# American Institute of Mathematical Sciences

• Previous Article
Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation
• KRM Home
• This Issue
• Next Article
Long-time behavior of second order linearized Vlasov-Poisson equations near a homogeneous equilibrium
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:

show all references

##### References:
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
 [1] Daniel Matthes, Giuseppe Toscani. Analysis of a model for wealth redistribution. Kinetic & Related Models, 2008, 1 (1) : 1-27. doi: 10.3934/krm.2008.1.1 [2] Dieter Armbruster, Christian Ringhofer, Andrea Thatcher. A kinetic model for an agent based market simulation. Networks & Heterogeneous Media, 2015, 10 (3) : 527-542. doi: 10.3934/nhm.2015.10.527 [3] Xiangyu Ge, Tifang Ye, Yanli Zhou, Guoguang Yan. Fiscal centralization vs. decentralization on economic growth and welfare: An optimal-control approach. Journal of Industrial & Management Optimization, 2016, 12 (2) : 487-504. doi: 10.3934/jimo.2016.12.487 [4] Dieter Armbruster, Matthew Wienke. Kinetic models and intrinsic timescales: Simulation comparison for a 2nd order queueing model. Kinetic & Related Models, 2019, 12 (1) : 177-193. doi: 10.3934/krm.2019008 [5] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [6] Yan Zhang, Yonghong Wu, Haixiang Yao. Optimal health insurance with constraints under utility of health, wealth and income. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021031 [7] Nicolas Vauchelet. Numerical simulation of a kinetic model for chemotaxis. Kinetic & Related Models, 2010, 3 (3) : 501-528. doi: 10.3934/krm.2010.3.501 [8] Claus Kirchner, Michael Herty, Simone Göttlich, Axel Klar. Optimal control for continuous supply network models. Networks & Heterogeneous Media, 2006, 1 (4) : 675-688. doi: 10.3934/nhm.2006.1.675 [9] Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006 [10] Martina Conte, Maria Groppi, Giampiero Spiga. Qualitative analysis of kinetic-based models for tumor-immune system interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2393-2414. doi: 10.3934/dcdsb.2018060 [11] Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041 [12] Kazuyuki Yagasaki. Optimal control of the SIR epidemic model based on dynamical systems theory. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021144 [13] Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067 [14] Heather Finotti, Suzanne Lenhart, Tuoc Van Phan. Optimal control of advective direction in reaction-diffusion population models. Evolution Equations & Control Theory, 2012, 1 (1) : 81-107. doi: 10.3934/eect.2012.1.81 [15] Martin Benning, Elena Celledoni, Matthias J. Ehrhardt, Brynjulf Owren, Carola-Bibiane Schönlieb. Deep learning as optimal control problems: Models and numerical methods. Journal of Computational Dynamics, 2019, 6 (2) : 171-198. doi: 10.3934/jcd.2019009 [16] Carsten Hartmann, Juan C. Latorre, Wei Zhang, Grigorios A. Pavliotis. Addendum to "Optimal control of multiscale systems using reduced-order models". Journal of Computational Dynamics, 2017, 4 (1&2) : 167-167. doi: 10.3934/jcd.2017006 [17] Hiroaki Morimoto. Optimal harvesting and planting control in stochastic logistic population models. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2545-2559. doi: 10.3934/dcdsb.2012.17.2545 [18] Maria do Rosário de Pinho, Filipa Nunes Nogueira. On application of optimal control to SEIR normalized models: Pros and cons. Mathematical Biosciences & Engineering, 2017, 14 (1) : 111-126. doi: 10.3934/mbe.2017008 [19] Carsten Hartmann, Juan C. Latorre, Wei Zhang, Grigorios A. Pavliotis. Optimal control of multiscale systems using reduced-order models. Journal of Computational Dynamics, 2014, 1 (2) : 279-306. doi: 10.3934/jcd.2014.1.279 [20] Gülden Gün Polat, Teoman Özer. On group analysis of optimal control problems in economic growth models. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2853-2876. doi: 10.3934/dcdss.2020215

2020 Impact Factor: 1.432