The transformation problem: a mathematical approach of its solution within Marx’s original framework

Abstract

This study demonstrates that the transformation problem concerns whether a contradiction exists between Marx’s concept of labor value and that of production price. In a narrow sense, this is just a mathematical problem. It can be boiled down to the construction of a mathematical model that can simulate the conversion from value to production price to conclude whether the constructed model meets certain constraints (so-called “the double invariance”). However, models of most researchers so far met only one of the two constraints, some of them couldn’t satisfy any one of them. So, they tried to explain this problem by modifying Marx’s conditions or conclusions, and hence doubted or even denied Marx’s conclusions. This was actually an act of self-deception. In contrast to related attempts that tampered with Marx’s transformation conditions or conclusions, this study is based on the solution of the transformation problem within Marx’s original framework. It extends Bortkiewicz (Third volume of Capital, Sweezy, 1907) and Samuelson (Am Econ Rev 47:884–912, 1957, Proc Natl Acad Sci 67:423–425, 1970, J Econ Lit 9:399–431, 1971) to overcome their shortcomings. This study provides the model of static transformation and that of dynamic transformation and illustrates them with examples of mathematical simulation: the static transformation is the result of the dynamic transformation and its end point. Both the general dynamic transformation model and the general static transformation model satisfy two constraints of Marx simultaneously. So, this study offers a new mathematical approach to the transformation problem. The author obtained a solution to the transformation problem in 2000 (a static version in line with Marx’s original intention), continued to conduct research, and published several research results. However, so far, the author’s publication was mainly in Chinese and Japanese and did not receive due attention in international academia. As I received the invitation to the Marx special feature of from this journal (*This plan was not realized. Some of contributed papers to this plan were published in this journal’s special feature “Promenade in the history of economic thought” (vol. 18–1). [Associate Editor]), I endeavored to describe my research in this topic deeper.

1 The transformation problem and the basic principles for the construction of its model

The so-called transformation problem concerns whether a contradiction exists between the concept of labor value and that of production price in Marx’s theoretical system.

Scholarly controversies over this problem have ranged very wide. Yet, in a narrow sense, the true nature of the transformation problem is a mathematical issue. It can be boiled down to the construction of a mathematical model that can simulate the conversion from value to production price, and then to conclude whether this model meets certain constraints (so-called “the double invariance”: (1) total production price = total value; (2) total average profit = total surplus value).

Hitherto, researchers of Marxian economics have constructed a large number of mathematical models to deal with this problem. Nevertheless, prior to Zhang (2000), most of their models satisfied only one of the two constraints—some satisfied none of two.¹ They had to modify the constraints, or conclusion of Marx, to get out of the impasse. As a result, the research on this problem became complicated whilst moving farther and farther away from Marx’s original intention.

We would not say that the efforts of researchers were totally wasted. The research has gradually approached to the correct path of the solution. Of the many literatures, the most important contributions were Bortkiewicz (1907), Samuelson (1957, 1970, 1971), and Zhang (2000). In the next section, we will follow them in chronological order to clarify the process to approach to the correct path.

From the perspective of mathematics, the construction of models of the transformation problem needs to observe a number of basic principles. Put specifically, the following questions need to be categorically answered:

• 1) Whether or not to adhere to Marx’s original framework for studying the transformation problem? A choice to be made, first of all.

• 2) How should mathematical conditions be set, as prerequisites for building a model of the transformation problem, without losing generality?

• 3) How to distinguish between endogenous variables and exogenous variables?

• 4) How to establish the mathematical relation between value and production prices?

• 5) How to construct a system of equations that reflects the relationship between value and production price?

• 6) How to deal with the relationship between the above equations and the constraint conditions of the transformation problem?

These questions will be answered below in Sect. 3 of the paper.

In Sect. 4, we will discuss several issues related to the transformation problem. In particular, we provide a model of inverse transformation. We will also mathematically give the static transformation model (the relation between value and production price in a certain year) and the dynamic transformation model (simulating the process of transformation from value to production price over several years²).

In Sect. 5, we provide a scheme to unify static transformation and dynamic transformation from a mathematical point of view. Section 6 gives a brief conclusion.

2 Explorations and misconceptions in the construction of transformation models

Regarding the conversion of value to production price, Marx provided a transitional calculation in the third volume of Capital.³ Marx’s starting point was the value system:

$$c_{i} + v_{i} + m_{i} = w_{i} \quad \left( {i = 1,2, \ldots ,n} \right)$$

(1)

Here, ${c}_{i}$, ${v}_{i}$, ${m}_{i}$, and ${w}_{i}$ represent constant capital, variable capital, surplus value, and total value of the i^th department under the value system, respectively.

Marx’s production price system is derived as

$$\left( {1 + r} \right)\left( {c_{i} + v_{i} } \right) = P_{i} \quad \left( {i = 1,2, \ldots ,n} \right)$$

(2)

Here, ${P}_{i}$ represents the total production price of the ith department, and r is the average profit rate. Marx wrote production price system as follows:

$$r = \frac{{\sum\limits_{i = 1}^{n} {m_{i} } }}{{\sum\limits_{i = 1}^{n} {\left( {c_{i} + v_{i} } \right)} }}$$

(3)

In this way, the calculation of r in formula (1) becomes extremely simple. Table 1 presents a specific example of this calculation. Because the cost price part (${c}_{i}+{v}_{i}$) in formula (2) is not converted into production price, this way of calculation is incomplete. Therefore, Zhang (2004) called (2) the “half- transformation formula”.

Table 1 Marx’s half-transformation: an example

In the last row of Table 1, we see that the sum of production price is equal to the sum of value, and the sum of average profit is equal to the sum of surplus value. Whether these two equivalences can be established is the condition for judging whether Marx’s concept of the transformation from value to production price is sustainable or not.

However, though Marx was conscious of the incompleteness of his transitional transformation (because the cost part is not converted into production price), he did not provide any further solution.

The mathematical refinement of Marx’s transformation method—i. e. the attempt to convert the cost price part into production prices—was initiated by Ladislaus von Bortkiewicz in 1907.

Bortkiewicz (1907) constructed a transformation model based on Marx’s simple reproduction theory. However, he erroneously expanded Marx’s two departments into three which resulted in incorrect balances. These errors reflected in his failure of the conversion of value into the production price.

One of Bortkiewicz’s main contributions was the establishment of a link between value and production price through the deviation rate of production price from value. Following the symbols in the translation by Sweezy (1949),⁴ we assume that the means of production, workers’ consumption goods, and capitalists’ consumption goods are produced in Department I, II, and III respectively.

Bortkiwicz believed that, under the condition of simple reproduction, following equilibrium relations must hold⁵:

$$\left. \begin{gathered} c_{1} + v_{1} + s_{1} = c_{1} + c_{2} + c_{3} \hfill \\ c_{2} + v_{2} + s_{2} = v_{1} + v_{2} + v_{3} \hfill \\ c_{3} + v_{3} + s_{3} = s_{1} + s_{2} + s_{3} \hfill \\ \end{gathered} \right\}$$

(4)

Suppose that the relationship between the price and the value of the products is (on average) $x$ for Department I, $y$ for Department II, and $z$ for Department III. Furthermore, let $\rho$ be the profit rate that is common to all departments; it is also called the average profit rate.

Then, the following equations should hold:

$$\left. \begin{gathered} \left( {1 + \rho } \right)\left( {c_{1} x + v_{1} y} \right) = \left( {c_{1} + c_{2} + c_{3} } \right)x \hfill \\ \left( {1 + \rho } \right)\left( {c_{2} x + v_{2} y} \right) = \left( {v_{1} + v_{2} + v_{3} } \right)y \hfill \\ \left( {1 + \rho } \right)\left( {c_{3} x + v_{3} y} \right) = \left( {s_{1} + s_{2} + s_{3} } \right)z \hfill \\ \end{gathered} \right\}$$

(5)

Currently, the number of unknowns is four, while there are only three equations. There are two ways to solve this problem: adding an equation or reducing an unknown. Hence, Bortkiwicz considered that, if we were to choose an appropriate price unit in such a way that the total price and total value become equal, we must set

$$Cx + Vy + Sz = C + V + S$$

(6)

where

$$C = c_{1} + c_{2} + c_{3} ,\quad V = v_{1} + v_{2} + v_{3} ,\quad S = s_{1} + s_{2} + s_{3} .$$

If, on the other hand, the price unit and the value unit are to be regarded as identical, then we have to consider in which of the three departments the good which serves as the value and price unit (numeraire) is produced. If gold is the good in question, then Department III is involved and in place of (6), we get $z = 1$. If we follow this procedure, the number of unknowns is reduced to three (x, y, and $\rho$).⁶

Therefore, the new equations are as follows:

$$\left. \begin{gathered} \left( {1 + \rho } \right)\left( {c_{1} x + v_{1} y} \right) = \left( {c_{1} + c_{2} + c_{3} } \right)x \hfill \\ \left( {1 + \rho } \right)\left( {c_{2} x + v_{2} y} \right) = \left( {v_{1} + v_{2} + v_{3} } \right)y \hfill \\ \left( {1 + \rho } \right)\left( {c_{3} x + v_{3} y} \right) = s_{1} + s_{2} + s_{3} \hfill \\ \end{gathered} \right\}$$

(7)

Then, he defined $\sigma = 1 + \rho ;\;f_{i} = \frac{{c_{i} }}{{v_{i} }},g_{i} = c_{i} + v_{i} + s_{i} \left( {i = 1,2,3} \right)$, from which he obtained the general solution of the equation system as

$$x = \frac{{\sigma f_{1} y}}{{g_{1} - \sigma }}\quad y = \frac{{g_{3} }}{{g_{2} + (f_{3} - f_{2} )\sigma }}.$$

However, when Bortkiewicz used data to test the model, he found that not all data can satisfy the two constraints (“double invariance”). Therefore, Bortkiewicz concluded that Marx’s transformation is valid only under certain special conditions and cannot be generalized. In fact, he denied the validity of the transformation problem.

We can find several defects in the Bortkiewicz model, but the largest consists in his confusion of the roles of constant and variable capital. This error was so subtle that even Morishima (1973) could not notice it.⁷ Seton (1957) went further in this mistake and no longer distinguished constant capital from variable capital.

After Bortkiewicz (1907), although many scholars made various attempts on the transformation model, no substantial progress was made untilSamuelson (1957).⁸ It was the first that pointed out the abovementioned error in the Bortkiewicz model.

To clarify this we take two departments model. We can represent the assumption of Bortkiewicz (1907), Morishima (1973, 1978), and others, in the style of the input–output Table 2 as follows:

Table 2 Misunderstanding of the exchange between two departments (Bortkiewicz, Morishima, etc.)

Samuelson (1957) noticed the problem mentioned above and made following corrections:

Based on the new table, Samuelson constructed equations in production price of his two-department model as follows:

$$\left. \begin{gathered} p_{1} K = \left( {wL_{1} + p_{1} K_{1} } \right)\left( {1 + r} \right) \hfill \\ p_{2} Y = \left( {wL_{2} + p_{1} K_{2} } \right)\left( {1 + r} \right) \hfill \\ \end{gathered} \right\}.$$

Here, Department I produces homogeneous capital goods $K$, and Department II produces homogeneous consumer goods $Y$. $p_{1}$,$p_{2}$ represent the production prices of the two departments. $r$ is the average profit rate, and $w$ is the wage rate (Table 3).

Table 3 Samuelson’s correction of the expression of the exchange relationship between two departments

Samuelson’s idea was correct. He expanded this logic into a system of production price equations of n departments. This system of principal equations is also correct. However, in his paper in 1970, he first constructed value equations of n departments before formulating price equations of n departments. He wrote the row vector ${\varvec{\uppi}}={({{\varvec{\uppi}}}_{j})}_{1\times n}=({{\varvec{\uppi}}}_{1},{{\varvec{\uppi}}}_{2},\dots ,{{\varvec{\uppi}}}_{n})$ to denote the commodity value vector,⁹${\mathbf{a}}_{0}={({\mathbf{a}}_{0j})}_{1\times n}=({\mathbf{a}}_{01},{\mathbf{a}}_{02},\dots ,{\mathbf{a}}_{0n})$ to denote the direct labor consumption vector, and $\mathbf{a}={({\mathbf{a}}_{ij})}_{n\times n}$ to denote the material (dead labor) consumption coefficient matrix. Therefore, he believed that Marx’s value formula in the first volume of Capital is written as.¹⁰

$${\varvec{\uppi}}=W{\mathbf{a}}_{0}+{\varvec{\uppi}}\mathbf{a}+sW{\mathbf{a}}_{0}.$$

(8)

Here, $s$ is the rate of surplus value, and W is the wage rate. Furthermore, he denoted ${\mathbf{A}}_{0} (0)$ as the vector of all labor consumption coefficients¹¹; then,

$${\mathbf{A}}_{0} (0) = {\mathbf{a}}_{0} ({\mathbf{I}} - {\mathbf{a}})^{ - 1} = \left( {{\mathbf{a}}_{01} ,{\mathbf{a}}_{02} , \ldots ,{\mathbf{a}}_{0n} } \right)\left[ {\begin{array}{*{20}c} {1 - {\mathbf{a}}_{11} } & { - {\mathbf{a}}_{12} } & \cdots & { - {\mathbf{a}}_{1n} } \\ { - {\mathbf{a}}_{21} } & {1 - {\mathbf{a}}_{22} } & \cdots & { - {\mathbf{a}}_{2n} } \\ \vdots & \vdots & \vdots & \vdots \\ { - {\mathbf{a}}_{n1} } & { - {\mathbf{a}}_{n2} } & \cdots & {1 - {\mathbf{a}}_{nn} } \\ \end{array} } \right]^{ - 1}$$

