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arXiv2008

Mathematical analysis of long tail economy using stochastic ranking processes

We present a new method of estimating the distribution of sales rates of, e.g., book titles at an online bookstore, from the time evolution of ranking data found at websites of the store. The method is based on new mathematical results on an infinite particle limit of the stochastic ranking process, and is suitable for quantitative studies of the long tail structure of online retails. We give an example of a fit to the actual data obtained from Amazon.co.jp, which gives the Pareto slope parameter of the distribution of sales rates of the book titles in the store.

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arXiv2008

A note on wealth in a volatile economy

I show that if the capital accumulation dynamics is stochastic a new term, in addition to that given by accounting prices, has to be introduced in order to derive a correct estimate of the genuine wealth of an economy. In a simple model with multiplicative accumulation dynamics I show that: 1) the value function is always a decreasing function of volatility 2) the accounting prices are affected by volatility 3) the new term always gives a negative contribution to wealth changes. I discuss results for models with constant elasticity utility functions. When the elasticity of marginal utility is larger than one, accounting prices increase with volatility whereas when it is less than one accounting prices decrease with volatility. These conclusions are not altered when adopting optimal saving rates.

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arXiv2008

The Problem of Modelling of Economic Dynamics in Differential Form

Traditional models of macroeconomic dynamics are fundamentally incorrect. The reason lies in a misunderstanding of peculiarities of the analysis of infinitesimal quantities. However, even those types of solutions that are envisaged by the above-mentioned models are nonrepresentative in the sense of the reflection of realities. It became obvious that the techniques of the theory of linear differential equations were insufficient here. Accordingly, the scientists' attention switched to the theory of nonlinear differential equations. At the same time, balance and, accordingly, the model with matrix properties are objectively inherent in the economic system. For the reduction of this model to a differential form, there exist rather elementary means that proved to be unclaimed. Macroeconomic rhetoric - the power of the accelerator, a lag on the part of demand, etc. - accompanied by the use of a lot of abstract coefficients prevailed. However, there is no organic interrelation between matrix and nonlinear differential equations. On the contrary, it can be said that linear theory of integral equations originated in matrix analysis. The Fredholm linear integral equation of the second kind with a parameter-dependent kernel proves to be rather representative with regard to the class of possible solutions. It seems that it can be used for the description of any zigzags of the economy. The price one has to pay for this is the nontriviality of existing theory.

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arXiv2008

Productivity Dispersion: Facts, Theory, and Implications

We study productivity dispersions across workers, firms and industrial sectors. Empirical study of the Japanese data shows that they all obey the Pareto law, and also that the Pareto index decreases with the level of aggregation. In order to explain these two stylized facts, we propose a theoretical framework built upon the basic principle of statistical physics. In this framework, we employ the concept of superstatistics which accommodates fluctuations of aggregate demand.

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arXiv2008

Convergence and cluster structures in EU area according to fluctuations in macroeconomic indices

The cluster analysis methods are used in order to perform a comparative study of 15 EU countries in relation with the fluctuations of some basic macroeconomic indicators. The statistical distances between countries are calculated for various moving time windows, and the time variation of the mean statistical distance is investigated. The decreasing of the mean statistical distance between EU countries is reflected in the correlated fluctuations of the basic ME indicators: GDP, GDP/capita, Consumption and Investments. This empirical evidence can be seen as an economic aspect of globalization. The Moving Average Minimal Length Path (MAMLP) algorithm allows to search for a cluster-like structures derived both from the hierarchical organization of countries and from their relative movement inside the hierarchy. It is found that the strongly correlated countries with respect to GDP fluctuations can be partitioned into stable clusters. Some of the highly correlated countries, with respect to GDP fluctuations, display strong correlations also in the Final Consumption Expenditure, while others are strongly correlated in Gross Capital Formation. On the other hand, one notices the similitude of the classifications regarding GDP and Net Exports fluctuations as concerns the squared sum of the correlation coefficients (so called country sensitivity). The final structure proves to be robust against the constant size time window moving over the scanned time interval. The policy implications of the above empirical results concern the economic clusters arising in the presence of Marshallian externalities and the relationships between trade barriers, R&D incentives and growth that must be accounted in elaborating a cluster-promotion policy.

