Abstract
Capital theory controversies and ‘paradoxes’ showed that, due to pricefeedback effects, the wage–production price–profit rate curves may display shapes inconsistent with the requirements of the neoclassical theory of value and distribution. Subsequent findings on a number of quite diverse actual singleproduct economies suggested that the impact of those effects is of limited empirical significance. This paper argues that, by focusing on the distributions of the eigenvalues and singular values of the system matrices, we can further study these issues and derive some meaningful theoretical results consistent with the available empirical evidence. Consequently, the real paradox, in the sense of knowledge vacuum and, thus, requiring further research, is the distributions of the characteristic values and not really the ‘paradoxes in capital theory’.
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Notes
 1.
Data limitations do not allow the treatment of the noncompetitive imports and, therefore, our investigation is based on a closed economy model. Future research should fill in the gap by treating imported means of production as a primary input (see Metcalfe and Steedman 1981, pp. 9–11).
 2.
The transpose of a \(1 \times n\) vector \({\mathbf{y}} \equiv [y_{j} ]\) is denoted by \({\mathbf{y}}^{\text{T}} ,\) and the diagonal matrix formed from the elements of \({\mathbf{y}}\) is denoted by \({\hat{\mathbf{y}}}.\) Furthermore, \({\mathbf{A}}_{j}\) denotes the \(j\)th column of a semipositive \(n \times n\) matrix \({\mathbf{A}} \equiv [a_{ij} ],\) \(\lambda_{{{\mathbf{A}}1}}\) the PF eigenvalue of \({\mathbf{A}}\) and \(({\mathbf{x}}_{{{\mathbf{A}}1}}^{\text{T}} ,{\mathbf{y}}_{{{\mathbf{A}}1}} )\) the corresponding eigenvectors, while \(\lambda_{{{\mathbf{A}}k}} ,\) \(k = 2, \ldots ,n\) and \(\left {\lambda_{{{\mathbf{A}}2}} } \right \ge \left {\lambda_{{{\mathbf{A}}3}} } \right \ge \cdots \ge \left {\lambda_{{{\mathbf{A}}n}} } \right\), denote the nondominant eigenvalues, and \(({\mathbf{x}}_{{{\mathbf{A}}k}}^{\text{T}} ,{\mathbf{y}}_{{{\mathbf{A}}k}} )\) the corresponding eigenvectors. Finally, \({\mathbf{I}}\) denotes the \(n \times n\) identity matrix, \({\mathbf{e}}\) the summation vector, i.e. \({\mathbf{e}} \equiv [1,1, \ldots ,1]\), and \({\mathbf{e}}_{j}\) the \(j\)th unit vector. It is also noted that, given any \({\mathbf{A}}\) and an arbitrary \(\varepsilon \ne 0,\) it is possible to perturb the entries of \({\mathbf{A}}\) by an amount less than \(\left \varepsilon \right\) so that the resulting matrix is diagonalizable (see, e.g. Aruka 1991, pp. 74–76).
 3.
If wages are paid ex ante, then \(\rho\) is no greater than the share of profits in the SSS.
 4.
One of the referees noted that the WPCs are “meromorphic functions […]. Such curves show strong curvatures in the neighbourhood of singularities, and these may be anywhere in the complex plane; they need not be real, let alone between zero and \(R\)”.
 5.
As we will see in the next sections of the present paper, this heuristic case is more geared towards reality than one at first sight might think.
 6.
For other heuristic cases, which are also interesting, both theoretically and empirically, and include more than one hyperbasic industry, see Mariolis and Tsoulfidis (2014, pp. 214–215, 2016, pp. 154–167).
 7.
See, e.g. Horn and Johnson (1991, Chap. 3) and take into account that \({\mathbf{J}}\) is similar to the column stochastic matrix \({\hat{\mathbf{y}}}_{{{\mathbf{J}}1}} {\mathbf{J}} {\hat{\mathbf{y}}}_{{{\mathbf{J}}1}}^{  1}\).
 8.
Nevertheless, C. Bidard, H. G. Ehrbar, U. Krause and I. Steedman have detected some ‘monotonicity (theoretical) laws’ for the relative prices (see Bidard and Ehrbar 2007, and the references therein).
 9.
Information on the sources of data and the construction of variables is available in the Appendix.
 10.
The price curve corresponding to industry 8 displays a maximum at \(\rho \cong 0.955,\), which is not visible in Fig. 2. It is deduced from more detailed data.
 11.
In effect, we found that, at \(\rho = 0,\) the vertically integrated capitalintensities in the circulating (fixed) capital case gave an arithmetic mean equal to 1.25 (to 3.87) and a standard deviation of 0.32 (of 2.52), with a coefficient of variation of 0.26 (of 0.65).
 12.
