A self regulated adaptive model for expenditure propensity and Gini index relationship

Over the past two decades, physicists have been devoted to the problem of income distribution $P(m)$. A key feature in this study is characterize
the inequalities implied by microeconomic models based on the mechanisms of exchange of goods and services. One way to quantify such inequalities
is based on the Gini index $0 \leqslant G \leqslant1$, a parameter that sets the maximum ($G=1$) and minimum ($G=0$) concentration of resources.
Current studies indicates that income distribution $P(m)$ has two distinct regimes separated by a scale $m_c$. The first one associated to a low-regime income ($m\leqslant m_c$) described by a gamma distribution $\Gamma(n,\beta)$ and a second one related to a high-income regime ($m>m_c$), mathematically represented by a power law function with a parameter $1\leqslant \nu \leqslant2$, usually called Pareto's exponent. More recently it has been pointed out the existence of a bimodality on this distribution. In close connection with microscopic models usually constructed to describe physical systems, two classes of models were introduced as a intent to mimic a closed economic system, compound by a fixed number of economic agents $N$ and resources $M$. At the first class (homogeneous) which describes the low-regime income, agents have the same consumption expenditure $\omega$ $(0<\omega\leqslant1)$, at the second (heterogeneous) providing a power law there is a single consumption expenditure $\omega_i$, specific to each agent, or in a more realistic case a probabilistic distribution $S(\omega)$. In both cases the expenditure consumption rate are set exogenously to the system. In this communication we introduce an adaptive heterogeneous model in order to describe quantitatively the relationship among the average expenditure rate $\langle\omega\rangle$ of economic agents, and the Gini index associated to the income distribution. In this approach a fraction $p_0$ of all economic agents $N$ do not modify their expenditure rates, a fraction $p_1$ are able to modify their consumption rate negatively correlated with their income and lastly a fraction $p_2$ positively. In this scenario the inertia $\gamma(m)$ associated with adaptation is self-regulated by the agent income level. Besides producing the distribution rates as an emerging feature the model are able to provide a bimodality on the income distribution and produce values $(\langle \omega \rangle,G) $ compatible with those available from real data obtained from the World Bank.