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Income Inequality Generates Chaos

The present paper is an attempt to bridge the gulf between economics and econophysics. That is, constructing a chaos-based theoretical model, we show the behavior of the goal-driven agents exhibits the behavior of the purpose-free agents.
Let us consider an intertemporal economy that consists of one representative firm and two classes of people, namely, working class people (i.e., laborers) and middle class people (i.e., consumers). The firm inputs the labor of the working class people to sell the products to the middle class people whose income source is the dividend from the firm and the capital gain from their assets. As in, we assume that consumers (i.e., middle class people), who purchase either 0 or 1 unit of products, are heterogeneous in that each of these consumers has different willingness-to-pay for the products, to generate the demand curve as aggregate of such demands. We assume the willingness-to-pay of each consumer relates positively to her/his income in period t so that the shape of the aggregate demand curve reflects the income distribution.
As for the decision making of the firm, we assume she/he determines the wage rate to maximize her/his profit in each period, anticipating, as in the efficiency wage model, that higher wage rate pulls up the efficiency of the labor, and hence the quality and the price of the product.
Based on the above analysis, this section shows that fixed income inequality among the middle class people, as well as the immobility between the two classes (i.e., the middle class and the labor class), generates chaos. For this purpose, let us assume that the firm’s profit in each period is allocated to the middle class people so as to maintain the distribution of the income, which is equivalent with the distribution of the willingness-to-pay
Main conclusion is: economy becomes chaos if 1) the capital gain of the middle class people is large enough for them to consume eight times as much as their income gain and 2) market for the middle class people is large enough and number of the middle class people is 16 times as large as the amount of products made by one producer.

Article by Yasunori Fujita from Keio University, Japan

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