On the non-naturalness of value:

A defense of Marx and Engels on
the transformation problem (part one)

by Joseph Green in CV #45, November 2010
(with errata corrected)

[Preface]

The labor theory of value has historically given a tremendous moral authority to the working class struggle, and it has provided activists with a scientific analysis of many features of the capitalist economy. The Marxist elaboration of the labor theory of value gives insight into the basic way in which capitalist firms operate, the exploitative nature of capitalism, the origin of profit in surplus-value, the constant need of the working class to fight over and over again to maintain gains it thought it had achieved, and that ensuring that goods are always priced at their value wouldn't emancipate the working class. So it's not surprising, given the desire of pro-capitalist economists to defend the capitalist system, that most bourgeois economists have sought to discredit the labor theory of value. Many thought they had found the Achilles heel of Marxist economics in the discussion in vol. III of Capital of how the tendency towards the equalization of the rate of profit causes the prices of goods to systematically deviate from their value. They claim that the mere fact of this deviation refuted Marx, and fuss over minor defects in Marx's presentation of the matter. This is the famous "transformation problem".

This article corrects a defect in the mathematical side of Marx's discussion of the transformation problem and modifies certain of the formulas he gave. In doing so, it doesn't undermine, but strengthens the case for Marx and Engels' overall view of the transformation process. Among other things, those mathematical results of the past, which have been taken as refuting Marx, turn out to be in line with the labor theory of value.

Most activists haven't paid much attention to the controversy over the transformation problem. It has appeared to them as an obscure secondary issue. And in fact, this common sense attitude is quite reasonable. The basic proof of the law of value lies in the repeated verification of the basic features of capitalism that it explains, not in the precise calculation of prices of production. Moreover, most of the literature on the transformation problem hasn't been very enlightening.

This article too will have some dry and technical matters. But I hope to express the main modification needed of Marx's calculations with regard to the transformation process briefly and clearly. And I will also seek to connect the transformation problem to some present-day issues of importance in their own right:

Introductory material

Some preliminaries

This section will run briefly through some basic formulas and concepts from the labor theory of value that are needed in discussing the transformation problem. The plan is to accustom the reader to the terminology and abbreviations used in this article, while leaving the reader to consult Capital for the precise definition of, or elaboration on, the various terms.

The value, or labor-content, of a product is the socially-necessary number of hours that go directly or indirectly go into its production. This means not just the immediate labor used in the final stage of production, but also the labor embodied in the raw materials, the machines and workplace buildings, etc.(1)

Thus, under very general conditions the exchange-value of a product (the average price over a period of time) will be its value, val:

val = c + v + s.

Here lowercase c stands for a certain part of "constant capital" or capital invested in material goods rather than immediate, living labor. It is that part of the constant capital which is used up during the production cycle that creates the final product and whose value passes into the product. It consists partly of capital invested in goods that are completely used up in production: I'll call this r, because raw materials are one example of it. This is the "circulating constant" capital.(2) And then there is the "fixed" capital (such as machinery, buildings, etc), or capital that lasts longer than a single production cycle and only gradually gets used up. I'll call the value of the part of the fixed capital that gets used up in a single cycle w, as it is the worn-out part of the machinery, buildings, etc. The value of the rest of the fixed capital is the "persistent fixed capital" which survives to serve in additional cycles of production, and I'll call it f for "fixed". So the total material capital, both fixed and circulating, used in production of the product is

f + w + r.

But only the circulating part, w + r, passes over into the value of the product, so

c = w + r.

v + s represents the hours of labor by the workers. The workers' do not get back in wages a money equivalent for this entire value; instead, this value divides into two parts. v stands for the variable capital, which represents the wages which are paid to the workers. If the workers are paid the value of their labor-power, it represents the amount of value or labor-time presented by the goods needed for the sustenance of the workers and their families. It's called variable capital as the capitalists see this part of their capital grow during the production process, in the sense that labor adds more to the value of the product than the value of the wages the workers receive from the capitalists. s stands for the surplus value, which is the excess of the value added to the product by the workers' labor over what the workers get paid; this excess is appropriated by the capitalist. It is the profit made by the capitalist, if everything including the workers' labor is bought and sold at its value.(3)

Marx calls s/v the rate of surplus-value: it is the ratio of the amount of money made by the capitalist to what the workers are paid. Or more simply, it is an index of exploitation.

The rate of profit is calculated differently. It is calculated by comparison to the total capital employed by the capitalist in producing the object. So this is

s/(f + c + v).(4)

These formulas give rise to a curious result. If everything is bought and sold at its value, a capitalist's profit is proportional to the amount of the variable capital, but has no relation to the amount of constant capital. Thus the more variable capital that is employed, the higher the total profit and the rate of profit, but with more constant capital, the profit stays the same and the rate of profit is lower. Thus the return on capital will differ from one enterprise to another. In this discussion, I'm assuming for simplicity that the rate of exploitation (s/v) is the same for all capitalists. Then comparing two capitalists in different fields, who both used the same amount of total capital, the one with more workers would have a higher rate of profit than the one with fewer workers.(5)

Thus the rate of profit will depend on the ratio between the constant and the variable capital used in production: this is what Marx calls the "organic composition" of the capital. A high organic composition is what is known in bourgeois economics as being "capital-intensive", although actually variable capital is capital too, and a low organic composition is known as being "labor-intensive". If prices averaged around their values, a high organic composition of capital would correspond to a lower than average rate of profit, and a low organic composition would correspond to a higher than average rate of profit.(6)

But if capital can flow from one field to another, then there will be a tendency for the rate of profit to equalize among capitalists. This requires that the prices be determined in a different way from what has just been described. For the rate of profit to tend to be equal, the prices of products should tend toward the so-called "prices of production" (pp). They are calculated as follows:

First one considers the cost-price (k) to the capitalist of producing some good. This is just c + v, which gives the cost of the materials used up in production plus the workers' wages. If the general rate of profit is R, then the profit should be R times the total capital employed in production, including all the fixed capital, or R(f + c + v), which can also be written as R(f + k). Hence

pp = k + R(f + k) = (c + v) + R(f + c + v)

=Rf + (1 + R)(c + v)

Well, this formula has to be taken over the entire output of that particular item during a production cycle. To see this, consider the case of a machine that costs a million dollars producing hundreds of thousands of items during a production cycle, with a general rate of profit of 10%. Rf is $100,000, and Rf is only part of the price of production. So if one applied this formula to one item, it would seem to say that a single item costs more than $100,000. But in fact, if 200,000 items are produced with the help of this machine in the course of a production cycle, then one has that the total price of production of all these 200,000 items is more than $100,000. Well, that means that the term Rf only adds $.50 to the course of each item. That explains why expensive machines can be used to produce cheap products.

Alternatively, let N be the number of units of the product that the machine produces in a year. If one wants to use the formula with respect to individual units of output, one lets

f=(value of machine)/N.

The important point about the formula for pp is that values and prices of production differ. This is the starting point of the transformation problem, as it shows that, if there is a tendency for the rate of return to equalize, the prices of products have to systematically deviate from their value or labor-content.

Adam Smith's and David Ricardo's transformation problems

The labor theory of value originated in the work of early bourgeois economists, such as Adam Smith and David Ricardo. Their work would at some points say that pricing was according to value, and at other points, that it was according to prices of production. This was only one of the contradictions that appeared in their work.

Since then, bourgeois economics has been unable to deal with this issue. Instead, it degenerated into apologetics for capitalist exploitation and gave up the labor theory of value.

Indeed, for a long time now bourgeois economists have mocked the theory of value. They claim one can avoid all the trouble and fuss surrounding the transformation problem if one throws away the concept of the labor value of a product and looks only at subjective preferences for products and their supply and demand. Accept this, and one will supposedly enter the realm of clear and straightforward economics. This claim might have a certain resonance among people sick of the widespread quibbling over the transformation problem.

Yet, despite such claims, bourgeois economics is extremely complicated and full of ever more elaborate and obscure mathematical formulae. Nothing was solved by throwing out value, and the issues involved were simply swept under the rug. As time went on, bourgeois economists discovered that their financial indices were subject to what they call the "aggregation problem", the "index problem", and even such an obscure term as "the Cambridge capital controversy". They wring their hands in many obscure and highly mathematical tomes about this, but when talking to the general public, they deal with these contradictions in a much simpler manner -- they ignore them. But this takes us too far ahead of ourselves in this story. We'll come back to the aggregation and index problems and the Cambridge capital controversy later in this article.

Marx and the transformation process

It was one of the strong points of Marx's approach that he noted this and other contradictions in the labor theory of value as developed by Smith and Ricardo, and developed a more scientific version of it. He pointed out that the tendency to the equalization of the rate of profit leads to a systematic deviation of prices from values.

Unlike what is pictured by critics of Marxism, this was not a particularly hard step for Marx to take. He had always noted that exchange-value and individual prices in the marketplace deviated, both because exchange-value represented an average price under general conditions, and because monopoly, shortages, absolute (but not differential) land rent, government regulations, and so forth caused deviations from value. Marxist economics analyzed and explained these deviations using the law of value, and reached useful conclusions about them. What was different with respect to prices of production was only that here was a systematic deviation of a more universal character.

