Measures of Production Price-Labour Value Deviation and Production Conditions

Abstract

Empirical studies indicate that the deviations of actual production prices from labour values are not too sensitive to the type of measure used for their evaluation. This chapter attempts to theorize this fact by focusing on the relationships between the ‘traditional’ and the numeraire-free measures of deviation. It also provides an illustration of these relationships using actual input-output data.

Appendix: Numeraire -Free Measures

Consider the price-wage-profit system described by

$$ \mathbf{p}=w\mathbf{l}+\left(1+r\right)\mathbf{p}\mathbf{A} $$

or

$$ \mathbf{p}=w\mathbf{v}+\rho \mathbf{p}\mathbf{J} $$

(4.23)

(see Eqs. 2.1 and 2.16). For ρ = 0, we get $\mathbf{p}(0)=w\mathbf{v}$, while for ρ = 1, we get $\mathbf{p}(1)=\mathbf{p}(1)\mathbf{J}$, i.e. p(1) is a left P-F eigenvector of J. Postmultiplying Eq. 4.23 by ${\left[\widehat{\mathbf{p}}\left(\overline{\rho}\right)\right]}^{-1}$, where $\mathbf{p}\left(\overline{\rho}\right)$ denotes the price vector associated with an arbitrarily chosen level, $\overline{\rho}$, of the relative profit rate, gives

$$ {\mathbf{p}}^{*}=w{\mathbf{v}}^{*}+\rho {\mathbf{p}}^{*}{\mathbf{J}}^{*} $$

(4.24)

where ${\mathbf{J}}^{*}\equiv \widehat{\mathbf{p}}\left(\overline{\rho}\right)\mathbf{J}{\left[\widehat{\mathbf{p}}\left(\overline{\rho}\right)\right]}^{-1}$ denotes the transformed (via a diagonal similarity matrix formed from the elements of $\mathbf{p}\left(\overline{\rho}\right)$) matrix of the system, the elements of which are independent of the normalization of p, and ${\mathbf{p}}^{*}\equiv \mathbf{p}{\left[\widehat{\mathbf{p}}\left(\overline{\rho}\right)\right]}^{-1},{\mathbf{v}}^{*}\equiv \mathbf{v}{\left[\widehat{\mathbf{p}}\left(\overline{\rho}\right)\right]}^{-1}$ the transformed vectors of prices and labour values, respectively.

Now suppose that one changes the units in which the various commodity quantities are measured. The shift of units converts [A, l] to $\left[\tilde{\mathbf{A}},\ \tilde{\mathbf{l}}\right]$, where $\tilde{\mathbf{A}}\equiv \mathbf{D}\mathbf{A}{\mathbf{D}}^{-1}$, $\tilde{\mathbf{l}}\equiv \mathbf{l}{\mathbf{D}}^{-1}$ and D is a diagonal matrix with positive diagonal elements. However, no element in Eq. 4.24 changes, since the original (non-transformed) vectors of labour values and prices change to $\mathbf{v}{\mathbf{D}}^{-1}$ and $\mathbf{p}{\mathbf{D}}^{-1}$, respectively. Thus, the angle, θ, between p ^* and v ^*, which is determined by

$$ \cos \theta ={\left({\left\Vert {\mathbf{p}}^{*}\right\Vert}_2{\left\Vert {\mathbf{v}}^{*}\right\Vert}_2\right)}^{-1}\left({\mathbf{p}}^{*}{\mathbf{v}}^{*}\right) $$

is independent of the choice of physical measurement units. Let $d\left(\overline{\rho},\rho \right)$ be the Euclidean distance between the unit vector s ${\mathbf{p}}^{**}\equiv {\left({\left\Vert {\mathbf{p}}^{*}\right\Vert}_2\right)}^{-1}{\mathbf{p}}^{*}$ and ${\mathbf{v}}^{**}\equiv {\left({\left\Vert {\mathbf{v}}^{*}\right\Vert}_2\right)}^{-1}{\mathbf{v}}^{*}$ as a function of ρ.¹² Then

$$ d\left(\overline{\rho},\rho \right)\equiv {\left\Vert {\mathbf{p}}^{**}-{\mathbf{v}}^{**}\right\Vert}_2=\sqrt{2\left(1- \cos \theta \right)} $$

constitutes a measure of price-value or, equivalently, value-price deviation, which is independent of any choice of numeraire and physical measurement units. If $\overline{\rho}=0$, then ${\mathbf{p}}^{*}=\mathbf{p}{\left[w\widehat{\mathbf{v}}\right]}^{-1}$, ${\mathbf{v}}^{*}={w}^{-1}\mathbf{e}$ and

$$ d\left(0,\rho \right)={d}_{\mathbf{p}}\left(\rho \right)\equiv {\left\Vert {\left({\left\Vert \mathbf{p}{\widehat{\mathbf{v}}}^{-1}\right\Vert}_2\right)}^{-1}\left(\mathbf{p}{\widehat{\mathbf{v}}}^{-1}\right)-{\left(\sqrt{n}\right)}^{-1}\mathbf{e}\right\Vert}_2 $$