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arXiv2008

Anomalous Returns in a Neural Network Equity-Ranking Predictor

Using an artificial neural network (ANN), a fixed universe of approximately 1500 equities from the Value Line index are rank-ordered by their predicted price changes over the next quarter. Inputs to the network consist only of the ten prior quarterly percentage changes in price and in earnings for each equity (by quarter, not accumulated), converted to a relative rank scaled around zero. Thirty simulated portfolios are constructed respectively of the 10, 20,..., and 100 top ranking equities (long portfolios), the 10, 20,..., 100 bottom ranking equities (short portfolios) and their hedged sets (long-short portfolios). In a 29-quarter simulation from the end of the third quarter of 1994 through the fourth quarter of 2001 that duplicates real-world trading of the same method employed during 2002, all portfolios are held fixed for one quarter. Results are compared to the S&P 500, the Value Line universe itself, trading the universe of equities using the proprietary ``Value Line Ranking System'' (to which this method is in some ways similar), and to a Martingale method of ranking the same equities. The cumulative returns generated by the network predictor significantly exceed those generated by the S&P 500, the overall universe, the Martingale and Value Line prediction methods and are not eroded by trading costs. The ANN shows significantly positive Jensen's alpha, i.e., anomalous risk-adjusted expected return. A time series of its global performance shows a clear antipersistence. However, its performance is significantly better than a simple one-step Martingale predictor, than the Value Line system itself and than a simple buy and hold strategy, even when transaction costs are accounted for.

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arXiv2010

How to grow a bubble: A model of myopic adapting agents

We present a simple agent-based model to study the development of a bubble and the consequential crash and investigate how their proximate triggering factor might relate to their fundamental mechanism, and vice versa. Our agents invest according to their opinion on future price movements, which is based on three sources of information, (i) public information, i.e. news, (ii) information from their "friendship" network and (iii) private information. Our bounded rational agents continuously adapt their trading strategy to the current market regime by weighting each of these sources of information in their trading decision according to its recent predicting performance. We find that bubbles originate from a random lucky streak of positive news, which, due to a feedback mechanism of these news on the agents' strategies develop into a transient collective herding regime. After this self-amplified exuberance, the price has reached an unsustainable high value, being corrected by a crash, which brings the price even below its fundamental value. These ingredients provide a simple mechanism for the excess volatility documented in financial markets. Paradoxically, it is the attempt for investors to adapt to the current market regime which leads to a dramatic amplification of the price volatility. A positive feedback loop is created by the two dominating mechanisms (adaptation and imitation) which, by reinforcing each other, result in bubbles and crashes. The model offers a simple reconciliation of the two opposite (herding versus fundamental) proposals for the origin of crashes within a single framework and justifies the existence of two populations in the distribution of returns, exemplifying the concept that crashes are qualitatively different from the rest of the price moves.

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arXiv2008

The Question of Relaxation in the Wealth Exchange Models

We look at the meaning of 'relaxation' in the wealth exchange models that are recently proposed in Econophysics to interpret the wealth distributions. To quantify and characterise the process of relaxation, we define an appropriate quantity and evaluate that numerically for the systems of many agents. Also, the numerical results have been supported heuristically by constructing a simple differential equation.

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arXiv2008

Global recessions as a cascade phenomenon with heterogenous, interacting agents

I examine global recessions as a cascade phenomenon. In other words, how recessions arising in one or more countries might percolate across a network of connected economies. A heterogeneous agent based model is set up in which the agents are Western economies. A country has a probability of entering a recession in any given year and one of emerging from it the next. In addition, the agents have a unique threshold propensity to import a recession from the agents with which they have the strongest connections. They are connected on a small world topology, and an agent's neighbours at any time are either in (state 1) or out (state 0) of recession. If the weighted sum exceeds the threshold, the agent goes into recession. Annual real GDP growth for 17 Western countries 1871-2006 is used as the data set. The distribution of the number of countries in recession in any given year is exponential, as is the duration of recessions within individual countries. The model is calibrated against these two facts, plus the 'wait time' between recessions. It is able to replicate them successfully. The network structure is essential for the agents to replicate the stylised facts. The country-specific probabilities of entering and emerging from recession by themselves give results very different to the actual data.

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arXiv2010

Cooperation Evolution in Random Multiplicative Environments

Most real life systems have a random component: the multitude of endogenous and exogenous factors influencing them result in stochastic fluctuations of the parameters determining their dynamics. These empirical systems are in many cases subject to noise of multiplicative nature. The special properties of multiplicative noise as opposed to additive noise have been noticed for a long while. Even though apparently and formally the difference between free additive vs. multiplicative random walks consists in just a move from normal to log-normal distributions, in practice the implications are much more far reaching. While in an additive context the emergence and survival of cooperation requires special conditions (especially some level of reward, punishment, reciprocity), we find that in the multiplicative random context the emergence of cooperation is much more natural and effective. We study the various implications of this observation and its applications in various contexts.

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arXiv2011

Agent-based model of competition in a social structure

Indirect competition emerged from the complex organization of human societies, and knowledge of the existing network topology may aid in developing effective strategies for success. Here, we propose an agent-based model of competition with systems co-existing in a `small-world' social network. We show that within the range of parameter values obtained from the model and empirical data, the network evolution is highly dependent on $k$, the local parameter describing the density of neighbors in the network. The model applied to language death and competition of telecommunication companies show strong correspondence with empirical data.