So it was natural for Marx to realize that the equalization of the rate of profit modified the way that the law of value was manifested in marketplace prices, but didn't overthrow it. The transformation to "prices of production" results in the surplus-value exploited from the working class being redistributed among the capitalists: some firms, those employing capital with a high organic composition, would appropriate to themselves not only the surplus-value they exploited from their workers, but also some of the surplus-value sweated out of the workers by other capitalists, while those firms employing capital with a low organic composition would give up to other capitalists some of the surplus value they exploited from their own workers. Only for those firms employing capital of an average organic composition would profit and surplus value coincide.

Marx held that, nevertheless, the labor-value of commodities dominated the formation of prices of production; surplus-value explained the origin and size of profits and the rate of profit; and changes in value were responsible for the main changes that took place in the prices of production. Thus volume III of Capital, which deals with the equalization of the rate of profit, showed that the conclusions reached in volumes I and II of Capital remained valid, while also bringing some additional features of capitalism into focus. Among other things, Marx pointed out that the sharing out of the pool of surplus-value among capitalists according to the size of their capital helps explain their class solidarity against the working class, as the extent of the profit obtained by an enterprise depends not only on what the individual capitalist exploits from the firm's workers, but also on what all the capitalists, as a class, have exploited from the working class as a whole.

The helper formulas for the transformation process

In Volume III of Capital, Marx gave some formulas concerning the transformation process. They provide an intuitive approach to seeing how the transformation process works. These include the following:

He also sets forward that the prices of production can be calculated from the values by the formula I have mentioned above, which if the persistent fixed capital is taken for simplicity to be zero, is:

pp = k + Rk = (1 + R) k = (1 + R) (c + v).

However, he also noted that ". . . We had originally assumed that the cost-price of a commodity equalled the value of the commodities consumed in its production. But for the buyer the price of production of a specific commodity is its cost-price, and may thus pass as cost-price into the prices of other commodities. Since the price of production may differ from the value of a commodity, it follows that the cost-price of a commodity containing this price of production of another commodity may also stand above and below that portion of its total value derived from the value of the means of production consumed by it. It is necessary to remember this modified significance of the cost-price, and to bear in mind that there is always the possibility of an error if the cost-price of a commodity in any particular sphere is identified with the value of the means of production consumed by it."(7)

This means that the formulas given above have to be modified. I have defined various things, such as the constant capital, the variable capital, and so forth, with respect to their values. Now it is necessary to consider the same categories, but calculated according to their prices of production. So, for example, I am using c to refer to the value of the constant capital. Let's use c to refer to how much the constant capital costs when calculated according to the prices of production of all its components. In general, I'll use underlining to indicate that a category should be calculated via the prices of production.(8)

Thus the formula for the price of production becomes

pp = (1 + R ) k = (1 + R) (c + v).

Marx held that the rate of profit is the same whether calculated in value terms or prices of production, i.e. that R = R, so the above formula reduces to

pp = (1 + R) k

(I'm using pp to indicate the approximate value for the price of production which results if one calculates with the values, and pp to indicate the precise price of production.)

Marx's formulas provided an appealing way to approach the transformation issue. However, some of these formulas turned out to be only approximate, and later in this article I will show how they have to be modified. This doesn't undermine Marx's overall view, because these approximate formulas are only helper formulas, not key assertions. If their more accurate versions also back the key assertions, as in fact is the case, then these modified formulas strengthen, rather than weaken, the Marxist view of the transformation issue.

Mathematical difficulties

The fact that the more accurate formulas for prices of production involve underlined quantities gives rise to two mathematical difficulties which were used to cast doubt on the Marxist view of the transformation process. The first difficulty concerns calculating the prices of production in terms of value, and the second concerns some of the helper formulas.

To begin with, pp can be calculated easily and directly from the values via the formula pp = (1 + R)k, but that is not so for the precise prices of production via the formula pp = (1 + R) k. This is because the latter formula involves relations between the prices of production of different products, rather than relating the price of production of a single product to the values of other products, and it also requires finding the transformed rate of profit. So the second formula doesn't directly give the price of production in terms of values.

In practice, one can probably obtain a suitable approximation fairly easily in most cases. Moreover, in any real economic situation, an approximation is indeed the best one can obtain. Marx, for example, noted repeatedly that there is only a tendency to achieve a uniform rate of profit, not an exact equalization of the rate of profit. And he also noted that various fields of production ended up left out altogether from the equalization of the rate of profit. These phenomena in themselves undermine the exactness of any formula based on assuming that the all rates of profit are equalized.

So it's not clear why a precise formula is that important. In practice, one needs to know the general way in which the transformation from values to prices of production affects the distribution of profits among firms and affects what is produced and what is not produced. One also needs to know in what type of economic problems one can directly apply values, leaving aside the transformation to prices of production as an irrelevant complexity, and in what type of problems one has to consider this transformation. But one rarely needs to know the precise number of abstract labor-hours represented by any product.

However, some economists wouldn't believe that value could determine the prices of production unless a more precise mathematical analysis was given. While I disagree with this, the point is moot, since it turns out that such an analysis would eventually be given.

But, as this analysis emerged, the transformation problem went into a new phase, because it turned out that some of the helper formulas were only approximations. Simple mathematical models of an economy were analyzed. Simultaneous equations were used to solve for prices of productions. It was determined that one could either set the total of the prices of production produced in all spheres or production to the total of the value of everything produced, or one could set the total of the profits in all spheres of production to the total of the surplus value in all spheres. But one couldn't, except in special cases, have both these helper formulas of Marx satisfied: that is, they both couldn't be completely satisfied -- it wasn't considered sufficient to have them both approximately satisfied. In this article, I will, unless otherwise noted, always take the total of the prices of production to be equal to the total of the values. This is always possible according to the mathematical models, and by doing this one avoids having to worry about defining the standard of money: the equating of the total prices to the total values accomplishes this automatically. The issue of defining the standard of money adds confusion and complexity to many discussions of the transformation problem, and yet is irrelevant to its solution.

The critics of Marxist economics took these developments as a refutation of the law of value, and a voluminous and obscure literature on this question has developed. Their point of view was that if the helper formulas weren't exact, then Marxist economics collapses. It didn't matter whether the formulas were a reasonably good approximation of economic life; such a question was not of interest to them. Instead they held that, unless these formulas were exact, the whole edifice of Marxist economics was without foundation. For example, if the sum of the profits in the whole economy wasn't equal to the sum of the surplus value, it would show that profit was created in some other way than exploitation via surplus value.

This brings me to the end of the introductory material. In the next part of the article I will put forward a refinement of some of the helper formulas that ensures that they all are exact. This modification follows from a closer look at the law of value, rather than contradicting it. This should remove a theoretical objection to the law of value that was bothering some activists, and vindicate the Marxist approach. It also has some useful theoretical implications with respect to current controversies concerning "true value" and financial calculation.

An overlooked feature of value

The money/value relationship and individual products

Marx pointed out that the equalization of the rate of profit required that products sell above or below their value, depending on the organic composition of the capital used to produce them.(9) So if two items, A and B, both represent the same value, A might sell for $100, while B sells for $200. But this means that when one spends $200 in the marketplace, if one spends it on B, one gets a product with a certain amount of value, but if one spends it on A, one could buy two A's for that $200 and thus take home products worth twice the value than if one were buying B's. Thus, if things are selling at their price of production, the amount of value represented by a sum of money depends on what product is bought with it. To be more precise, it depends on the organic composition of the capital used in producing the item.

It's useful to express this in mathematical symbols. If everything sold at its value, then the amount of value represented by a certain amount of money would be equal to

vallh = L · m

where vallh is the value measured in labor-hours, m is the amount of money in dollars, and L is the ratio of value to the price of an item. So L is how much value, measured in units of socially-average labor (abstract labor-time), is represented by $1. So if $1 represents 2 minutes of labor-time (1/30th of an hour), then L = 1/30 labor-hour per dollar; and if some product costs $15, then the value in labor-hours of that product is 15(1/30) = ½ an hour.

This formula has the inverse

m = D · vallh

i.e. the amount of money spent on items is so much times their value, where D is the ratio of the price to the value, with the value measured in labor-hours. D = 1/L, represents how many dollars a product worth a socially-average labor-hour will sell for. Recalling that $1 represents 2 minutes, and L = 1/30 labor-hour per dollar, then D = 30 dollars per labor-hour, so a product with a value of 2 labor-hours would cost 2 · 30 = $60.

When things are sold at their price of production, these formulas change, and in particular, they break up into many formulas. In these new formulas, m will be underlined to indicate that it refers to prices of productions. Depending if one is buying A's or B's, one has

vallh = LA  · m or vallh = LB  · m.

More generally, one has

vallh = Lproduct · m

where Lproduct is a number depending on the organic composition of the capital used to produce that particular product. L is the ratio of the value of product, measured in labor-hours, to the price of production. And similarly,

m = Dproduct  · vallh,

where Dproduct is the ratio of the price of production to the value of a product, measured in labor hours.

