Abstract
This chapter follows a recently developed line of research, which focuses on the spectral analysis of the wage-price-profit rate relationships in actual single-product economies. It shows that main aspects of those relationships could be connected to the skew distribution of both the eigenvalues and the singular values of the system matrices. The results finally suggest that there is room for using low-dimensional models as surrogates for actual economies.
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- 1.
This chapter draws on Mariolis and Tsoulfidis (2011, 2014), Iliadi et al. (2014), Mariolis (2011, 2013, 2015) and Mariolis et al. (2015). In the present line of research, there have also been contributions by Schefold (2008, 2013) and Mariolis and Tsoulfidis (2009), while Bienenfeld (1988, p. 255) has already shown that, in the extreme case where the non-dominant eigenvalues of the matrix of vertically integrated technical coefficients equal zero, the production prices in terms of Sraffa’s Standard commodity (SSC) are strictly linear functions of the profit rate, and Shaikh (1998) has noted that ‘[a] large disparity between first and second eigenvalues is another possible source of linearity.’. (p. 244; also see p. 250, note 9).
- 2.
Steedman’s (1999) numeraire entails that
$$ {\left({\displaystyle \sum_{i=1}^n{\Pi}_i}{d}_i\right)}^{-1}=1 $$and, therefore, w = ∏0 and
$$ \mathbf{p}={\displaystyle \sum_{i=1}^n{\Pi}_i{\mathbf{y}}_{\mathbf{J}i}} $$(see Eqs. 5.9a and 5.10a). Thus, the w–ρ and p–ρ relationships take on simpler forms in the sense that the former is expressed solely in terms of the eigenvalues of J, while the latter is expressed in terms of powers of ρ up to ρ n−1.
- 3.
If, for instance, \( \mathbf{T}=\left[{\mathbf{x}}_{\mathbf{J}1}^{\mathrm{T}},{\mathbf{e}}_2^{\mathrm{T}},\dots, {\mathbf{e}}_n^{\mathrm{T}}\right] \), then \( {\tilde{\mathbf{J}}}_{12}={\left({\mathbf{y}}_{\mathbf{J}1}{\mathbf{x}}_{\mathbf{J}1}^{\mathrm{T}}\right)}^{-1}\left[{y}_{2\mathbf{J}1},{y}_{3\mathbf{J}1},\dots, {y}_{n\mathbf{J}1}\right] \).
- 4.
Consider the theorem mentioned in footnote 31 of Chap. 2. It may be added that, if, for instance, x > 0 and \( {y}_{1i}={x}_{i1}^{-1}+{\delta}_i \), where δ i ≥ 0, then
$$ \lambda \approx {\left(1+n+{\displaystyle \sum_{i=1}^n{x}_{i1}{\delta}_i}\right)}^{-1}\le {\left(1+n\right)}^{-1} $$If x > 0 and \( {y}_{1i}=-{x}_{i1}^{-1}+{\delta}_i \), where δ i ≤ 0, then
$$ \lambda \approx {\left(1-n+{\displaystyle \sum_{i=1}^n{x}_{i1}{\delta}_i}\right)}^{-1}\ge {\left(1-n\right)}^{-1} $$ - 5.
It is noted that if we adopt Steedman’s numeraire (see footnote 2 in this chapter) and \( {\lambda}_{\mathbf{J}k}=\lambda =\alpha \), then
$$ \mathbf{p}={\left(1-\rho \lambda \right)}^{n-2}\left\{\mathbf{p}(0)+\rho \left[{\left(1-\lambda \right)}^{-n+2}\mathbf{p}(1)-\mathbf{p}(0)\right]\right\} $$where p(1) is now equal to (1–λ)n−1 y J1. Hence, the p j –ρ curves are not necessarily monotonic.
- 6.
Any complex number is an eigenvalue of a positive 3 × 3 ‘circulant matrix’ (Minc 1988, p. 167). To the best of our knowledge, however, the problem of determining necessary and sufficient conditions for a list of numbers to be the spectrum of a nonnegative matrix (‘nonnegative inverse eigenvalue problem ’) remains unsolved (for a recent contribution, see Laffey and Šmigoc 2006).
- 7.
Setting \( b={\left(1+\mathbf{y}{\mathbf{x}}^{\mathrm{T}}\right)}^{-1},\mathbf{S}=\mathbf{I},{\boldsymbol{\upchi}}^{\mathrm{T}}={\left(\mathbf{y}{\mathbf{x}}^{\mathrm{T}}\right)}^{-1}{\mathbf{x}}^{\mathrm{T}} \) and replacing e by y, we obtain Case 3.
