اقتصاد خرد مدل های بازار گاز مانند ایده آل

We develop a framework based on microeconomic theory from which the ideal gas like market models can be addressed. A kinetic exchange model based on that framework is proposed and its distributional features have been studied by considering its moments. Next, we derive the moments of the CC model (Eur. Phys. J. B 17 (2000) 167) as well. Some precise solutions are obtained which conform with the solutions obtained earlier. Finally, an output market is introduced with global price determination in the model with some necessary modifications.

Starting with an early attempt by Angle [1] and [2], a number of models based on the kinetic theory of gases have been proposed to understand the emergence of the universal features of income and wealth distributions (see e.g. Refs. [3], [4] and [5]). The main focus of those models was to develop a framework that would give rise to gamma function-like behavior for the bulk of the distribution and a power-law for the richer section of the population. The CC-CCM models [6] and [7] have both of these features. The kinetic exchange model proposed by Dragulescu and Yakovenko [8] and later studied in more detail by Guala [9], produces the gamma function-like behavior for the income distribution. We note that all of these models are generally based on some ad-hoc stochastic asset evolution equations with little theoretical foundations for it. Our primary aim in this paper is to develop a consistent framework from which we can address this type of market model. Here we propose a model based on consumers’ optimization which can give rise to those particular forms of asset exchange equations used in Refs. [6] and [8] as special cases. We then focus exclusively on the asset exchange equations and an analytically simple kinetic exchange model is proposed. Its distributional features are analyzed by considering its moments. The same technique is then applied to derive the moments of the distribution of income in the CC model [6] as well. We find that it provides a rigorous justification for the values of the parameters of the distribution, conjectured earlier in Ref. [10]. A possible extension of the microeconomic settings of the basic model is also studied where we consider the output market explicitly with global price determination.

Our primary focus was to develop a minimal microeconomic framework to derive the asset equations used in the ideal gas like market models. We see that the framework considered above can very easily reproduce the exchange equations used in the CC model (with fixed savings parameter). In a certain limit, it also produces the exchange equations with complete random sharing of monetary assets. Based on this model we have proposed an ideal gas like model of income distribution and we have shown that it captures the gamma function-like behavior of the real income distribution quite well. As discussed above, the framework considered here and the resulting exchange equations differ significantly from those considered in [12] and [13]. The utility function (in the basic microeconomic model considered above) deals with the behavior of the agents in an exchange economy. However, it also captures the behavior of traders of put and call options of the same stock in a stock market. The price of call and put options of a particular stock generally vary inversely, depending on strike prices and expiration dates. An exception to this generalization is periods of symmetric volatility in the stock’s price, when the simultaneous purchase of call and put options, a straddle, may be profitable. An option’s price (particularly the log of proportional return) is readily identified with its utility. Further, λλ may be slightly re-interpreted from determining the utility of savings to determining the utility of protecting savings from risky trading. An interesting question would be whether the stationary distribution of the CC model returns in this model of option trading or not. If not, what might the modification of the CC model be? Next, we have analyzed the Monte Carlo simulation results by considering the first two moments of the income distribution. The same has been done to analyze the income distributions produced by the CC model. Only the moment considerations in both the models show the transition from exponential to delta function with changes in the parameter values of the respective models (the rate of transfer in case of the transfer model and the rate of savings in case of CC model). Moreover, the values of the income distribution parameters, conjectured in Ref. [10], have been derived here only by considering those moments. Next, the initial microeconomic model is generalized by incorporating an output market (with global price determination) explicitly. The asset evolution equation in this context is seen to be represented well by an autoregressive process which very easily produces a power law distribution of assets. It has already been discussed in Ref. [17] in detail how such a process can generate insightful results regarding the distribution of monetary assets. In the same context, an aggregative equation is derived which is analogous to the Fisher equation. From this equation, a power law in price fluctuation is also derived. Taken together, these models provide a link between the standard microeconomic settings (individual optimization and output market) and the asset exchange equations used in the ideal gas like market models.