Notes

This is a commented and modified extract from pages 109-130 of The Theory of Capitalist Development - Paul M. Sweezy. The explanation provided there is very good.

TLDR

The solution doesn’t work because:

1. The LTV predicts that the organic composition of capital should converge across industries. This evidently doesn’t happen. The proposed solution hides this fact behind arbitrary value->price mappings.

2. The proposed solution leads to the interesting conclusion that the organic composition of capital and the surplus value in the goods department (food, clothes, etc) doesn’t affect the rate of profit.

3. The proposed solution uses a different definition of rate of profit that not surplus value. The new definition is proportional to constant capital + variable capital.

The framework

The solution divides the economy in three departments:

1. Department 1 produces constant capital. The output of this department is equal to all constant capital used up in all departments.

2. Department 2 produces consumer goods. The output of this department is equal to all wages payed.

3. Department 3 produces luxury goods. The output of this department equals all surplus value extracted.

4. An unusual definition of organic composition of capital is used, namely \(\omega = \frac{c}{c+v}\) instead of the usual \(\omega = \frac{c}{v}\).

The problem

The problem Bortkiewicz’s tried to solve is that of the convergence of the organic composition of capital across all industries in an economy.

Table I - Value Calculation

Everything is evidently in order. All commodities sell at their values. The conditions of simple reproduction are fulfilled: the amount of constant capital laid out (400) just equals the amount of constant capital produced (400); total wages (200) are just sufficient to buy the quantity of wage goods produced (200); and the surplus value of all departments (200) covers the output of the luxury-goods department (200). Finally all capitalists are enjoying the same rate of profit 33.3% and hence none has an incentive to shift from one line of production to another.

In the real world, however, the organic composition of capital is not the same in all industries. For example, it is relatively high in the electric-power industry and relatively low in the clothing industry. In order to bring this fact to light we must alter our assumptions. In Table II, Department III has been left unchanged, but the organic com position of capital in Department I is assumed to be higher and in Department II to be lower.

This here should be enough to set the matter. The LTV makes a prediction but the data disagrees. Therefore the theory is wrong.

Table II - Value Calculation

As before, total production is 800, and the conditions of Simple Reproduction are still satisfied as far as the output of the three departments is concerned. But the effect of changing the organic compositions of capital is clearly seen in the new rates of profit. Whereas before the rates of profit were all equal at 33.3%, they now stand at 23%, 60% and 33.3% in the three departments respectively.

Obviously this could not be a position of equilibrium. The capitalists would all want to go into the production of wage goods in order to share in the higher rate of profit obtainable there. And such a migration of capital out of some industries and into others would clearly upset the whole scheme. A position of equilibrium must be characterized by equality in the rates of profit yielded by all the industries in the system. Marx put it strongly when he wrote that ‘there is no doubt that, aside from unessential, accidental, and mutually compensating distinctions, a difference in the average rate of profit of the various lines of industry does not exist in reality and could not exist without abolishing the entire system of capitalist production.’ *

Proposed Solution

This method doesn't actually solve the problem. It only takes an arbitrary mapping of one unit of gold (which is considered a luxury good) to one unit of labour in the Department III. For the sake of the argument assume such mapping is possible and meaningful, as you will see it doesn't solve the problem of convergent organic compositions of capital, instead it creates a new problem.

As a first step, let us assume that the price of a unit of constant capital is \( x \) times its value, the price of a unit of wage goods is \( y \) times its value, and the price of a unit of luxury goods is \( z \) times its value. Further let us call the general rate of profit \( r \) - it is important to understand that \( r \) is not defined as Marx defined the rate of profit and hence it seems wise not to use the same symbol for both concepts. Now in value calculation the following three equations describe the conditions of Simple Reproduction:

These equations represent the relations between inputs and outputs of the three departments.

\[ (I) \hspace{1em} c_{1} + v_{1} + s_{1} = c_{1} + c_{2} + c_{3} \] \[ (II) \hspace{1em} c_{2} + v_{2} + s_{2} = v_{1} + v_{2} + v_{3} \] \[ (III) \hspace{1em} c_{3} + v_{3} + s_{3} = s_{1} + s_{2} + s_{3} \]

These equations, when transformed into price terms, become:

\[ (I) \hspace{1em} c_{1}*x + v_{1}*y + r(c_{1}*x + v_{1}*y) = (c_{1} + c_{2} + c_{3})x \] \[ (II) \hspace{1em} c_{2}*x + v_{2}*y + r(c_{2}*x + v_{2}*y) = (v_{1} + v_{2} + v_{3})y \] \[ (III) \hspace{1em} c_{3}*x + v_{3}*y + r(c_{3}*x + v_{3}*y) = (s_{1} + s_{2} + s_{3})z \]

Note that the new definition of rate of profit is not equivalent to surplus value. Surplus value is explicitly stated to be worker's labour hours appropriated by the capitalist, while the new definition is merely "a quantity proportional to constant and variable capital".

