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arXiv2009

Agent-Based Model Approach to Complex Phenomena in Real Economy

An agent-based model for firms' dynamics is developed. The model consists of firm agents with identical characteristic parameters and a bank agent. Dynamics of those agents is described by their balance sheets. Each firm tries to maximize its expected profit with possible risks in market. Infinite growth of a firm directed by the "profit maximization" principle is suppressed by a concept of "going concern". Possibility of bankruptcy of firms is also introduced by incorporating a retardation effect of information on firms' decision. The firms, mutually interacting through the monopolistic bank, become heterogeneous in the course of temporal evolution. Statistical properties of firms' dynamics obtained by simulations based on the model are discussed in light of observations in the real economy.

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arXiv2009

Structure and temporal change of the credit network between banks and large firms in Japan

We present a new approach to understanding credit relationships between commercial banks and quoted firms, and with this approach, examine the temporal change in the structure of the Japanese credit network from 1980 to 2005. At each year, the credit network is regarded as a weighted bipartite graph where edges correspond to the relationships and weights refer to the amounts of loans. Reduction in the supply of credit affects firms as debtor, and failure of a firm influences banks as creditor. To quantify the dependency and influence between banks and firms, we propose a set of scores of banks and firms, which can be calculated by solving an eigenvalue problem determined by the weight of the credit network. We found that a few largest eigenvalues and corresponding eigenvectors are significant by using a null hypothesis of random bipartite graphs, and that the scores can quantitatively describe the stability or fragility of the credit network during the 25 years.

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arXiv2009

Information geometries and Microeconomic Theories

More than thirty years ago, Charnes, Cooper and Schinnar (1976) established an enlightening contact between economic production functions (EPFs) -- a cornerstone of neoclassical economics -- and information theory, showing how a generalization of the Cobb-Douglas production function encodes homogeneous functions. As expected by Charnes \textit{et al.}, the contact turns out to be much broader: we show how information geometry as pioneered by Amari and others underpins static and dynamic descriptions of microeconomic cornerstones. We show that the most popular EPFs are fundamentally grounded in a very weak axiomatization of economic transition costs between inputs. The strength of this characterization is surprising, as it geometrically bonds altogether a wealth of collateral economic notions -- advocating for applications in various economic fields --: among all, it characterizes (i) Marshallian and Hicksian demands and their geometric duality, (ii) Slutsky-type properties for the transformation paths, (iii) Roy-type properties for their elementary variations.

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arXiv2009

Mathematical analysis of Soros's theory of reflexivity

The mathematical model proposed by George Soros for his theory of reflexivity is analyzed under the framework of discrete dynamical systems. We show the importance of the notion of fixed points for explaining the behavior of a reflexive system governed by its cognitive and manipulative functions. The interrelationship between these two functions induces fixed points with different characteristics, which in turn generate various system behaviors including the so-called "boom then bust" phenomenon in Soros's theory.

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arXiv2009

The Reality Game

We introduce an evolutionary game with feedback between perception and reality, which we call the reality game. It is a game of chance in which the probabilities for different objective outcomes (e.g., heads or tails in a coin toss) depend on the amount wagered on those outcomes. By varying the `reality map', which relates the amount wagered to the probability of the outcome, it is possible to move continuously from a purely objective game in which probabilities have no dependence on wagers to a purely subjective game in which probabilities equal the amount wagered. We study self-reinforcing games, in which betting more on an outcome increases its odds, and self-defeating games, in which the opposite is true. This is investigated in and out of equilibrium, with and without rational players, and both numerically and analytically. We introduce a method of measuring the inefficiency of the game, similar to measuring the magnitude of the arbitrage opportunities in a financial market. We prove that convergence to equilibrium is is a power law with an extremely slow rate of convergence: The more subjective the game, the slower the convergence.