Since $W{\mathbf{a}}_{0}{(\mathbf{I}-\mathbf{a})}^{-1}\left(1+s\right)=W{\mathbf{A}}_{0}(0)\left(1+s\right)$, the following system is obtained by combining (8):

$$\left. \begin{gathered} {{\varvec{\uppi}}} = W{\mathbf{A}}_{0} \left( 0 \right)\left( {1 + s} \right) \hfill \\ {\mathbf{\pi m}} = W \hfill \\ \end{gathered} \right\}$$

(9)

Here, ${\mathbf{m^{\prime}}} = \left[ {m_{i} } \right] = \left( {m_{1} ,m_{2} , \ldots ,m_{n} } \right)^{\prime }$ is the column vector of minimum-subsistence goods needed as real wage to cover the cost of production and reproduction of labor.¹²

Since ${\varvec{\uppi}}\mathbf{m}=W$, ${\mathbf{A}}_{0} (0)(1 + s){\mathbf{m}} = 1$, and, thus,

$$s = \frac{1}{{{\mathbf{A}}_{0} (0){\mathbf{m}}}} - 1$$

Then, because ${\varvec{\uppi}}={\varvec{\uppi}}\mathbf{m}{\mathbf{A}}_{0}(0)\left(1+s\right)$, and $\frac{1}{1+s}$ is the eigenvalue of $\mathbf{m}{\mathbf{A}}_{0}(0)$, ${\varvec{\uppi}}$ is its eigenvector, and (9) is solvable. Hence, Samuelson believed that the physical quantities ${\mathbf{a}}_{0}$, $\mathbf{a}$, and $\mathbf{m}$ determine the rate of surplus value s and the commodity value vector ${\varvec{\uppi}}$ through ${\varvec{\uppi}}=W{\mathbf{a}}_{0}+{\varvec{\uppi}}\mathbf{a}+sW{\mathbf{a}}_{0}$ (although he did not discuss the uniqueness of the solution).

After using the physical quantity system to solve the problem of value determination, he constructed his n-department production price equations as follows:

$$\mathbf{P}=(W{\mathbf{a}}_{0}+\mathbf{P}\mathbf{a})(1+r)$$

(10)

Here, $r$ is the average profit rate, and ${\mathbf{P}} = \left[ {P_{i} } \right] = \left( {P_{1} ,P_{2} , \ldots ,P_{n} } \right)$ is the production price vector.

Regarding the solution of (10). Samuelson first considered how to solve for r. For this reason, he then he defined ${\mathbf{A}}_{0} (r)$ to be the vector of all labor consumption coefficients under the condition of average profit, namely,

$${\mathbf{A}}_{0} (r) = {\mathbf{a}}_{0} \left( {1 + r} \right)\left[ {{\mathbf{I}} - {\mathbf{a}}\left( {1 + r} \right)} \right]^{ - 1} = \left( {1 + r} \right)\left( {{\mathbf{a}}_{01} ,{\mathbf{a}}_{02} , \ldots ,{\mathbf{a}}_{0n} } \right)\left[ {\begin{array}{*{20}c} {1 - \left( {1 + r} \right){\mathbf{a}}_{11} } & { - \left( {1 + r} \right){\mathbf{a}}_{12} } & \cdots & { - \left( {1 + r} \right){\mathbf{a}}_{1n} } \\ { - \left( {1 + r} \right){\mathbf{a}}_{21} } & {1 - \left( {1 + r} \right){\mathbf{a}}_{22} } & \cdots & { - \left( {1 + r} \right){\mathbf{a}}_{2n} } \\ \vdots & \vdots & \vdots & \vdots \\ { - \left( {1 + r} \right){\mathbf{a}}_{n1} } & { - \left( {1 + r} \right){\mathbf{a}}_{n2} } & \cdots & {1 - \left( {1 + r} \right){\mathbf{a}}_{nn} } \\ \end{array} } \right]^{ - 1}$$

Since $W{\mathbf{a}}_{0}\left(1+r\right){[\mathbf{I}-\mathbf{a}\left(1+r\right)]}^{-1}=W{\mathbf{A}}_{0}(r)$, $\mathbf{P}\mathbf{m}=W$, so

$$\left. \begin{gathered} {\mathbf{P}} = W{\mathbf{A}}_{0} (r) \hfill \\ {\mathbf{Pm}} = W \hfill \\ \end{gathered} \right\}$$

(11)

Then, he used $\mathbf{P}\mathbf{m}=W$ to get ${\mathbf{A}}_{0} (r){\mathbf{m}} = 1$, so

$${\mathbf{a}}_{0} \left( {1 + r} \right)\left[ {{\mathbf{I}} - {\mathbf{a}}(1 + r)} \right]^{ - 1} {\mathbf{m}} = 1$$

(12)

Equation (12) is a one-dimensional higher-order equation with unknown $r$. By substituting its solution into $\mathbf{P}=W{\mathbf{A}}_{0}(r)$, ${\mathbf{P}}$ can be obtained.¹³ This shows that the production price is determined by a physical quantity system.

In short, Samuelson thought “the transformation problem” was merely “the problem of comparing and contrasting the mutually exclusive alternative of ‘value’ and ‘price’”.

His conclusion was that the so-called “transformation problem” is nothing but a problem of the transformation of “two alternative and discordant systems”: “Write down one. Now, transform by taking an eraser and rubbing it out. Then, we fill in the other one. Voila! You have completed your transformation algorithm. By this technique, one can ‘transform’ from phlogiston to entropy; from Ptolemy to Copernicus; from Newton to Einstein; from Genesis to Darwin—and from entropy to phlogiston”.¹⁴ Samuelson proved that the production price system can be separated from the value system and determined independently by the physical quantity system. Therefore, he believed that the transformation of value into production price is nothing but a “returning from the unnecessary detour”,¹⁵ thus completely denying the necessity and significance of the transformation of value into production price.

Samuelson then began to use his method to analyze the Bortkiewicz model. He rewrote the Bortkiewicz model as¹⁶

$$\left. \begin{gathered} y_{j} \pi_{j} = \left( {W{\mathbf{a}}_{0j} + \sum\limits_{i = 1}^{n} {y_{i} \pi_{i} {\mathbf{a}}_{ij} } } \right)\left( {1 + r} \right)\quad \left( {j = 1,2, \ldots ,n} \right) \hfill \\ \sum\limits_{j = 1}^{n} {y_{j} \pi_{j} m_{j} } = W \hfill \\ \end{gathered} \right\}.$$

(14)

However, (14) is precisely that of (11) with ${P}_{i}={y}_{i}{\pi }_{i}$, ${y}_{i}{={P}_{i}/\pi }_{i} \left(i=\mathrm{1,2},\dots ,n\right)$.