For example, recall that a single unit of A and a single unit of B both have the same value. Suppose that value is 5 labor-hours. Then, recalling that a single A costs $100, 5 = LA · 100, so LA = 1/20=.05 labor-hour per dollar. Similarly, recalling that a single B costs $200, 5 = LB  · 200, so LB = 1/40 = .025 labor-hour per dollar. Thus L, the amount of value represented by a $1, might average out at 1/30=.033 in general, but LA, the amount of value represented by $1 spent on item A's, is .05, and LB, the amount of value represented by $1 spent on item B's, is .025.

Similarly, D, the amount of money which an item with a value of 1 labor hour costs, might be $30 on the average. But when one is buying A's, m = DA · vallh, so 100 = DA · 5. Thus DA = 20, and one can similarly see that DB = 40.

Thus in place of a single L, good for all products, there are a large number of Lproduct's, one for each product. And similarly for D.

Well, one might be buying quite a few different types of items with a sum of money, in which case one has

vallh = LA · mA + LB · mB + LC · mC + and so on,

where mX is the amount of money spent buying item X's, and the total amount of money m = mA + mB + mC + and so on. This could also be expressed also

vallh = Lshopping basket · m

where Lshopping basket is the average of the L's for different items that is bought with the money, weighted according to how much money was spent on them.

For example, suppose one spends $300 to buy two items: one A and one B. Then the resulting value vallh=.05 · 100 +.025 · 200 = 5 + 5 = 10 labor hours. This could be expressed as 10 =
Lshopping basket · 300, so Lshopping basket = 1/30 = .0333 labor-hour per dollar, where Lshopping basket. thus represents some kind of average of .50 and .025. But suppose one had a different shopping basket of $500 which is to be used to buy three items: one A and two B's. One could do a similar calculation and end up with 15 = Lshopping basket 2 · 500, so Lshopping basket 2 = 15/500 = .03, instead of .033. Thus Lshopping basket depends not just on which commodities are in the shopping basket, but on how much of each commodity is there.

Lall would be the L when the shopping basket includes everything. This is an L which is averaged out for the entire economy, so I might also call it Laverage. The other L's, or LX's, would vary, some being higher than L = Laverage and some lower. Similarly, the original D in this section is the same as Dall=Daverage.

Marx and Engels pointed out that a capitalist economy never directly estimates values in terms of the number of labor-hours they represent, but instead makes this estimate indirectly in terms of exchanges between different products and money. And in discussing the transformation problem, in Capital and various other places, the amount of value is measured in money rather than hours. So it will be of use to discuss the variations in how much value is represented by a sum of money in terms of val, which differs from vallh in that it is expressed in dollars. This is done by measuring labor hours by using the average amount of dollars represented by a labor hour. Thus

val = Daverage · vallh.

Recall that I have been using, as an example, that Daverage=Dall = $30 per labor-hour. So, if something has a value of three-labors, then it can also be measured as a value of 30 · 3 = $90 dollars. The difference between measuring value in abstract labor-hours or in the average amount of dollars represented by a labor-hour is like the difference between measuring distance in yards or feet. A certain length might be described as either 3 yards or 9 feet, and a certain value might be described as either 3 labor-hours or $90.

Note that one uses the same D=Daverage= Dall as the conversion factor no matter what product's value is being measured. The difference in dollars per labor-hour which occurs in prices of production are not reflected in val, as val is a measure of value, not of the price of production. In order to deal with the deviations caused by prices of production with respect to val another formula is needed.

To get this formula, recall that vallh = LX · m. And so val = Dall · vallh = Dall (LX · m) = (Dall · Lx) m. Now let UX = Dall · Lx. The result is that

val = UX  · m

where val is the amount of value, measured in dollars, represented by the sum of money, m, used to purchase item X at its price of production, and UX is the ratio between the value and the price of production of the product X.

For example, recall that item A has a value of 5 abstract labor-hours. Its price of production, m, the amount of money needed to purchase it, is $100. Also $100=DA · 5. So DA = 20. But its value measured in dollars is Daverage · 5, not DA · 5, and Daverage was taken above as $30 per abstract labor hour. So the value measured in dollars would be 30 · 5 = $150, not $100. Now, from the equation $150 = UA · $100, it turns out that UA=1.5. UA being greater than 1, as it is here, means that the value of A is greater than its price of production. Thus UA is a measure of the deviation between prices of production and values; if UA is greater than one, the value is higher.

Now, item B also has a value of 5 abstract labor-hours, but its price of production m is $200 = DB · 5. Thus DB = 40. But its value measured in dollars is Daverage · 5, not DB · 5, and Daverage = $30 per abstract labor hour. So the value measured in dollars is 30 · 5=$150, not $200. And, from the equation $150=UB · $200, it turns out that UB is .75. This illustrates that UB being less than one corresponds to the value of B being less than its price of production.

If UX = 1, then val = m, i.e. the value and the price of production are identical.

The inverse of the formula for UX (the ratio of value to price of production) would be a formula that gave the amount of m for a given amount of val, rather than the amount of val for a given amount of m. Instead of val = UX · m, one would have m = (1/UX) · val. Define TX as equal to 1/UX, and the following formula results:

m = TX · val

where TX, the ratio of the price of production to the value (measured in dollars), shows how much the price of an item is changed when one passes from values to costs of production. I use the letter T here, for transformation, since the transformation problem is often regarding as finding the formula for the prices of production of things, given their values.

Well, the total of the prices of production for all spheres of production is equal to the total value, so mall = valall. But also mall  = Tall · valall , so Tall = 1. In general, TA will vary according to the organic composition of the capital used in producing item A. TA is less than one when A is produced in a labor-intensive sphere of production, and greater than one in the capital-intensive situation.

By now, the reader may well be getting impatient. All this may appear as much ado about nothing. Surely, the reader may think, just about anyone who did much work on the transformation problem must have been aware of these simple formulas. Perhaps. Didn't these theorists refer to the price-value deviations for various individual products? Of course. But they viewed the gist of the transformation problem as finding a way around these deviations, a way to aggregate them out of existence in the helper formulas by considering whole sectors of production rather than individual products. They generally didn't want to ponder the significance of the fact, reflected in these formulas, that the value of a sum of money remained indefinite until it was exchanged for a product; they wanted to brush this aside.

There are, indeed, some things that might lead one to overlook this significance. For one thing, the different L's (ratio of value to price) wouldn't usually vary anywhere near as much as they do for the hypothetical items A and B above, where A had same value as B, though B costs twice as much. Moreover, usually a sum of money is spent buying many items, so the L for the entire sum of money (the aggregate L, so to speak), as the average of many constituent L's, would come close to Lall, and the aggregate T (ratio of price to value), as the average of many constituent T's, would come close to Tall = 1. For this and other reasons, in most practical problems one can brush aside all these L's, D's, U's and T's.

Also, although this article is inspired by Marx and Engels's work and vindicates their approach to the transformation problem, they didn't talk about the relationship of price and value in quite this way. This article brings out an aspect of the Marxist analysis of value, namely a certain indeterminacy and vagueness in value, that was implicit in Marxism from the start, but Marx expressed it in different ways from what is said here. One way he did this was by stressing that value, the abstract labor-hour, was a category that glossed over the qualitative differences between different sectors of production and different products, differences which had to be taken into account in the economic planning of a classless society. I will come back to this point later in the article when I discuss Marx's view that value is a "non-natural" category.

The relation of the total profits to the total surplus value

Now let's apply these relations to the transformation problem. The equality of the total surplus value and the total profits is one of Marx's helper formulas, and it is a formula which was challenged by subsequent mathematical work. A good deal of the literature on the transformation problem revolves around this question.

The mathematical models which are used to calculate the price of production from values specify that the total physical quantity of goods bought, when everything is priced at their value, by the capitalists with their profits remains the same when things are priced at prices of production. Each enterprise and sphere of production continues to produce the exact same products, and in the same physical amounts, as before. But these models allow the amount of the profits which any individual capitalist obtains to vary (which corresponds to a redivision of the surplus value among the capitalists). However, they don't allow any variation in the total amount of goods which are bought by the capitalists as whole with these profits. But although the total physical amount of goods purchased by the profits are the same before or after the transformation from pricing at value to pricing at prices of production, the total price of this physical amount of goods changes (except in special cases).(10) This result for the total profits expressed in dollars was obtained over and over again by mathematicians and economists.

Now, when goods are bought and sold at their value, the profit obtained by any firm is identical with the surplus-value which it extracts from its workers. So in that case, the total profits equals the total surplus-value. Thus the change in the total profits, from the situation where goods sold at their values to that where goods are sold at their prices of production, means that the total surplus value doesn't equal the total profit (calculated at prices of production), except in special cases. And this directly contradicts Marx's helper formula.

