric Society, vol. 38(3), pages 387-392, May.
    5. Debreu, Gerard, 1976. "Regular Differentiable Economies," American Economic Review, American Economic Association, vol. 66(2), pages 280-287, May.
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    Cited by:

    1. Rubinchik, Anna, 2015. "The chase of a multi-armed economist for the elusive social discount rate," Working Papers WP2015/8, University of Haifa, Department of Economics, revised 18 Nov 2015.

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