Although Samuelson partially corrected the main equations in the Bortkiewicz model, a fatal error remained—that is, Samuelson unreasonably assumes that the wage rate $W$ remained unchanged after the transformation of value into production price. This means that ${\varvec{\uppi}}\mathbf{m}=W=\mathbf{P}\mathbf{m}$. However, unless the vector ${\varvec{\uppi}}=\mathbf{P}$, ${\varvec{\uppi}}\mathbf{m}=\mathbf{P}\mathbf{m}$ is usually impossible.

Moreover, Samuelson forcibly replaced Marx’s two constraints (double invariance) with one constraint $\left( {\sum\limits_{j = 1}^{n} {y_{j} \pi_{j} m_{j} } = W} \right)$. Although this replacement has an approximate effect, it is not equivalent. Therefore, the production price derived from value using the Samuelson model cannot be accurate unless the exogenous data of $\mathbf{m}$ can meet certain conditions.

Therefore, except under special circumstances, the production price calculated according to the Samuelson model could not meet the two constraints of Marx. However, ironically, this calculation error made by Samuelson let him deny the Marxian transformation problem.

Despite the abovementioned errors, Samuelson’s model was the closest to the Zhang (2000) model. Let us discuss the Zhang model in detail below.

3 Construction and examples of static transformation model

In the first section of this paper, we listed 6 problems that need to be solved to construct a static transformation model. Zhang (2000) provides answers to them as follows:

• (1) We adhere to Marx’s original framework for studying the transformation problem.

• (2) The precondition of the static transformation model is to assume that the turnover rate of constant and variable capital is 1 year and that the technology remains unchanged.

• (3) The mathematical relation between value and production price is reflected by the deviation rate of the production price from the value.

• (4) The number of endogenous variables (unknowns) is n + 2, including the n deviation rates of production price to value, the deviation rate of wage rate under production price to the wage rate under value, and the average profit rate.

• (5) The main equations reflecting the relationship between value and production price after the transformation is constructed with reference to the input–output analysis.

• (6) The two constraints (the sum of production prices = the sum of values; the sum of average profit = the sum of surplus value) of Marx and the main equations mentioned above need to be combined to form a complete transformation model.

The symbols are specified below. First, let’s set signs concerning the value system.

Exogenous variables of the static transforming model:${c}_{i}$, ${v}_{i}$, ${m}_{i}$, and ${w}_{i}$ represent the constant capital, variable capital, surplus value, and total value of the ith department under the value system, respectively. Here, ${c}_{i}+{v}_{i}+{m}_{i}={w}_{i}(i=\mathrm{1,2},\dots ,n)$, which is Marx’s value Eq. (1). However, what ${c}_{i}$ represents here is only the sum of the constant capital used by the ith department. In reality, constant capital comes from various departments—that is, $c_{i} = \sum\limits_{j = 1}^{n} {c_{ij} } \quad \left( {i = 1,2, \ldots ,n} \right)$. Measuring unit of all exogenous variables in labor time.

Marx’s value system is presented in Table 4.

Table 4 Value system of n departments

Second, there are signs concerning the production price system. ${C}_{i}$, ${V}_{i}$, ${R}_{i}$, and ${P}_{i}$ represents the constant capital, variable capital, average profit, and total production price of the department under the production price system, respectively. Here, ${C}_{i}+{V}_{i}+{R}_{i}={P}_{i}(i=\mathrm{1,2},\dots ,n)$. Further, $C_{i} = \sum\limits_{j = 1}^{n} {C_{ij} } \quad \left( {i = 1,2, \ldots ,n} \right)$. Let $r$ represent the average profit rate; then, ${R}_{i}=r({C}_{i}+{V}_{i})(i=\mathrm{1,2},\cdots ,n)$. Thus,$\left(1+r\right)\left({C}_{i}+{V}_{i}\right)={P}_{i} (i=\mathrm{1,2},\dots ,n)$. This is the production price equation commonly used by Marx. Marx’s production price system is presented in the Table 5 below.

Table 5 The production price system of n departments

Then, we establish a mathematical connection between value and production price.

Endogenous variables of the static transforming model: the deviation rate of production price of the ith department to value is ${x}_{i}$, the deviation rate of variable capital under production price to variable capital under the value is y.¹⁷ Add the average profit rate $r$.and we obtain a total of $n+2$ endogenous variables. Obviously, ${C}_{ij}={x}_{j}{c}_{ij}$ $(i,j=\mathrm{1,2},\dots ,n)$, ${V}_{i}=y{v}_{i}$, ${P}_{i}={x}_{i}{w}_{i} (i=\mathrm{1,2},\dots ,n)$; thus,

$${x}_{i}{w}_{i}={P}_{i}=\left(1+r\right)\left({C}_{i}+{V}_{i}\right)=\left(1+r\right)\left(\sum_{j=1}^{n}{C}_{ij}+{V}_{i}\right)=\left(1+r\right)\left(\sum_{j=1}^{n}{x}_{j}{c}_{ij}+y{v}_{i}\right)$$

Here, ${P}_{i}={x}_{i}{w}_{i}$ clearly shows the conversion relationship of value ${w}_{i}$ to production price ${P}_{i}$.

If we consider ${x}_{i}$ $(i=\mathrm{1,2},\dots ,n)$, $y$, and $r$ as unknowns, we obtain the following n equations with n + 2 unknowns:

$$\left( {1 + r} \right)\left( {\mathop \sum \limits_{j = 1}^{n} c_{ij} x_{j} + v_{i} y} \right) = w_{i} x_{i} \quad \left( {i = 1,2, \ldots ,n} \right)$$

(15)

This is the main equation set of the general static transformation model proposed by Zhang at the 48th annual conference of the Japan Society of Political Economy in October 2000.¹⁸

According to the analysis above, as long as Marx’s two constraints (double invariance) are added to this main equation system, a general static transformation model is obtained. The first constraint (the sum of production prices = the sum of values) is relatively simple and can be expressed as:

$$\sum_{i=1}^{n}{w}_{i}{x}_{i}=\sum_{i=1}^{n}{w}_{i}$$

(16)

However, the second constraint can be expressed in three ways. The first shows that the sum of the average profit is the sum of the surplus value. This can be expressed as follows:

$$r\sum_{i=1}^{n}\left(\sum_{j=1}^{n}{c}_{ij}{x}_{j}+{v}_{i}y\right)=\sum_{i=1}^{n}{m}_{i}$$

(17)

The second shows that the total cost price under the value is equal to the total cost price under the production price. This can be expressed as follows:

$$\sum_{i=1}^{n}\left(\sum_{j=1}^{n}{c}_{ij}{x}_{j}+{v}_{i}y\right)=\sum_{i=1}^{n}\left(\sum_{j=1}^{n}{c}_{ij}+{v}_{i}\right)$$

(18)

Third, the average profit rate is determined by Marx’s formula for the average profit rate. This can be derived from formulas (17) and (18). The formula is as follows:

$$r=\sum_{i=1}^{n}{m}_{i}/\sum_{i=1}^{n}\left(\sum_{j=1}^{n}{c}_{ij}+{v}_{i}\right)$$

