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February  2020, 13(1): 169-185. doi: 10.3934/krm.2020006

## Kinetic models of conservative economies with need-based transfers as welfare

 1 Department of Mathematics and Actuarial Science, Otterbein University, Westerville, OH 43081, USA 2 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA 3 Institut für Geometrie und Praktische Mathematik (IGPM), RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany

Received  March 2019 Revised  August 2019 Published  December 2019

Fund Project: D. A. and K. K. gratefully acknowledge support through NSF grant DMS-1515592 and travel support through the KI-Net grant, NSF RNMS grant No. 1107291.

Kinetic exchange models of markets utilize Boltzmann-like kinetic equations to describe the macroscopic evolution of a community wealth distribution corresponding to microscopic binary interaction rules. We develop such models to study a form of welfare called need-based transfer (NBT). In contrast to conventional centrally organized wealth redistribution, NBTs feature a welfare threshold and binary donations in which above-threshold individuals give from their surplus wealth to directly meet the needs of below-threshold individuals. This structure is motivated by examples such as the gifting of cattle practiced by East African Maasai herders or food sharing among vampire bats, and has been studied using agent-based simulation. From the regressive to progressive kinetic NBT models developed here, moment evolution equations and simulation are used to describe the evolution of the community wealth distribution in terms of efficiency, shape, and inequality.

Citation: Kirk Kayser, Dieter Armbruster, Michael Herty. Kinetic models of conservative economies with need-based transfers as welfare. Kinetic & Related Models, 2020, 13 (1) : 169-185. doi: 10.3934/krm.2020006
##### References:
 [1] A. Aktipis, R. De Aguiar, A. Flaherty, P. Iyer, D. Sonkoi and L. Cronk, Cooperation in an uncertain world: For the Maasai of East Africa, need-based transfers outperform account-keeping in volatile environments, Human Ecology, 44 (2016), 353-364.  doi: 10.1007/s10745-016-9823-z.  Google Scholar [2] C. A. Aktipis, L. Cronk and R. de Aguiar, Risk-pooling and herd survival: An agent-based model of a Maasai gift-giving system, Human Ecology, 39 (2011), 131-140.  doi: 10.1007/s10745-010-9364-9.  Google Scholar [3] J. Angle, The surplus theory of social stratification and the size distribution of personal wealth, Social Forces, 65 (1986), 293-326.  doi: 10.2307/2578675.  Google Scholar [4] M. Bisi, Some kinetic models for a market economy, Bollettino dell'Unione Matematica Italiana, 10 (2017), 143-158.  doi: 10.1007/s40574-016-0099-4.  Google Scholar [5] M. Bisi, G. Spiga, G. Toscani et al., Kinetic models of conservative economies with wealth redistribution, Communications in Mathematical Sciences, 7 (2009), 901-916. doi: 10.4310/CMS.2009.v7.n4.a5.  Google Scholar [6] G. G. Carter and G. S. Wilkinson, Food sharing in vampire bats: Reciprocal help predicts donations more than relatedness or harassment, Proc. R. Soc. B, 280 (2013), 20122573.  doi: 10.1098/rspb.2012.2573.  Google Scholar [7] B. K. Chakrabarti, A. Chakraborti, S. R. Chakravarty and A. Chatterjee, Econophysics of Income and Wealth Distributions, Cambridge University Press, 2013. doi: 10.1017/CBO9781139004169.  Google Scholar [8] A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: How saving propensity affects its distribution, The European Physical Journal B-Condensed Matter and Complex Systems, 17 (2000), 167-170.  doi: 10.1007/s100510070173.  Google Scholar [9] A. Chatterjee, B. K. Chakrabarti and S. Manna, Pareto law in a kinetic model of market with random saving propensity, Physica A: Statistical Mechanics and its Applications, 335 (2004), 155-163.  doi: 10.1016/j.physa.2003.11.014.  Google Scholar [10] S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, Journal of Statistical Physics, 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0.  Google Scholar [11] B. Düring, D. Matthes and G. Toscani, A boltzmann-type approach to the formation of wealth distribution curves, 2008. Google Scholar [12] B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Physical Review E, 78 (2008), 056103.  doi: 10.1103/PhysRevE.78.056103.  Google Scholar [13] B. Düring, L. Pareschi and G. Toscani, Kinetic models for optimal control of wealth inequalities, The European Physical Journal B, 91 (2018), Paper No. 265, 12 pp. doi: 10.1140/epjb/e2018-90138-1.  Google Scholar [14] J. L. Gastwirth, The estimation of the lorenz curve and gini index, The Review of Economics and Statistics, 54 (1972), 306-316.  doi: 10.2307/1937992.  Google Scholar [15] Y. Hao, D. Armbruster, L. Cronk and C. A. Aktipis, Need-based transfers on a network: A model of risk-pooling in ecologically volatile environments, Evolution and Human Behavior, 36 (2015), 265-273.  doi: 10.1016/j.evolhumbehav.2014.12.003.  Google Scholar [16] K. Kayser and D. Armbruster, Social optima of need-based transfers, Physica A: Statistical Mechanics and its Applications, 536 (2018), 121011, URL http://www.sciencedirect.com/science/article/pii/S037843711930620X. doi: 10.1016/j.physa.2019.04.247.  Google Scholar [17] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, OUP Oxford, 2013. Google Scholar [18] V. Pareto, Cours D'économie Politique, Lausanne and Paris, 1897. doi: 10.3917/droz.paret.1964.01.  Google Scholar [19] A. C. Silva, Temporal evolution of the "thermal" and "superthermal" income classes in the USA during 1983–2001, EPL (Europhysics Letters), 69 (2004), 304.  doi: 10.1209/epl/i2004-10330-3.  Google Scholar [20] F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Physical Review E, 69 (2004), 046102.  doi: 10.1103/PhysRevE.69.046102.  Google Scholar [21] G. Toscani, Wealth redistribution in conservative linear kinetic models, EPL (Europhysics Letters), 88 (2009), 10007.  doi: 10.1209/0295-5075/88/10007.  Google Scholar [22] G. Toscani and C. Brugna, Wealth redistribution in boltzmann-like models of conservative economies, in Econophysics and Economics of Games, Social Choices and Quantitative Techniques, Springer, 2010, 71–82. doi: 10.1007/978-88-470-1501-2_9.  Google Scholar [23] G. Wäscher and T. Gau, Heuristics for the integer one-dimensional cutting stock problem: A computational study, Operations-Research-Spektrum, 18 (1996), 131-144.   Google Scholar [24] G. S. Wilkinson, Reciprocal food sharing in the vampire bat, Nature, 308 (1984), 181-184.  doi: 10.1038/308181a0.  Google Scholar [25] G. S. Wilkinson, Reciprocal altruism in bats and other mammals, Ethology and Sociobiology, 9 (1988), 85-100.  doi: 10.1016/0162-3095(88)90015-5.  Google Scholar

show all references

##### References:
 [1] A. Aktipis, R. De Aguiar, A. Flaherty, P. Iyer, D. Sonkoi and L. Cronk, Cooperation in an uncertain world: For the Maasai of East Africa, need-based transfers outperform account-keeping in volatile environments, Human Ecology, 44 (2016), 353-364.  doi: 10.1007/s10745-016-9823-z.  Google Scholar [2] C. A. Aktipis, L. Cronk and R. de Aguiar, Risk-pooling and herd survival: An agent-based model of a Maasai gift-giving system, Human Ecology, 39 (2011), 131-140.  doi: 10.1007/s10745-010-9364-9.  Google Scholar [3] J. Angle, The surplus theory of social stratification and the size distribution of personal wealth, Social Forces, 65 (1986), 293-326.  doi: 10.2307/2578675.  Google Scholar [4] M. Bisi, Some kinetic models for a market economy, Bollettino dell'Unione Matematica Italiana, 10 (2017), 143-158.  doi: 10.1007/s40574-016-0099-4.  Google Scholar [5] M. Bisi, G. Spiga, G. Toscani et al., Kinetic models of conservative economies with wealth redistribution, Communications in Mathematical Sciences, 7 (2009), 901-916. doi: 10.4310/CMS.2009.v7.n4.a5.  Google Scholar [6] G. G. Carter and G. S. Wilkinson, Food sharing in vampire bats: Reciprocal help predicts donations more than relatedness or harassment, Proc. R. Soc. B, 280 (2013), 20122573.  doi: 10.1098/rspb.2012.2573.  Google Scholar [7] B. K. Chakrabarti, A. Chakraborti, S. R. Chakravarty and A. Chatterjee, Econophysics of Income and Wealth Distributions, Cambridge University Press, 2013. doi: 10.1017/CBO9781139004169.  Google Scholar [8] A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: How saving propensity affects its distribution, The European Physical Journal B-Condensed Matter and Complex Systems, 17 (2000), 167-170.  doi: 10.1007/s100510070173.  Google Scholar [9] A. Chatterjee, B. K. Chakrabarti and S. Manna, Pareto law in a kinetic model of market with random saving propensity, Physica A: Statistical Mechanics and its Applications, 335 (2004), 155-163.  doi: 10.1016/j.physa.2003.11.014.  Google Scholar [10] S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, Journal of Statistical Physics, 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0.  Google Scholar [11] B. Düring, D. Matthes and G. Toscani, A boltzmann-type approach to the formation of wealth distribution curves, 2008. Google Scholar [12] B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Physical Review E, 78 (2008), 056103.  doi: 10.1103/PhysRevE.78.056103.  Google Scholar [13] B. Düring, L. Pareschi and G. Toscani, Kinetic models for optimal control of wealth inequalities, The European Physical Journal B, 91 (2018), Paper No. 265, 12 pp. doi: 10.1140/epjb/e2018-90138-1.  Google Scholar [14] J. L. Gastwirth, The estimation of the lorenz curve and gini index, The Review of Economics and Statistics, 54 (1972), 306-316.  doi: 10.2307/1937992.  Google Scholar [15] Y. Hao, D. Armbruster, L. Cronk and C. A. Aktipis, Need-based transfers on a network: A model of risk-pooling in ecologically volatile environments, Evolution and Human Behavior, 36 (2015), 265-273.  doi: 10.1016/j.evolhumbehav.2014.12.003.  Google Scholar [16] K. Kayser and D. Armbruster, Social optima of need-based transfers, Physica A: Statistical Mechanics and its Applications, 536 (2018), 121011, URL http://www.sciencedirect.com/science/article/pii/S037843711930620X. doi: 10.1016/j.physa.2019.04.247.  Google Scholar [17] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, OUP Oxford, 2013. Google Scholar [18] V. Pareto, Cours D'économie Politique, Lausanne and Paris, 1897. doi: 10.3917/droz.paret.1964.01.  Google Scholar [19] A. C. Silva, Temporal evolution of the "thermal" and "superthermal" income classes in the USA during 1983–2001, EPL (Europhysics Letters), 69 (2004), 304.  