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 agent could 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.
We provide an exact solution to the ideal-gas-like models studied in econophysics to understand the microscopic origin of Pareto-law. In these class of models the key ingredient necessary for having a self-organized scale-free steady-state distribution is the trading or collision rule where agents or particles save a definite fraction of their wealth or energy and invests the rest for trading. Using a Gibbs ensemble approach we could obtain the exact distribution of wealth in this model. Moreover we show that in this model (a) good savers are always rich and (b) every agent poor or rich invest…
This paper analyzes the equilibrium distribution of wealth in an economy where firms' productivities are subject to idiosyncratic shocks, returns on factors are determined in competitive markets, dynasties have linear consumption functions and government imposes taxes on capital and labour incomes and equally redistributes the collected resources to dynasties. The equilibrium distribution of wealth is explicitly calculated and its shape crucially depends on market incompleteness. In particular, a Paretian law in the top tail only arises if capital markets are incomplete. The Pareto exponent de…
We study the finite-size effects in some scaling systems, and show that the finite number of agents N leads to a cut-off in the upper value of the Pareto law for the relative individual wealth. The exponent $α$ of the Pareto law obtained in stochastic multiplicative market models is crucially affected by the fact that N is always finite in real systems. We show that any finite value of N leads to properties which can differ crucially from the naive theoretical results obtained by assuming an infinite N. In particular, finite N may cause in the absence of an appropriate social policy extreme we…
In simulations of some economic gas-like models, the asymptotic regime shows an exponential wealth distribution, independently of the initial wealth distribution given to the system. The appearance of this statistical equilibrium for this type of gas-like models is explained in a rigorous analytical way.
In this short paper we define the wealth process in a spin model for market microstructure, for individual agents and in aggregate. The agents in our model try to balance their desire to belong to the local majority (herding behavior), defined over random network neighborhoods, and the occasional advantage of belonging to the global minority (contrarian trading). We arrive at a classification of the martingale properties of this wealth process and use it to determine the strategic stability of the agents' interactions. Our goal is to add a behavioral interpretation to this stochastic agent-bas…
We analyze the ideal gas like models of markets and review the different cases where a `savings' factor changes the nature and shape of the distribution of wealth. These models can produce similar distribution of wealth as observed across varied economies. We present a more realistic model where the saving factor can vary over time (annealed savings) and yet produces Pareto distribution of wealth in certain cases. We discuss the relevance of such models in the context of wealth distribution, and address some recent issues in the context of these models.
The LLS stock market model is a model of heterogeneous quasi-rational investors operating in a complex environment about which they have incomplete information. We review the main features of this model and several of its extensions. We study the effects of investor heterogeneity and show that predation, competition, or symbiosis may occur between different investor populations. The dynamics of the LLS model lead to the empirically observed Pareto wealth distribution. Many properties observed in actual markets appear as natural consequences of the LLS dynamics: truncated Levy distribution of s…
Wealth inequality is an important matter for economic theory and policy. Ongoing debates have been discussing recent rise in wealth inequality in connection with recent development of active financial markets around the world. Existing literature on wealth distribution connects the origins of wealth inequality with a variety of drivers. Our approach develops a minimalist modelling strategy that combines three featuring mechanisms: active financial markets; individual wealth accumulation; and compound interest structure. We provide mathematical proof that accumulated financial investment return…
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…
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 the majority of exchange models, th…
We give a microscopic representation of the stock-market in which the microscopic agents are the individual traders and their capital. Their basic dynamics consists in the auto-catalysis of the individual capital and in the global competition/cooperation between the agents mediated by the total wealth invested in the stock (which we identify with the stock-index). We show that such systems lead generically to (truncated) Pareto power-law distribution of the individual wealth. This, in turn, leads to intermittent market (short time) returns parametrized by a (truncated) Levy distribution. We re…
We find the wealth distribution for an economic agent in the financial market, in analogy with standard derivation of generaliz Boltzman (Tsallis) factor in statistical mechanics. In this respect, Tsallis entropic index separates two different regimes, the large and small size market. The Pareto like wealth distribution is obtained in the case of small size market. A method for computing the premium is suggested based on the surplus average vanishing.
We find the explicit expression for the equilibrium wealth distribution of the Directed Random Market process, recently introduced by Martínez-Martínez and López-Ruiz, which turns out to be a Gamma distribution with shape parameter $\frac{1}{2}$. We also prove the convergence of the discrete-time process describing the evolution of the distribution of wealth to the equilibrium distribution.