(19)

It should be noted that (19) is the same formula as (3) in this paper, but the denominator is expressed in a different way. Zhang’s (2000) model adopted the third mode of expression for the second constraint, while Zhang (2008) argued that the second constraint of the model is more standardized in the second mode of expression, because it enables us to omit the exogenous variable ${m}_{i}$ as well as make $r$ endogenous. Hence, the Zhang’s present transformation model is as follows:

$$\left. \begin{gathered} (1 + r)\left( {\sum\limits_{j = 1}^{n} {c_{ij} x_{j} + } v_{i} y} \right) = w_{i} x_{i} \begin{array}{*{20}c} {} & {(i = 1,2, \cdots ,n)} \\ \end{array} \hfill \\ \sum\limits_{i = 1}^{n} {w_{i} x_{i} } = \sum\limits_{i = 1}^{n} {w_{i} } \hfill \\ \sum\limits_{i = 1}^{n} {\left( {\sum\limits_{j = 1}^{n} {c_{ij} x_{j} + } v_{i} y} \right) = } \sum\limits_{i = 1}^{n} {\left( {\sum\limits_{j = 1}^{n} {c_{ij} + } v_{i} } \right)} \hfill \\ \end{gathered} \right\}$$

(20)

It can be proved that, under the premise that the exogenous variables can satisfy $\left( {1 + r} \right)c_{i} = \left( {1 + r} \right)\sum\limits_{j = 1}^{n} {c_{ij} < w_{i} } \quad \left( {i = 1,2, \ldots ,n} \right)$, the solutions of model (20) are unique and greater than zero.¹⁹

Need to emphasize, in Japan, scholars such as Makoto Itoh, one of the representatives of the Uno School, have always insisted that value and price belong to different dimensions. A fundamental divergence arises from this: is the transformation problem the problem of converting value (measured as labor time) to production price (measured in labor time) to production price (measured also in labor time) to price (measured in money)²⁰? In this regard, we think that, although “production price is the result of deviation from value”, its deviation is that of quantitative nature, and its essence is not changed. The essence of production price and value are the same. Both are measured in labor time.

It needs to be emphasized that the problem of the conversion of value to pecuniary price is another problem that is different from “the transformation problem”. Originally, price is the pecuniary expression of value, so the conversion of value to price is very simple. When value is transformed into production price, then the price becomes the pecuniary expression of production price. At this time, the problem of “the transformation of value to price” becomes the problem of value transformation to price (of the pecuniary expression of production price). If we import the concept of unit commodity value, we can also include these two different aspects of the transformation process through model (20) demonstrations or simulations. In particular, we can simulate that the matter of value transformation to price (of the pecuniary expression of production price), through the model (20).²¹ This result may be able to fit the disagreement between this paper and Itoh et al.

To avoid unnecessary misunderstandings, we emphasize that in Capital profit and its rate are terms applicable to both value and production price. In model (20), the average profit rate r exists as the solution of the model. It is an endogenous variable determined by the model and is not an exogenous independent variable.²²

Next, let us examine a famous example on the transformation problem that Bortkiewicz (1907) and Sweezy (1942) have both analyzed. Samuelson (1970, 1971) adopted Eqs. (9) and (11) under the assumption $r=\frac{1}{4}$, $W=1$ to derive the value system and the production price system, respectively. The data of the value system for this example are listed in Table 6.

Table 6 Bortkiewicz’s value system (Example)

Samuelson (1971) uses formula (11) to derive the production price system that is consistent with Sweezy’s (1942) calculation result of this example. For the sake of convenience, Samuelson reduced it by 15 /16. In the calculation result of Samuelson’s model, the total production price is 937.5, which is not equal to the total value of 875, and the total average profit of 187.5 is not equal to the total surplus value of 200 (as shown in Table 7).²³

Table 7 Samuelson’s production price system derived from Bortkiewicz’s value system

There is no doubt that Samuelson’s production price calculation was wrong, as we have already determined in the previous section.

Now, we use Zhang’s model and formula (11) to calculate the same example to see the result. For the convenience of calculation, Zhang first calculates the average profit rate $r=\frac{200}{375+300}=\frac{8}{27}$ according to formula (6) (note: r is not $\frac{1}{4}$ subjectively stipulated by Samuelson). In this way, by substituting the data into formula (11), the following equations are obtained:

$$\left(1+\frac{8}{27}\right)\left(225{x}_{1}+100y\right)=375{x}_{1}$$

$$\left(1+\frac{8}{27}\right)\left(90{x}_{1}+120y\right)=300{x}_{2}$$

$$\left(1+\frac{8}{27}\right)\left(60{x}_{1}+80y\right)=200{x}_{3}$$

$$375{x}_{1}+300{x}_{2}+200{x}_{3}=875$$

The solution of this equation system is ${x}_{1}=1.145$, ${x}_{2}=0.919$, ${x}_{3}=0.848$, and $y=0.818$. Thus, we obtain Table 8. Here, we see that the total production price 875.000 is equal to the total value 875, and the average total profit 200.000 is equal to the total surplus value of 200.

Table 8 Use formula (11) to derive the production price system from Bortkiewicz’s value system

Let us examine another generalized example. The value system data are shown in Table 9.

Table 9 A general example of value system

Now, we use formula (11) to calculate this example. After calculating the average profit rate $r=\frac{1}{4}$ according to formula (6) and substituting the data into formula (11), we obtain the following equations:

$$\left(1+\frac{1}{4}\right)\left(105{x}_{1}+205{x}_{2}+70{x}_{3}+100{x}_{4}+300y\right)=1050{x}_{1}$$

$$\left(1+\frac{1}{4}\right)\left(250{x}_{1}+130{x}_{2}+235{x}_{3}+75{x}_{4}+150y\right)=960{x}_{2}$$

$$\left(1+\frac{1}{4}\right)\left(60{x}_{1} +110{x}_{2} + 55{x}_{3} + 210{x}_{4}+200y\right)=785{x}_{3}$$

$$\left(1+\frac{1}{4}\right)\left(135{x}_{1}+75{x}_{2} + 95{x}_{3} + 80{x}_{4} +160y\right)=705{x}_{4}$$

$$1050{x}_{1}+960{x}_{2}+785{x}_{3}+705{x}_{4}=3500$$

The solution of this equation system is ${x}_{1}=0.939$, ${x}_{2}=1.088$, ${x}_{3}=1.009$, ${x}_{4}=0.960$, $y=1.002$. Thus, we obtain Table 10. Here, we see that the total production price 3500.000 is equal to the total value 3500, and the average total profit 700.000 is also equal to the total surplus value 700.