- 8.
It is easily checked that
$$ {\dot{f}}_{\mu}\left(\rho \right)=-\left(1-{\lambda_{\mathbf{J}}}_{\mu}\right){\left(1-\rho {\lambda}_{\mathbf{J}\mu}\right)}^{-2}<0 $$since \( \left|{\lambda_{\mathbf{J}}}_{\mu}\right|<1 \), and
$$ {\ddot{f}}_{\mu}\left(\rho \right)=-2\left(1-{\lambda}_{\mathbf{J}\mu}\right){\lambda}_{\mathbf{J}\mu }{\left(1-\rho {\lambda}_{\mathbf{J}\mu}\right)}^{-3} $$ - 9.
- 10.
It is easily checked that
$$ \dot{g}\left(\rho \right)={\left(1-\rho \right)}^{-2}\left(1-\alpha \right) $$and
$$ \dot{h}\left(\rho \right)={\left[\left(1-\rho \right)\left(1-\rho \alpha \right)\right]}^{-2}\rho {\beta}^2\left[2-\rho \left(1+\alpha \right)\right] $$Hence, \( \dot{g}\left(\rho \right)>0 \) and \( \dot{h}\left(\rho \right)>0 \), since \( \left|\alpha \right|<1 \) and ρ(1 + α) < 2.
- 11.
It is easily checked that the first derivative of \( {\left({\left|{\kappa}_{\mu}\right|}^{-1}{\kappa}_{\mathrm{S}}\right)}^2 \) with respect to ρ equals
$$ 2\left[\left(1-\rho \alpha \right)\left(1-\alpha \right)+\rho {\beta}^2\right]{\left[{\left(1-\rho \right)}^3\sqrt{\alpha^2+{\beta}^2}\right]}^{-1} $$ - 12.
It may be said that this is not unanticipated on the basis of Goodwin’s (1976, 1977) contribution to the linear value and distribution theory. By following an approach which is closer to our, Bidard and Ehrbar (2007, pp. 203–204) show that |κ μ | decrease with ρ, and if κ μ is complex, then the derivative of its argument does not change sign, i.e. κ μ moves monotonically, either clockwise or counterclockwise, across the complex plane . Since there are statements in the foreign trade theory (e.g. Stolper-Samuelson effect, ‘factor price’ equalization theorem) that depend crucially on the existence of monotonic price-profit rate relationship s, our conclusion would seem to be of some importance for that theory (also see Mariolis 2004).
- 13.
- 14.
See footnote 3 in this chapter.
- 15.
It is noted that, in this example, M is ‘circulant ’ (since A is cyclic ) and, therefore, ‘normal ’ (MM T = M T M) and doubly stochastic : \( {\sigma}_{\mathbf{M}1}=\left|{\lambda}_{\mathbf{J}1}\right|=1 \), \( {\sigma}_{\mathbf{M}2}={\sigma}_{\mathbf{M}3}=\left|{\lambda}_{\mathbf{J}2,3}\right| \) and \( {\varepsilon}_{\mathbf{M}1}\cong 0.193 \), \( {\varepsilon}_{\mathbf{M}2}\cong 0.097 \) (see, e.g. Meyer 2001, pp. 379 and 555).
- 16.
The dimensions of those SIOTs vary from 19 industries (Greece, 1988–1997) to 39 industries (USA). The tables of China and Japan are available from the OECD STAN database. Those of Greece and Korea are provided by the National Statistical Service of Greece and the Bank of Korea, respectively (also see Chap. 3). Finally, those of USA are from the Bureau of Economic Analysis (BEA) and have been compiled by Juillard (1986) (the data used in the studies by Ochoa 1984; Bienenfeld 1988 and Shaikh 1998 are from the same source although at 71 × 71 industry detail).
- 17.
Finkelstein and Friedberg (1967) discuss E and EN and apply them to studies of industrial competition and concentration, while Jasso (1982) and Bailey (1985) discuss SF and RE, respectively, and apply them to studies of income distribution . It may also be noted that there is a connection between SF and entropy : using π k , the former can be expressed as
$$ SF=\left(n-1\right){\displaystyle \prod_{k=2}^n{\pi_k}^{{\left(n-1\right)}^{-1}}} $$or, taking the logarithm of both sides,
$$ \log SF={E}_{\max }-\left[-{\left(n-1\right)}^{-1}{\displaystyle \sum_{k=2}^n \log }{\pi}_k\right] $$where log SF is known as the Wiener entropy and the term in brackets can be conceived as a ‘cross-entropy’ expression.