But Marx's derivation of this helper formula implicitly relied on the idea that the variation of the T's can be ignored. The idea is presumably that as the total profits come from all spheres of production, one can assume that Ttotal profits = 1, as Tall = 1. But while the profits may come from the factories and other workplaces in all spheres of production, enterprises that produce everything in the whole economy, the profits are spent only on a part of the output. The sum of goods indicated by the subscript "total profits" is not the same as the total economic output indicated by the subscript "all". Thus there is no reason to assume that Ttotal profits = Tall.

For example, consider the following simple but often-used model of an economy with three sectors or spheres of production: one sector produces means of production, a second produces means of consumption, and a third produces luxury goods that are bought only by capitalists. Assume that the capitalists spend all their profits on luxury goods (this is a model of a static economy, which continues unchanged from year to year as the capitalists never invest in expanding production), and that only capitalists buy these goods. Then the mass of profits will correspond to the total output of these goods, and only these goods.

Thus the profits will be spent on one sector of production only, the third or luxury sector, and not on either of the other two sectors. Therefore Ttotal profits will depend on the organic composition of simply one sector, that of luxury goods; in this model, Ttotal profits = Tluxury sector. And there is no reason that the sector producing luxury goods would have an average organic composition. True, in practice, in most real economies of any substantial size and complexity, there might be good reason to believe that its organic composition didn't differ that much from the other spheres. But there is no reason to believe that it would be precisely the same. So, except in special cases, Ttotal profits wouldn't be equal to one. This is the crucial point. But to express this clearly, a few additional formulas will be useful.

Let S be the total surplus value produced in the entire economy, and in general I'll use capitalized categories to indicate those that refer to the entire economy or to large branches of it. So, similarly, let P be the total profits produced in the entire economy. In the economic models used in discussing the transformation problem, the total surplus value and the total profit refers to the same physical amount of goods, only the surplus value represents the total value of these goods, while the profits refer to the total of the prices of production of these goods. (In the case of the three-sector model I have been discussing, these goods are the total output of the luxury sector.) So P and S, the total profits and the prices of production of the goods representing the surplus value, are the exact same thing: P = S.(11)

Now, recalling that T refers to a ratio between prices of production and value, the amount of total profits, measured in dollars, is given by the following formula:

P = S = Tluxury sector · S .

Or, to express it in a form which generalizes better,

P = Ttotal profits · S.

Now, since Ttotal profits is not equal to 1 in this model, except for the special case in which the sphere of luxury good production has the average organic composition, the total profits and the total surplus value differ when expressed in dollar terms.

But wait! How can the same physical amount of goods, the output of the luxury sector, be expressed by two different prices? It's because the surplus value represents these goods priced as if all goods were priced at their values. But the total profits represents these same goods, when they are priced at their prices of production. And the whole point of the transformation process is that the price of production of commodities usually differs from their value.

In physical terms, and also in terms of value, the capitalists as a whole (not the individual capitalist) get the same total amount of profits before or after the transformation to prices of production. Individual capitalists may get more or less profits, whether in physical terms, value, or price, as the profits are redivided in order to obtain an equalization of the rate of profit. But the total profits remain the same in physical terms and value, and the difference in price reflects only the change from evaluating a certain quantity of goods by its value or by its price of production.

Marx's view was that the total surplus value or total profits remained the same but was redistributed in a different way. That is so, as expressed in both value and physical terms. It is not exactly so when one measures by prices (except in the special case when Utotal profits = 1). But this modification of Marx's helper formula doesn't affect the overall deductions which Marx made with regard to the transformation issue.

Actually, since Utotal surplus value (and Ttotal profits = 1/Utotal surplus value) are probably usually both close to 1, the total profits and the total surplus value are probably approximately equal in most cases. But the issue raised in the transformation problem was that the slightest difference would, in principle, undermine the Marxist theory of surplus value by proving that some profits didn't come from surplus value. That objection is overcome by the fact that this difference only reflects that the price of production and the value differ for the physical amount of goods in which the profits are realized, since the organic composition of the capital in the sphere of production producing these goods is not the same as the average organic composition for the whole economy.

This discussion has proceeded on the basis of a simple division of the economy into three sectors of production. However, the point being made is true in general. In any economy undergoing simple reproduction, the mass of goods can be divided into those which replace the means of production used up in the course of a cycle of production, those which are means of consumption for the workers during that cycle, and those which are purchased by the capitalists with their profits. But in a more general situation, those three masses of goods might not represent entirely distinct sectors of production: for example, the capitalists might buy with their profits, not just luxury goods, but means of production and/or consumption in order to expand production. Thus the organic composition of the capital that produces the goods bought by the profits won't be simply the organic composition of the luxury industries, but a weighted average of the organic composition of the different spheres of production involved in producing the goods representing the mass of profits. That is the only change needed in generalizing from the simple model of an economy to a more realistic model. Even if the profits were spent on some goods from every sector of production, they still wouldn't represent the total output of all these spheres, but only part of the output. Thus the weighted average of the organic composition of capital used to produce the goods represented by the profits still would only by accident equal the average organic composition of the entire economy.

Thus the law of value provides, in principle, a clear, precise, and simple relationship of the total profits and total surplus value. To get the exact formula for the relationship between the two expressed in dollars, one has to calculate Ttotal surplus value. The precise formula turns out to be complex, and finding it requires careful mathematical calculation. But the overlooked property of value, the fact that the same amount of dollars can represent different values, what I call a certain vagueness or indeterminacy of value, clearly explains why the dollar figures for the total profits and total surplus value usually differ.

The rate of profit

The same considerations that apply to the total profits also apply to the total constant capital and the total variable capital. Just as the total profits only represents a fraction of the mass of products of the economy, the same goes for the total constant capital and total variable capital. We thus have that, measured in dollars, the price of the total constant capital differs depending on whether the goods making up that capital are priced at their value, or at their prices of production. The same goes for the variable capital. Hence, letting V stand for the total variable capital and C for the total constant capital, we have, not only

P = Ttotal profits · S.

but

V = Ttotal variable capital · V

and

C = Ttotal constant capital C.

And the rate of profit calculated in value terms is,

R = Slh /(Vlh + Clh ) = (Laverage S)/(Laverage C + Laverage V) = S/(C+V).

Taking E to be the value of the entire mass of goods of the economy,

E = C + V + S, and so

R = S/(E - S).

The rate of profit calculated when using prices of production is, when one abbreviates Ttotal profits as TP, Ttotal variable capital as TV, Ttotal constant capital as TC, and Ttotal surplus value as TS(12)

R = P /(V + C) = TP S/(TC C + TVV)

Now, as the sum of the values of all products equals the sum of the prices of production,

E = E = TCC + TVV + TSS

and so another formula for R is

R = (TSS)/(E - TSS).

The formulas for R and R are different, and so the rate of profit calculated via prices of production usually differs from the rate of profit calculated via values. The two rates of profit will generally be reasonably close, since TS won't usually be that far from 1. But they will only be exactly the same in special cases.

The modified helper formulas

So the following formulas replace the helper formulas, modifying all those listed except the first one:

E = E (the sum of the value of everything equals the sum of all the prices of production)

P = TP S, not S.

V = TV V, not V.

C = TC C, not C.

R = (TS S)/(E - TS S), not R = S/(E - S).

According to these formulas, there is no mysterious gain or loss of profits in going from the description of the economy via value to the description via prices of production, just a change in how much value equals how many dollars depending on the organic composition of the goods comprising the total profits. These formulas provide a suitable basis for the transformation process that Marx mapped out in Vol. III of Capital.

Moreover, for any large and complex economy, the various T's are likely to be close to 1, so that the modified formulas are quite close to the original ones. The organic composition of any one product may differ from that of the average, but the organic composition of a gigantic sector of production, such as the sector of all means of production, is likely to have an organic composition rather close to the average for the entire economy.

But in any case, the objection to Marx's formulas wasn't that the observed aggregate quantities differed substantially from Marx's formulas, but that any difference at all would supposedly undermine the logical basis of the theory of value. Thus the fact that differences would appear in the mathematical models of an economy were regarded as a refutation of the theory of value. The modified formulas, however, show that a certain deviation should be expected on the basis of the law of value. They therefore eliminate the contradiction between the past mathematical calculations and the theory of value.

Caveats

Tcollection=Tprod1 (m1/(m1 + m2 + m3)) + Tprod2 (m2/(m1 + m2 + m3)) + Tprod3(m3/(m1 + m2 + m3)).

Thus, one could say that it depends on the full organic composition of the product. This is one reason why the formula for Tproduct can be quite complicated.

However, if one is concerned simply with how far one capitalist, due to the equalization of the rate of profit, obtains more or less profits than one might expect from his own exploitation of labor, then what matters is the organic composition expressed as the ratio of constant capital to the variable capital, evaluated in prices of production. And it seems to me that in practical problems, this is more likely to be what one is concerned about.

Nevertheless, there are three different organic compositions might end up being considered:

(1) the organic composition evaluated in value terms,

cproduct/vproduct,

(2) the organic composition evaluated in prices of production,

cproduct/vproduct = (Tc for that product/Tv for that product) (cproduct/vproduct,),

and (3) the full organic composition, represented by Tproduct.

