Abstract
In economics, there have been found few fundamental theories as complying with statistical laws, really examined by observation. Econometrics still clings to a very special structure, patronized with favoritism by the traditional economic theory. In 1990s, Hidenbrand, Grandmont, Grodal and others have tried to formulate consumer demand by a procedure definitely different from the traditional way of using an individualistic utility function. An alternative approach is to make assumptions on the population of the households as a whole – the macroscopic microeconomic approach. If the households’ demand functions are not identical then one needs a certain form of heterogeneity of the population of households. This approach really also requires the empirical tests. The method of nonparametric test on income distribution to estimate covariances on the households spending may be applied. In this article, Japanese Family Expenditure Data is used for estimation. The law of consumer demand is one of the most important topics since a birth of economics. Our statistical economics should, first of all, demand innovation in this area.
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Notes
- 1.
In view of an individualistic utility function, h can be derived from the problem.
$$ \eqalign{ h^i(\,p, x^i): = {\rm arg\ min}_{u(z)\,=\,\xi} \ pz,\cr h(q,x(q)):q\rightarrow f (q,x(q)).} $$ - 2.
Ell means the ellipsoid of dispersion in (5.19) of Sect. 5.2.5.
- 3.
\({R^\prime}(x) = \frac{\displaystyle{\partial R(x)}}{{\partial x}} = \frac{\displaystyle{\partial E(\,{f^2})}}{{\partial x}} \). Then \( R(x) = E(\,{f^2}) \).
- 4.
As h = 1.5 is chosen, the eigenvalues of the covariance matrices remarkably holds all positive during almost the periods of 1979 to1998. See Aruka (2000).
- 5.
Since this may be considered \( {C_\rho } = \int {\rm cov}\;\nu (x,p)\rho dx \) by identifying \( {\partial_x} \) with \( {\rm cov}\;\nu (x + \Delta ) - {\rm cov}\;\nu (p|x) \), Property 1 virtually is equivalent to say that \( {C_\rho } \) is positive semidefinite. Thus the statistical tests of Property 1 requires to check whether the eigenvalues of \( {C_\rho } \) are all semipositive or not. According to Hildenbrand (1994), and Hildenbrand and Kneip(1993), these eigenvalues should be subject to the bootstrap test. Our matrix of \( {C_\rho } \) derived by the average derivative method has produced all strictly positive eignvalues of 10 distinct roots all in 1990s, if the bandwidth h is specified \( 1.5 \). Over a half of the eigenvalues, we may have the eigenvalues of nearly zero. If we should apply the bootstrap test on our covariance matrices \( {C_\rho } \), there could appear many negative eigenvalues. See Aruka (2000).
- 6.
\( {\rm The\ cov}(1,2) \) denotes (1,2)-th component of the covariance matrix.
- 7.
This may be regarded as the form \( {C_\rho } \) appreciated at all \( x = \bar{x} \).
References
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Aruka, Y. (2011). The Law of Consumer Demand in Japan: A Macroscopic Microeconomic View. In: Aruka, Y. (eds) Complexities of Production and Interacting Human Behaviour. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-2618-0_5
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DOI: https://doi.org/10.1007/978-3-7908-2618-0_5
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