Already in 1907 the labour theory of value of Marx got somewhat into discredit, after the economist L. von Bortkiewicz had found a problem in the marxist transformation of labour values into product prices. In the sixties this critique has been stated in a more clear and understandable way by P. Sraffa and his adherents. The marxist theoreticians did not succeed in finding an appropriate response to the critique, despite the enormous marxist-scientific apparatus at the time in the Leninist states. Their usual answer was that the theory of Sraffa (also called the neoricardian theory) is based on a number of assumptions, that conflict with the economic vision of Marx.

The assumptions that underlie the neoricardian theory, have been explained in a previous column. The marxist economist Alfred Müller, incidentally a West-German, gives a number of reasons, why the neoricardian theory is irreconcilable with the analysis of Marx^{1}:

- In the neoricardian model the available production techniques are fixed and unchangeable. The system is in a static state, and does not consider the technological progress. The state of the technique is determined in an exogenous way
^{2}. The economic system develops itself as it were in a perfect social equilibrium, without radical revolutions. On the other hand in the vision of Marx the production is subjected to continuous technical innovations, because due to the competition the producers are forced to diminish their production costs. For this purpose they accumulate their profits, and keep investing these in the renewal of their production process. - The neoricardian model consists of linear equations, so that it is neutral with respect to increasing scales. The increase of the production does not change the production process, and does not lead to a reduction of the costs. In the English language this is called the
*constant returns to scale*. On the other hand, according to Marx the capitalist economy is characterized by a forced concentration of the production capacity, in order to reduce the production costs. The producers can increase the labour productivity by an ongoing mechanization and automation of the production, and by doing so they enlarge the quantities of the products and they reduce the production costs. - According to the neoricardian theoreticians the income distribution is also an exogenous factor, just like the available quantity of labour. On the other hand Marx believes that the mechanisation leads to a structural excess of workers. For that reason labour is not scarce, and the workers miss the power to improve their wage level. This system pushes their wage at the minimal level, that is needed to just survive in the then common social circumstances. Therefore the wage is determined in an endogenous way.
- In addition Müller objects to the absence of money as a means of exchange in the neoricardian model.

Undoubtedly Müller is right, but still the situation remains unsatisfactory. For even if the neoricardian theory presents a wrong interpretation of the marxist price theory, then still the question remains what interpretation does apply. In the third volume of *Das Kapital* Marx and Engels write: *So the product price of a good equals its cost price and the added profit as a percentage, according to the average profit rate. Or, it equals the cost price and the average profit*^{3}. And further on: *Now the formula, that equals the product price of the good to K+π, to the cost price and profit, is determined in detail by π=r ^{*}×K (where r^{*} is the general profit rate), and thus by the product price K + r^{*}×K*

At first sight the argument of Marx and Engels is crystal-clear. In an earlier column their explanation is translated into the price formula

(1) p_{i} = (c_{i} + v_{i}) × (1 + r^{*}) / Q_{i}

In the formula 1 p_{i} is the product price of good i, c_{i}+v_{i} represents its cost price (in labour values), and Q_{i} is the produced quantity. In that column it is shown by means of the neoricardian theory, that unfortunately the formula generates contradicting results, as long as the calculation is performed with constant product prices. In other words, a problem emerges, when the goods with value c_{i} have the same product prices as the Q_{i} end products. In the preceding text it has already been stated, how Alfred Müller objects to this simultaneous price setting.

In 1988 Andrew Kliman and Ted McGlone discovered an alternative interpretation of the text in *Das Kapital*. Their interpretation, which they called the *Temporal Single System* interpretation, in short TSSI, does get around the inner contradictions. In the following decades especially Kliman has perfected the new exegesis^{5}, and finally he has explained her in his book *Reclaiming Marx's Capital*^{6}. His work deserves admiration, especially because it meets much resistance from the established economic science. Apparently these economists are not taken with a marxist renaissance. So one has reason to make known the work of Kliman to a wide audience, like in the present column.

