Statistical mechanics of money: how saving propensity affects its distribution

We consider a simple model of a closed economic system where the total money is conserved and the number of economic agents is fixed. Analogous to statistical systems in equilibrium, money and the average money per economic agent are equivalent to energy and temperature, respectively. We investigate the effect of the saving propensity of the agents on the stationary or equilibrium probability distribution of money. When the agents do not save, the equilibrium money distribution becomes the usual Gibb's distribution, characteristic of non-interacting agents. However with saving, even for individual self-interest, the dynamics becomes cooperative and the resulting asymmetric Gaussian-like stationary distribution acquires global ordering properties. Intriguing singularities are observed in the stationary money distribution in the market, as functions of the marginal saving propensity of the agents. Received 2 May 2000     

Revealing of the simplest mechanisms of a structure formation

In this paper, results of investigations of the simplest mechanisms of a structure formation are presented. In frameworks of the suggested model the main attention was focused on such characteristics as wiring of the system, clusters formation, dynamics of the wiring. The idea to take into account an influence of the environment factor is employed in the proposed model. Investigations of systems with such principle of a structure formation reveal that the system's dynamics has typical features of self-organized criticality phenomenon. In the avalanche-like processes, which occur in the wiring dynamics, a power law was found with the index close to 1.4. It is independent on the environment factor (which in a sense can be considered as system parameter). The system wiring is approximated pretty well by the Gaussian distribution. The size of the system does not play any role in the dynamics of the model. Received 10 March 1999 and Received in final form 24 May 1999     

Clustering of volatility as a multiscale phenomenon

The dynamics of prices in financial markets has been studied intensively both experimentally (data analysis) and theoretically (models). Nevertheless, a complete stochastic characterization of volatility is still lacking. What is well known is that absolute returns have memory on a long time range, this phenomenon is known as clustering of volatility. In this paper we show that volatility correlations are power-laws with a non-unique scaling exponent. This kind of multiscale phenomenology has some analogies with fully developed turbulence and disordered systems and it is now pointed out for financial series. Starting from historical returns series, we have also derived the volatility distribution, and the results are in agreement with a log-normal shape. In our study, we consider the New York Stock Exchange (NYSE), daily composite index closes (January 1966 to June 1998) and the US Dollar/Deutsche Mark (USD-DM) noon buying rates certified by the Federal Reserve Bank of New York (October 1989 to September 1998). Received 1 February 2000     

Gambling on the Budapest stock exchange

The statistical properties of the total yield are analyzed for an assembly of gamblers in an erratic period on the Budapest stock exchange. Random trading results in a log-normal limit distribution of a surprisingly large width, while the simplest profit realizing strategy narrows down the peak around a positive average value. The effect of transaction costs, the statistics of extremes, and patterns of successful trading are also investigated. In spite of the very simple approach, we present strong indications that large trading activity (e.g. day trading) is a rather risky way of capital investment. A comparison with the yield distribution of 32 public investment funds in the given period does not reflect the presence of a sophisticated investment strategy in the background. Received 5 May 2000     

The Nasdaq crash of April 2000: Yet another example of log-periodicity in a speculative bubble ending in a crash

The Nasdaq Composite fell another % on Friday the 14'th of April 2000 signaling the end of a remarkable speculative high-tech bubble starting in spring 1997. The closing of the Nasdaq Composite at 3321 corresponds to a total loss of over 35% since its all-time high of 5133 on the 10'th of March 2000. Similarities to the speculative bubble preceding the infamous crash of October 1929 are quite striking: the belief in what was coined a “New Economy” both in 1929 and presently made share-prices of companies with three digits price-earning ratios soar. Furthermore, we show that the largest draw downs of the Nasdaq are outliers with a confidence level better than 99% and that these two speculative bubbles, as well as others, both nicely fit into the quantitative framework proposed by the authors in a series of recent papers. Received 3 May 2000     

Self-organized model for information spread in financial markets

A self-organized model with social percolation process is proposed to describe the propagations of information for different trading ways across a social system and the automatic formation of various groups within market traders. Based on the market structure of this model, some stylized observations of real market can be reproduced, including the slow decay of volatility correlations, and the fat tail distribution of price returns which is found to cross over to an exponential-type asymptotic decay in different dimensional systems. Received 15 March 2000     

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.     