And these can be rewritten as:

\[ (I) \hspace{1em} (r+1)(c_{1}*x + v_{1}*y) = (c_{1} + c_{2} + c_{3})x \] \[ (II) \hspace{1em} (r+1)(c_{2}*x + v_{2}*y) = (v_{1} + v_{2} + v_{3})y \] \[ (III) \hspace{1em} (r+1)(c_{3}*x + v_{3}*y) = (s_{1} + s_{2} + s_{3})z \]

In these three equations there are four unknown quantities, namely, \(x\), \(y\) , \(z\), and \(r\). For an unique solution it is necessary to have the same number of equations and unknowns. Hence we ought to have either one more equation or one less unknown. We might proceed as Marx did by setting total value equal to total price. This would give us the following fourth equation:

\[ (c_{1} + c_{2} + c_{3})*x + (v_{1} + v_{2} + v_{3})*y + (s_{1} + s_{2} + s_{3})*z = \]

\[ (c_{1} + c_{2} + c_{3}) + (v_{1} + v_{2} + v_{3}) + (s_{1} + s_{2} + s_{3}) \]

The economic meaning of this equation can be easily seen. So far in our value schemes we have reckoned everything in terms of hours of labor; in other words, one hour of labor has been the unit of account. By assuming that total output in value terms is equal to total output in price terms, we should simply be retaining the same unit of account in the price scheme. There is no logical objection to this way of proceeding, but from a mathematical point of view there is an alternative method which is simpler and hence more attractive.

What is meant here is that price should be measured in labour hours.

The original text uses gold as the monetary unit and assumes one unit of labour in the luxury department maps to one thirty-fifth of an ounce of gold. This assumption is as ludicrous and arbitrary as it seems but the author still goes on to claim it shows how the luxury sector is not productive and how it corroborates Marxism.

Instead I will take the equally valid mapping of one unit of labour in the goods department to one bag of 1kg of rice, then arrive at the conclusion that food does not affect production. We don't even need to change the derivation of the equations, just the interpretation at the end.

Instead of calculating the value scheme in terms of units of labor time we might have put it in money terms. Thus the value of each commodity would not be expressed in units of labor but in terms of the number of units of the money commodity for which it would exchange. The number of units of labor necessary to produce one unit of the money commodity would provide a direct link between the two systems of accounting. Let us assume that the value scheme has been cast in money terms and that gold, which we will classify as a luxury good, has been selected as the money commodity. Then one unit of gold (say one thirty-fifth of an ounce) is the unit of value. For the sake of simplicity we will also suppose that the units of other luxury goods have been so chosen that they all exchange against the unit of gold on an one-to-one basis: in other words the unit value of all luxury goods, including gold, is equal to one. Now in going from a value to a price scheme we wish to retain one thirty-fifth of an ounce of gold as the unit of account. The unit of gold will therefore be equal to one in both schemes, and under the assumed conditions the same must be true of all luxury goods. Since we have already assumed that the price of a unit of luxury goods is \(z\) times its value, this amounts to setting

\[ z = 1 \]

and this, in turn, reduces the number of unknowns to three. Since we have three equations the system is now completely determined.

If we now set \( 1 + r = m \), our three equations finally look as follows:

\[ (I) \hspace{1em} m(c_{1}*x + v_{1}*y) = (c_{1} + c_{2} + c_{3})x \] \[ (II) \hspace{1em} m(c_{2}*x + v_{2}*y) = (v_{1} + v_{2} + v_{3})y \] \[ (III) \hspace{1em} m(c_{3}*x + v_{3}*y) = (s_{1} + s_{2} + s_{3})z \]

The actual solution of the equations is, of course, a matter of algebra; what concerns us is the outcome. To express the result most conveniently, the following six expressions are formed:

The equations below are not derived from anything, they simply make the result easier to read.

\[ f_{1} = \frac{v_{1}}{c_{1}} \hspace{5em} g_{1} = \frac{v_{1} + c_{1} + s_{1}}{c_{1}} \]

\[ f_{2} = \frac{v_{2}}{c_{2}} \hspace{5em} g_{2} = \frac{v_{2} + c_{2} + s_{2}}{c_{2}} \]

\[ f_{3} = \frac{v_{3}}{c_{3}} \hspace{5em} g_{3} = \frac{v_{3} + c_{3} + s_{3}}{c_{3}} \]

Remembering that

\[ c_{1} + c_{2} + c_{3} = c_{1} + v_{1} + s_{1} \]

\[ v_{1} + v_{2} + v_{3} = c_{2} + v_{2} + s_{2} \]

\[ s_{1} + s_{2} + s_{3} = c_{3} + v_{3} + s_{3} \]

our equations can be rewritten

\[ (I) \hspace{1em} m(x + f_{1} * y) = g_{1} * x \] \[ (II) \hspace{1em} m(x + f_{2} * y) = g_{2} * y \] \[ (III) \hspace{1em} m(x + f_{3} * y) = g_{3} \]

The solutions * which emerge are then as follows:

\[ y = \frac{g_{3}}{g_{2} + (f_{3} - f_{2})m} \]

\[ x = \frac{f_{1}ym}{g_{1} - m} \]