Contrary to the neoricardian theory the TSSI includes the ongoing innovation of the production process in the capitalist society. The labour productivity keeps increasing, and consequently the product prices fall. Suppose that the production process requires a time period τ, then the product prices during the sale of the end produrcts are different from those at the purchase of the means of production. Due to this time dependence the formula 1 takes on the form

(2) p_{i}(t+τ) = (γ_{i}(t) + φ_{i}(t)) × (1 + r^{*}(t,τ)) / Q_{i}(t)

In the formula 2 r^{*}(t,τ) depends on τ, because the surplus value m(t,τ) is created during the production process, and is realized only in the sale. In the TSSI the formula 2 is a real price equation. The lefthand and righthand side both represent a monetary sum, expressed in a monetary unit, like the euro. The costs of the means of production are

(3) γ_{i}(t) = Σ_{j=1}^{N} p_{j}(t) × q_{ji}(t)

The summation includes all N products available in the society. In the formula 3 q_{ji} indicates what quantity of means of production j is needed for the production of a quantity Q_{i} of the product i. Therefore γ_{i}(t)/Q_{i}(t) takes on the form Σ_{j=1}^{N} p_{j}(t) × a_{ji}(t), where a_{ji}(t) are the production coefficients. Or in matrix notation __γ/Q__(t) = __p__(t) · __A__(t), where __p__(t) is a horizontal vector. The wage sum is

(4) φ_{i}(t) = w(t) × ω × *l*_{i}(t)

In the formula 4 w(t) is the real wage level, and *l*_{i}(t) is the amount of labour, that is expended on the production of a quantity Q_{i}(t) of the product i. The variable ω is a conversion factor that transforms a unit of labour value into a unit of money. She is a kind of exchange rate, which for instance could indicate the value in euros of a working-hour. In the English language ω is called the *monetary expression of labour time* (in short MELT). In the TSSI it is common to express the MELT as a time-dependent function, but in the present column she is a constant.

Indeed often the MELT will change as a function of time, for instance as a result of a creeping inflation or of a devaluation of the monetary unit. The time-dependent behaviour of the MELT has even become an important subject in the scientific discussion about the TSSI^{7}. Since that discussion is rather confusing, and the time-dependent MELT would complicate the formulas 2 and following, for conveniences sake the present column is limited to the case ω(t) = ω. This case is already quite interesting. The reader who wants to study the time-dependent MELT, can take up for instance the book of Kliman^{8}.

The wage level w(t) determines the pay that the workers receive in exchange for their supplied labour time. The surplus in the branch i is

(5) m_{i}(t,τ) = *l*_{i} − v_{i} =

= *l*_{i} × (1 − w(t))

If the formula 5 is used in a summation for all branches i, then one obtains the total social surplus value m(t,τ). Apparently the real wage level w(t) is an alternative measure for the degree of exploitation^{9}. In the labour theory of value the worker is exploited, and he receives as his wage only a part of the total value, that he adds. Note, that φ_{i}(t)/Q_{i}(t) takes on the form w(t)×ω×a_{i}, where a_{i} is the labour coefficient is. In matrix notation __φ/Q__(t) = w(t)×ω×__a__(t). In a society without profit w=1 holds, so that ω coincides with the nominal wage.

In the formula 2 the size of the general profit rate r^{*}(t,τ) is defined in a marxist way by r^{*}(t,τ) = m(t,τ) / C(t). However, in the TSSI it is more convenient to calculate with prices, instead of with labour values. The profit rate in prices is

(6) r^{*}(t,τ) = π(t,τ) / (γ(t) + φ(t))

In the formula 6 π(t,τ) can be obtained by a multiplication of the social surplus value m(t,τ) with the MELT ω. Thus the formal framework of the TSSI is completed.

It seems that the TSSI gives a satisfactory representation of the theory of Marx and Engels, such as is put into words in *Das Kapital*. In particular the explicit introduction of the time variable t guarantees, that the TSSI describes a dynamic system. And the wage formula 4 is perhaps not really endogenous, but she is clearly coupled to the degree of exploitation through w(t). Incidentally w(t) in the TSSI can quite well be interpreted as the minimal wage, that maintains the existence of the workers. Thus it is somewhat surprising, that the TSSI is ill received within the global economic community. Certainly, it is the duty of each new theory (or exegesis) to prove itself against any initial scepticism. But the publication of *Reclaiming Marx's Capital* has answered so many questions, that by now some acceptance of the TSSI seems justified. The present scientific resistance against the TSSI seems to originate mainly from a misplaced conservatism.

