Abstract
In the preceding two chapters, relationships for individual categories of consumption expenditures vis-a-vis one another have been investigated through a sequence of least-squares regression equations in which each of 14 categories of expenditure is regressed on the expenditures for the other 13. The resulting (14 × 14) matrix of “intra-budget” coefficients is then seen as representing an ex post summarization of relationships amongst separate categories of consumption that arise from a confrontation of tastes and preferences with income and prices. Not only are effects associated with income and prices reflected in these coefficients but also any “internal” influences that expenditures, independent of income and prices, may have on one another. In the conventional (i.e., neoclassical) theory of consumer behavior, interrelationships amongst goods are defined in terms of effects that depend upon the signs of cross-price elasticities. Goods are deemed to be substitutes with one another if the sign of the (income-compensated) cross elasticity is positive and complements if the sign is negative. Substitution and complementarity that might be associated with income are not part of the picture.
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Notes
- 1.
The total expenditure generated by these expenditures is $13,593. As this represents a quarterly total, the implied total annual expenditure is about $54,000.
- 2.
The difference of $3,599 between the mean total expenditure and the base calculated from the mean intercepts is a consequence of this nonadditivity and, as will be discussed in Sect. 4.5 below, can be interpreted as representing an internal “self-energy” that gives rise to a “force of expansion” in consumption.
- 3.
Thus, for food in Table 4.4, z1 will have 782 for its first element rather than 732, while for housing, its intercept will be 1,204 instead of 1,154.
- 4.
In expression (4.13), the budget constraint is imposed through adjustments to intercepts. Results in which the adjustments are made to Ay and to both z and Ay will be presented in Sect. 4.6 below. The entries in Tables 4.6, 4.7, 4.8 below have been obtained using equation (4.13) through sufficient iterations to achieve convergence to a vector y*. In general, however, this vector implies a total expenditure, Y* (say), that is not necessarily equal to Y0. This being the case, once convergence is achieved, the budget constraint is finally imposed by multiplying y* by Y0/Y*. The same procedure is employed for the simulations reported in Tables 4.12, 4.13, 4.14, 4.15, 4.16 and 4.20.
- 5.
In Sect. 4.6, it will be seen that the skew asymmetry in this case is a consequence of the budget constraint.
- 6.
That this is the case can be seen analytically as follows: At convergence, yk = w kz + Ayk, so that [I–A]yk = w kz and yk = w k[I–A]− 1z. Because [I–A]− 1z is the vector of base expenditures (which sum to $13,593), the final result says that expenditures at convergence for each category are a multiple of base expenditures. When as a final step, total expenditures are constrained to equal total base expenditures plus the increment added in the endogenous simulations, the end result is that the increment gets spread over the categories in proportion to relative base expenditures, independent of the category in which the endogenous change occurs. I am grateful to Timothy Tardiff for with this insight.
- 7.
An interpretation of this result in terms of substitution and complementation will be offered in Sect. 4.6.
- 8.
$8.44 = $50.00–$41.56; $2.59 = $50–$47.41.
- 9.
A strong caveat is in order regarding interpretation of this result, for, as will be seen in Sect. 4.7, the smallness of the entries in Tables 4.13, 4.14, 4.15, 4.16 is in great part a consequence of the imposition of the total-expenditure budget constraint (of $13,593) immediately after the second iteration.
- 10.
- 11.
An interesting illustration of the irrelevance of the initial value of y0 in expression (4.13) to final equilibrium is as follows. Out of curiosity, I did a simulation using expression (4.13)—with w k always equal to 1—starting from the base intercepts for z and a zero vector for y0. Convergence of yk to the vector in the third column of Table 4.3 was finally achieved after 52 iterations. The reason that the endogenous changes of ± $50 in Tables 4.6, 4.7, 4.8 are not transitory is because of imposition of the budget constraint.
- 12.
See Lancaster [8].
- 13.
“Size” of the eigenvalues associated with principal components is calculated with respect to the trace of A′A (i.e., to the sum of the diagonal elements). The usefulness of principal-component analysis in a context such as this is that it can give a good indication of the number of clearly identifiable dimensions of variation.
- 14.