A certain part of the literature on the transformation problem consists, essentially, of making a big fuss about the difference between the full organic composition and the organic composition. Oh horrors, it might occur in some special case that product A has a higher organic composition than product B, but a lower full organic composition. That's conceivable, but not something of any special significance.

It might conceivably have been useful if the economists who worked on the transformation problem had considered finding useful approximations to the T's; considered examples of when products had exceptionally high or low T's, or examples of where the organic composition and full organic composition differed significantly; and looked into whether this had some useful significance in analyzing real economies. But the belief that the very existence of the T's cast doubt on the labor theory of value resulted in the attention being focused simply on such things as whether, in principle, the discrepancy between the organic composition and the full organic composition overthrew Marxist economics.

Relation to some past results on the transformation problem

The recognition of the overlooked property of value makes sense of the previous results obtained on the transformation problem. Below I remark on a few of them.

The Bortkiewicz-Sweezy results

In 1907 the neo-Ricardian economist Ladislaus Bortkiewicz published a paper that showed, in the case of a simple economic model and by use of simultaneous equations, how to obtain prices of production from values. He also showed that, in general, either the total prices of production wouldn't equal the total value, or the total of the profits (calculated according to prices of production) wouldn't equal the total surplus value. He regarded this as an important part of his criticism of Marx and defense of Ricardo.

In his book The Theory of Capitalist Development (1942), Paul M. Sweezy popularized Bortkiewicz's calculations. He used the three-sector model of the economy used above, where profits  profits were spent on the luxury sector and only on the luxury sector. He held that the ability to obtain the prices of production from the values was an important verification of Marx's transformation process.

But it wasn't clear what his view was towards what I call the helper formulas. He appears to have thought it important to ensure that the total profits were equal to the total surplus value, but he let the total of the prices of production deviate from the total value by using a gold standard for money.(13) He asserted correctly that in his system "only in the special case where the organic composition of capital in the gold industry is exactly equal to the social average organic composition of capital is it true that total price and total value will be identical."(14) This makes it appear as if he didn't think the helper formulas (such as the equality of the total prices and the total value) would usually be satisfied.

However, he also claimed that one could overcome the deviations in the helper formulas, writing that "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 [i.e. the standard for money] in both the value and the price schemes, the totals would have been the same. Since we elected to use the unit of gold (money) as the unit of account, the totals diverge."(15)

Sweezy's claim that he could simultaneously achieve the equality of total prices and total values, and total profits and total surplus value, was wrong. What he failed to realize, or at least he certainly failed to point out, was that, in his system, if he had switched the money standard in order to ensure that the total prices equal the total value, then this would have upset the equality of the total profits to the total surplus value.

However, Sweezy immediately goes on to add that it doesn't matter whether the total prices equals the total value, saying "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."(16) Sweezy is correct that it is not necessary to have all the helper formulas satisfied, but his reasoning is wrong. For one thing, he doesn't prove, and it isn't true, that all the relations (ratios) between the various elements of the system will remain the same. That depends on the organic composition of the different sectors of the system.

So it is rather confusing whether Sweezy thought that all the helper formulas could be satisfied, or whether he thought it wasn't important to have them satisfied. In any case, it seems to me that what his calculations actually showed (as opposed to what he said about his calculations) was essentially that, for the simple three-sector economic model he and Bortkiewicz used, both the total prices would equal the total values, and the total profits equaled the total surplus value, if the luxury sector (on which, in the model he was using, profits, and only profits, were spent) had an average organic composition.

A similar view of his calculations (if not of his claims) is put forward in a survey of the transformation problem in the New Palgrave. Here it is stated that "Sweezy went beyond Bortkiewicz, and claimed that his solution would satisfy both of Marx's claims. . . . Unfortunately, Sweezy's success is a result of his assumptions. First, since surplus value is equal to the output of the luxury sector, setting this output equal to one in both prices and values ensures that total surplus value will equal total profit. The assumption of a socially average organic composition in the third sector [luxury goods] obtains the second condition [total prices of production equals total value]."(17)

Thus the result of Sweezy's calculations appears to be in line with the formula I have given above, namely,

P = TP · S,

which says that the total profits equals the total surplus value if and only if TP = 1, i.e., if the capital producing the goods the profits are spent on has an average organic composition. (Since I always set the total prices equal the total value, the above formula says that both conditions -- the equality of the total prices and total value, and of total profit and total surplus value -- are satisfied if and only if TP = 1.) Moreover, by deriving this result directly from the fact that a certain sum of money may represent different values, depending on the product it is spent on, I have shown that this result has nothing to do with playing with different monetary standards. Nor does it have anything to do with other special features of the Sweezy/Bortkiewicz calculations.

Funny money, or the search for the golden numeraire

The Sweezy/Bortkiewicz calculations are relatively complex, and Sweezy's claims about what they showed are rather obscure or even contradictory. So the thought seems to have arisen that he had satisfied the various helper formulas in the situation where the luxury sector had an average organic composition, and perhaps one could go further and satisfy them all in more general situations. This was particularly because Sweezy, following Bortkiewicz's example, brought into the calculations the issue of setting this or that standard of money. In fact, the issue of trying different "numeraires" (standard basis for measuring money or value) introduces numerous mind-numbing complexities into the argument, while obscuring its essential features. Yet, for some academic economists, finding the proper numeraire took on something in the nature of the search for the Holy Grail.

As I have shown above, the basic feature of value that explains the modifications needed in the helper formulas has nothing to do with what standard one takes for money. Let's look at some additional reasons why that's so. Consider the two products, A and B, which were considered earlier in this article, which have the same value but different prices, A costing $100 and B costing $200. If we change the numeraire for calculating prices, if the standard of value is, say, reduced in half, then A will cost $200 and B will cost $400. The prices change, but the ratio of these prices remains the same. Similarly, if one changes the numeraire for values, the ratio of the values of two products remains the same as it was before.

Now what is the issue in the transformation problem? Ultimately, it is that A and B might have the same value, but different prices. Or, more generally, given two products X and Y, the ratio of their values, valX/valy, differs from the ratio of their prices of production, x/y. This is the fundamental issue that gives rise to the need to modify the helper formulas. But the change in numeraires can have no effect on either valX/valy or x/y. No matter how they change, it is always going to be the case that

x/y = (Tx · valX) / (Ty · valy) = (Tx/Ty) (valX/valy).

So even though changing the numeraire may seem to make certain formulas work right, it is bound to do so at the expense of creating a problem elsewhere with other formulas.

But when the numeraires are changed in the midst of calculations, what is happening gets obscured. It becomes easy to make such errors as inadvertently defining the standard of money twice, thus introducing inconsistency into the calculations.

The so-called "new solution"

The so-called "new solution" was developed in the 1980s by a number of academic economists. Its focus is in ensuring that certain formulas, such as that the total profits equals the total surplus value, be maintained without modification.(18) To do this, it makes use of two methods.

On one hand, it searches for a new numeraire. But, as noted above, this can't by itself suffice. So on the other hand, the "new solution" redefines pricing for variable capital, and -- in some variants -- for constant capital. By having different pricing mechanisms for different categories of things, it can avoid the problem that setting a different numeraire doesn't affect the ratio of the prices of different things. So the "new solution" involved arguing that its way of looking at the prices and values of variable and constant capital is better than the ordinary Marxist way.

Thus the "new solution" doesn't look into the significance of the same sum of money representing different values, the issue for which I have introduced the L's (ratio of value measured in labor-hours to price) and U's (ratio of value measured in dollars to price), but continues the old path to hell of seeking to brush them aside. As a result, it has been subject to the criticism, among other things, that "in the set of 'new solution' prices of production the sum of the values of constant capital does not equal the total sum of its prices."(19) Of course, from the point of view of this article, the value of the total constant capital C = UC · C, so it's clear why C, the sum of the values of the constant capital, doesn't usually equal C, the sum of the prices of production of the constant capital. But for the "new solution", it's would be a mystery why the value and price of production of the total constant capital should differ.

Altogether, the "new solution" is a complex system that is obscure, arbitrary, and even differs among its advocates on important points such as how to deal with constant capital. One Marxist category after another is reinterpreted, supposedly in the name of Marx's real intention. It saves one or two helper formulas by, in essence, sacrificing the content of the Marxist theory of value.

Anwar Shaikh and the transfer between two circuits of capital

Anwar Shaikh has some useful contributions to the transformation problem, such as his analysis of the iterative method by which prices of production can arise from values, which I hope to discuss in a continuation of this article, but he has also sought to explain the discrepancy between total profits and total surplus value through the idea of transfers taking place between "the circuit of capital and the circuit of capitalist revenue".

In what Shaikh calls the circuit of capital, profit is reinvested to form new capital, while in the circuit of capitalist revenue, it is serves as "revenue", something to be consumed by the capitalists.(20) He wrongly believes that it is the diversion of profit to revenue that gives rise to the possibility of the discrepancy between total profits and total surplus-value.

