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March  2017, 10(1): 117-140. doi: 10.3934/krm.2017005

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

Martin Burger 1, , Alexander Lorz 2,3,4,5, and  Marie-Therese Wolfram 6,7,

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  January 2016 Revised  September 2016 Published  November 2016

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.

Keywords: Mean-field games, Boltzmann-type equations, Hamilton-Jacobi equations, travelling wave solutions.
Mathematics Subject Classification: Primary: 49J20, 49N90, 35Q20, 70H20, 35Q91.
Citation: Martin Burger, Alexander Lorz, Marie-Therese Wolfram. Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth. Kinetic & Related Models, 2017, 10 (1) : 117-140. doi: 10.3934/krm.2017005
References:
[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. Google Scholar

[2]

F. E. Alvarez, F. J. Buera and R. E. Lucas Jr, Models of Idea Flows, Technical report, National Bureau of Economic Research, 2008.Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

M. Burger, L. Caffarelli, P. 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. Google Scholar

[6]

M. Burger, A. 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. Google Scholar

[7]

C. Cercignani, The Boltzmann Equation, Springer, 1988.Google Scholar

[8]

P. Degond, J.-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. Google Scholar

[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. Google Scholar

[10]

B. Düring, P. Markowich, J.-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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

R. E. Lucas Jr and B. Moll, Knowledge growth and the allocation of time, Journal of Political Economy, 122.Google Scholar

[16]

E. G. Luttmer, Eventually, Noise and Imitation Implies Balanced Growth, Technical report, Federal Reserve Bank of Minneapolis, 2012.Google Scholar

[17]

E. G. Luttmer, Four Models of Knowledge Diffusion and Growth, Technical report, Federal Reserve Bank of Minneapolis, 2015.Google Scholar

[18]

V. Mahajan and R. A. Peterson, Models for Innovation Diffusion, vol. 48, Sage, 1985.Google Scholar

[19]

P. Markowich, Mathematical model for the diffusion of innovation, IEEE Trans. Sys., Man, and Cyber., 11 (1981), 504-509. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[23]

G. Toscani, Kinetic models of opinion formation, Communications in mathematical sciences, 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[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. Google Scholar

show all references

References:
[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. Google Scholar

[2]

F. E. Alvarez, F. J. Buera and R. E. Lucas Jr, Models of Idea Flows, Technical report, National Bureau of Economic Research, 2008.Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

M. Burger, L. Caffarelli, P. 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. Google Scholar

[6]

M. Burger, A. 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. Google Scholar

[7]

C. Cercignani, The Boltzmann Equation, Springer, 1988.Google Scholar

[8]

P. Degond, J.-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. Google Scholar

[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. Google Scholar

[10]

B. Düring, P. Markowich, J.-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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

R. E. Lucas Jr and B. Moll, Knowledge growth and the allocation of time, Journal of Political Economy, 122.Google Scholar

[16]

E. G. Luttmer, Eventually, Noise and Imitation Implies Balanced Growth, Technical report, Federal Reserve Bank of Minneapolis, 2012.Google Scholar

[17]

E. G. Luttmer, Four Models of Knowledge Diffusion and Growth, Technical report, Federal Reserve Bank of Minneapolis, 2015.Google Scholar

[18]

V. Mahajan and R. A. Peterson, Models for Innovation Diffusion, vol. 48, Sage, 1985.Google Scholar

[19]

P. Markowich, Mathematical model for the diffusion of innovation, IEEE Trans. Sys., Man, and Cyber., 11 (1981), 504-509. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[23]

G. Toscani, Kinetic models of opinion formation, Communications in mathematical sciences, 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[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. Google Scholar

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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