Table 10 The production price system transformed from the general example of the value system

4 Several related issues

From a mathematical perspective, the static transformation model that satisfies Marx’s two constraints can be constructed in several ways. For example, Zhang (2004) indirectly derived the following transformation model²⁴ from the three diagrams of Goodwin (1983)²⁵:

$$\left. \begin{gathered} c_{i} + v_{i} x_{i} + m_{i} = w_{i} y_{i} \begin{array}{*{20}c} {} & {(i = 1,2, \ldots ,n)} \\ \end{array} \hfill \\ \frac{{m_{i} }}{{c_{i} + v_{i} x_{i} }} = \frac{{m_{j} }}{{c_{j} + v_{j} x_{j} }}\begin{array}{*{20}c} {} & {} & {} & {(i \ne j)} \\ \end{array} \hfill \\ \sum {v_{i} x_{i} } = \sum {v_{i} } \hfill \\ \end{gathered} \right\}$$

(23)

Although (23) meets Marx’s two constraints, it fails to reflect Marx’s requirement for redistribution of surplus value, because this model is completed through the redistribution of variable capital. In addition, Zhang (2004) also introduced the construction of a transformation model similar to Goodwin (1983) in 1997²⁶:

$$\left. \begin{gathered} (c_{i} + v_{i} )x_{i} + m_{i} = w_{i} y_{i} \begin{array}{*{20}c} {} & {(i = 1,2, \ldots ,n)} \\ \end{array} \hfill \\ \frac{{m_{i} }}{{(c_{i} + v_{i} )x_{i} }} = \frac{{m_{j} }}{{(c_{j} + v_{j} )x_{j} }}\begin{array}{*{20}c} {} & {} & {} & {(i \ne j)} \\ \end{array} \hfill \\ \sum {(c_{i} + v_{i} )x_{i} } = \sum {(c_{i} + v_{i} )} \hfill \\ \end{gathered} \right\}$$

(24)

Although (24) also meets Marx’s two constraints, it fails to reflect Marx’s requirement for redistribution of surplus value, because the model is obtained through the redistribution of cost prices.

In general, the problem of converting value into production price can be called the direct transformation problem, and the problem of converting production price into value is called the inverse transformation problem. Mathematically, the inverse transformation is equivalent to the inverse function of the direct transformation.

Zhang (2004), under the premise that the rate of surplus value is equal, first provided a static inverse transformation model that meets Marx’s two constraints as follows²⁷:

$$\left. \begin{gathered} \sum\limits_{j = 1}^{n} {C_{ij} X_{j} + (1 + e)V_{i} Y = P_{i} X_{i} } \quad (i = 1,2, \ldots ,n) \hfill \\ \sum\limits_{i = 1}^{n} {P_{i} } X_{i} = \sum\limits_{i = 1}^{n} {P_{i} } \hfill \\ e = \sum\limits_{i = 1}^{n} {S_{i} /\sum\limits_{i = 1}^{n} {V_{i} Y} } \hfill \\ \end{gathered} \right\}$$

(25)

Here, $X_{i}$ represents the deviation rate of the value $w_{i}$ from the production price $P_{i}$, that is $w_{i} = P_{i} X_{i}$ ($i$ = 1,2,…,n), and $Y$ represents the deviation rate of the variable capital $V_{i}$ under the production price, that is, $v_{i} = YV_{i}$ (i = 1,2,…,n). The average profit rate r is still given by formulas (19).

In a mathematical sense, Zhang (2004) directly derived the reversal model from the aforementioned three diagrams of Goodwin (1983), which satisfies Marx’s two constraints²⁸:

$$\left. \begin{gathered} C_{i} + V_{i} X_{i} + R_{i} = P_{i} Y_{i} \begin{array}{*{20}c} {} & {(i = 1,2, \ldots ,n)} \\ \end{array} \hfill \\ \frac{{R_{i} }}{{V_{i} X_{i} }} = \frac{{R_{j} }}{{V_{j} X_{j} }}\begin{array}{*{20}c} {} & {(i \ne j)} \\ \end{array} \hfill \\ \sum {V_{i} X_{i} } = \sum {V_{i} } \hfill \\ \end{gathered} \right\}$$

(26)

Regarding the dynamic transformation model, Zhang (2003) provided two department transformation models as follows²⁹:

$$\left. \begin{gathered} \frac{{w_{1} }}{{c_{1} + v_{1} \frac{{\beta_{1}^{t} }}{{\lambda^{t} }}}} = 1 + r \hfill \\ \frac{{w_{2} }}{{\left( {c_{2} \frac{1}{{\beta_{1}^{t} }} + v_{2} \frac{1}{{\lambda^{t} }}} \right)\beta_{2}^{t} }} = 1 + r \hfill \\ c_{1} \beta_{1}^{t} + c_{2} \beta_{2}^{t} - w_{1} \beta_{1}^{t - 1} = 0 \hfill \\ (w_{1} + w_{2} )\beta^{t} - w_{1} \beta_{1}^{t} - w_{2} \beta_{2}^{t} = 0 \hfill \\ \end{gathered} \right\}$$

(27)

It is assumed here that the two departments grow with deviation according to $\beta_{1}$ and $\beta_{2}$,³⁰ respectively, starting from the first year. The expanded reproduction system in the tth year is ${c}_{1}{\beta }_{1}^{t}+{v}_{1}{\beta }_{1}^{t}+{m}_{1}{\beta }_{1}^{t}={w}_{1}{\beta }_{1}^{t}$, ${c}_{2}{\beta }_{2}^{t}+{v}_{2}{\beta }_{2}^{t}+{m}_{2}{\beta }_{2}^{t}={w}_{2}{\beta }_{2}^{t}$.

Let ${\beta }^{t}$ denote the growth index of the whole society in the tth year; the growth deviation degree of the ith department value in the $t\mathrm{th}$ year is $\frac{{\beta^{t} }}{{\beta_{1}^{t} }}$, and the growth deviation degree of variable capital is represented by $\frac{{\beta_{2}^{t} }}{{\lambda^{t} }}$. The model (27) is solvable; therefore,³¹ it is unnecessary to reinvestigate it here.

5 The unity of static transformation and dynamic transformation

In this section, we first extend Zhang’s (2003) 2-department dynamic transformation model to the n-department to build a more general dynamic transformation model. However, prior to this, we must emphasize a theoretical premise. When we construct a static transformation model, we follow the premise that the transformation of the value to the production price is the result of the residual value redistribution. Because Marx described it in Chapter 9 of vol. III of Capital, Okishio (1977) clearly recognized this.³² Now, we build a dynamic transformation model of the n-department based on these principles.