The formulas 2-6 demonstrate why Kliman speaks about a *Temporal Single System* interpretation. The word *temporal* is the opposite of *simultaneous*. In the formulas 2-3 the prices of the end products differ from the prices of the means of production. The prices are a function of the time. The driving force behind the price variations is the technological progress, that is characteristic for the human society. The general average profit rate r^{*} is just a snapshot, and thus time-dependent. He does not originate, like in the neoclassical theory, from economic forces, that should move the system towards an equilibrium. In this respect the TSSI distinguishes herself also from the neoricardian theory. The expression *Single System* is opposed to *Dual System*. The dual system distinghuishes explicitly between the labour values η_{i} and the product prices p_{i}. The earlier column about the transformation problem shows how those two separate systems of labour values and prices are coupled. In the TSSI the labour values and the prices coincide, except for a conversion factor (namely the MELT). In the TSSI there is only one system. The price system is not the externally visible appearance of a higher substance, that represents the system of labour values. Kliman states that Marx has never suggested the existence of a dual system. This idea is mainly due to Léon Walras and von Bortkiewicz, who introduced a separate price system. In the TSSI the labour theory of value is concentrated in the formula 5, that attributes the nett product completely to the efforts of the factor labour (*l*). Profit exists, because the workers do not get paid the full value, that thet have added to the product.

Just like previous columns in *Heterodox Gazette Sam de Wolff* the present column will conclude with a calculation, in order to illustrate the application of the TSSI. The calculation is copied from paragraph 8.9.2 in the proof-text *Vooruitgang der economische wetenschap*. Although the TSSI is most fruitful in dynamic situations with economic growth, for simplicity the chosen example describes a static situation^{10}. This is what Marx calls the simple reproduction. In a static situation the produced quantities Q_{i} and the production technique are fixed. Therefore the technical coefficients __A__ and __a__ in the formulas 2-3-4 are constants. For convenience it is assumed that ω=1 (for instance in euro per minute working time). Note that in this case the real and nominal wages are identical. The economic system consists of two branches, namely the agriculture and the industry. The agriculture produces corn (measured in bales), and the industry produces metal (measured in tons). The quantities are the same as those in the calculation in the column about the transformation problem. It is assumed that the degree of exploitation m' equals 1. Then the real wage is w=0.5. That is to say, the workers obtain half the nett product: 0.05 bales of corn and 0.015 tons of metal per worker. The relevant quantities at the time t=0 are summarized in the table 1. Because of ω=1 the price sums in the table are equal to the labour values. The quantity O is the yield, consisting of γ and ω×*l*. In other words, she is the total monetary value present in the system.

branch | γ(0) | φ(0) | m(0,τ) | O(0) |
---|---|---|---|---|

corn (12 bales) | 21.74 | 10 | 10 | 41.74 |

metal (3.1 tons) | 57.40 | 5 | 5 | 67.40 |

sum | 79.14 | 15 | 15 | 109.14 |

The profit rates in the two branches differ, namely 31.50% in the agriculture and 8.01% in the industry. In the theory of Marx the capital markets level the profit rates, and thus change the product prices in agreement with the formula 2. There is a shift of value between the branches. It turns out that the corn prices fall by 11.8%, and the metal prices rise by 7.3%. The monetary yields change also, according to the formula

(7) O(τ) = p_{g}(τ) × Q_{g} + p_{m}(τ) × Q_{m}

Note that the quantities in the example remain unchanged, while the product prices are time-dependent. The results of this redistribution are shown in the table 2. In the redistribution the social profit sum π and the total yield O remain unchanged.

branch | γ(0) | φ(0) | m(0,τ) | O(0) | r(0,τ) [%] | π(τ) | O(τ) |
---|---|---|---|---|---|---|---|

corn (12 bales) | 21.74 | 10 | 10 | 41.74 | 31.50 | 5.06 | 36.80 |

metal (3.1 tons) | 57.40 | 5 | 5 | 67.40 | 8.01 | 9.94 | 72.34 |

sum | 79.14 | 15 | 15 | 109.14 | 15.93 | 15 | 109.14 |

Since in the theory of Marx the prices and the general profit rate usually do not move towards an equilibrium, the next product cycle (called revolution in the marxist terminology, *Umschlag* in German) begins in a different initial condition than the previous one. The system remains dynamic, even though it is in a state of simple reproduction, like in the example. The corn producers have a yield of 36.80, which they employ for a capital advancement in order to repeat the production process. The metal producers do the same thing, from their yield of 72.34. Though a time θ will pass before the production factors have again been bought, the absence of technological progress and of inflation guarantee, that their purchasing power remains intact. At the beginning of the second revolution (on t=τ+θ) the prices of the production factors are still equal to the prices during the sale of the products of the previous revolution (on t=τ). The costs in the second revolution can be written in matrix notation as

