Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth

Martin Burger; Alexander Lorz; Marie-Therese Wolfram

doi:10.3934/krm.2017005

Article Contents

2017, Volume 10, Issue 1: 117-140. Doi: 10.3934/krm.2017005

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Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth

1.
Institute for Computational and Applied Mathematics, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany
2.
CEMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia
3.
Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
4.
CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
5.
INRIA-Paris-Rocquencourt, EPC MAMBA, Domaine de Voluceau, BP105, 78153 Le Chesnay Cedex, France
6.
University of Warwick, Coventry CV4 7AL, UK
7.
RICAM, Austrian Academy of Sciences, Altenbergerstr. 69,4040 Linz, Austria

Received: December 31, 2015

Revised: August 31, 2016

Published: September 2017

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Abstract

In this paper we study balanced growth path solutions of a Boltzmann mean field game model proposed by Lucas and Moll [15] to model knowledge growth in an economy.Agents can either increase their knowledge level by exchanging ideas in learning events or by producing goods with the knowledge they already have.The existence of balanced growth path solutions implies exponential growth of the overall production in time. We prove existence of balanced growth path solutions if the initial distribution of individuals with respect to their knowledge level satisfiesa Pareto-tail condition. Furthermore we give first insights into the existence of such solutions if in addition to production and knowledge exchange theknowledge level evolves by geometric Brownian motion.

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Mathematics Subject Classification: Primary: 49J20, 49N90, 35Q20, 70H20, 35Q91.

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Figure 1. Solution of the time dependent solver converging to a non-trivial BGP

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Figure 2. Balanced growth path solutions for different diffusivities $\nu$

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Figure 3. Comparison of the solvers in the case of a knowledge shock

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