Standard approaches to the theory of financial markets are based on equilibrium and efficiency. Here we develop an alternative based on concepts and methods developed by biologists, in which the wealth invested in a financial strategy is like the abundance of a species. We study a toy model of a market consisting of value investors, trend followers and noise traders. We show that the average returns of strategies are strongly density dependent, i.e. they depend on the wealth invested in each strategy at any given time. In the absence of noise the market would slowly evolve toward an efficient …
We briefly review results on nonlinear kinetic equation of Boltzmann type which describe the evolution of wealth in a simple agents market. The mathematical structure of the underlying kinetic equations allows to use well-known techniques of wide use in kinetic theory of rarefied gases to obtain information on the process of relaxation to a stationary profile, as well as to identify simple interaction rules which are responsible of the formation of Pareto tails.
We introduce and discuss a nonlinear kinetic equation of Boltzmann type which describes the influence of knowledge in the evolution of wealth in a system of agents which interact through the binary trades introduced in Cordier, Pareschi, Toscani, J. Stat. Phys. 2005. The trades, which include both saving propensity and the risks of the market, are here modified in the risk and saving parameters, which now are assumed to depend on the personal degree of knowledge. The numerical simulations show that the presence of knowledge has the potential to produce a class of wealthy agents and to account …
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 commo…
Behavioral Finance has become a challenge to the scientific community. Based on the assumption that behavioral aspects of investors may explain some features of the Stock Market, we propose an agent based model to study quantitatively this relationship. In order to approximate the simulated market to the complexity of real markets, we consider that the investors are connected among them through a small world network; each one has its own psychological profile (Imitation, Anti-Imitation, Random); two different strategies for decision making: one of them is based on the trust neighborhood of the…
We study a system of $N$ agents, whose wealth grows linearly, under the effect of stochastic resetting and interacting via a tax-like dynamics -- all agents donate a part of their wealth, which is, in turn, redistributed equally among all others. This mimics a socio-economic scenario where people have fixed incomes, suffer individual economic setbacks, and pay taxes to the state. The system always reaches a stationary state, which shows a trivial exponential wealth distribution in the absence of tax dynamics. The introduction of the tax dynamics leads to several interesting features in the sta…
We develop a model for the evolution of wealth in a non-conservative economic environment, extending a theory developed earlier by the authors. The model considers a system of rational agents interacting in a game theoretical framework. This evolution drives the dynamic of the agents in both wealth and economic configuration variables. The cost function is chosen to represent a risk averse strategy of each agent. That is, the agent is more likely to interact with the market, the more predictable the market, and therefore the smaller its individual risk. This yields a kinetic equation for an ef…
Several problems arising in Economics and Finance are analyzed using concepts and quantitative methods from Physics. Here is the abridged abstact: Chapter 1: By analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. A thermal machine which extracts a monetary profit can be constructed between two economic systems with different temperatures. Chapter 2: Using data from several sources, it is found that the distribution of income is de…
Aiming to describe the wealth distribution evolution, several models consider an ensemble of interacting economic agents that exchange wealth in binary fashion. Intriguingly, models that consider an unbiased market, that gives to each agent the same chances to win in the game, are always out of equilibrium until the perfect inequality of the final state is attained. Here we present a rigorous analytical demonstration that any system driven by unbiased binary exchanges are doomed to drive the system to perfect inequality and zero mobility.
We undertake a study of markets from the perspective of a financial agent with limited access to information. The set of wealth processes available to the agent is structured with reasonable economic properties, instead of the usual practice of taking it to consist of stochastic integrals against a semimartingale integrator. We obtain the equivalence of the boundedness in probability of the set of terminal wealth outcomes (which in turn is equivalent to the weak market viability condition of absence of arbitrage of the first kind) with the existence of at least one strictly positive deflator t…
We consider the ideal-gas models of trading markets, where each agent is identified with a gas molecule and each trading as an elastic or money-conserving (two-body) collision. Unlike in the ideal gas, we introduce saving propensity $λ$ of agents, such that each agent saves a fraction $λ$ of its money and trades with the rest. We show the steady-state money or wealth distribution in a market is Gibbs-like for $λ=0$, has got a non-vanishing most-probable value for $λ\ne 0$ and Pareto-like when $λ$ is widely distributed among the agents. We compare these results with observations on wealth distr…
We run experimental asset markets to investigate the emergence of excess trading and the occurrence of synchronised trading activity leading to crashes in the artificial markets. The market environment favours early investment in the risky asset and no posterior trading, i.e. a buy-and-hold strategy with a most probable return of over 600%. We observe that subjects trade too much, and due to the market impact that we explicitly implement, this is detrimental to their wealth. The asset market experiment was followed by risk aversion measurement. We find that preference for risk systematically l…
A generalized continuous economic model is proposed for random markets. In this model, agents interact by pairs and exchange their money in a random way. A parameter controls the effectiveness of the transactions between the agents. We show in a rigorous way that this type of markets reach their asymptotic equilibrium on the exponential wealth distribution.