It will be recalled that we defined \(m\) as equal to \(r + 1\), and hence \(r\) (the rate of profit) is given by

\[ r = m - 1 \]

These formulas look rather formidable, but actually they are not difficult to apply. As an example of how prices can be derived from values, let us perform the necessary operations on the basic data presented in Table II. The value scheme is as follows:

\[ (I) \hspace{1em} 250(c_{1}) + 75(v_{1}) + 75(s_{1}) = 400 \] \[ (II) \hspace{1em} 50(c_{2}) + 75(v_{2}) + 75(s_{2}) = 200 \] \[ (III) \hspace{1em} 100(c_{3}) + 50(v_{1}) + 50(s_{3}) = 200 \]

Using the formulas for \(x\), \(y\) and \(m\) we get

\[ x = \frac{9}{8} \]

\[ y = \frac{3}{4} \]

\[ m = \frac{4}{3} \]

This implies a rate of profit \( (m - 1) \) of \( 1/3 \). All that remains to be done now is to substitute the actual figures in the final set of price equations. The result is shown in Table IIIb.

Numerical example

There was another example (Table IIIb) where the total prices after and before the transformation didn't change. I don't think it added much to the discussion and therefore didn't include it.

To apply the transformations simply obtain \(x\), \(y\) and \(z\) (remember \(z=1\) by hypothesis) and multiply respectively by \(c\), \(v\) and \(s\). Profit is given by \(r\). Price is the sum of constant capital, variable capital and profit.

Table IVa is derived from Table IV in the same way that Table IIIb was derived from Table II. We see once again that all the conditions of Simple Reproduction are fully satisfied by this method of transformation. But there is one difference between this case and the earlier one. In Table IVa total price (1000) diverges from total value in Table IV (875); whereas in the previous example the two totals were the same. A brief explanation of this difference will show that the earlier example is a special case while the later example must be regarded as possessing general validity.

Table IV - Value Calculation

Table IVa - Price Calculation

The problem turns on the organic composition of capital in the gold industry relative to the organic composition of the total social capital before the transformation to price terms has been carried through. This can be readily demonstrated. It is clear, first, that if in the gold industry a relatively high organic composition of capital obtains, the price of gold will be greater than its value. This follows from the fact that in price calculation profit is proportional to total capital whereas in value calculation it is proportional to variable capital alone. Consequently if all other commodities are expressed in terms of gold, their total price must be less than their total value. This can be put otherwise as follows: since ex hypothesis the price and the value of a unit of gold are both numerically equal to one, the fact that its price is ‘higher’ than its value can be expressed only by the fact that the average price of all other commodities is lower than their average value. Put still otherwise, if the organic composition of capital is relatively high in the gold industry, the transformation from value to price will raise the purchasing power of gold. The same reasoning applies, mutatis mutandis, to the case where the organic composition of capital in the gold industry is relatively low. In this case total price will be greater than total value. Only in the special case where the organic composition of capital in the gold industry is exactly equal to the social average organic com position of capital is it true that total price and total value will be identical.

These principles can be tested by reference to the numerical examples already presented. In Table II the organic composition of capital in the luxury-goods department (and hence in the gold industry) was 100/150 or 66.6%, while the organic composition of the total capital was 400/600, also 66.6%. Hence the transformation to price (Table IIIb) resulted in a total price equal to total value. In the example taken from Bortkiewicz, however, the organic composition of capital in the luxury-goods department was originally 50/140 compared to an organic composition of the social capital of 375/675 or 55% per cent. Since in this case the organic composition of capital in the gold industry was relatively low , the transformation from value to price resulted in a total price greater than total value.

Since there is no reason to assume that the organic composition of capital in the gold industry is equal to the average organic composition of the social capital, it follows that in general the Bortkiewicz method leads to a price total differing from the value total.

It is important to realize that no significant theoretical issues are involved in this divergence of total value from total price. It is simply a question of the unit of account. If we had used the unit of labor time as the unit of account in both the value and the price schemes, the totals would have been the same.*

The divergence does lead to an empirical issue. Before we had the problem of having the same organic composition of capital across all industries and now we have the problem of finding a mapping from units of labour to prices that is correct across all industries.

Since we elected to use the unit of gold (money) as the unit of account, the totals diverge. But in either case the proportions of the price scheme (ratio of total profit to total price, of output of constant capital to output of wage goods, et cetera) will come out the same, and it is the relations existing among the various elements of the system rather than the absolute figures in which they are expressed which are important.

With the help of the Bortkiewicz method we have shown that a system of price calculation can be derived from a system of value calculation. This is the problem in which Marx was really interested. He believed he could solve it by using an average rate of profit calculated directly from the value magnitudes. This was an error, but it was an error which pales into insignificance when compared with his profoundly original achievement in posing the problem correctly. For, by this accomplishment, Marx set the stage for a final vindication of the labor theory of value, the solid foundation of his whole theoretical structure.*