Anirban Chakraborti

Statistical mechanics of money: how saving propensity affects its distribution

Abstract: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 Gibbs 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.

Dynamic asset trees and Black Monday

The minimum spanning tree, based on the concept of ultrametricity, is constructed from the correlation matrix of stock returns. The dynamics of this asset tree can be characterised by its normalised length and the mean occupation layer, as measured from an appropriately chosen centre called the ‘central node’. We show how the tree length shrinks during a stock market crisis, Black Monday in this case, and how a strong reconfiguration takes place, resulting in topological shrinking of the tree.

Econophysics review: II. Agent-based models

This article is the second part of a review of recent empirical and theoretical developments usually grouped under the heading Econophysics. In the first part, we reviewed the statistical properties of financial time series, the statistics exhibited in order books and discussed some studies of correlations of asset prices and returns. This second part deals with models in Econophysics from the point of view of agent-based modeling. Of the large number of multi-agent-based models, we have identified three representative areas. First, using previous work originally presented in the fields of behavioral finance and market microstructure theory, econophysicists have developed agent-based models of order-driven markets that we discuss extensively here. Second, kinetic theory models designed to explain certain empirical facts concerning wealth distribution are reviewed. Third, we briefly summarize game theory models by reviewing the now classic minority game and related problems.

Statistical model with a standard Gamma distribution

We study a statistical model consisting of N basic units which interact with each other by exchanging a physical entity, according to a given microscopic random law, depending on a parameter lambda. We focus on the equilibrium or stationary distribution of the entity exchanged and verify through numerical fitting of the simulation data that the final form of the equilibrium distribution is that of a standard Gamma distribution. The model can be interpreted as a simple closed economy in which economic agents trade money and a saving criterion is fixed by the saving propensity lambda. Alternatively, from the nature of the equilibrium distribution, we show that the model can also be interpreted as a perfect gas at an effective temperature T(lambda), where particles exchange energy in a space with an effective dimension D(lambda).

DISTRIBUTIONS OF MONEY IN MODEL MARKETS OF ECONOMY

We study the distributions of money in a simple closed economic system for different types of monetary transactions. We know that for arbitrary and random sharing with locally conserving money transactions, the money distribution goes to the Gibbs distribution of statistical mechanics. We then consider the effects of savings, etc. and see how the distribution changes. We also propose a new model where the agents invest equal amounts of money in each transaction. We find that for short time-period, the money distribution obeys a power-law with an exponent very close to unity, and has an exponential tail; after a very long time, this distribution collapses and the entire amount of money goes to a tiny fraction of the population.

Statistical mechanics of competitive resource allocation using agent-based models

Demand outstrips available resources in most situations, which gives rise to competition, interaction and learning. In this article, we review a broad spectrum of multi-agent models of competition (El Farol Bar problem, Minority Game, Kolkata Paise Restaurant problem, Stable marriage problem, Parking space problem and others) and the methods used to understand them analytically. We emphasize the power of concepts and tools from statistical mechanics to understand and explain fully collective phenomena such as phase transitions and long memory, and the mapping between agent heterogeneity and physical disorder. As these methods can be applied to any large-scale model of competitive resource allocation made up of heterogeneous adaptive agent with non-linear interaction, they provide a prospective unifying paradigm for many scientific disciplines. (This abstract was borrowed from another version of this item.)

Influence of saving propensity on the power-law tail of the wealth distribution

Some general features of statistical multi-agent economic models are reviewed, with particular attention to the dependence of the equilibrium wealth distribution on the agents’ saving propensities. It is shown that in a finite system of agents with a continuous saving propensity distribution a power-law tail with Pareto exponent α=1 can appear also when agents do not have saving propensities distributed over the whole interval between zero and one. Rather, a power-law can be observed in a finite interval of wealth, whose lower and upper ends are shown to be determined by the lower and upper cutoffs, respectively, of the saving propensity distribution. It is pointed out that a cutoff of the power-law tail can arise also through a different mechanism, when the number of agents is small enough. Numerical simulations have been carried out by implementing a procedure for assigning saving propensities homogeneously, which results in a smoother wealth distributions and correspondingly wider power-law intervals than other procedures based on random algorithms.

Gibbs versus non-Gibbs distributions in money dynamics

We review a simple model of closed economy, where the economic agents make money transactions and a saving criterion is present. We observe the Gibbs distribution for zero saving propensity, and non-Gibbs distributions otherwise. While the exact solution in the case of zero saving propensity is already known to be given by the Gibbs distribution, here we provide the explicit analytical form of the equilibrium distribution for the general case of nonzero saving propensity. We verify it through comparison with numerical data and show that it can be cast in the form of a gamma-distribution.

Opinion formation in kinetic exchange models: spontaneous symmetry-breaking transition.

We propose a minimal multiagent model for the collective dynamics of opinion formation in the society by modifying kinetic exchange dynamics studied in the context of income, money, or wealth distributions in a society. This model has an intriguing spontaneous symmetry-breaking transition to polarized opinion state starting from nonpolarized opinion state. In order to analyze the model, we introduce an iterative map version of the model, which has very similar statistical characteristics. An approximate theoretical analysis of the numerical results is also given, based on the iterative map version.

First principles calculations of the optical properties of CxNy single walled nanotubes

The optical properties of (8, 0) single walled carbon nanotubes alloyed with nitrogen (N) have been examined using relaxed carbon-carbon (C-C) bond length ab initio density functional theory (DFT) calculations in the long wavelength limit. The maximum value of the absorption coefficient is shown to depend strongly on the concentration of N in a non-linear way as well as on the direction of polarization. The reflectivity at normal incidence vanishes at some unique concentration of N. It is also observed that the peak of the loss function (in parallel polarization and unpolarized cases) shifts to a higher frequency indicating the enhanced metallic character. The observed variation of the plasma resonance frequencies with N concentration indicates the existence of a unique maximum for parallel polarization and a step function like behavior for the unpolarized situation with concentration.