Only a single curve, corresponding to industry 2 in the circulating capital case, crosses the horizontal axis (at \(\rho \cong 0.639\)).
 13.
If \(p_{j}^{\text{S}}\) denotes the price of commodity \(j\) in terms of SSC, and \(w^{j}\) denotes the money wage rate corresponding to the normalization equation \(p_{j} = v_{j} ,\) then \((w^{j} )^{  1} p_{j} = (w^{\text{S}} )^{  1} p_{j}^{\text{S}}\) or \(w^{j}  w^{\text{S}} = w^{j} (1  p_{j}^{\text{S}} v_{j}^{  1} )\) or, recalling Eq. (15), \(w^{j}  w^{\text{S}} = w^{j} \rho R(R^{  1}  k_{j} ).\) It then follows that (i) \(p_{j}^{\text{S}} v_{j}^{  1} = 1\) implies \(w^{j} = w^{\text{S}} ;\) and (ii) \(w^{j}  w^{\text{S}}\) is directly related to \(R^{  1}  k_{j}\) (compare the outer curves in Fig. 9 with the relevant differences \(R^{  1}  k_{j}\) in Figs. 3 and 6).
 14.
This statement per se refers to the economically relevant interval of \(\rho\) and relies on the detected monotonicity of the price curves and inflection points of the WPCs; not on the nearlinearity of the latter curves. Even in \(2 \times 2\) corntractor systems the WPC may deviate considerably form the straight line (see Eqs. (9a–b)).
 15.
As is well known, the geometric mean is more appropriate for detecting the central tendency of an exponential set of numbers. In our case, it can be written as
\(GM = \left {\det {\mathbf{J}}} \right^{{(n  1)^{  1} }}\)
We define the index of inseparability associated with the eigenvalues as: \(\varepsilon_{{{\mathbf{J}}1}}^{\lambda } \equiv 1  \lambda_{{{\mathbf{J}}1}} \left( {\sum\nolimits_{i = 1}^{n} {\left {\lambda_{{{\mathbf{J}}i}} } \right} } \right)^{  1} = 1  \left( {1 + \sum\nolimits_{k = 2}^{n} {\left {\lambda_{{{\mathbf{J}}k}} } \right} } \right)^{  1}\)
The research to date suggests that there are statistically significant regressions, but not onetoone relationships, between alternative measures of the characteristic value distributions and of the errors in approximate curves (see Mariolis 2015b; Mariolis and Tsoulfidis 2016, pp. 167–198). Thus, our analysis is rather phenomenological.
 16.
There is, however, a caveat for the SIOTs as adequate representations of actual economies. It is well known that these tables can be derived from the ‘System of National Accounts’ framework of Supply and Use Tables (SUTs), where the latter tables could be considered as the counterpart of joint production systems à la v. Neumann (1937) and Sraffa (1960). The hitherto evidence from the SUTs indicates that actual economies do not necessarily have the usual properties of singleproduct systems (Mariolis and Soklis 2010; Soklis 2011, 2015); for instance, there are cases in which elements in the (positive) vector of ‘labourcommanded prices’, \(w^{  1} {\mathbf{p}},\) decrease with the profit rate and, therefore, the monotonicity of the WPC depends on the numeraire choice.
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Acknowledgments
The material in the manuscript has been acquired according to modern ethical standards and does not contain material copied from anyone else without their written permission and there is no conflict of interest. We are indebted to two anonymous referees of this journal for helpful remarks and hints. An earlier version of this paper was presented at the 19th European Society for the History of Economic Thought Conference, 14–16 May 2015, Roma Tre University, Italy. We thank the participants of this conference and especially Heinz D. Kurz, Fabio Petri, Bertram Schefold, Persefoni Tsaliki and Stefano Zambelli for their comments, criticisms and discussions. The usual caveat applies.
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Appendix: Data sources and construction of variables
Appendix: Data sources and construction of variables
The input–output table of UK for the year 1990 is available from the OECD STAN database (http://www.oecd.org), and the degree of disaggregation is such that 33 product/industry groups are identified (see Table 1).
The market prices of all products are taken to be equal to one; that is to say, the physical unit of measurement of each product is that unit which is worth of a monetary unit (in the SIOT of the UK economy, the unit is set to one million pounds). The various variables used in our estimations were constructed as follows:

1.
The matrix of direct technical coefficients, \({\mathbf{A}}\), is obtained from the SIOT by dividing each industry’s inputs by their respective gross output.

2.
The matrix of capital stocks, \({\mathbf{S}}\), is constructed by using the capital flows table, which allocates the gross investment flows, \(I_{ij}\), of each industry to itself and to the other industries. With the aid of the capital flows table we form a matrix of weights by dividing the elements of each column by the respective column sum, i.e. \({\mathbf{I}}_{\text{w}} \equiv [I_{ij} (\sum\nolimits_{i = 1}^{n} {I_{ij} )^{  1} } ]\). By assuming that the capital stocks are allocated amongst industries in a way similar to that of investment flows, we can write \({\mathbf{S}} \equiv {\mathbf{I}}_{\text{w}} {\hat{\mathbf{\varvec\kappa }}}\), where \({\hat{\mathbf{\varvec\kappa }}}\) denotes the diagonal matrix formed from the vector of net capital stocks, \({\varvec{\upkappa}}\) (both the capital flows table for the year 1990 and \({\varvec{\upkappa}}\) are available in the OECD STAN data base). The so derived matrix of capital stocks is subsequently divided by the actual gross output vector, \({\bar{\mathbf{x}}}\), to obtain the matrix of capital stock coefficients, i.e. \({\mathbf{K}} \equiv {\mathbf{S}}{\hat{\bar{\mathbf{x}}}}^{  1}\). In similar fashion we could construct the matrix of depreciation coefficients, but the lack of depreciation data at the required industry detail did not allow the construction of such matrix.

3.
The vector of direct labour coefficients, \({\mathbf{l}}\), is estimated using the wage bill (the product of annual wage times the number of employees) of each of our 33 industries. The problem with this estimation is that the selfemployed population is not accounted for. Fortunately, the OECD data base provides information on both the total employment and the number of employees for each of our 33 industries. From the available data, we estimate the average industry wage and we divide it by the economywide minimum wage, the soestimated relative industry wages are subsequently multiplied by the total employment (employed plus selfemployed) and so we derive the homogenized industry employment. This reduction, of course, is only meaningful when the relative wages express with sufficient precision the differences in skills and intensity of labour. The adjusted for skills total employment is divided by the industry total output to obtain the vector of direct labour coefficients.

4.
By assuming that all wages are consumed and that consumption out of wages has the same composition as the vector of the final consumption expenditures of the household sector, \({\mathbf{c}}^{\text{T}}\), directly available in the SIOT, the commodity vector defining the ‘actual’ real wage rate is estimated as \({\mathbf{b}}^{\text{T}} = [\mathop {\hbox{min} }\nolimits_{j} \{ w_{{{\text{m}}j}} \} ({\mathbf{p}}_{\text{m}} {\mathbf{c}}^{\text{T}} )^{  1} ]{\mathbf{c}}^{\text{T}}\), where \(\mathop {\hbox{min} }\nolimits_{j} \{ w_{{{\text{m}}j}} \}\) denotes the economywide minimum money wage rate in terms of market prices, and \({\mathbf{p}}_{\text{m}}\) the vector of market prices, which is identified with \({\mathbf{e}}\).

5.
Substituting \(w = {\mathbf{pb}}^{\text{T}}\) in \({\mathbf{p}} = w{\mathbf{v}} + \rho {\mathbf{pJ}}\) yields \({\mathbf{p}} = \rho {\mathbf{pJ}}[{\mathbf{I}}  {\mathbf{b}}^{\text{T}} {\mathbf{v}}]^{  1}\). Thus the ‘actual’ relative profit rates for both models are estimated as the reciprocals of the PF eigenvalues of matrices \({\mathbf{J}}[{\mathbf{I}}  {\mathbf{b}}^{\text{T}} {\mathbf{v}}]^{  1}\).
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Mariolis, T., Tsoulfidis, L. Capital theory ‘paradoxes’ and paradoxical results: resolved or continued?. Evolut Inst Econ Rev 13, 297–322 (2016). https://doi.org/10.1007/s4084401600434
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Keywords
 Capital theory
 Characteristic value distributions
 Hyperbasic industry
 Spectral decompositions
 Wage–price–profit rate curves
JEL Classification
 B21
 B51
 C67
 D57