Let us consider a general dynamic transformation model, i. e. a transformation process that takes $k$ years to complete. To simplify the explanation, we assume that, in the $t\mathrm{th }(0<t\le k)$ year, the growth rate of the jth department’s total value is ${\delta }_{j}^{(t)}$ (exogenous parameter)³³; when $t=0$, there is no economic growth, so${\delta }_{j}^{(0)}=0$; under the same premise, the growth rate of constant capital and variable capital of the jth department is ${\theta }_{j}^{(t)}$ (exogenous parameter), and, in the process of transforming value to production price, the deviation rate of the value of this department is ${x}_{j}^{(t)}$, and the deviation rate of the variable capital is${y}^{(t)}$. ${k}_{j}^{(t)}$ (exogenous parameter) represents the redistribution ratio of the surplus value of the jth department in the tth year, and ${k}_{j}^{(0)}=0$, $\sum\limits_{s = 0}^{k} {k_{j}^{(s)} } = 1$. In the tth year, the redistribution of the surplus value of the ${i}^{th}$ department is:

$${\sum }_{s=0}^{t}{k}_{i}^{(s)}\left\{r\left({\sum }_{j=1}^{n}\left[{\prod }_{s=0}^{t}(1+{\delta }_{j}^{(s)}){c}_{ij}{x}_{j}^{(t)}\right]+{\prod }_{s=0}^{t}(1+{\theta }^{(s)}){v}_{i}{y}^{(t)}\right)-{m}_{i}\right\}$$

In this way, we get the general dynamic transformation model $f(t)(\forall 0\le t\le k)$ as follows:

$$f(t) = \left\{ \begin{gathered} \left\{ {\sum\limits_{j = 1}^{n} {\left[ {\prod\limits_{s = 0}^{t} {\left( {1 + \delta_{j}^{(s)} } \right)} c_{ij} x_{j}^{(t)} } \right] + \prod\limits_{s = 0}^{t} {\left( {1 + \theta^{(s)} } \right)} v_{i} y^{(t)} } } \right\} + \lambda^{(t)} S_{i}^{(t)} = \prod\limits_{s = 0}^{t} {\left( {1 + \delta_{j}^{(s)} } \right)} w_{i} x_{i}^{(t)} \begin{array}{*{20}c} {} & {(i = 1,2, \cdots ,n)} \\ \end{array} \hfill \\ \sum\limits_{i = 1}^{n} {\left[ {\mathop \Pi \limits_{s = 0}^{t} \left( {1 + \delta_{j}^{(s)} } \right)w_{i} x_{i}^{(t)} } \right]} = \sum\limits_{i = 1}^{n} {\left[ {\left( {1 + \delta_{j}^{(s)} } \right)w_{i} } \right]} \hfill \\ \sum\limits_{i = 1}^{n} {\left\{ {\sum\limits_{j = 1}^{n} {\left[ {\prod\limits_{s = 0}^{t} {\left( {1 + \delta_{j}^{(s)} } \right)} c_{ij} x_{j}^{(t)} } \right] + \prod\limits_{s = 0}^{t} {\left( {1 + \theta^{(s)} } \right)} v_{i} y^{(t)} } } \right\}} = \sum\limits_{i = 1}^{n} {\left\{ {\sum\limits_{j = 1}^{n} {\left[ {\prod\limits_{s = 0}^{t} {\left( {1 + \delta_{j}^{(s)} } \right)} c_{ij} } \right] + \prod\limits_{s = 0}^{t} {\left( {1 + \theta^{(s)} } \right)} v_{i} } } \right\}} \hfill \\ \end{gathered} \right\}$$

(28)

Here, the variable ${\lambda }^{(t)}$ is used to adjust the redistribution of surplus value in the ${t}^{th}(0<t<k)$ year.³⁴ Additionally,

$${S}_{i}^{\left(t\right)}={m}_{i}+{\sum }_{s=0}^{t}{k}_{i}^{(s)}\left\{r\left({\sum }_{j=1}^{n}\left[{\prod }_{s=0}^{t}(1+{\delta }_{j}^{(s)}){c}_{ij}{x}_{j}^{(t)}\right]+{\prod }_{s=0}^{t}(1+{\theta }^{(s)}){v}_{i}{y}^{(t)}\right)-{m}_{i}\right\}$$

which reflects the change in the surplus value in the process of profit averaging.

Note that, since ${S}_{i}^{(t)}$ contains ${\lambda }^{(t)}$, we now have n + 3 unknowns: ${x}_{i}$ ($i=\mathrm{1,2},\cdots ,n$), $y$, $r$, and ${\lambda }^{(t)}$, but as we have only n + 2 equations there is no way to solve them. To solve this problem, let us consider how model (28) will change in the kth year.

The transformation process is completed in the kth year. At this point, the redistribution of surplus value ends, and ${S}_{i}^{(k)}$ transforms itself to average profit, that is, ${S}_{i}^{(k)}={R}_{i}^{\left(k\right)}(i=\mathrm{1,2},\cdots ,n)$. Here, ${R}_{i}^{(k)}$ represents the average profit of the ith department.

$${R}_{i}^{(k)}= r\left({\sum }_{j=1}^{n}\left[{\prod }_{s=0}^{k}(1+{\delta }_{j}^{(s)}){c}_{ij}{x}_{j}^{(t)}\right]+{\prod }_{s=0}^{k}(1+{\theta }^{(s)}){v}_{i}{y}^{(t)}\right)$$

Thereby,

$$S_{i}^{(k)} = r\left( {\sum\limits_{j = 1}^{n} {\left[ {\prod\limits_{s = 0}^{t} {\left( {1 + \delta_{j}^{(s)} } \right)} c_{ij} x_{j}^{(t)} } \right] + \prod\limits_{s = 0}^{t} {\left( {1 + \theta^{(s)} } \right)} v_{i} y^{(t)} } } \right)$$

This conclusion can also be derived from $\sum\limits_{s = 0}^{k} {k_{j}^{(s)} } = 1$ and ${\lambda }^{(k)}=1$. From this, we get

$$f(k) = \left\{ \begin{gathered} (1 + r^{ * } )\left\{ {\sum\limits_{j = 1}^{n} {\left[ {\prod\limits_{s = 0}^{k} {\left( {1 + \delta_{j}^{(s)} } \right)} c_{ij} x_{j}^{(k)} } \right] + \prod\limits_{s = 0}^{k} {\left( {1 + \theta^{(s)} } \right)} v_{i} y^{(k)} } } \right\} = \prod\limits_{s = 0}^{k} {\left( {1 + \delta_{j}^{(s)} } \right)w_{i} x_{i}^{(k)} } \begin{array}{*{20}c} {} & {(i = 1,2, \cdots ,n)} \\ \end{array} \hfill \\ \sum\limits_{i = 1}^{n} {\left[ {\mathop \Pi \limits_{s = 0}^{k} \left( {1 + \delta_{j}^{(s)} } \right)w_{i} x_{i}^{(k)} } \right]} = \sum\limits_{i = 1}^{n} {\left[ {\mathop \Pi \limits_{s = 0}^{k} \left( {1 + \delta_{j}^{(s)} } \right)w_{i} } \right]} \hfill \\ \sum\limits_{i = 1}^{n} {\left\{ {\sum\limits_{j = 1}^{n} {\left[ {\prod\limits_{s = 0}^{k} {\left( {1 + \delta_{j}^{(s)} } \right)} c_{ij} x_{j}^{(k)} } \right] + \prod\limits_{s = 0}^{k} {\left( {1 + \theta^{(s)} } \right)} v_{i} y^{(k)} } } \right\}} = \sum\limits_{i = 1}^{n} {\left\{ {\sum\limits_{j = 1}^{n} {\left[ {\prod\limits_{s = 0}^{k} {\left( {1 + \delta_{j}^{(s)} } \right)} c_{ij} } \right] + \prod\limits_{s = 0}^{k} {\left( {1 + \theta^{(s)} } \right)} v_{i} } } \right\}} \hfill \\ \end{gathered} \right\}$$

