Abstract
Recently, Schefold proposed a new approach to the transformation problem consisting of two contentions. One is that input matrices of large-scale economies can be approximated by positive rank one matrices. The second is that additional original assumptions about the relationship among input coefficients, labor coefficients, gross outputs, and surplus outputs can establish the equality of the total profits and total surplus value under a numéraire equalizing the total production prices and total value. The purpose of this study is to critically examine the second part of Schefold’s argument. First, we will confirm that it is an attempt to give plausible grounds to a condition that has been known as sufficient for the successful solution of the transformation problem but considered to hold only by chance. Next, we will indicate that, except for a supposition about the rank of input matrices, his key assumptions depend on the measurement units of each product. Because this dependence implies that these assumptions hold only when measurement units fill particular conditions, it naturally casts a serious doubt on the generality of the analysis based on them. While it is possible to combine these assumptions into a unit-independent form, such a reformulation deprives them of the meaning attached to their original form. Thus, in spite of its unique viewpoint, Schefold’s new approach does not succeed in bringing the value system closer to the production price system.
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Notes
On the concept and properties of “random system”, see also Schefold (2013, pp. 1171–1179).
Our notation of variables mostly follows that of Schefold. In Schefold (2013), he calls vector f “the composition of capital” and vector c “the distribution of capital over industries” (pp. 1176, 1187).
Let n be the number of sectors, then the characteristic equation of matrix A = cf is given by \( \lambda^{n - 1} \left( {\lambda - \mu } \right) = 0 \).
Here we disregard problems related to heterogeneous labor, joint production, fixed capital, and choice of techniques and so on. None of them is relevance to our argument below.
If wage is assumed to be paid after production and prices are normalized by net-product vector, then we have Sraffa’s famous formula \( w = 1 - r/R \) (Sraffa 1960).
Under general input matrices, \( P = M \) holds if and only if \( \left( {{\mathbf{yt}}} \right)\left( {{\mathbf{sq}}} \right) - \left( {{\mathbf{st}}} \right)\left( {{\mathbf{yq}}} \right) = 0 \), where \( {\mathbf{q}} = \left( {{\mathbf{I}} - \left( {1 + r} \right){\mathbf{A}}} \right)^{ - 1} {\mathbf{l}} = {\mathbf{p}}/\left( {\left( {1 + r} \right)w} \right) \). In order for q to be proportional to \( {\mathbf{p}} \) for \( r > 0 \), l must be \( {\mathbf{A}} \)’s right-hand Frobenius vector. If \( w = {\mathbf{dp}} \), then \( P = M \) is equivalent to \( {\mathbf{y}}\left( {{\mathbf{A}} + {\mathbf{ld}}} \right)\left( {{\mathbf{p}} - {\mathbf{p}}_{0} } \right) = 0 \). This can be satisfied even if \( {\mathbf{y}}\left( {{\mathbf{A}} + {\mathbf{ld}}} \right) \) is not proportional to \( {\mathbf{y}} \) so long as \( {\mathbf{y}}\left( {{\mathbf{A}} + {\mathbf{ld}}} \right) \) is orthogonal to \( {\mathbf{p}} - {\mathbf{p}}_{0} \).
Schefold defines vector \( {\mathbf{m}} \) by \( {\mathbf{m}} = {\mathbf{y}} - {\mathbf{q}}_{1} = {\mathbf{q}}_{2} + \cdots + {\mathbf{q}}_{n} \), where \( {\mathbf{q}}_{1} \) is \( {\mathbf{A}} \)’s left-hand Frobenius vector and \( {\mathbf{q}}_{2} \), …, \( {\mathbf{q}}_{n} \) are other eigen vectors corresponding to eigenvalue 0 (Schefold 2016, p. 172). Since \( {\mathbf{yA}} = {\mathbf{q}}_{1} {\mathbf{A}} = \mu {\mathbf{q}}_{1} \), we have \( {\mathbf{q}}_{1} = \left( {{\mathbf{yc}}/\mu } \right){\mathbf{f}} \), \( {\mathbf{m}} = {\mathbf{y}} - \left( {{\mathbf{yc}}/\mu } \right){\mathbf{f}} = {\mathbf{yB}}/\mu \). In a similar manner, \( {\mathbf{v}} = {\mathbf{Bl}}/\mu \) is derived from \( {\mathbf{v}} = {\mathbf{l}} - {\mathbf{x}}_{1} = {\mathbf{x}}_{2} + \cdots + {\mathbf{x}}_{n} \).
This definition of \( {\text{cov }}\left( {{\mathbf{m}},{\mathbf{v}}} \right) \) is given in Schefold (2013, p. 1172).
In Schefold (2013, p. 1172), he explicitly supposes that “the components of the m and v as independent random variables with small means” (emphasis in original), and derives \( {\text{cov }}\left( {{\mathbf{m}},{\mathbf{v}}} \right) = 0 \) from this supposition.
Note that Schefold does not assume \( {\bar{\text{m}}} = 0 \). In fact, he emphasizes that “our solution of the transformation problem should not be confused with that relying on standard proportions. It is therefore essential that the deviations m are not assumed to vanish” (Schefold 2016, p. 172).
Measurement units of products should not be confused with units of time, distance, weight etc. When a product is measured by its length (like the case of sewing thread), the adoption of the metric system does not determine one unit of a particular kind of thread.
Although Schefold thinks “it seems more plausible that \( {\bar{\text{v}}} \) tends to zero than \( {\bar{\text{m}}} \)” (2013, p. 1172), as we have shown in the text, under a positive rank one input matrix, it is always possible to choose a set of measurement units satisfying \( {\bar{\text{m}}} = 0 \).
Both Marx and Sraffa paid great attention to the difference of positions occupied by each product (commodity) in the input–output structure. In the case of Sraffa, this concern is reflected in his concept of basic and non-basic commodities (Sraffa 1960). Assumption \( {\mathbf{f}} > 0 \) implies that there is no such difference of positions. In this economy, products can be different only in that how much they are demanded for final consumption.
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Morioka, M. The transformation problem under positive rank one input matrices: on a new approach by Schefold. Evolut Inst Econ Rev 16, 303–313 (2019). https://doi.org/10.1007/s40844-019-00144-2
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DOI: https://doi.org/10.1007/s40844-019-00144-2