Thus he holds that this discrepancy can't occur if all profits are reinvested as capital. He writes that this discrepancy "is the combined result of two factors. First, it depends on the extent to which the prices of capitalists' articles of consumption deviate from the values of these articles. . . And second, it depends on the extent to which this surplus-value is consumed by capitalists as revenue . . . Where all surplus-value is consumed (as in simple reproduction), then the relative deviation of actual profits from direct profits [surplus-value] will be at its maximum. When, on the other hand, all surplus-value is re-invested (as in maximum expanded reproduction), then there is no circuit of capitalist revenue and consequently no transfer at all. Total actual profits must, in this case, equal total direct profits, regardless of the size and nature of individual price-value deviations." (Emphasis added)(21)

By way of contrast, the formulas I have given above make no distinction about whether the profit is re-invested or consumed as revenue. Those formulas attribute the discrepancy between total profits and total surplus value entirely to the organic composition of the goods represented by the profits differing from the average organic composition. It makes no difference whether the profits are used to expand the means of production or as revenue: if the organic compositions differ, then there will be a deviation between the total profit and the total surplus value. Moreover, these formulas also say that there are no transfers in physical or value terms among the total constant capital, total variable capital, and total surplus value (although there is a redistribution of surplus value in physical and value terms among individual capitalists): the difference between total profits and total surplus value only reflects different ways of measuring the same amount of goods.

Shaikh didn't simply present a theoretical argument for his view of the two circuits of capital, but conscientiously sought to verify his argument about the transfer between two circuits by using a mathematical model of an economy and calculating the difference between the total surplus value and total profits. But the model he chose had some special properties. It assumed that the new investment in means of production and consumption was exactly proportional to the already existing means.

Shaikh points out that, in this model, when all the surplus value is devoted to reinvestment, and there is no revenue at all, then there is no deviation between total profits and total surplus value. And that's right, but not for the reason Shaikh says. It's not simply because there isn't any capitalist revenue. It's because, in his model, in the case where there is no revenue (a) this model would have only means of production and consumption, and (b) the goods purchased by the profits would be means of production and consumption in exact proportion to the already existing means of production and consumption. So, for example, if the economy grows 10%, then every constituent part of the economy grows 10%: so, in particular, the total constant capital grows 10%, and the total variable capital grows 10%. In this case, the surplus value, which consists solely of the added 10% in means of production and consumption, has the exact same organic composition as the economy as a whole. In this case, TS = 1, and so total profits and total surplus value would be equal.

But suppose, while still assuming that all the surplus value was devoted to reinvestment, Shaikh's assumption of proportional growth is dropped. Then, even though all of the surplus value was reinvested, if it was invested in an assortment of means of production and consumption that wasn't proportional to the already existing means, then there would be a total profits/total surplus value deviation by an amount equal to the price-value deviation of the new means of production and consumption coming from the surplus value. I give an example of this in appendix 2. This refutes the claim that the total profits/total surplus value deviation can only come from the use of profits as revenue. It shows that even when there is no capitalist revenue at all, and hence no "circuit of capitalist revenue", the total surplus value can deviate from the total profits.

Now, Shaikh used his model not just in the case when all profits went to reinvestment, but also when the profits were divided between reinvestment (which, in his model, was to be strictly proportional to the existing means of production and consumption) and capitalist revenue. Shaikh obtained a formula for the deviation between total surplus value and total profit that only referred to the revenue and not to the part of the surplus value that is realized as means of production and consumption.

Nevertheless, in actuality, even in this case, the total deviation between the surplus value and profits comes from the sum of two deviations -- that coming from the amount of profits devoted to capitalist revenue (call this REV) and the amount of profits that is invested in expanding the means of production and consumption (call this SMPC). True, Shaikh's formula doesn't refer explicitly to SMPC. But with a little algebraic manipulation of the formula, this can be seen as follows:

To begin with, restating the results of Shaikh's model with the symbols used in this article, he obtained the result that P - S, the difference between the total profits and the total surplus value, was

(REV - REV)/(1+g)

where g is the growth rate of the economy.(22)

Here, at first sight, the total profits/total surplus value deviation depends only on the deviation resulting from REV, the capitalist revenue. This seems to verify Shaikh's view. But note that the total profits/total surplus value deviation isn't equal to the deviation between the price of production and value of REV. It is, as Shaikh himself notes, only equal to a fraction of it, to that deviation divided by (1 + g). This means they're unequal. This means that the total profits/total surplus value deviation isn't composed simply of the price/value deviation of the REV, but that there is also another factor involved. And, with some minor algebra, we can see that this other factor involves the price/value deviation of the surplus means of production and consumption, SMPC.

Let's see this in formulas. The total surplus value is composed of capitalist revenue, plus the surplus means of production: S = REV+SMP. And so the total profits equals

P = REV + SMPC. Subtracting one from the other, the result is

P-S = (REV-REV) + (SMPC - SMPC).

That is, the total profits/total surplus value deviation is the sum of the price/value deviation of the revenue and that of the surplus means of production and consumption.

Now, Shaikh obtained the result that

P-S = (REV -REV)/(1+g). This can be rewritten as

REV - REV = (1+ g)(P - S).

And so

SMPC - SMPC = (P - S) - (REV - REV) = (P - S) - (1+ g)(P - S)

= - g(P - S). Dividing both sides by -g, the result is

P - S = - (SMPC - SMPC)/g.

Thus the total profits/total surplus value deviation can be expressed by a formula that involves only the surplus means of production and consumption, SMPC. Shaikh's formula for P-S only involved the capitalist revenue REV, but this formula for P-S only involves SMPC. And both formulas are right.

What's happening is that, in Shaikh's model of proportional growth, REV-REV and SMPC-SMPC aren't independent of each other. Instead, if you know the numerical value of one of these terms, you can calculate the numerical value of the other. In fact, REV -REV = (1+g)(P - S) = - ((1 + g)/g) (SMPC - SMPC).

This is not always true. Usually, knowing the numerical value of SMP -SMP doesn't tell one the value of REV-REV. But in the special economy that Shaikh considered, it does. And therefore, when considering this special economy, there is no significance to the fact that one of the formulas for P-S contains only REV and not SMPC. One can express the total profits/total surplus value deviation either in a formula containing only REV or in a formula containing only SMPC, as one chooses. The total profits/total surplus value deviation ­ in the special case of proportional growth ­ is proportional to the price/value deviation in revenue, but it is also proportional to the price/value deviation of the reinvested profits (surplus means of production and consumption), so there isn't a special role for revenue, not even in Shaikh's model.

The vagueness and indeterminacy of money

Above I have shown that the mathematical objections to the Marxist transformation process can be overcome by taking systematic account of the fact that the amount of value represented by a sum of money depends on what products are bought with it. This property of value could be described as a certain vagueness or indeterminacy of value: a sum of money might represent any of a range of values depending on what it is going to be spent on. On the average a sum of money -- provided one doesn't get cheated in the marketplace or cheat others -- s a definite value. So it appears that money should always have a definite and precise value. And in practice, for many economic problems, one can take it as always having a certain value. But when one looks closely, it turns out that a certain sum of money can represent different values.

The idea that value has some inherent vague and indeterminate features might be a shocking concept to those who aren't familiar with it. The Marxist concept of value is often misunderstood, as a result of which it is widely felt that value can serve as a corrective to the ills of financial transactions. Indeed, some left-wing trends see socialist planning as planning in labor-hours.(23) And a prominent left-wing economist has advocated that the Venezuelan government shift money in the direction of being denominated in labor-hours as the way to deal with inflation and move towards ending exploitation.(24) The idea that value can be somewhat vague and indeterminate goes sharply against this. But it seems to be widely felt that to admit any vagueness and indeterminacy in value is not to vindicate Marxist economics and the labor theory of value, but to undermine it.

Yet value is not a socialist alternative to financial calculation, but a category that explains the underlying laws of the marketplace and financial calculation. The vagueness of value turns out to be a reflection of the fact that money and financial calculation have a similar vagueness. Indeed, bourgeois economics has had its hands full trying to shove this back under the rug, and seeks to hide the indeterminacy of its calculations in obscure terminology and complex mathematics. Once one understands the vagueness and indeterminacy of money, it makes it easier to understand the properties of value and the Marxist view of the labor theory of value. In contrast to the bourgeois economists, Marx directly referred to value and price as "non-natural" properties of products.

Inflation

Common sense might at first seem to lead to the conclusion that if an economic category, such as value, has some vague and indeterminate features, then it must be a mistaken category, a chimera that doesn't really exist. So let's look at inflation. Surely no one will deny that inflation is a real phenomena, something that affects everyone. Even today, when unemployment, speedup, and wage-cutting are ever-more-terrible causes of growing insecurity and mass misery, no one can forget inflation in health care, education, and food costs either.

But how does one measure inflation? If there were only one product on the market, it would be easy. The cost-of-living index would simply track how far that product increased or decreased in price.

But there are many products on the market. They don't all change their prices in the same way and to the same amount. Some may even go down as others go up. The cost-of-living index has to be an "aggregate" index, that lumps together the different changes that take place in the cost of different products into a single, averaged-out figure. But one can't just give equal weight to all the products: a product that is rarely used shouldn't count as much as something that one needs a lot of. So one has to use a weighted average.