(8a) __γ/Q__(τ+θ) = __p__(τ+θ) · __A__

(8b) __φ/Q__(τ+θ) = (__p__(τ+θ) · __w___{r}) × __a__

The values of the technical coefficients can be found in the earlier column about the transformation problem. As a reminder they are again summarized in the table 3. In the formula 8b the vertical vector __w___{r} is introduced, that represents the real wage. In the preceding text it has already been stated, that for a surplus rate of 1 it equals [0.05, 0.015]. Apparently the quantity w(t) in the formula 4 is replaced in the formula 8b by the inner product __p__·__w___{r}. This implies that in the present example of a static economy the wage level does become time-dependent, and consequently also the rate of surplus value (degree of exploitation)! Since the prices fluctuate, and in the TSSI there exists only one value system, also the labour values fluctuate. This may surprise a marxist who is used to thinking in traditional frames.

agriculture | industry | |
---|---|---|

corn | a_{gg}=0.4167 | a_{gm}=1.290 |

metal | a_{mg}=0.01667 | a_{mm}=0.6452 |

workers | a_{g}=1.667 | a_{m}=3.226 |

The monetary value of the production factors at a time t=τ+θ is calculated by means of the set of formulas 8a-b. The table 1 is filled in again, now with the monetary value sums for the time t=τ+θ. The result is presented in the table 4. The table shows that due to the price changes also the costs of production have changed. That is to say, at the time t=τ+θ the producers must advance a different cost price than at t=0. It is instructive to see how the wage level has risen somewhat, due to the rise of the price of metal. Since during a production revolution the workers still add the value __ l__, in this way the surplus value has decreased a bit. An interesting quantity with respect to the changing production costs is the residue (for conveniences sake again in the matrix notation)

(9)

This residue

branch | γ(τ+θ) | φ(τ+θ) | δ(τ+θ) | m(τ+θ,τ) | O(τ+θ) |
---|---|---|---|---|---|

corn (12 bales) | 20.00 | 10.07 | 6.73 | 9.93 | 40.00 |

metal (3.1 tons) | 58.94 | 5.03 | 8.37 | 4.97 | 68.94 |

sum | 78.94 | 15.10 | 15.10 | 14.90 | 108.94 |

Now everything in the economic system is ready for the executing of a new revolution. Again the profit rates in the branches differ, so that the profits have to be levelled by means of the formula 2. In this way the price development can be calculated for the whole sequence of revolutions, if desired until the end of times. However such a calculation would be at odds with the assumptions in the TSSI. For the TSSI assumes an ongoing dynamics in the economy, which hardly leaves room for a static system, that keeps reproducing itself. For instance the producers in the calculation apparently consume their entire profit. This is unthinkable in the social vision of Marx, where the producers invest most of their profit in the production, in order to gain a competitive advantage with respect to the other enterprises. In fact the assumption of constant technical coefficients is not reconcilable with the character of the capitalist system. Also regular monetary shocks occur in the capitalism, that will change the value of the MELT.

Because of the dynamic nature of the capitalist system the practical economist will use the TSSI mainly as a conceptual scheme, a way of thinking, in order to explain the real developments. The TSSI is not a mathematical model, that can be used for the planning or bridling of the economy. Nevertheless the theoretician will investigate what tendencies in the economic development can be expected on the basis of the formulas of the TSSI. An important feature of the TSSI is the analytic decoupling of the physical quantities of produced goods on the one hand, and the production of labour value on the other hand. For instance, in the neoricardian theory the physical production and the production of labour value coincide, simply because each product represents a fixed amount of labour value. This phenomenon occurs in all theories, that assume *simultaneism*. For the time-dependent changes in the labour values of the products are not taken into account. Kliman calls the basic idea behind such theories *physicalism*. He states that they are lacking a sense of reality.

It is a pity that yet hardly any research has been done with respect to the consequences, that can be expected in the economic thinking, when the assumption of the physicalism is abandoned. For exactly here the dynamic character of the TSSI unfolds, which may give a deeper insight into the chaotic behaviour of our modern economic world^{12}.

- See p.408 and further in
*Die Marxsche Konjunkturtheorie*(2009, Papy Rossa Verlag GmbH). (back) - Here the expression
*exogenous*means: coming from outside of the system. In that vision the technological progress is a random process, that depends on scientific discoveries and on individual creativity. The expression*endogenous*means, that the system itself is the driving force. The properties of the system compel an ongoing innovation of the production process and of the products. (back) - See p.167 of the third volume of
*Das Kapital*(1974, Dietz Verlag). (back) - See p.175 of the third volume of
*Das Kapital*(1974, Dietz Verlag). (back) - The word exegesis means explanation of the text. Kliman prefers this word to alternatives like "model" or "theory". For the theory is developped by Marx and Engels. Kliman just gives an explanation, that guarantees the analytic consistency in the work of Marx and Engels. He refuses to answer the question whether the theory of Marx is actually a convincing description of the real world. (back)
- See
*Reclaiming Marx's Capital*(2007, Lexington Books). (back) - See for instance p.170 and further in
*Die Aktualität der Arbeitswerttheorie*(2009, Metropolis-Verlag), by N. Fröhlich. Apparently the discussion concerns the question whether the time dependent MELT in the TSSI can be defined in a convincing manner. (back) - See for instance p.185 and further in
*Reclaiming Marx's Capital*. Or paragraph 8.9.1 in the proof-text*Vooruitgang der economische wetenschap*(2011, uitgeverij E. de Bibelude). An inflation or deflation i(t,τ) during a time interval τ can be expressed in terms of the MELT by i(t,τ) = (ω(t+τ)−ω(t)) / ω(t). In the case of a time-dependent MELT for instance the formula 2 changes into p_{i}(t+τ) = (i(t,τ)+1) × (γ_{i}(t)+φ_{i}(t)) × (1+r^{*}(t,τ)) / Q_{i}(t). For the inflation or deflation must be included in the product prices at the time t+τ. One also sees, that in the case of a constant MELT i(t,τ) equals zero. (back) - To be precise: one has
*l*= v+m = m × (1+1/m'). Thus w(t) = 1 / (1+m'). (back) - Calculations of economic growth can be found for instance in
*Reclaiming Marx's Capital*and*Vooruitgang der economische wetenschap*. (back) .
- The reader should keep in mind, that a simple reproduction takes place, This implies that the price changes do not influence the physical (material) exchanges, that occur in the economic system. In other words, the producers do have a larger yield in total at the start of the second revolution (namely 15.10) than they had at the end of the preceding revolution as a total profit sum (namely 15.00). However this difference is purely monetary. In terms of physical quantities of corn and metal their income remains unchanged. This also holds for the workers, who do receive a larger money wage, but in reality still get 0.05 bales of corn and 0.015 tons of metal. Despite the fluctuating incomes the markets are still cleared physically. Apparently the physical quantities have been detached from the price sums, and thus also from the labour values. This somewhat remarkable phenomenon is characteristic for the TSSI, and will be explained at the end of this column. (back)
- Your columnist had made an insignificant attempt. The following text is copied from the paragraph 8.9.2 in the proof-text
*Vooruitgang der economische theorie*. The argument start with the consideration that in the theory of Marx there is always a transfer of surplus value from labour intensive branches towards the capital intensive branches. In the case of simple reproduction with a constant MELT there are four characteristic developments:- In the departments or branches with the highest organic composition (γ/φ) the product prices have the tendency to rise. In general that will be the department I of the production of the means of production. This is also clear in the example, where the industry prices turn out to increase.
- In the department or branches with the lowest organic composition the product prices have the tendency to fall. In general that is the department II of the production of consumer goods, and/or the department III of the production of luxury goods. This is also clear in the example. If the real (material, physical) wage is based on only the products of the department II, then the monetary wage (expressed in money) will eventually approach the value of zero.
- The price increases in the department I cause a continuous rise of the value of the constant capital γ and consequently also of the total product Q. In other words, besides the changing price ratios (see the two characteristic developments described just now) also the general price level will rise. This can not be called inflation, because in the common economy she results from the devaluation of money as a result of the shortage of goods on the market. This is not the case here, for the rising price is caused by the mechanism of price formation (formula 2, the temporal application of the formula 1). Therefore in these conditions, with a constant MELT, the TSSI inflation rate (see a previous foot note) remains zero.
- The profit rate exhibits a falling tendency and will finally approach zero. For the value of the constant capital γ, and thus of the advanced capital, exhibits a continuous rise, whereas the surplus value can never exceed
*l*. Indeed also the example exhibits this behaviour. Namely if the calculation of the second revolution in the table 4 is continued, then one finds after the levelling of the profit rates in the branches a general profit rate r^{*}(τ+θ,τ) of 15.84%. This is lower than the general profit rate of 15.93% during the first revolution. This process continues, and after for instance thirty revolutions the general profit rate turns out to be only a few per cent.