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arXiv2008

Modelling interest rates by correlated multi-factor CIR-like processes

We investigate the joint description of the interest-rate term stuctures of Italy and an AAA-rated European country by mean of a --here proposed-- correlated CIR-like bivariate model where one of the state variables is interpreted as a benchmark risk-free rate and the other as a credit spread. The model is constructed by requiring the strict positivity of interest rates and the asymptotic decoupling of the joint distribution of the two state variables on a long time horizon. The second condition is met by imposing the reversibility of the process with respect to a product measure, the first is then implemented by using the tools of potential theory. It turns out that these conditions select a class of non-affine models, out of which we choose one that is quadratic in the two state variables both in the drift and diffusion matrix. We perform a numerical analysis of the model by investigating a cross section of the term structures comparing the results with those obtained with an uncoupled bivariate CIR model.

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arXiv2008

Taxes in a simple wealth distribution model by inelastically scattering particles

In this work we use an inelastic scattering process of particles to propose a model able to reproduce the salient features of the wealth distribution in an economy by including taxes to each trading process and redistributing that collected among the population according to a given criterion. Additionally, we show that different optimal levels of taxes may exist depending on the redistribution criterion.

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arXiv2008

Theory of Zipf's Law and of General Power Law Distributions with Gibrat's law of Proportional Growth

We summarize a book under publication with his title written by the three present authors, on the theory of Zipf's law, and more generally of power laws, driven by the mechanism of proportional growth. The preprint is available upon request from the authors. For clarity, consistence of language and conciseness, we discuss the origin and conditions of the validity of Zipf's law using the terminology of firms' asset values. We use firms at the entities whose size distributions are to be explained. It should be noted, however, that most of the relations discussed in this book, especially the intimate connection between Zipf's and Gilbrat's laws, underlie Zipf's law in diverse scientific areas. The same models and variations thereof can be straightforwardly applied to any of the other domains of application.

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arXiv2008

A housing-demographic multi-layered nonlinear model to test regulation strategies

We propose a novel multi-layered nonlinear model that is able to capture and predict the housing-demographic dynamics of the real-state market by simulating the transitions of owners among price-based house layers. This model allows us to determine which parameters are most effective to smoothen the severity of a potential market crisis. The International Monetary Fund (IMF) has issued severe warnings about the current real-state bubble in the United States, the United Kingdom, Ireland, the Netherlands, Australia and Spain in the last years. Madrid (Spain), in particular, is an extreme case of this bubble. It is, therefore, an excellent test case to analyze housing dynamics in the context of the empirical data provided by the Spanish National Institute of Statistics and other sources of data. The model is able to predict the mean house occupancy, and shows that i) the house market conditions in Madrid are unstable but not critical; and ii) the regulation of the construction rate is more effective than interest rate changes. Our results indicate that to accommodate the construction rate to the total population of first-time buyers is the most effective way to maintain the system near equilibrium conditions. In addition, we show that to raise interest rates will heavily affect the poorest housing bands of the population while the middle class layers remain nearly unaffected.

nlin.CG↗

arXiv2008

Hedging and production decisions under uncertainty: A survey

This paper synthesizes and analyzes some important current and recent contributions to the theory of the firm under uncertainty. In so doing, it examines the production and hedging decisions of the competitive firm under a single source and multiple sources of uncertainty.

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arXiv2008

Trust! Why it Has Been Lost and How to Regain It

This essay suggests that a proper assessment of the presently unfolding financial crisis, and its cure, requires going back at least to the late 1990s, accounting for the cumulative effect of the ITC, real-estate and financial derivative bubbles. We focus on the deep loss of trust, not only in Wall Street, but more importantly in Main Street, and how to recover it on the short and long terms. A multi-disciplinary approach is needed to deal with the nonlinear complex systems of the present world, in order to develop a culture of fairness, and of upside opportunities associated with a risky world.

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arXiv2008

Evolution of the personal income distribution in the USA: High incomes

The personal income distribution (PID) above the Pareto threshold is studied and modeled. A microeconomic model is proposed to simulate the PID and its evolution below and above the Pareto income threshold. The model balances processes of income production and dissipation for any person above 15 years of age. The model accurately predicts the observed dependence of the number of people reaching the Pareto threshold on work experience and the functional dependence of the relationship on the per capita real GDP growth for the period from 1994 to 2002. Predictions of the income distribution depending on age are given for past and future. In future, relatively less rich people are observed in the younger age groups and the peak of the relative number shifts to older ages with time. The effect of the power law distribution extending itself to very high incomes is speculated to be the cause of low performance of socialist countries.

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