(29)

Thus, in (29), the average profit rate $r^{ * }$ is determined as follows³⁵:

$$r^{ * } = \frac{{\sum\limits_{i = 1}^{n} {\left[ {\mathop \Pi \limits_{s = 0}^{k} \left( {1 + \delta_{j}^{(s)} } \right)w_{i} } \right]} }}{{\sum\limits_{i = 1}^{n} {\left\{ {\sum\limits_{j = 1}^{n} {\left[ {\prod\limits_{s = 0}^{k} {\left( {1 + \delta_{j}^{(s)} } \right)} c_{ij} } \right] + \prod\limits_{s = 0}^{k} {\left( {1 + \theta^{(s)} } \right)} v_{i} } } \right\}} }} - 1$$

(30)

If we make $c_{ij}^{ * } = \prod\limits_{s = 0}^{k} {\left( {1 + \delta_{j}^{(s)} } \right)} c_{ij}$, $v_{i}^{ * } = \prod\limits_{s = 0}^{k} {\left( {1 + \theta^{(s)} } \right)} v_{i}$, $w_{i}^{ * } = \mathop \Pi \limits_{s = 0}^{k} \left( {1 + \delta_{j}^{(s)} } \right)w_{i}$, then (30) can also be rewritten as

$$r^{ * } = \frac{{\sum\limits_{i = 1}^{n} {w_{i}^{ * } } }}{{\sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\left( {c_{ij}^{ * } + v_{i}^{ * } } \right)} } }} - 1$$

(31)

Therefore, we can further rewrite (29) as

$$f(k) = \left\{ \begin{gathered} \left( {1 + r^{ * } } \right)\left\{ {\sum\limits_{j = 1}^{n} {c_{ij}^{ * } x_{j}^{(k)} + v_{i}^{ * } y^{(k)} } } \right\} = w_{i}^{ * } x_{i}^{(k)} \begin{array}{*{20}c} {} & {(i = 1,2, \cdots ,n)} \\ \end{array} \hfill \\ \sum\limits_{i = 1}^{n} {w_{i}^{ * } x_{i}^{(k)} } = \sum\limits_{i = 1}^{n} {w_{i}^{ * } } \hfill \\ \sum\limits_{i = 1}^{n} {\left\{ {\sum\limits_{j = 1}^{n} {c_{ij}^{ * } x_{j}^{(k)} + v_{i}^{ * } y^{(k)} } } \right\}} = \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\left( {c_{ij}^{ * } + v_{i}^{ * } } \right)} } \hfill \\ \end{gathered} \right\}$$

(32)

Thus, (32) seems to degenerate into a general static transformation model (20). In fact, the general static transformation model (20) should be regarded as the completion of the general dynamic transformation model (28). Compared with (20), (29) or (32) is more general. In this way, we can harmonize static transformation with dynamic transformation. Moreover, this end point satisfies “the double invariance”.

Now, we use model (28) to re-examine the famous example of Bortkiewicz (1907), which we discussed in Sect. 3 of this paper (the data of the value system are in Table 5). We analyzed two situations.

• 1) Assuming that the initial year is the 0th year and the transformation is completed in the 2nd year. It is also assumed that the transformation is performed under conditions of simple reproduction such that ${\delta }_{j}^{(t)}=0$ and ${\theta }^{(t)}=0$. Assume that ${k}_{j}^{(1)}=0.5\left(j=\mathrm{1,2},3\right)$. Therefore, according to model (28), we obtain Table 11 as follows. Here, we see that the scale of production remains unchanged, and the value of the 2nd year is transformed into the production price. The total production price of 875.00 is equal to the total value of 875, and the average total profit of 200.00 is also equal to the total surplus value of 200.

• 2) We still assume that the initial year is the 0th year and that the transformation is completed in the 2nd year. However, it is assumed that the transformation is carried out under conditions of expanded reproduction. Here, it is assumed that ${\delta }_{j}^{(1)}=3\%(j=\mathrm{1,2},3)$, ${\theta }^{(1)}=3.5\%$, and ${k}_{j}^{(1)}=0.5(j=\mathrm{1,2},3)$. Therefore, we obtain Table 12 as follows, according to Model (28). Here, the scale of production has expanded, and the value has been converted into production price in the 2nd year. The total production price 933.45 in the 2nd year is equal to the total value 933.45 and the total average profit 214.25 is equal to the total surplus value 214.25.

Table 11 The dynamic transformation results of Bortkiewicz’s value system ($k=2$, simple reproduction)

Table 12 The dynamic transformation results of Bortkiewicz’s value system ($k=2$, expanded reproduction)

Let us look at a more general example and use the data in Table 9. Assuming that the initial year is the 0th year, the transformation is completed in the 3rd year. It is assumed that the transformation is carried out under expanded reproduction. The assumed exogenous parameters are listed in Table 13.

Table 13 Assumed values of exogenous parameters

Therefore, according to Model (28), we obtain Table 14 as follows. Here, we see that the scale of production has expanded, and the value is converted into production price in the 3rd year.

Table 14 The dynamic transformation results of the value system of 4 departments ($k=3$, expanded reproduction)

The solutions of various variables used in calculating Table 14 for each year are shown in Table 15:

Table 15 Solutions of various variables in each year

6 Conclusion

From a mathematical perspective, the transformation problem in the narrow sense is mainly about how to construct a mathematical model that can reflect the transformation process of value to production price under certain constraints.

From a historical perspective, according to Marx’s logic, there is value first and then it is followed by production price. When the commodity economy reaches the point where surplus value is redistributed according to the law of “equal sums of capital demand equal profits”, the value is transformed into production price. This conversion process may take a long time. Thus, the conversion of value to production price is represented as a process, and the mathematical description of this process should assume the form of a dynamic model.

After being transformed into production price, value of commodities is stabilized in the form of production price. The production price becomes prominent whereas the value (labor value) retreats behind the scenes. However, the link between value and production price remains. The static transformation model reflects the relation between the value and production price.

Using simulations, this study attempts to construct a computable mathematical model that connects dynamic transformation and static transformation. This simulation may have many flaws, but it can demonstrate the unity of two modes of the transformation. Hopefully, our study will stimulate the construction of a more complete transformation model.

In short, from a mathematical perspective, not only can the transformation problem be solved, but there may also be multiple solutions. The general static transformation model given by Zhang (2000) and the general dynamic transformation model given in this paper are not necessarily the best, but they are both effective. Regardless of whether a better solution should appear in the future, we are able to declare that the transformation problem can be solved mathematically.