Kinetic market models with single commodity having price fluctuations

We study here numerically the behavior of an ideal gas like model of markets having only one non-consumable commodity. We investigate the behavior of the steady-state distributions of money, commodity and total wealth, as the dynamics of trading or exchange of money and commodity proceeds, with local (in time) fluctuations in the price of the commodity. These distributions are studied in markets with agents having uniform and random saving factors. The self-organizing features in money distribution are similar to the cases without any commodity (or with consumable commodities), while the commodity distribution shows an exponential decay. The wealth distribution shows interesting behavior: gamma like distribution for uniform saving propensity and has the same power-law tail, as that of the money distribution, for a market with agents having random saving propensity.     

Gamma-distribution and wealth inequality

We discuss the equivalence between kinetic wealth-exchange models, in which agents exchange wealth during trades, and mechanical models of particles, exchanging energy during collisions. The universality of the underlying dynamics is shown both through a variational approach based on the minimization of the Boltzmann entropy and a microscopic analysis of the collision dynamics of molecules in a gas. In various relevant cases, the equilibrium distribution is well-approximated by a gamma-distribution with suitably defined temperature and number of dimensions. This in turn allows one to quantify the inequalities observed in the wealth distributions and suggests that their origin should be traced back to very general underlying mechanisms, for instance, the fact that smaller the fraction of the relevant quantity (e.g. wealth) that agent can exchange during an interaction, the closer the corresponding equilibrium distribution is to a fair distribution.     

An almost linear stochastic map related to the particle system models of social sciences

We propose a stochastic map model of economic dynamics. In the past decade, an array of observations in economics has been investigated in the econophysics literature, a major example being the universal features of inequality in terms of income and wealth. Another area of enquiry is the formation of opinion in a society. The model proposed attempts to produce positively skewed distributions and power law distributions as has been observed in the real data of income and wealth. Also, it shows a non-trivial phase transition in the opinion of a society (opinion formation). A number of physical models also generate similar results. In particular, kinetic exchange models have been successful especially in this regard. Therefore, we compare the results obtained from these two approaches and discuss a number of new features and drawbacks of this model.     

Emergence of power-law in a market with mixed models

We investigate the problem of wealth distribution from the viewpoint of asset exchange. Robust nature of Pareto's law across economies, ideologies and nations suggests that this could be an outcome of trading strategies. However, the simple asset exchange models fail to reproduce this feature. A Yardsale (YS) model in which amount put on the bet is a fraction of minimum of the two players leads to condensation of wealth in hands of some agent while theft and fraud (TF) model in which the amount to be exchanged is a fraction of loser's wealth leads to an exponential distribution of wealth. We show that if we allow few agents to follow a different model than others, i.e., there are some agents following TF model while rest follow YS model, it leads to distribution with power-law tails. Similar effect is observed when one carries out transactions for a fraction of one's wealth using TF model and for the rest YS model is used. We also observe a power-law tail in wealth distribution if we allow the agents to follow either of the models with some probability.     

Economic exchanges in a stratified society: End of the middle class?

We study the effect of the social stratification on the wealth distribution on a system of interacting economic agents that are constrained to interact only within their own economic class. The economical mobility of the agents is related to its success in exchange transactions. Different wealth distributions are obtained as a function of the width of the economic class. We find a range of widths in which the society is divided in two classes separated by a deep gap that prevents further exchange between poor and rich agents. As a consequence, the middle wealth class is eliminated. The high values of the Gini indices obtained in these cases indicate a highly unequal society. On the other hand, lower and higher widths induce lower Gini indices and a fairer wealth distribution.     

Modeling wealth distribution in growing markets

We introduce an auto-regressive model which captures the growing nature of realistic markets. In our model agents do not trade with other agents, they interact indirectly only through a market. Change of their wealth depends, linearly on how much they invest, and stochastically on how much they gain from the noisy market. The average wealth of the market could be fixed or growing. We show that in a market where investment capacity of agents differ, average wealth of agents generically follow the Pareto-law. In few cases, the individual distribution of wealth of every agentcould also be obtained exactly. We also show that the underlying dynamics of other well studied kinetic models of markets can be mapped to the dynamics of our auto-regressive model.     

Inequality reversal: Effects of the savings propensity and correlated returns

In the last decade, a large body of literature has been developed to explain the universal features of inequality in terms of income and wealth. By now, it is established that the distributions of income and wealth in various economies show a number of statistical regularities. There are several models to explain such static features of inequality in a unifying framework, and the kinetic exchange models in particular provide one such framework. Here we focus on the dynamic features of inequality. In the process of development and growth, inequality in an economy in terms of income and wealth follows a particular pattern of rising in the initial stage followed by an eventual fall. This inverted U-shaped curve is known as the Kuznets Curve. We examine the possibilities of such behavior of an economy in the context of a generalized kinetic exchange model. It is shown that under some specific conditions, our model economy indeed shows inequality reversal.     

On a Kinetic Model for a Simple Market Economy

In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasi-stationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker–Planck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis.     