But different people buy different market-baskets of goods; people use different goods in different areas (and they are often priced differently in different areas); and as goods become more expensive, people shift from goods they can no longer afford to cheaper ones. Does one calculate a weighted average based on the assortment of goods people bought in the earlier years, or the later years? When things were cheaper or when things were more expensive? All these things, and more, cause problems in preparing the proper average for the cost-of-living.

Perhaps the reader thinks that I am making a mountain out of a molehill, and that really, for crying out loud, all these complexities can be overcome. But take a look at the New Palgrave Dictionary of Economics, a massive reference work prepared by eminent bourgeois economists.(25) Its entry on inflation states that "Since there are many different ways of measuring prices, there are also many different measures of inflation."(26)

In other words, there is no one accepted way of defining inflation. Thus vagueness and indeterminacy creep into so basic and clear a concept as inflation. As an example of this, even after decades of preparing the Consumer Price Index, it continues to be revised. Some of the pressures to revise the CPI are political, as now when the ruling bourgeoisie doesn't want to pay cost-of-living raises to workers or Social Security recipients, and so wants to minimize the cost-of-living index. But it's also true that there are legitimate questions about how to maintain the CPI.

Continuing with The New Palgrave on inflation: "The most commonly used measures in the modern world are the percentage rate of change in a country's Consumer Price Index or in its Gross National Product deflator." If one follows up on this by looking at the entries for national income, creating an index, inflation accounting, and similar topics, one will find references to more and more ambiguities in the concept of inflation, and to more and more competing and complicated mathematical formulas.

Despite these complexities, it's clear that not only is inflation a real phenomenon, but it's possible to prepare price indices that are good enough for many practical purposes. This is true for comparing prices over a relatively short period of time, and in an economy whose overall structure hasn't changed substantially during this period. But, and this goes against common sense intuition, one will need different price indices in different situations, or even for measuring different aspects of inflation in just one situation.

From the point of view of mechanical materialism, any category which doesn't have a precise value -- in principle, even if one only knows the value approximately in practice -- is suspect. But from the point of view of dialectical materialism, such categories exist and are widespread. Social behavior, such as marketplace behavior, is arbitrary and indeterminate with regard to an individual's decision, but has an iron logic of its own when a mass of people take part. And such things also take place in the physical world. In quantum mechanics, categories such as position, velocity, mass, energy and even time lose some of their precision and become, in a sense, vague and indeterminate except during times of "collapse of the wave function", when they are precisely measured. Ironically, it's only by taking account of this indeterminacy that quantum mechanics is able to achieve great precision in its calculations.

The index problem

The problem of creating a price index and defining inflation is a special case of what's called the "index problem" -- the problem of finding a single numerical figure that represents the reality of several qualitatively different things. One can easily measure the increase or decrease of price of a single product in a single market: it's when one has to construct an index to keep track of all of them combined, that the problem arises. And the index problem is theoretically unsolvable. By that I mean, one can construct indices that are useful within limits, but one can't construct a perfect index. If one needs precise enough information, one will end up having to use many indices, such as the inflation indices in different cities, or the inflation for producer goods as opposed to consumer goods, or -- as one sometimes sees in the newspaper -- the figure for the core inflation minus energy costs, etc.

This problem is not peculiar to inflation, but comes up in the preparation of index numbers in general. Take a look at the entry for "index numbers" in The New Palgrave: it refers to a variety of competing indices; goes on for fourteen pages; refers to the most abstruse mathematics; and includes a huge bibliography of over a page.(27) However voluminous the literature on the transformation problem may have been, the literature on the index problem dwarfs it; however obscure the material on the transformation problem may have been, the index problem, as discussed by bourgeois economics, reaches similar depths of obscurity; and the index problem will never go away, because while indices are necessary and useful for certain purposes, there never will be one perfect index, or perfect way of preparing indices, good for all situations and completely accurate. A single number (or scalar quantity) simply can't reflect the full reality of inflation, or productivity, or other economic categories. This isn't simply because the statisticians lack sufficient knowledge of the economy: it's because in principle, even if the statisticians knew everything, any single index they prepared could only be approximately accurate, and even that only within a limited range. Reality is multi-dimensional; indices are one-dimensional. The New Palgrave doesn't say in so many words that the index problem is, in principle, unsolvable, but that's what the huge length of the entry on index numbers testifies to.

In practice, this problem comes up with respect to the most common economic categories, including measuring the size of the national economy, measuring efficiency, and so forth. The entries of The New Palgrave on these subjects describe competing systems used for various measurements or even refer directly back to the problem of index numbers.

The aggregation problem

If measuring inflation is one aspect of the index problem, the index problem in turn is one aspect of the so-called aggregation problem, that of combining qualitatively different things into a single category. For example, such categories as "capital" or "consumer goods" group together many different products. When such aggregate categories are created, there is generally an attempt to measure them by adding together the cost of all their parts, or by using some other way to create an index.

The entry in The New Palgrave for the "aggregation problem" raises the issue of whether such overall concepts have a real meaning at all:

"Microeconomic theory elegantly treats the behaviour of optimizing individual agents in a world with an arbitrarily long list of individual commodities and prices. However, the desire to analyse the great aggregates of macroeconomics -- gross national product, inflation, unemployment, and so forth -- leads to theories that treat such aggregates directly. What is the relation of such theory (or empirical work) to the underlying theory of the individual agent? When is it possible to speak of 'food', rather than of 'apples, bananas, carrots, etc.' When can one treat the investment decisions of all firms together as though there were a single good called 'capital' and all firms were a single firm?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

"Such results show that the analytic use of such aggregates as 'capital', 'output', 'labour' or 'investment' as though the production side of the economy could be treated as a single firm is without sound foundation. This has not discouraged macroeconomists from continuing to work in such terms."(28)

This discusses the aggregation problem from the standpoint of an establishment economist who is in love with bourgeois microeconomics.(29) It also displays the standpoint of mechanical materialism, according to which general categories such as "food", "capital", and "investment" aren't meaningful if they can't be handled as one-dimensional mathematical entities.

The Cambridge capital controversy

A special case of the aggregation problem, the validity of the concept of capital itself, was debated in the so-called "Cambridge capital controversy". It is referred to in a subsection of the entry on "capital theory: debates" in The New Palgrave. At one point, in discussing the neo-Ricardian Piero Sraffa's view of the matter, it points out that he believed he had "destroy[ed] the foundations of those versions of the traditional theory that attempted to define the conditions of production in terms of production functions with 'capital' as a factor. Moreover, as regards the concept of the 'capital endowment' of the economy conceived as a value magnitude, the same 'real' capital may assume different values depending on the level of r [rate of profit -- JG]. Sraffa concludes that these findings 'cannot be reconciled with any notion of capital as a measurable quantity independent of distribution and prices'".(30)

Thus Sraffa held that the usual aggregate measure of the total capital was faulty, because its numerical value would differ depending on the general rate of profit in the economy and the division of wealth between workers and capitalidoes sts. So in Sraffa's view, any real measure of the total capital in an economy was of a somewhat vague and indeterminate nature (these were probably not the terms he used) until the rate of profit and other issues were specified.

Marx, the aggregation problem, and value as "non-natural"

It is one of the strong points of Marx's version of the labor theory of value that, although he didn't use the present-day terms of "index and aggregation problems", he raised the basic issues behind them. He did this via making the distinction between abstract and concrete labor a key point of the theory of value. Concrete labor is the labor of this or that individual, performed at a certain time and place, with a certain level of skill, and a certain intensity. By way of contrast, abstract labor is human labor in general, an aggregate category that encompasses the individual labor of different individuals, in different branches of industry, performed in different locations, and with different levels of skill. One hour of concrete labor is different qualitatively from another, and produces a product which is qualitatively different from that produced by another hour. Such hours are not interchangeable: a particular type of labor is needed for a particular purpose. But abstract labor-hours are identical and interchangeable: one can be exchanged for another, and in fact is so exchanged in the form of money.

Marx pointed out that the marketplace, by equating concrete labors, turns them into abstract labor, and strips them of their particular properties. He wrote in Capital that

". . . As use-values, commodities are, above all, of different qualities, but as exchange values they are merely different quantities, and consequently do not contain an atom of use-value.