We analyze a conservative market model for the competition among economic agents in a close society. A minimum dynamics ensures that the poorest agent has a chance to improve its economic welfare. After a transient, the system self-organizes into a critical state where the wealth distribution have a minimum threshold, with almost no agent below this poverty line, also, very few extremely rich agents are stable in time. Above the poverty line the distribution follows an exponential behavior. The local solution exhibits a low Gini index, while the mean field solution of the model generates a wea…
The contradiction between physical and economical sciences concerning the growth of the production/consumption mechanism is analyzed. It is then shown that if one wishes to keep the security level stable or to enhance it in a growing economy the cost of security grows faster than the gross wealth. The result is a typical evolution in which the net wealth increases up to a maximum, then abruptly collapses. Besides this, any system of relations based on a growing volume of exchanges is bound to go progressively out of control. The voluntary blindness of the ruling classes toward these facts is l…
This paper studies the trading volumes and wealth distribution of a novel agent-based model of an artificial financial market. In this model, heterogeneous agents, behaving according to the Von Neumann and Morgenstern utility theory, may mutually interact. A Tobin-like tax (TT) on successful investments and a flat tax are compared to assess the effects on the agents' wealth distribution. We carry out extensive numerical simulations in two alternative scenarios: i) a reference scenario, where the agents keep their utility function fixed, and ii) a focal scenario, where the agents are adaptive a…
Masanao Aoki developed a new methodology for a basic problem of economics: deducing rigorously the macroeconomic dynamics as emerging from the interactions of many individual agents. This includes deduction of the fractal / intermittent fluctuations of macroeconomic quantities from the granularity of the mezo-economic collective objects (large individual wealth, highly productive geographical locations, emergent technologies, emergent economic sectors) in which the micro-economic agents self-organize. In particular, we present some theoretical predictions, which also met extensive validation f…
We argue that the present crisis and stalling economy continuing since 2007 are rooted in the delusionary belief in policies based on a "perpetual money machine" type of thinking. We document strong evidence that, since the early 1980s, consumption has been increasingly funded by smaller savings, booming financial profits, wealth extracted from house price appreciation and explosive debt. This is in stark contrast with the productivity-fueled growth that was seen in the 1950s and 1960s. This transition, starting in the early 1980s, was further supported by a climate of deregulation and a massi…
Social and economic inequality is a plague of the XXI Century. It is continuously widening, as the wealth of a relatively small group increases and, therefore, the rest of the world shares a shrinking fraction of resources. This situation has been predicted and denounced by economists and econophysicists. The latter ones have widely used models of market dynamics which consider that wealth distribution is the result of wealth exchanges among economic agents. A simple analogy relates the wealth in a society with the kinetic energy of the molecules in a gas, and the trade between agents to the e…
In this paper we test computationally the performance of CAPM in an evolutionary setting. In particular we study the stability of wealth distribution in a financial market where some traders invest as prescribed by CAPM and others behave according to different portfolio rules. Our study is motivated by recent analytical results that show that, whenever a logarithmic utility maximiser enters the market, traders who either ``believe'' in CAPM and use it as a rule of thumb for their portfolio decisions, or are endowed with genuine mean-variance preferences, vanish in the long run. Our analysis pr…
We consider the ideal-gas models of trading markets, where each agent is identified with a gas molecule and each trading as an elastic or money-conserving (two-body) collision. Unlike in the ideal gas, we introduce saving propensity $λ$ of agents, such that each agent saves a fraction $λ$ of its money and trades with the rest. We show the steady-state money or wealth distribution in a market is Gibbs-like for $λ=0$, has got a non-vanishing most-probable value for $λ\ne 0$ and Pareto-like when $λ$ is widely distributed among the agents. We compare these results with observations on wealth distr…
The dynamics of generalized Lotka-Volterra systems is studied by theoretical techniques and computer simulations. These systems describe the time evolution of the wealth distribution of individuals in a society, as well as of the market values of firms in the stock market. The individual wealths or market values are given by a set of time dependent variables $w_i$, $i=1,...N$. The equations include a stochastic autocatalytic term (representing investments), a drift term (representing social security payments) and a time dependent saturation term (due to the finite size of the economy). The $w_…
Sharpe et al. proposed the idea of having an expected utility maximizer choose a probability distribution for future wealth as an input to her investment problem instead of a utility function. They developed a computer program, called The Distribution Builder, as one way to elicit such a distribution. In a single-period model, they then showed how this desired distribution for terminal wealth can be used to infer the investor's risk preferences. We adapt their idea, namely that a risk-averse investor can choose a desired distribution for future wealth as an alternative input attribute for inve…