i.e. d(0, ρ) equals the d – distance , proposed by Steedman and Tomkins (1998, p. 382), and measures the deviation of the original prices from the original labour values, and not vice versa, in the sense that, in general,

$$ {d}_{\mathbf{p}}\left(\rho \right)\ne {d}_{\mathbf{v}}\left(\rho \right)\equiv {\left\Vert {\left({\left\Vert \mathbf{v}{\widehat{\mathbf{p}}}^{-1}\right\Vert}_2\right)}^{-1}\left(\mathbf{v}{\widehat{\mathbf{p}}}^{-1}\right)-{\left(\sqrt{n}\right)}^{-1}\mathbf{e}\right\Vert}_2 $$

(except for the case where n = 2).¹³ If $\overline{\rho}=1$, then ${\mathbf{p}}^{*}=\mathbf{p}{\left[\widehat{\mathbf{p}}(1)\right]}^{-1}$, ${\mathbf{v}}^{*}=\mathbf{v}{\left[\widehat{\mathbf{p}}(1)\right]}^{-1}$ and, in general, $d\left(1,\rho \right)\ne {d}_{\mathbf{p}}\left(\rho \right)$. Finally, at $\rho =\overline{\rho}$ we obtain ${\mathbf{p}}^{*}=\mathbf{e}$ and, therefore, $d\left(\overline{\rho},\overline{\rho}\right)={d}_{\mathbf{v}}\left(\overline{\rho}\right)$ measures the deviation of the original labour values from the original prices corresponding to $\overline{\rho}$ (and not vice versa).¹⁴

The empirical results provided by Mariolis and Soklis (2011, pp. 616–617) are associated with input-output data from the Swedish economy for the year 2005 (n = 50), where the actual value of ρ is approximately equal to 0.520 or, if wages are paid ex ante, 0.368 ($R\cong 0.807$), and the distances (i) $d\left(0,\rho \right)={d}_{\mathbf{p}}\left(\rho \right)$, (ii) d(1, ρ), (iii) $d\left(\overline{\rho},\rho \right),\ \overline{\rho}=0.8$ (for instance) and (iv) d _v (ρ). Those results suggest that:

• (i) The distances increase with $\rho, 0\le \rho \le 1$. Nevertheless, they tend to be close to each other for ‘low’ values of ρ. For instance, at the actual value of the relative profit rate, ρ ^a, they are in the range of 0.184–0.216, i.e.

  $$ {d}_{\mathbf{p}}\left({\rho}^{\mathrm{a}}\right)\cong 0.184<{d}_{\mathbf{v}}\left({\rho}^{\mathrm{a}}\right)\cong 0.202<d\left(0.8,{\rho}^{\mathrm{a}}\right)\cong 0.215<d\left(1,{\rho}^{\mathrm{a}}\right)\cong 0.216 $$

  or, if wages are paid ex ante, in the range of 0.127–0.144, i.e.

  $$ {d}_{\mathbf{p}}\left({\rho}^{\mathrm{a}}\right)\cong 0.127<{d}_{\mathbf{v}}\left({\rho}^{\mathrm{a}}\right)\cong 0.134<d\left(1,{\rho}^{\mathrm{a}}\right)\cong 0.141<d\left(0.8,{\rho}^{\mathrm{a}}\right)\cong 0.144 $$

• (ii) As expected, the curve d _v (ρ) intersects the curves d(1, ρ) and d(0.8, ρ) at ρ = 1 and ρ = 0.8, respectively. The curve d(1, ρ) also intersects the curves d(0.8, ρ), d _v (ρ) and d _p (ρ) at $\rho \cong 0.470$, $\rho \cong 0.075$ and $\rho \cong 0.057$, respectively.

It emerges, therefore, that the ranking of these numeraire -free measures is a priori unknown (even when the measures are monotonic functions of the relative profit rate).

It has been shown that in order to construct a measure of price-value deviation, which does not depend on the choice of numeraire and physical measurement units, it suffices to transform the price-wage-profit system via a diagonal similarity matrix formed from the elements of the price vector corresponding to a value of the relative profit rate. This implies that there exists an infinite number of such measures, whose ranking is a priori unknown, and the choice between them depends either on the theoretical viewpoint or the aim of the observer. Schematically speaking, we could say that observers thinking in Marxian terms would prefer to use d _p (ρ), i.e. the d – distance , while those thinking in ‘Marx after Sraffa’ terms would prefer to use d _v (ρ), since the determination of prices is ‘logically prior to any determination of value magnitudes’ (Steedman 1977, p. 65), or, more generally, $d\left(\overline{\rho},\rho \right)$, with $\overline{\rho}>0$.