- 18.
In fact, we tried an optimization procedure to find the best possible form, and from the many possibilities, we opted for a simple but, at the same time, general enough to fit the moduli of the eigenvalues of all countries and years.
- 19.
It should be noted that we have also experimented with the flow SIOTs of Canada (1997, 34 × 34; source: OECD STAN database), Japan (1995–1997, 41 × 41; source: OECD STAN database), UK (1998, 40 × 40; source: OECD STAN database) and USA (1997, 40 × 40; source: BEA, compilation through the OECD STAN database), and the results were quite similar, i.e. SF, 0.359 (USA)–0.500 (UK); π 2, 8 % (UK)–18 % (Canada); RE, 0.811 (Canada)–0.888 (UK); and REN, 52 % (Canada)–67 % (UK).
- 20.
The original input-output data comprised 108 industries and are published by the Statistical Service of Japan. The problem with this data set is that eight of the industries have zero rows (i.e. they do not deliver any output to the other sectors and to themselves), which give rise to an input-output structure with non-basic sectors and, therefore, zero eigenvalues corresponding to each of these eight industries. To sidestep this problem, we aggregated each of these eight industries to corresponding similar industries so as the resulting input-output structure consists of dimensions 100 × 100 basic industries. Finally, it should be noted that the results displayed in Table 5.2 are not comparable with those displayed in Table 5.1, since the 33 industries SIOTs of Japan are constructed using different sources and also methodology.
- 21.
- 22.
It may be recalled that
$$ rank\left[{\mathbf{A}}^{\mathrm{C}}\right]+ rank\left[\mathbf{I}-\mathbf{A}\right]-n\le rank\left[{\mathbf{A}}^{\mathrm{C}}{\left[\mathbf{I}-\mathbf{A}\right]}^{-1}\right]\le \min \left\{ rank\left[{\mathbf{A}}^{\mathrm{C}}\right],\ rank\left[\mathbf{I}-\mathbf{A}\right]\right\} $$(see, e.g. Meyer 2001, p. 211). We also experimented with an aggregation in a 3 × 3 SIOT for the USA (1977): in the flow version, the modulus of the subdominant (complex) eigenvalue equals 0.146; in the stock version, the subdominant eigenvalue equals 0.031, while the third eigenvalue equals –0.0001. The aggregation in a 3 × 3 SIOT for Greece (1970) did not give any different results: in the flow version, the modulus of the subdominant (complex) eigenvalue equals 0.087; in the stock version, the subdominant eigenvalue equals –0.027, while the third eigenvalue equals zero (see Tsoulfidis 2010, pp. 150–155). Also consider the evidence provided by Steenge and Thissen (2005).
- 23.
The same holds true for the German economy, while the functions associated with all the other economies of this sample are strictly decreasing.
- 24.
It is noted that (i) p is identified with e; (ii) for the construction of [A,l] and the estimation of w, we follow the usual procedure; and (iii) the sectoral ‘profit factor s’ are estimated from
$$ 1+{\overline{r}}_j=\left(1-w{l}_j\right){\left({a}_{1j}+\dots +{a}_{nj}\right)}^{-1} $$ - 25.
For instance, the ‘root-mean-square-percent-error ’ or the distances à la Steedman-Tomkins (consider Chap. 4) lead to more complicated expressions. Thus, it is much more convenient to focus on the MAED.
- 26.
Τhe effects of technical changes on the eigenvalues of A (and, therefore, on λ J k ) are unknown a priori. It could be noted that the first partial derivatives of λ A j with respect to the elements of A are given by
$$ \partial {\lambda}_{\mathbf{A}j}/\partial \mathbf{A} \equiv \left[\partial {\lambda}_{\mathbf{A}j}/\partial {a}_{ij}\right]={\left({\mathbf{y}}_{\mathbf{A}j}{\mathbf{x}}_{\mathbf{A}j}^{\mathrm{T}}\right)}^{-1}\left({\mathbf{y}}_{\mathbf{A}j}^{\mathrm{T}}{\mathbf{x}}_{\mathbf{A}j}\right) $$ - 27.
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Mariolis, T., Tsoulfidis, L. (2016). Spectral Decompositions of Single-Product Economies. In: Modern Classical Economics and Reality. Evolutionary Economics and Social Complexity Science, vol 2. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55004-4_5
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