A stability criterion for financial markets

Using the theory of random cluster models, we give a stability criterion for financial markets with random communications between agents. Received 25 September 1999 and Received in final form 2 October 1999     

A kinetic approach to some quasi-linear laws of macroeconomics

Some previous works have presented the data on wealth and income distributions in developed countries and have found that the great majority of population is described by an exponential distribution, which results in idea that the kinetic approach could be adequate to describe this empirical evidence. The aim of our paper is to extend this framework by developing a systematic kinetic approach of the socio-economic systems and to explain how linear laws, modelling correlations between macroeconomic variables, may arise in this context. Firstly we construct the Boltzmann kinetic equation for an idealised system composed by many individuals (workers, officers, business men, etc.), each of them getting a certain income and spending money for their needs. To each individual a certain time variable amount of money is associated - this meaning him/her phase space coordinate. In this way the exponential distribution of money in a closed economy is explicitly found. The extension of this result, including states near the equilibrium, give us the possibility to take into account the regular increase of the total amount of money, according to the modern economic theories. The Kubo-Green-Onsager linear response theory leads us to a set of linear equations between some macroeconomic variables. Finally, the validity of such laws is discussed in relation with the time reversal symmetry and is tested empirically using some macroeconomic time series. Received 25 February 2002 / Received in final form 11 July 2002 Published online 19 November 2002     

How popular is your paper? An empirical study of the citation distribution

Numerical data for the distribution of citations are examined for: (i) papers published in 1981 in journals which are catalogued by the Institute for Scientific Information (783,339 papers) and (ii) 20 years of publications in Physical Review D, vols. 11-50 (24,296 papers). A Zipf plot of the number of citations to a given paper versus its citation rank appears to be consistent with a power-law dependence for leading rank papers, with exponent close to -1/2. This, in turn, suggests that the number of papers with x citations, N(x), has a large-x power law decay , with . Received: 12 May 1998 / Accepted: 12 May 1998     

On the physical relevance of max- and log-max-selfsimilar distributions

This work emphasizes the special role played by max-semistable and log-max-semistable distributions as relevant statistical models of various observable and “internal” variables in Physics. Some of their remarkable properties (chiefly self-similarity) are displayed in some detail. One of their characteristic features is a log-periodic variation of the scale parameter which appears in the stable extreme value distributions. Received 29 November 1999 and Received in final form 24 March 2000     

Economic returns of research: the Pareto law and its implications

At what level should government or companies support research? This complex multi-faceted question encompasses such qualitative bonus as satisfying natural human curiosity, the quest for knowledge and the impact on education and culture, but one of its most scrutinized component reduces to the assessment of economic performance and wealth creation derived from research. Many studies report evidences of positive economic benefits derived from basic research [#!Martin!#,#!NAS!#]. In certain areas such as biotechnology, semi-conductor physics, optical communications [#!Ehrenreich!#], the impact of basic research is direct while, in other disciplines, the path from discovery to applications is full of surprises. As a consequence, there are persistent uncertainties in the quantification of the exact economic returns of public expenditure on basic research. This gives little help to policy makers trying to determine what should be the level of funding. Here, we suggest that these uncertainties have a fundamental origin to be found in the interplay between the intrinsic “fat tail” power law nature of the distribution of economic returns, characterized by a mathematically diverging variance, and the stochastic character of discovery rates. In the regime where the cumulative economic wealth derived from research is expected to exhibit a long-term positive trend, we show that strong fluctuations blur out significantly the short-time scales: a few major unpredictable innovations may provide a finite fraction of the total creation of wealth. In such a scenario, any attempt to assess the economic impact of research over a finite time horizon encompassing only a small number of major discoveries is bound to be highly unreliable. New tools, developed in the theory of self-similar and complex systems [#!Dubrulleetal!#] to tackle similar extreme fluctuations in Nature [#!Mandelbrot!#], can be adapted to measure the economic benefits of research, which is intimately associated to this large variability. Received 26 October 1998 and Received in final form 27 October 1998     

Are citations of scientific papers a case of nonextensivity?