As four of the eigenvalues of A are complex (two conjugate pairs), “size” in this case is calculated as proportions of the sum of eigenvalue “lengths,” where the “lengths” of eigenvalues 1 and 2 are given by the square root of (0.2415)2 + (0.0226)2 and similarly for eigenvalues 10 and 11. Since the principal components of A are calculated from the symmetric positive definite matrix A′A, the associated eigenvalues are all real and positive. That the eigenvalues of A all have modulus less than 1 guarantees that iterations using expression ultimately converge to a finite vector.
- 15.
Cf. The large coefficients for personal care or reading in Table 4.2 in the columns headed by food, housing, apparel, transportation, health, entertainment, and personal insurance.
- 16.
The actual amounts vary from $66 for tobacco to nearly $2,600 for education.
- 17.
Note that, from the second to third iteration, convergence does not appear to be monotonic. This would appear to be a consequence of the fact that four of the eigenvectors of A are associated with complex eigenvalues.
- 18.
As with expression (4.13), increases are commutative and total-expenditure imposed decreases are the negative of increases (i.e., budget-constrained increases and decreases are skew-symmetric).
- 19.
By “mandated” in this context is meant the $50 higher total expenditure underlying the iterations. Why deviations for this weighting scheme turn out always to be half the mandated amount was a mystery until Tim Tardiff provided a proof as follows:
At convergence, the unconstrained amounts in Table 4.20 are given by
$$ {\mathrm{Y}}^k={w}^k{\left[\mathrm{I}-\mathrm{A}\right]}^{-1}\mathrm{z}={w}^k{\mathrm{y}}^0, $$while the “constrained” amounts are given by
$$ {\mathrm{y}}^{k+1}={w}^k{\mathrm{y}}^k={w}^{2k}{\mathrm{y}}^0. $$Consequently, when y0 is subtracted from both the constrained and unconstrained amounts, their ratio is
$$ \left({w}^{2k}-1\right)/\left({w}^k-1\right)={w}^k+1. $$Since w k is essentially equal to 1, the constrained results will necessarily be very close to double the unconstrained results. Tim also notes that the value of w k at convergence would appear to be (initial budget + increment)/budget)0.5.
- 20.
For an approach as to how these ephemeral factors might be modeled in a state- and flow-adjustment framework, see Taylor and Houthakker [19], pp. 232–238).
- 21.
“Spontaneous fluctuations” and “self-energy” suggest a parallel to vacuum fluctuations in quantum theory. Scales aside, while I think the parallel to be apt, it is intended only metaphorically.
- 22.
- 23.
Because the budget constraint in all of the simulations is total expenditure, income has received only fleeting attention. While income is clearly the constraint on spending over long periods of time, this is not necessarily the case over short periods, for expenditures can be financed through a reduction in current saving or by drawing from past savings (including borrowing or use of credit cards). Consequently, accounting-wise, any actual changes in total expenditure, whether from exogenous or endogenous sources, will be reflected in offsetting changes in saving or net wealth.
- 24.
From the third column of Table 4.3, it is to be noted that these four categories have the smallest base expenditures ($115, $130, $70, and $78, respectively). In addition, the estimated least-squares coefficients for these categories [in Eq. (4.1)] are among the least stable of the 14 categories over the 40 BLS samples underlying in the analysis.
- 25.
- 26.
In passing, it is useful to note that substitution and complementarity in this framework, though they may be initiated by price changes, actually are what they are because of income (or total-expenditure) effects that diffuse through the system of intra-budget coefficients.
- 27.
- 28.
“Force of expansion” can also be seen as reflective of a continual desire for novelty in consumption behavior (see Chap. 2 of Taylor and Houthakker [19]).
References
Lancaster K (1971) Consumer demand: a new approach. Columbia University Press, New York, NY
Taylor LD, Houthakker HS (2010) Consumer demand in the United States: prices, income, and consumption behavior, 3rd edn. Springer, New York
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Taylor, L.D. (2014). Effects of a Change in Expenditures for One Good on Expenditures of Other Goods. In: The Internal Structure of U. S. Consumption Expenditures. Springer, Cham. https://doi.org/10.1007/978-3-319-02225-3_4
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