Wealth condensation in a multiplicative random asset exchange model

Random Asset Exchange (RAE) models, despite a number of simplifying assumptions, serve the purpose of establishing direct relationships between microscopic exchange mechanisms and observed economical data. In this work a conservative multiplicative RAE model is discussed in which, at each timestep, two agents “bet” for a fraction f of the poorest agent's wealth. When the poorest agent wins the bet with probability p, we show that, in a well defined region of the (p,f) phase space, there is wealth condensation. This means that all wealth ends up owned by only one agent, in the long run. We derive the condensation conditions analytically by two different procedures, and find results in accordance with previous numerical estimates. In the non-condensed phase, the equilibrium wealth distribution is a power law for small wealths. The associated exponent is derived analytically and it is found that it tends to -1 on the condensation interface. I turns out that wealth condensation happens also for values of p much larger than 0.5, that is under microscopic exchange rules that, apparently, favor the poor. We argue that the observed “rich get richer” effect is enhanced by the multiplicative character of the dynamics.     

Class formation in a social network with asset exchange

We study two kinds of economic exchanges, additive and multiplicative, in a system of N agents. The work is divided into two parts. In the first one, the agents are free to interact with each other. The system evolves to a Boltzmann-Gibbs distribution with additive exchange and condenses with a multiplicative one. If bankruptcy is introduced, both types of exchange lead to condensation. Condensation times have been studied. In the second part, the agents are placed in a social network. We analyze the behavior of wealth distributions in time, and the formation of economic classes is observed for certain values of network connectivity.     

Entropy and equilibrium state of free market models

Many recent models of trade dynamics use the simple idea of wealth exchanges among economic agents in order to obtain a stable or equilibrium distribution of wealth among the agents. In particular, a plain analogy compares the wealth in a society with the energy in a physical system, and the trade between agents to the energy exchange between molecules during collisions. In physical systems, the energy exchange among molecules leads to a state of equipartition of the energy and to an equilibrium situation where the entropy is a maximum. On the other hand, in a large class of exchange models, the system converges to a very unequal condensed state, where one or a few agents concentrate all the wealth of the society while the wide majority of agents shares zero or almost zero fraction of the wealth. So, in those economic systems a minimum entropy state is attained. We propose here an analytical model where we investigate the effects of a particular class of economic exchanges that minimize the entropy. By solving the model we discuss the conditions that can drive the system to a state of minimum entropy, as well as the mechanisms to recover a kind of equipartition of wealth.     

Wealth distributions in asset exchange models

A model for the evolution of the wealth distribution in an economically interacting population is introduced, in which a specified amount of assets are exchanged between two individuals when they interact. The resulting wealth distributions are determined for a variety of exchange rules. For “random” exchange, either individual is equally likely to gain in a trade, while “greedy” exchange, the richer individual gains. When the amount of asset traded is fixed, random exchange leads to a Gaussian wealth distribution, while greedy exchange gives a Fermi-like scaled wealth distribution in the long-time limit. Multiplicative processes are also investigated, where the amount of asset exchanged is a finite fraction of the wealth of one of the traders. For random multiplicative exchange, a steady state occurs, while in greedy multiplicative exchange a continuously evolving power law wealth distribution arises. Received: 13 August 1997 / Revised: 31 December 1997 / Accepted: 26 January 1998     

The distribution of wealth in the presence of altruism in simple economic models

We study the effect of altruism in two simple asset exchange models: the yard sale model (winner gets a random fraction of the poorer player's wealth) and the theft and fraud model (winner gets a random fraction of the loser's wealth). We also introduce in these models the concept of bargaining efficiency, which makes the poorer trader more aggressive in getting a favorable deal thus augmenting his winning probabilities. The altruistic behavior is controlled by varying the number of traders who behave altruistically and by the degree of altruism that they show. The resulting wealth distribution is characterized using the Gini index. We compare the resulting values of the Gini index at different levels of altruism in both models. It is found that altruistic behavior does lead to a more equitable wealth distribution but only for unreasonable high values of altruism that are difficult to expect in a real economic system.     

Buying on margin,selling short in an agent-based market model

Credit trading, or leverage trading, which includes buying on margin and selling short, plays an important role in financial markets, where agents tend to increase their leverages for increased profits. This paper presents an agent-based asset market model to study the effect of the permissive leverage level on traders’ wealth and overall market indicators. In this model, heterogeneous agents can assume fundamental value-converging expectations or trend-persistence expectations, and their effective demands of assets depend both on demand willingness and wealth constraints, where leverage can relieve the wealth constraints to some extent. The asset market price is determined by a market maker, who watches the market excess demand, and is influenced by noise factors. By simulations, we examine market results for different leverage ratios. At the individual level, we focus on how the leverage ratio influences agents’ wealth accumulation. At the market level, we focus on how the leverage ratio influences changes in the asset price, volatility, and trading volume. Qualitatively, our model provides some meaningful results supported by empirical facts. More importantly, we find a continuous phase transition as we increase the leverage threshold, which may provide a further prospective of credit trading.     

