On steady distributions of kinetic models of conservative economies. (English) Zbl 1138.91020
Summary: We analyze the large-time behavior of various kinetic models for the redistribution of wealth in simple market economies introduced in the pertinent literature in recent years. As specific examples, we study models with fixed saving propensity introduced by Chakraborti and Chakrabarti [Eur. Phys. J. B 17, 167–170 (2000)], as well as models involving both exchange between agents and speculative trading as considered by S. Cordier et al. [J. Stat. Phys. 120, No. 1–2, 253–277 (2005; Zbl 1133.91474)]. We derive a sufficient criterion under which a unique non-trivial stationary state exists, and provide criteria under which these steady states do or do not possess a Pareto tail. In particular, we prove the absence of Pareto tails in pointwise conservative models, like the one of Chakraborti and Chakrabarti [loc. cit.], while models with speculative trades introduced in [J. Stat. Phys. 120, No. 1–2, 253–277 (2005; Zbl 1133.91474)] develop fat tails if the market is “risky enough”. The results are derived by a Fourier-based technique first developed for the Maxwell-Boltzmann equation [G. Gabetta et al., J. Stat. Phys. 81, No. 5–6, 901–934 (1995; Zbl 1081.82616); M. Bisi et al., J. Stat. Phys. 118, No. 1–2, 301–331 (2005; Zbl 1085.82008); L. Pareschi and G. Toscani, J. Stat. Phys. 124, No. 2–4, 747–779 (2006; Zbl 1134.82037)] and from a recursive relation which allows to calculate arbitrary moments of the stationary state.
MSC:
| 91B24 | Microeconomic theory (price theory and economic markets) |
| 91B30 | Risk theory, insurance (MSC2010) |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
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