In this work, an ensemble of economic interacting agents is considered. The agents are arranged in a linear array where only local couplings are allowed. The deterministic dynamics of each agent is given by a map. This map is expressed by two factors. The first one is a linear term that models the expansion of the agent's economy and that is controlled by the {\it growth capacity parameter}. The second one is an inhibition exponential term that is regulated by the {\it local environmental pressure}. Depending on the parameter setting, the system can display Pareto or Boltzmann-Gibbs behavior in the asymptotic dynamical regime. The regions of parameter space where the system exhibits one of these two statistical behaviors are delimited. Other properties of the system, such as the mean wealth, the standard deviation and the Gini coefficient, are also calculated.
Deep Dive into An Economic Model of Coupled Exponential Maps.
In this work, an ensemble of economic interacting agents is considered. The agents are arranged in a linear array where only local couplings are allowed. The deterministic dynamics of each agent is given by a map. This map is expressed by two factors. The first one is a linear term that models the expansion of the agent’s economy and that is controlled by the {\it growth capacity parameter}. The second one is an inhibition exponential term that is regulated by the {\it local environmental pressure}. Depending on the parameter setting, the system can display Pareto or Boltzmann-Gibbs behavior in the asymptotic dynamical regime. The regions of parameter space where the system exhibits one of these two statistical behaviors are delimited. Other properties of the system, such as the mean wealth, the standard deviation and the Gini coefficient, are also calculated.
Nowadays it is well established that the wealth distribution in western societies presents essentially two phases. This means that the whole society can be split in two disjoint parts in which the richness distribution is different in each of them. Thus, Dragulescu and Yakovenko [1] have found, over real economic data from UK and USA societies, that one phase presents an exponential (Boltzmann-Gibbs → BG) distribution which covers about 95% of individuals, those with low and medium incomes, and that the other phase, which is integrated by the high incomes, i.e., the 5% of individuals, shows a power (Pareto) law. Different dynamical mechanisms for the interaction among agents in multi-agent economic models have been proposed in the literature in order to reproduce these two types of statistical behavior.
In the Dragulescu and Yakovenko [2] model, a set of N economic agents exchange their own amount of money u i , with i = 1, 2, • • • , N , under random binary (i, j) interactions, (u i , u j ) → (u ′ i , u ′ j ), by the following exchange rule:
with ǭ = (1 -ǫ), and ǫ a random number in the interval (0, 1).
Starting the system from an initial state with equity among all the agents, it evolves towards an asymptotic dynamical state where the richness follows a BG distribution [3]. This model is equivalent to the saving propensity model introduced by Chakraborti and Chakrabarti [4] for the particular case in which the agents do not save any fraction of money before carrying out each transaction. A similar model was introduced by Angle [5]. In this case, the random pair (i, j) of economic agents, (u i , u j ), exchanges money under the rule:
where
with ǫ a random number in the interval (0, 1). The exchange parameter, ω, with 0 < ω < 1, represents the maximum fraction of wealth lost by the agent u i in the transaction. When ω = 0.75, the exponential distribution of the wealth is also found for this model.
In order to recover a power law behavior, these models can be reformulated in a nonhomogeneous way. These are models in which each agent i can have assigned by some random function a different value ω i of the exchange parameter ω in the Angle model [5] or a different saving propensity in the Chakraborti and Chakrabarti model [4]. Under quite general conditions, robust Pareto distributions have been found at large u-values in these reformulations [4], [5].
Randomness is an essential ingredient in all the former models. Thus, agents interact by pairs chosen at random, and these pairs exchange a random quantity of money in each transaction. Moreover, the transition from the BG to the Pareto behavior requires the change of the structural properties of the system. It can be reached, for instance, by introducing a strong inhomogeneity that breaks the initial indistinguishability among the units of the ensemble. From a practical point of view, this could seem an unrealistic approach since we need to conform very different setups in order to mimic the two statistical behaviors. On the other hand, interactions among the economic or social agents of a real collectivity are not fully random. In fact, they are driving in the majority of transactions by some kind of mutual interest or rational forces. Hence, it would be useful to dispose of a multi-agent model with the ability of displaying BG and Pareto behaviors emerging from an asymptotic dynamics where determinism would play some role in the evolution of the system. Now, we proceed to show with some detail the properties of one of these models [6] recently introduced in the literature.
The model [6] consists of a linear array of N interacting agents with periodic boundary conditions. Each agent, which can represent a company, country or other economic entity, is identified by an index i, with i = 1 • • • N and N being the system size. Its actual state is characterized by a real number, x i , denoting the strength, wealth or richness of the agent, with x i ∈ [0, ∞). The system evolves in time synchronously, and only interactions among nearest-neighbors are allowed. Thus, the state of the agent, x t+1 i , at time t+1 is given by the product of two terms at the precedent time t; the natural growth of the agent, r i x t i , with its own growth capacity parameter, r i , and a control term that limits this growth with respect to the local field
) through a negative exponential function with local environmental pressure a i :
The parameter r i represents the capacity of the agent to become richer and the parameter a i describes some kind of local pressure [7] that saturates its exponential growth. This means that the largest possibility of growth for the agent is obtained when x i ≃ a i Ψ t i , i.e., when the agent has reached some kind of adaptation to the local environment. In this note, for the sake of simplicity, we concentrate our interest in a homogeneous system with a constant capacity r and a constant selection pressure a for the whole array of sites.
If all t
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