Testing Böhm-Bawerk’s theory of capital: Some evidence from the Finnish economy

Abstract

This paper considers the pure labour theory of value and Böhm-Bawerk’s theory of capital as approximations of Sraffa’s model of single production, and tests them with data from the Symmetric Input-output Tables of the Finnish economy. The results show that (i) in comparison with the labour values, the actual Böhm-Bawerkian production prices are ‘equally’ good or even better approximations of the actual Sraffian production prices and market prices; and (ii) the Sraffian production price-profit rate relationship is, by and large, governed by the differences in the Böhm-Bawerkian average periods of production.

Appendix: A Note on the Steedman-Tomkins distance

Consider two price vectors, ${{\chi }^{{\text{T}}}} \equiv \left[ {{{\chi }_{i}}} \right]$ and ${{\psi }^{{\text{T}}}} \equiv \left[ {{{\psi }_{i}}} \right]$, i = 1, 2,…,n, corresponding to the same production technique [A, L]. The angle, θ, between ${{{\text{X}}}^{{\text{T}}}} \equiv {{\chi }^{{\text{T}}}}{{\widehat{\psi }}^{{ - 1}}} = \left[ {{{\chi }_{i}}{{\psi }_{i}}^{{ - 1}}} \right]$ and ${{\Psi }^{{\text{T}}}} \equiv {{\psi }^{{\text{T}}}}{{\widehat{\psi }}^{{ - 1}}} = {{{\text{e}}}^{{\text{T}}}}$, is determined by

$$ \cos \theta = {{{\left( {{{\text{{\text X}}}^{\text{T}}}{\text{e}}} \right)}} \left/ {{\left( {\left\| {\text{{\text X}}} \right\|\left\| {\text{e}} \right\|} \right)}} \right.} $$

where ||∙|| denotes the Euclidean norm of ∙. When, for example, χχ^T, ψ ^T represent the vectors of Sraffian production prices and labour values, respectively, θ measures the degree to which relative Sraffian production prices deviate from relative labour values. Now, let d be the Steedman-Tomkins distance, i.e., the Euclidean distance between the unit vectors ${\text{X}}\prime \equiv {\text{{\text X}}}{\left\| {\text{{\text X}}} \right\|^{{ - 1}}}$ and $\Psi \prime \equiv \Psi {\left\| \Psi \right\|^{{ - 1}}} = {\text{e}}{\left( {\sqrt {n} } \right)^{{ - 1}}}$. Then

$$ d \equiv \left\| {{\text{X}}\prime - \Psi \prime } \right\| = \sqrt {{2\left( {1 - \cos \theta } \right)}} $$

(Α.1)

constitutes a measure of the deviation between χχ^T and ψ ^T, which is independent of the choice of numeraire and physical measurement units.

On this basis, we may note the following:

• (i). When all but one of the elements of X tend to zero, cosθ tends to its theoretically minimum value of ${\left( {\sqrt {n} } \right)^{{ - 1}}}$ and the Steedman-Tomkins distance tends to its maximum value of $D \equiv \sqrt {{2\left[ {1 - {{\left( {\sqrt {n} } \right)}^{{ - 1}}}} \right]}}$. Thus, in our case, where n = 57 and, therefore, $D \cong 1.317$, the ‘normalized Steedman-Tomkins distances’ (Mariolis and Soklis 2010, p. 94), defined as $d{D^{{ - 1}}}$, are in the range of 0.052–0.256, for the year 1995, and 0.053–0.279, for the year 2004 (consider Table 2).

• (ii). As it has recently been pointed out, there exists an infinite number of distances à la Steedman-Tomkins, whose ranking is a priori unknown, and the choice between them depends either on the theoretical viewpoint or on the aim of the observer (Mariolis and Soklis 2011). Nevertheless, if one, in accordance with most of the empirical studies on this topic, pay regard to measure the extent to which an actual (or accepted) price vector (e.g., p ^aT) deviates from its approximation (e.g., ${\text{p}}_{\text{A}}^{\text{aT}}$ or v(0)^T), then the Steedman-Tomkins distance (defined by the equation (A.1)) is adequate (see Mariolis and Soklis 2011, p. 615).

• (iii). The traditional measures of price deviations, such as the ‘mean absolute deviation’, i.e., ${n^{{ - 1}}}\sum\limits_{{i = 1}}^n {\left| {\left( {{x_i}{y_i}^{{ - 1}}} \right) - 1} \right|}$, the ‘mean absolute weighted deviation’ and the ‘root-mean-square-percent-error’ (see, e.g., Petrović 1987, and Ochoa 1989), are unit-free. However, they are not numeraire-free. Thus, there is a good reason for preferring the Steedman-Tomkins distance (for an exploration, both theoretical and empirical, of the relationships between all these measures, see Mariolis and Tsoulfidis 2010).