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

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

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  • 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.

    Mathematics Subject Classification: 35Q93, 82C40, 91B15, 91B80.


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  • Figure 1.  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

    Figure 2.  (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

    Figure 3.  (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

    Figure 4.  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 $

    Figure 5.  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

    Figure 6.  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

    Figure 7.  Steady state distributions and data for parameterized kinetic NBT policies with initial condition $ f_0(w) \sim $ Gamma

    Figure 8.  Steady state distributions and data for parameterized kinetic NBT policies with initial condition $ f_0(w) \sim $ Uniform

    Figure 9.  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)

    Figure 10.  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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