The distribution N(x) of citations of scientific papers has recently been illustrated (on ISI and PRE data sets) and analyzed by Redner (Eur. Phys. J. B 4, 131 (1998)). To fit the data, a stretched exponential () has been used with only partial success. The success is not complete because the data exhibit, for large citation count x, a power law (roughly for the ISI data), which, clearly, the stretched exponential does not reproduce. This fact is then attributed to a possibly different nature of rarely cited and largely cited papers. We show here that, within a nonextensive thermostatistical formalism, the same data can be quite satisfactorily fitted with a single curve (namely, [0pt] for the available values of x. This is consistent with the connection recently established by Denisov (Phys. Lett. A 235, 447 (1997)) between this nonextensive formalism and the Zipf-Mandelbrot law. What the present analysis ultimately suggests is that, in contrast to Redner's conclusion, the phenomenon might essentially be one and the same along the entire range of the citation number x. Received 13 April 1999     

Stretched exponential distributions in nature and economy: “fat tails” with characteristic scales

To account quantitatively for many reported “natural” fat tail distributions in Nature and Economy, we propose the stretched exponential family as a complement to the often used power law distributions. It has many advantages, among which to be economical with only two adjustable parameters with clear physical interpretation. Furthermore, it derives from a simple and generic mechanism in terms of multiplicative processes. We show that stretched exponentials describe very well the distributions of radio and light emissions from galaxies, of US GOM OCS oilfield reserve sizes, of World, US and French agglomeration sizes, of country population sizes, of daily Forex US-Mark and Franc-Mark price variations, of Vostok (near the south pole) temperature variations over the last 400 000 years, of the Raup-Sepkoski's kill curve and of citations of the most cited physicists in the world. We also discuss its potential for the distribution of earthquake sizes and fault displacements. We suggest physical interpretations of the parameters and provide a short toolkit of the statistical properties of the stretched exponentials. We also provide a comparison with other distributions, such as the shifted linear fractal, the log-normal and the recently introduced parabolic fractal distributions. Received: 20 January 1998 / Received in final form: 27 January 1998 / Accepted: 6 February 1998     

Evidence for the exponential distribution of income in the USA

Using tax and census data, we demonstrate that the distribution of individual income in the USA is exponential. Our calculated Lorenz curve without fitting parameters and Gini coefficient 1/2 agree well with the data. From the individual income distribution, we derive the distribution function of income for families with two earners and show that it also agrees well with the data. The family data for the period 1947-1994 fit the Lorenz curve and Gini coefficient 3/8 = 0.375 calculated for two-earners families. Received 21 August 2000     

Complexity of two-dimensional patterns

To describe quantitatively the complexity of two-dimensional patterns we introduce a complexity measure based on a mean information gain. Two types of patterns are studied: geometric ornaments and patterns arising in random sequential adsorption of discs on a plane (RSA). For the geometric ornaments analytical expressions for entropy and complexity measures are presented, while for the RSA patterns these are calculated numerically. We compare the information-gain complexity measure with some alternative measures and show advantages of the former one, as applied to two-dimensional structures. Namely, this does not require knowledge of the “maximal” entropy of the pattern, and at the same time sensitively accounts for the inherent correlations in the system. Received 12 November 1999     

How the financial crash of October 1997 could have been predicted

From the analysis of (closing value) stock market index like the Dow Jones Industrial average and the S&P500 it is possible to observe the precursor of a so-called crash. This is shown on the Oct. 1987 and Oct. 1997 cases. The data analysis indicates that the index divergence has followed twice a “universal” behavior, i.e. a logarithmic dependence, superposed on a well defined oscillation pattern. The prediction of the crash date is remarkable and can be done two months in advance. In the spirit of phase transition phenomena, the economic index is said to be analogous to a signal signature found in a two dimensional fluid of vortices. Received: 23 March 1998 / Revised and Accepted: 23 April 1998     

On the estimation of the power-law exponent in the mean-field Bouchaud-Mézard model

In this article, our primary objective is to develop an estimator of the power-law exponent based on the observable individual wealth in the mean-field Bouchaud-Mézard model. As a result, a simple and strongly consistent estimator of the power-law exponent in the mean-field Bouchaud-Mézard model has been established and performs well on simulated data.     

On some natural and model 2D bimodal random cellular structures

The topological and metric properties of a few natural 2D random cellular structures, namely an armadillo shell structure and young soap froths, which are formed from two classes of cells, large and small, have been characterized. The topological properties of a model generated from a Kagome tiling, which mimics such random binary structures, have also been exactly calculated. The distribution of the number of cell sides is bimodal for the structures investigated. In contrast to the classical Aboav-Weaire law for homogeneous 2D random cellular structures, nm(n), the mean total number of edges of neighbouring cells of cells with n sides does not vary linearly with n. Only the nm(i, n) (i=1,2) determined separately for every class of cells are linear in n for all investigated structures. Topological properties and correlations between metric and topological properties are finally compared with the predictions of various literature models. Received: 24 December 1997 / Revised: 7 April 1998 / Accepted: 20 April 1998