Essential properties of $L_p,q$ spaces (the amalgams) and the implicit function theorem for equilibrium analysis in continuous time
(2013) Journal of Mathematical Economics — Vol. 50, p. 187-196 (2014)
Files
Rep2549.pdf
Closed Access - Adobe PDF
- 765.63 KB
Details
- 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 $l_p$ 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. 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.
- Affiliations
Citations
Mertens, J.-F., & Rubinchik, A. (2013). Essential properties of $L_p,q$ spaces (the amalgams) and the implicit function theorem for equilibrium analysis in continuous time. Journal of Mathematical Economics, 50, 187-196. https://doi.org/10.1010/j.jmateco.2013.06.002 (Original work published 2014)