Advanced Search
Article Contents
Article Contents

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

Abstract Full Text(HTML) Figure(3) Related Papers Cited by
  • 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.

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


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Solution of the time dependent solver converging to a non-trivial BGP

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

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

  • [1] Y. Achdou, F. J. Buera, J. -M. Lasry, P. -L. Lions and B. Moll, Partial differential equation models in macroeconomics, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130397, 19 pp. doi: 10.1098/rsta.2013.0397.
    [2] F. E. Alvarez, F. J. Buera and R. E. Lucas Jr, Models of Idea Flows, Technical report, National Bureau of Economic Research, 2008.
    [3] L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekulen, Wien. Ber, 66 (1872), 275-370.  doi: 10.1007/978-3-322-84986-1_3.
    [4] L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation, M2AN Math. Model. Numer. Anal., 43 (2009), 507-522, URL http://dx.doi.org/10.1051/m2an/2009004. doi: 10.1051/m2an/2009004.
    [5] M. BurgerL. CaffarelliP. Markowich and M.-T. Wolfram, On a Boltzmann-type price formation model, Proc. R. Soc. A, 469 (2013), 20130126.  doi: 10.1098/rspa.2013.0126.
    [6] M. BurgerA. Lorz and M.-T. Wolfram, On a Boltzmann mean field model for knowledge growth, SIAM J. Appl. Math., 76 (2016), 1799-1818.  doi: 10.1137/15M1018599.
    [7] C. Cercignani, The Boltzmann Equation, Springer, 1988.
    [8] P. DegondJ.-G. Liu and C. Ringhofer, Large-scale dynamics of mean-field games driven by local Nash equilibria, Journal of Nonlinear Science, 24 (2014), 93-115.  doi: 10.1007/s00332-013-9185-2.
    [9] R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Annals of Mathematics, 130 (1989), 321-366.  doi: 10.2307/1971423.
    [10] B. DüringP. MarkowichJ.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, (), rspa20090239. 
    [11] B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Physical Review E, 78 (2008), 056103, 12pp. doi: 10.1103/PhysRevE.78.056103.
    [12] D. Hilhorst and Y.-J. Kim, Diffusive and inviscid traveling waves of the Fisher equation and nonuniqueness of wave speed, Applied Mathematics Letters, 60 (2016), 28-35.  doi: 10.1016/j.aml.2016.03.022.
    [13] J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.  doi: 10.1007/s11537-007-0657-8.
    [14] R. E. Lucas, Ideas and growth, Economica, 76 (2009), 1-19, URL http://dx.doi.org/10.1111/j.1468-0335.2008.00748.x. doi: 10.1111/j.1468-0335.2008.00748.x.
    [15] R. E. Lucas Jr and B. Moll, Knowledge growth and the allocation of time, Journal of Political Economy, 122.
    [16] E. G. Luttmer, Eventually, Noise and Imitation Implies Balanced Growth, Technical report, Federal Reserve Bank of Minneapolis, 2012.
    [17] E. G. Luttmer, Four Models of Knowledge Diffusion and Growth, Technical report, Federal Reserve Bank of Minneapolis, 2015.
    [18] V. Mahajan and R. A. Peterson, Models for Innovation Diffusion, vol. 48, Sage, 1985.
    [19] P. Markowich, Mathematical model for the diffusion of innovation, IEEE Trans. Sys., Man, and Cyber., 11 (1981), 504-509. 
    [20] L. Pareschi and G. Toscani, Wealth distribution and collective knowledge: A Boltzmann approach, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130396, 15 pp, URL http://rsta.royalsocietypublishing.org/content/372/2028/20130396.abstract. doi: 10.1098/rsta.2013.0396.
    [21] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013, URL http://EconPapers.repec.org/RePEc:oxp:obooks:9780199655465.
    [22] M. Staley, Growth and the diffusion of ideas, Journal of Mathematical Economics, 47 (2011), 470-478.  doi: 10.1016/j.jmateco.2011.06.006.
    [23] G. Toscani, et al., Kinetic models of opinion formation, Communications in mathematical sciences, 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.
    [24] C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, 1 (2002), 71-305.  doi: 10.1016/S1874-5792(02)80004-0.
  • 加载中



Article Metrics

HTML views(262) PDF downloads(208) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint