A Macroscopic Order of Consumer Demand Due to Heterogenous Consumer Behaviors on Japanese Household Demand Tested by the Random Matrix Theory

Abstract

So far the consumer theory was microscopically too restrictive to overlook many important scenes of the whole consumption activities. This view is deliberately dropping the inter-correlated factors between different income classes and household demands. The household demands must have a certain bias toward either common or different directions among different income classes. In some sense, the traditionally narrow interest may be dangerous because other decisive factors contributing to the consumption activities may be missed. This article argued to choose a particular scene where some natural or social correlative relations i.e., some dominant forces, may work in the consumption activities over the different income classes. By introducing the different income classes, we can just analyze a new facet of interactive correlations among the heterogeneous consumers. Here we can find any correlative relation, irrespective of price variations. Such a way of thinking may lead us observing another hidden force of the consumption activities.

Appendices

Appendix 1

We denote a demand at price p by f(p). We then have a next formula between two distinct price vector p ¹ and p ²: ?

$$ \bigl(p^1 - p^2\bigr) \bigl(f\bigl(p^1 \bigr)-f\bigl(p^2\bigr)\bigr) \leq0 $$

(10.11)

i.e.,

$$ \mathrm{d}p\mathrm{d}f \leq0. $$

(10.12)

This is the demand law. The law does not state any analytical confirmation with respect of price when an income variation x derived by price changes is taken account into. The tentative formulation is called the Pareto-Slutsky equation:

$$\begin{aligned} \frac{\partial f_j}{\partial p_k}=\frac{\partial h_j}{\partial p_k}-\frac{\partial f_j}{\partial x} f_k. \end{aligned}$$

(10.13)

In other words,

• demand change = substitution effects + income effects

Here j and k are indices of good. f _j is a demand of good k. h _j is a compensated demand of good j. x is an income level. Demand for good depend on prices of goods p _k including itself (p _j ) and also depend on income. Income x is to be measured in terms of goods. So income may be changeable depending on price variations Δp. A change of income naturally induces a change of demand. But the sign of a change of income is not decisive in general. This is a complicated factor for establishing the demand law. Consequently, economists never were successful to confirm the sign of income effect until a new assumption was invented by Hildenbrand [3]). It took about 90 years to solve this problem.

Compensated demand is a sophisticated idea. This demand h always has a negative sign respect with a price rise. Actually, h is supposed to be a special form only reactive to price variation but inactive to income level to guarantee the same level of satisfaction by supplementing a new injection of income if short, or reducing if long.

We introduce into the Pareto-Slutsky equation a very small footnote-sized perturbation of Δp and h:Δp and Δh. A variation of dpdf caused by Δp and Δh i.e., ΔpΔf may be approximately estimated as $\Delta p \frac{\partial f_{j}}{\partial p_{k}} \Delta p$ In other words, it holds:

$$\begin{aligned} \Delta p \frac{\partial f_j}{\partial p_k \partial p_k} = \Delta p \frac{\partial h_j}{\partial p_k \partial p_k} - \Delta p \frac {\partial f_j}{\partial x_k} \Delta p f. \end{aligned}$$

(10.14)

We cannot find any reason that the first item is always equal to the second item. If the second item should be negative, the total variation caused by Δp _k could be positive, leading the contrary to the demand law. Thus, the Pareto-Slutsky equation, as it is, is not successful to guarantee a definite analytical relation in general. A so-called Giffen effect could not be removed.

Appendix 2

Most of the items of expenditure have strong seasonal dependence as easily expected. The excepted items are Housing, Medical and Transport. Removal of seasonal components out of the original data is thus a critical procedure to elucidate possible correlations embedded in expenditure of Japanese consumers.

Here we adopt two seasonal adjustment methods. One of them is the X-12-ARIMA,⁵ developed and used by the U.S. Census Bureau. It is also a standard seasonal adjustment method for the Statistic Bureau in Japan. The X-12-ARIMA program, having a long history in the development, is full of experimental knowledge with a number of degrees of freedom left for users to optimize the procedure. However, we allowed the program to determine a best regARIMA model by itself.

To assess the reliability of such an automatic seasonal adjustment, we applied another program called DECOMP [6] to the same data. The DECOMP based on state-space-modeling is free from the moving average procedure that plays an important role in X-12-ARIMA. As indicated in its name, the DECOMP decomposes a given time series data into trend, seasonal and irregular components in a transparent way. In return, there is no much room for us to play with the program for optimization of the procedure. The parameter set for the program that we used is

• Log Transformed: Yes

• Seasonal frequency: 12

• Trend order: 1

• AR order: 0

• Trading Day Effects: Yes

As will be shown later, the X-12-ARIMA and DECOMP bring about no fundamentally different results for the principal component analysis. The RMT (random matrix theory) tells us that there are two statistically meaningful principal components for both of the seasonally adjusted data. And the characteristic features of the principal components so obtained are essentially the same between the two alternative data.