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arXiv2010

A Paradigm Shift from Production Function to Production Copula: Statistical Description of Production Activity of Firms

Heterogeneity of economic agents is emphasized in a new trend of macroeconomics. Accordingly the new emerging discipline requires one to replace the production function, one of key ideas in the conventional economics, by an alternative which can take an explicit account of distribution of firms' production activities. In this paper we propose a new idea referred to as production copula; a copula is an analytic means for modeling dependence among variables. Such a production copula predicts value added yielded by firms with given capital and labor in a probabilistic way. It is thereby in sharp contrast to the production function where the output of firms is completely deterministic. We demonstrate empirical construction of a production copula using financial data of listed firms in Japan. Analysis of the data shows that there are significant correlations among their capital, labor and value added and confirms that the values added are too widely scattered to be represented by a production function. We employ four models for the production copula, that is, trivariate versions of Frank, Gumbel and survival Clayton and non-exchangeable trivariate Gumbel; the last one works best.

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arXiv2009

Time and symmetry in models of economic markets

These notes discuss several topics in neoclassical economics and alternatives, with an aim of reviewing fundamental issues in modeling economic markets. I start with a brief, non-rigorous summary of the basic Arrow-Debreu model of general equilibrium, as well as its extensions to include time and contingency. I then argue that symmetries due to similarly endowed individuals and similar products are generically broken by the constraints of scarcity, leading to the existence of multiple equilibria. This is followed by an evaluation of the strengths and weaknesses of the model generally. Several of the weaknesses are concerned with the treatments of time and contingency. To address these we discuss a class of agent based models. Another set of issues has to do with the fundamental meaning of prices and the related question of what the observables of a non-equilibrium, dynamic model of an economic market should be. We argue that these issues are addressed by formulating economics in the language of a gauge theory, as proposed originally by Malaney and Weinstein. We review some of their work and provide a sketch of how gauge invariance can be incorporated into the formulation of agent based models.

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arXiv2009

A dynamic nonlinear model for saturation in industrial growth

A general nonlinear logistic equation has been proposed to model long-time saturation in industrial growth. An integral solution of this equation has been derived for any arbitrary degree of nonlinearity. A time scale for the onset of nonlinear saturation in industrial growth can be estimated from an equipartition condition between nonlinearity and purely exponential growth. Precise predictions can be made about the limiting values of the annual revenue and the human resource content that an industrial organisation may attain. These variables have also been modelled to set up an autonomous first-order dynamical system, whose equilibrium condition forms a stable node (an attractor state) in a related phase portrait. The theoretical model has received close support from all relevant data pertaining to the well-known global company, IBM.

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arXiv2009

Financial Atoms and Molecules

Atoms and molecules are important conceptual entities we invented to understand the physical world around us. The key to their usefulness lies in the organization of nuclear and electronic degrees of freedom into a single dynamical variable whose time evolution we can better imagine. The use of such effective variables in place of the true microscopic variables is possible because of the separation between nuclear/electronic and atomic/molecular time scales. Where separation of time scales occurs, identification of analogous objects in financial markets can help advance our understanding of their dynamics. To detect separated time scales and identify their associated effective degrees of freedom in financial markets, we devised a two-stage statistical clustering scheme to analyze the price movements of stocks in several equity markets. Through this two-time-scale clustering analysis, we discovered a hierarchy of levels of self-organization in real financial markets. We call these statistically robust self-organized dynamical structures financial atoms, financial molecules, and financial supermolecules. In general, the detailed compositions of these dynamical structures cannot be deduced based on raw financial intuition alone, and must be explained in terms of the underlying portfolios, and investment strategies of market players. More interestingly, we find that major market events such as the Chinese Correction and the Subprime Crisis leave many tell-tale signs within the correlational structures of financial molecules.

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arXiv2009

Price Impact

We define what "Price Impact" means, and how it is measured and modelled in the recent literature. Although this notion seems to convey the idea of a forceful and intuitive mechanism, we discuss why things might not be that simple. Empirical studies show that while the correlation between signed order flow and price changes is strong, the impact of trades on prices is neither linear in volume nor permanent. Impact allows private information to be reflected in prices, but by the same token, random fluctuations in order flow must also contribute to the volatility of markets.

