Appendix 5.A: Further testing of the randomness hypothesis

The lack of proximity between the actual output vector x with the standard output, s, the employment coefficients, 1 and the l.h.s. eigenvector, Я, of the matrix A pave the way to examine the extent to which the matrices A or H under study are random or not. Following Schefold (2019, 2020) and also his discussion with Mori (2019) and Morioka (2019), the randomness hypothesis requires the following condition: the correlation and covariance coefficients of the vectors m=s—x and v = n — 1 are equal to each other and both are equal to zero. That is,

The testing was for the years 2007 and 2014 in the case of circulating capital model using data from the WIOD (2016). We did not test the fixed capital model because the matrix of the capital stock coefficients, in our case, by construction has a rank equal to one, and so all eigenvalues except the first are zero. This is a special case of randomness, because random matrices ge- nerically have full rank and their eigenvalues are small, but not all zero. This means that in our fixed capital model, the paths of PP are linear, and the same is true with the capital—output ratios.

Our data gave the following correlation coefficients and covariance of vectors m and v, along with the t-ratio, for the year 2007

Similar were the results for the year 2014; in particular, we got

Clearly, the correlation coefficients and the covariances are not the same; hence, the requirement for the randomness hypothesis of matrix A is violated. The intuitive idea is that if m and v are totally unrelated, then it follows that the elements of the matrix A or H are randomly distributed. If, however, they are related, it follows that the elements of the matrix A or H are characterized by some patterns indicative of the presence of regularities. For example, the columns of these matrices are close to each other or they are multiples, as a result the row vectors n and 1 are too close to each other and the same is true with the column vectors s and x. If there is no correlation then there is no covariance; however, the converse is not true. The idea is that the correlation coefficient is independent of the normalization condition, or the scaling of the vectors in comparison, whereas the covariance depends on both the normalization and the scaling factor.

In the next Figure 5.A1, we display the m and v vectors along with the R-square provided that the slopes are statistically significant, and the intercept in both cases are no different than zero.

At OLS regressions between m and v, 2007 and 2014

Figure 5.At OLS regressions between m and v, 2007 and 2014.

The results indicate a rather weak but certainly not zero correlation between the two vectors in comparison and certainly, the covariance although is not different from zero, nevertheless, is dependent on the normalization condition. In any case, since correlation and covariance are not equal, they cast doubt on the randomness hypothesis. Perhaps there is need for more testing from other countries and years before something more definitive is stated.

It is worth noting that the randomness hypothesis is also under question by the fact that the empirical findings show that the second and the third eigenvalues of the system matrices that we examine in the next chapter get closer rather than further from the maximal eigenvalue. In our view, the quasilinearities of PRP and WRP are of paramount importance in this research and the explanation for that is to be found in the structural characteristics and the repeated patterns, to the extent that they exist, in the economic system matrices. Torres-Gonzalez and Yang (2018) argue that intertemporal changes of the technical coefficients are not random but follow deterministic trends. We grapple with this issue along with related others in the next chapters.


1 The capital stock, in principle, should be augmented to become capital advanced, that is, to include circulating capital and wages advanced. Both require the estimation of turnover times. The interested reader in these estimations may find details in Ochoa (1984, 1989) and Shaikh (1998, 2016). Finally, the fixed capital, in principle, must be adjusted by the degree of capacity utilization. We need not argue how demanding are the data for the estimation of both, turnover times and capacity utilization for all industries in a single year, let alone many years and countries. However, the difficulties in estimations should not discourage the empirical research whose results, in our view, would be strengthened by using the missing variables.

  • 2 It is important to note that J. Robinson in her response to a question by Sche- fold indicated precisely this scenario (Kersting and Schefold 2020). Salvadori and Steedman (1988) have argued that if all techniques are [near] linear, then one of them will necessarily dominate over all the others. In this case, the frontier is a single linear system with relative prices equal to relative labor values (Shaikh 201б" p. 431).
  • 3 The same vectors have been invoked to support the hypothesis of randomness (Schefold 2013, 2019, 2020) that we subject to a preliminary empirical testing in the appendix to this chapter.
  • 4 For the methods of construction of the matrix of fixed capital per unit of output coefficients, see Appendix 4.A.
  • 5 Capital deepening paradox in the capital theory refers to the property according to which it might be efficient to have a lower (higher) capital—labor ratio and a lower (higher) rate of profit. This is inconsistent with the neoclassical theory according to which prices reflect relative scarcity and so if the capital—labor ratio or intensity decreases, it follows that the rate of profit increases and vice versa.
  • 6 The rate of profit in Han and Schefold (2006) varies reasonably within the low range, whereas in Zambelli et al. (2017), the maximum rate of profit is 250%! Which is quite distant from the reality of the standard ratio for circulating capital in the range of 100%.
  • 7 After all, we should bear in mind that the marginal physical productivity theory of income distribution, as advanced by J.B. Clark and other major neoclassical economists, sought to sanctify the origins of profit income on sources other than exploitation of labor. The marginal physical productivity of income distribution was particularly appealing as a first-rate explanation of profit income derived from capital’s marginal physical productivity. In short, capital is placed on par with labor as being a factor of production contributing to the total output produced and its reward for that is equal to its marginal physical contribution (see Chapter 3 for details).
  • 8 The use of value of output together with constant income shares renders the production function to a tautological equation. For further discussions on this issue, and the “humbug” production function, see Shaikh (1974, 1981, 1990, 2005, 2016) as well as Solow’s (1974) response while his article with the estimation of production function (1957). The issues with the estimations of production functions are also discussed by Felipe and McCombie (2013).
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