doi: 10.1209/epl/i2004-10330-3.  Google Scholar [20] F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Physical Review E, 69 (2004), 046102.  doi: 10.1103/PhysRevE.69.046102.  Google Scholar [21] G. Toscani, Wealth redistribution in conservative linear kinetic models, EPL (Europhysics Letters), 88 (2009), 10007.  doi: 10.1209/0295-5075/88/10007.  Google Scholar [22] G. Toscani and C. Brugna, Wealth redistribution in boltzmann-like models of conservative economies, in Econophysics and Economics of Games, Social Choices and Quantitative Techniques, Springer, 2010, 71–82. doi: 10.1007/978-88-470-1501-2_9.  Google Scholar [23] G. Wäscher and T. Gau, Heuristics for the integer one-dimensional cutting stock problem: A computational study, Operations-Research-Spektrum, 18 (1996), 131-144.   Google Scholar [24] G. S. Wilkinson, Reciprocal food sharing in the vampire bat, Nature, 308 (1984), 181-184.  doi: 10.1038/308181a0.  Google Scholar [25] G. S. Wilkinson, Reciprocal altruism in bats and other mammals, Ethology and Sociobiology, 9 (1988), 85-100.  doi: 10.1016/0162-3095(88)90015-5.  Google Scholar
Simple examples of the different cases of steady states for Equation (2) where $\theta = 0$; the area under the probability density curve is shaded for visibility
(a) A few different initial wealth distributions with $M_1 = 14$, that when evolving according to Equation (2) with $\theta = 0$ approach the steady states shown in (b). The second and third moment evolutions are shown in (c) and (d) respectively
(a) A few different initial wealth distributions with $M_1 = 14$, that when evolving according to Equation (6) with $\theta = 0$ approach an attractor manifold (b). Note that three curves are present in (b), but they are overlapping. The second and third moment evolutions are shown in (c) and (d) respectively
Numerical steady state solution to Equation (7) as well as analytical steady state solution from (8) with initial condition $f_0(w)$ a Normal distribution $\mathcal{N}(\mu = 10, \sigma^2 = 20^2)$ and parameters $\theta = 0, \epsilon_0 = 10$
Probability densities for probability of choosing donor threshold $\theta + \epsilon$ for regressive, flat, and progressive policies with $\theta = 0$ and maximal wealth $L = 100$. The equation for these parameterized donor threshold probability distributions is given in Equation 10
Flat policy comparison with agent-based simulation. A gamma initial condition is used for $f_0(w)$ and $10^4$ agents are sampled from this distribution as well. Equation (9) is used with $\alpha = 0$ to find the steady state solution of the Boltzmann-like equation; for the agents, interactions are randomly generated and transfers are conducted according to the microscopic description of equation (1) until all $10^4$ agents are at or above threshold
Steady state distributions and data for parameterized kinetic NBT policies with initial condition $f_0(w) \sim$ Gamma
Steady state distributions and data for parameterized kinetic NBT policies with initial condition $f_0(w) \sim$ Uniform
Wealth distributions at $t = 200$ for various policies. The notation used in the legend is such that $fb_p$: 0.2918 means that for the progressive policy (p), the fraction of the population below threshold ($fb_p$) is equal to 0.2918. The initial condition is chosen to be a Gamma distribution. $fb_o$ identifies the optimal policy corresponding to (11) and (12)
A simple example density $f(w)$ to illustrate why the optimal control policy leads to a uniform distribution of surpluses. Choosing a donor threshold of 1 maximizes the product of matching donor-recipient densities
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