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

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