Modeling interactions of trading volumes in financial dynamics

A dynamic herding model with interactions of trading volumes is introduced. At time t, an agent trades with a probability, which depends on the ratio of the total trading volume at time t−1 to its own trading volume at its last trade. The price return is determined by the volume imbalance and number of trades. The model can reproduce the power-law distributions of the trading volume, number of trades and price return, and the probable relation between them. The exponents are tunable by adjusting the values of the parameters, but show slight deviation from those revealed in empirical studies. Moreover, the time series generated are long-range correlated. We demonstrate that the results are rather robust, and do not depend on the particular form of the trading probability.     

Study of a market model with conservative exchanges on complex networks

Many models of market dynamics make use of the idea of conservative wealth exchanges among economic agents. A few years ago an exchange model using extremal dynamics was developed and a very interesting result was obtained: a self-generated minimum wealth or poverty line. On the other hand, the wealth distribution exhibited an exponential shape as a function of the square of the wealth. These results have been obtained both considering exchanges between nearest neighbors or in a mean field scheme. In the present paper we study the effect of distributing the agents on a complex network. We have considered archetypical complex networks: Erdös–Rényi random networks and scale-free networks. The presence of a poverty line with finite wealth is preserved but spatial correlations are important, particularly between the degree of the node and the wealth. We present a detailed study of the correlations, as well as the changes in the Gini coefficient, that measures the inequality, as a function of the type and average degree of the considered networks.     

Hydrodynamics from kinetic models of conservative economies

In this paper, we introduce and discuss the passage to hydrodynamic equations for kinetic models of conservative economies, in which the density of wealth depends on additional parameters, like the propensity to invest. As in kinetic theory of rarefied gases, the closure depends on the knowledge of the homogeneous steady wealth distribution (the Maxwellian) of the underlying kinetic model. The collision operator used here is the Fokker-Planck operator introduced by J.P. Bouchaud and M. Mezard [Wealth condensation in a simple model of economy, Physica A 282 (2000) 536-545], which has been recently obtained in a suitable asymptotic of a Boltzmann-like model involving both exchanges between agents and speculative trading by S. Cordier, L. Pareschi and one of the authors [S. Cordier, L. Pareschi, G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys. 120 (2005) 253-277]. Numerical simulations on the fluid equations are then proposed and analyzed for various laws of variation of the propensity.     

Empirical behavior of a world stock index from intra-day to monthly time scales

Most of the papers that study the distributional and fractal properties of financial instruments focus on stock prices or foreign exchange rates. This typically leads to mixed results concerning the distributions of log-returns and some multi-fractal properties of exchange rates, stock prices, and regional indices. This paper uses a well diversified world stock index as the central object of analysis. Such index approximates the growth optimal portfolio, which is demonstrated under the benchmark approach, it is the ideal reference unit for studying basic securities. When denominating this world index in units of a given currency, one measures the movements of the currency against the entire market. This provides a least disturbed observation of the currency dynamics. In this manner, one can expect to disentangle, e.g., the superposition of the two currencies involved in an exchange rate. This benchmark approach to the empirical analysis of financial data allows us to establish remarkable stylized facts. Most important is the observation that the repeatedly documented multi-fractal appearance of financial time series is very weak and much less pronounced than the deviation of the mono-scaling properties from Brownian-motion type scaling. The generalized Hurst exponent H(2) assumes typical values between 0.55 and 0.6. Accordingly, autocorrelations of log-returns decay according to a power law, and the quadratic variation vanishes when going to vanishing observation time step size. Furthermore, one can identify the Student t distribution as the log-return distribution of a well-diversified world stock index for long time horizons when a long enough data series is used for estimation. The study of dependence properties, finally, reveals that jumps at daily horizon originate primarily in the stock market while at 5min horizon they originate in the foreign exchange market. The principal message of the empirical analysis is that there is evidence that a diffusion model without multi-scaling could reasonably well model the dynamics of a broadly diversified world stock index.     

Hadronization from color interactions

A quark coalescence model, based on semi-relativistic molecular dynamics with color interactions among quarks, is presented and applied to pp collisions. A phenomenological potential with two tunable parameters is introduced to describe the color interactions between quarks and antiquarks. The interactions drive the process of hadronization that finally results in different color neutral clusters, which can be identified as hadrons based on some criteria. A Monte Carlo generator PYTHIA is used to generate quarks in the initial state of hadronization, and different values of tunable parameters are used to study the final state distributions and correlations. Baryon-to-meson ratio, transverse momentum spectra, pseudorapidity distributions and forward-backward multiplicity correlations of hadrons produced in the hadronization process, obtained from this model with different parameters, are compared with those from PYTHIA.     

Entanglement control in one-dimensional s= 1/2 random XY spin chain

The entanglement in one-dimensional random XY spin systems where the impurities of exchange couplings and the external magnetic fields are considered as random variables is investigated by solving the different spin-spin correlation functions and the average magnetization per spin. The entanglement dynamics near particular locations of the system is also studied when the exchange couplings （or the external magnetic fields） satisfy three different distributions （the Gaussian distribution, double-Gaussian distribution, and bimodal distribution）. We find that the entanglement can be controlled by varying the strength of external magnetic field and the distributions of impurities. Moreover, the entanglement of some nearest-neighbouring qubits can be increased for certain parameter values of the three different distributions.