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arXiv2009

The Transfer Pricing Problem with Non-Linearities

A number of approaches to solving the well-known transfer pricing problem are known. However, few models satisfactorily resolve the core problem of allowing both the source and receiving divisions to earn a profit on transfers during a period in such a way that sub-optimal output levels are avoided. In 1969, Samuel proposed to use a transfer price schedule instead of just a single transfer price. An essential improvement of Samuels' model was given by Tomkins (1990) in his pragmatic-analytical transfer pricing approach, which is a combination of a single cost-plus transfer price and the pragmatic process of negotiation. This fundamental approach was developed under the assumption that the net average revenue curve for the final product is linear. In this paper, Tomkins' pragmatic-analytical model is further developed for non-linear net average revenue curves. In particular, typical quadratic functions are considered and corresponding transfer price schedules are determined. A similar technique can be used for the transfer pricing problem with any net average revenue curve.

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arXiv2009

Behavioural and Dynamical Scenarios for Contingent Claims Valuation in Incomplete Markets

We study the problem of determination of asset prices in an incomplete market proposing three different but related scenarios. One scenario uses a market game approach whereas the other two are based on risk sharing or regret minimizing considerations. Dynamical schemes modeling the convergence of the buyer's and of the seller's prices to a unique price are proposed.

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arXiv2009

Unemployment and inflation in Western Europe: solution by the boundary element method

Using an analog of the boundary element method in engineering and science, we analyze and model unemployment rate in Austria, Italy, the Netherlands, Sweden, Switzerland, and the United States as a function of inflation and the change in labor force. Originally, the model linking unemployment to inflation and labor force was developed and successfully tested for Austria, Canada, France, Germany, Japan, and the United States. Autoregressive properties of neither of these variables are used to predict their evolution. In this sense, the model is a self-consistent and completely deterministic one without any stochastic component (external shocks) except that associated with measurement errors and changes in measurement units. Nevertheless, the model explains between 65% and 95% of the variability in unemployment and inflation. For Italy, the rate of unemployment is predicted at a time horizon of nine years with pseudo out-of-sample root-mean-square forecasting error of 0.55% for the period between 1973 and 2006. One can expect that the u nemployment will be growing since 2008 and will reach 11.4% near 2012. After 2012, unemployment in Italy will start to descend.

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arXiv2009

Does economics need a scientific revolution?

Economics does not need a scientific revolution. Economics needs accurate measurements according to high standards of natural sciences and meticulous work on revealing empirical relationships between measured variables.

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arXiv2009

The Problem of Modeling of Economic Dynamics

The correctness of Harrods model in the differential form is studied. The inadequacy of exponential growth of economy is shown; an alternative result is obtained. By example of Phillips model, an approach to correction of macroeconomic models (in terms of initial prerequisites) is generalized. A methodology based on balance relations for modelling of economic dynamics, including obtaining forecast estimates, is developed. The problems thus considered are reduced to the solution of Volterra and Fredholm integral equations of the second kind.

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arXiv2009

The (unfortunate) complexity of the economy

This article is a follow-up of a short essay that appeared in Nature 455, 1181 (2008) [arXiv:0810.5306]. It has become increasingly clear that the erratic dynamics of markets is mostly endogenous and not due to the rational processing of exogenous news. I elaborate on the idea that spin-glass type of problems, where the combination of competition and heterogeneities generically leads to long epochs of statis interrupted by crises and hyper-sensitivity to small changes of the environment, could be metaphors for the complexity of economic systems. I argue that the most valuable contribution of physics to economics might end up being of methodological nature, and that simple models from physics and agent based numerical simulations, although highly stylized, are more realistic than the traditional models of economics that assume rational agents with infinite foresight and infinite computing abilities.

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arXiv2009

Risk Measures in Quantitative Finance

This paper was presented and written for two seminars: a national UK University Risk Conference and a Risk Management industry workshop. The target audience is therefore a cross section of Academics and industry professionals. The current ongoing global credit crunch has highlighted the importance of risk measurement in Finance to companies and regulators alike. Despite risk measurement's central importance to risk management, few papers exist reviewing them or following their evolution from its foremost beginnings up to the present day risk measures. This paper reviews the most important portfolio risk measures in Financial Mathematics, from Bernoulli (1738) to Markowitz's Portfolio Theory, to the presently preferred risk measures such as CVaR (conditional Value at Risk). We provide a chronological review of the risk measures and survey less commonly known risk measures e.g. Treynor ratio.

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