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Essential properties of \(L_{p,q}\) spaces (the amalgams) and the implicit function theorem for equilibrium analysis in continuous time. (English) Zbl 1296.46027

Authors’ abstract: To extend the analysis of continuous-time general-equilibrium macro models, we study 2 parameter variants \(L_{p,q}\) 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 \(\ell _p\) and w.r.t.  \(q\) like the spaces \(L_q\) 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., [J.-F. Mertens and A. Rubinchik, “Regularity and stability of equilibria in an overlapping generations growth model”, Working paper (2011), http://econ.haifa.ac.il/~arubinchik/papers/Reg_Stab_OLG.pdf]). 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.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47J07 Abstract inverse mapping and implicit function theorems involving nonlinear operators
91B50 General equilibrium theory
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
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