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Solution of the time dependent solver converging to a non-trivial BGP
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2017, 10(1): 117-140.
doi: 10.3934/krm.2017005
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 |
In this paper we study balanced growth path solutions of a Boltzmann mean field game model proposed by Lucas and Moll [
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 and 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. |
| [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. 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. |
| [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. |
| [7] |
C. Cercignani,
The Boltzmann Equation, Springer, 1988. |
| [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. |
| [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. 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.
|
| [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,
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. |
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. |
| [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. 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. |
| [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. |
| [7] |
C. Cercignani,
The Boltzmann Equation, Springer, 1988. |
| [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. |
| [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. 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.
|
| [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,
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. |
Figure 2.
Balanced growth path solutions for different diffusivities $\nu$
Figure 3.
Comparison of the solvers in the case of a knowledge shock
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