"If then we leave out of consideration the use-value of commodities, they have only one common property left, that of being products of labour. But even the product of labour itself has undergone a change in our hands. If we make abstraction from its use-value, we make abstraction at the same time from the material elements and shapes that make the product a use-value; we see in it no longer a table, a house, yarn, or any other useful thing. Its existence as a material thing is put of out sight. . . . there is nothing left but what is common to them all; all are reduced to one and the same sort of labour, human labour in the abstract."(31)

Marx pointed out that abstract labor has a purely social existence: it is not a material entity, but is created by marketplace exchange. He pointed out that when, by exchange, one equates, say, a certain quantity of iron to a certain quantity of sugar-loaf, the result "represents a non-natural property of both, something purely social, namely, their value."(32)

Thus measuring things in abstract labor, or aggregating a group of things by adding together their cost (the quantity of abstract labor they contain), eliminates the specific nature of things. Thus the total cost, the financial index, is not a "natural" property of things, and it obscures the qualitative features of things that must be taken account in natural planning. Neither price nor value are natural properties of material objects, but social properties, in particular, marketplace properties. Marx referred to the difference between planning taking account of qualitative differences on one hand and marketplace exchange via abstract labor (money) on the other, as follows:

"...Thus, economy of time, along with the planned distribution of labour time among the various branches of production, remains the first economic law on the basis of communal production [production in a classless and moneyless society -- JG]. . . . However, this is essentially different from a measurement of exchange values (labour or products) by labour time. The labour of individuals in the same branch of work, and the various kinds of work, are different from one another not only quantitatively but also qualitatively. What does a solely quantitative difference between things presuppose? The identity of their qualities. Hence, the quantitative measure of labours presupposes the equivalence, the identity of their quality."(33)

So Marx saw that measuring things according to a single index (which is the same as seeing nothing but the quantitative difference between things) results in slurring over and overlooking their qualitative differences. This is a clearer and more general presentation of the index and aggregation problems than is common in present-day economics.

A social and non-natural category is still a real category

Marx elaborated on the social character of value in his famous analysis of commodity fetishism. He pointed out that price and value represent social relationships between people disguised as relations between objects. This is important because if value were a relationship between objects, it would be something eternal, something that will exist so long as humanity needs to deal with material objects. But if value is a relationship between people, then its role will last only so far as the particular social conditions giving rise to this relationship, namely marketplace relationships, exist.

But Marx, as a dialectical rather than mechanical materialist, didn't write off social relationships as something that didn't really exist. The fact that money and value represent social relationships and that they are non-natural doesn't meant that they are arbitrary categories or fraudulent ones (although fraud does play a big role in the accumulation of many capitals). Marxism doesn't hold that abstract labor, though subject to the aggregation problem (the blurring of qualitative properties), doesn't exist. On the contrary, the goal of capitalist production is to produce surplus value and increase capital. The fact that value and capital are subject to the index and aggregation problems doesn't destroy their use as categories for certain purposes: on the contrary, it's the strong point of Marxist economics that it points out the key role that these aggregate quantities play in capitalism, and it's the rule of these aggregate quantities that is the law of value, the law of the devastation of the working class and of the environment. Marx both pointed to the central role of these aggregate quantities, and analyzed their particular nature, the particular contradictions that were inherent in them.

Money illusion and value

But for most of those working on the transformation problem, value was implicitly a natural -- almost a material -- category. The modification of the helper formulas requires explicitly dealing with a certain indeterminacy of value, and this goes against the strong feeling that the value of a sum of money should be as well-defined as the mass of a particle.

The New Palgrave has an entry on what it calls "money illusion": "The term money illusion is commonly used to describe any failure to distinguish monetary from real magnitudes. It seems to have been coined by Irving Fisher, who defined it as 'failure to perceive that the dollars, or any other unit of money, expands or shrinks in value'. . ."(34)

But the widespread money illusion in capitalist society goes way beyond simply forgetting at times to correct prices for inflation. It's the belief that monetary indices have a real, essentially physical meaning. Bourgeois economics restricts the idea of "money illusion" to some technicalities, while promoting money illusion overall. For example, take the work of William Nordhaus, an eminent neo-liberal economist working on environmental models. He's confident that he can evaluate the costs and benefits of environment action for decades in advance via setting discount rates and elasticities in a financial spreadsheet. It never strikes him that financial indices are impotent with respect to major changes in the infrastructure and the environment. Money illusion has reached the point where bourgeois economists think that financial fantasy can make up for their lack of knowledge about future technology as well as the limits of our knowledge about how the global climate works.(35)

But money illusion doesn't exist only among the neo-liberals. It gets carried over into the transformation problem in the belief that value, which is simply the essence of pricing, has such a meaning.

Living under capitalism, we have to buy and sell all the time. We need to be vigilant to buy and sell things at their value: we don't want to be cheated, and we don't want to cheat other workers who we may be dealing with. The idea that commodities have a definite and proper value is beaten into us 24/7, by everyday practice. And this suggests that things would be fine if only everything, our labor as well as the things we buy, were bought and sold at their proper value.

But Marxist economics says otherwise. The law of buying and selling things at their value is the law of enslaving people to the marketplace; it is the law of an obsolete economic system that must be replaced by something new. Thus true Marxist economics uses value to show the contradictions of capitalism and money, not as a model of what prices should be to have good things happen. From this point of view, it is not surprising that if money is subject to the index and aggregation problems, then these contradictions should be reflected in value as well.

The transformation problem is in essence a form of the aggregation problem: the simple formulas for value work properly if all spheres of production have the same organic content, but one has to aggregate with spheres of other organic content. And I have shown that the modifications needed to the helper formulas involve recognizing that a sum of money has an indefinite value unless the products which will be bought with it are specified. This in turn is a reflection of the fact that the aggregation and index problems show that money has indefinite and indeterminate features. It is a reflection of a certain vagueness and indeterminacy of money, as well as of value, that a sum of money has an indefinite value until the products which will be bought with it are specified.

True-value pricing and the non-natural nature of value

At one time the idea that pricing things at their true value would liberate the working class was common. Today the idea of true cost pricing is promoted most often with respect to the environment. The idea is that if only carbon fuels were priced at their true cost, then marketplace forces would take care of restricting their use and providing for alternatives.

Marx's idea was quite different. He held that it was the lack of overall economic planning that resulted in the devastation of the environment. He didn't look to a reformed marketplace as the way to deal with either environmental devastation or working-class misery, but to conscious planning by a humanity which was liberated from the marketplace and from the private ownership of the means of production.(36)

The aggregation and index problems strongly suggest that prices, no matter how they are adjusted, can't deal with the environment. An aggregate index, such as price, slurs over the particular features of each individual thing that it is supposed to measure. If a price measures the amount of carbon emissions, then it can't also measure the socially-necessary labor needed to produce a product. If it tries to measure both, then it is subject to the aggregation problem, and it can't really measure either adequately. This is not the only reason why relying on market measures to solve environmental issues won't work; it might not even be the most important reason; but it does help undermine as "money illusion" and commodity fetishism the search for the "true prices" that will supposedly result in marketplace forces respecting the environment.

Marx and Engels analyzed the contradictions in value, and showed how the law of value leads to class exploitation and environmental devastation. But the widespread misunderstanding of value that existed in Marx's day and still today, is that value overcomes the contradictions of capitalism, and that the marketplace has contradictions because it departs from value. From that point of view, the idea that value could be vague and indeterminate in any sense seems like a slap in the face to the honor of value, a denial of its importance for analyzing the capitalist economy. But from the Marxist point of view, it means that value accurately reflects the contradictions inherent in money and marketplace exchange.

Appendix 1: List of abbreviations and formulas

(roughly in order of appearance)

val stands for value, the socially-necessary amount of labor to make a product or, as the case may be, to make the total amount of products in some sector of production.

c stands for "constant capital", that is, capital invested in other things than immediate, productive, living labor. This is the material means of production, such as raw materials, machinery, buildings, etc. However, the constant capital is divided into two parts: circulating constant capital and fixed capital. Depending on context, in this article and in Marx's Capital, c can mean either circulating constant capital or the total constant capital.

r stands for the part of capital that is invested in goods that are completely used up in the production cycle, the circulating constant capital. I'll call it r because raw materials are one example of it.

Fixed capital is the part of the constant capital that isn't completely used up during a production cycle, such as machinery, buildings, etc. These things usually deteriorate somewhat in a single cycle. So the value of the fixed capital has two parts: the amount that has worn out in a production cycle and thus passed its value to the product, and the part that remains unchanged, the "persistent fixed capital".

w is the part of the fixed capital that gets worn-out in a single cycle ­ -- the part of the machinery, buildings etc that gets worn out.

f is the persistent fixed capital, the part of fixed capital that isn't used up during a production cycle. Note that most formulas that include f have to take account of the entire production of a commodity during a single production cycle.

c = f + w + r. The total constant capital consists of the fixed capital plus the circulating constant capital.

w + r is the part of the constant capital used up during a production cycle.

v + s represents the socially-necessary hours of labor by the workers during a production cycle.

v represents the variable capital, which is used to pay wages.

s represents the surplus-value.

s/v is the rate of surplus-value (rate of exploitation).

R = s/(f + c + v) is the rate of profit. When one considers f, the formula has to be calculated not over an individual product, but for all the products during a single production cycle.

s/(c + v) is sometimes given as the formula for the rate of profit. This could be because f is taken to be zero, for simplicity, when the fixed capital isn't relevant to the problem under discussion. Or it could be because c is taken to include f.

c/v is the ratio of the constant to the variable capital, the so-called organic composition of capital. When all technicalities are taken into consideration, there are three slightly different definitions of the organic composition of capital. (1) There is c/v, with c and v measured in value. (2) There is c/v, with c and v measured according to prices of production. And there is (3) the "full organic composition", which takes account of the organic composition of the branches of industry that produce the goods (machinery, raw materials, etc.) representing the constant capital, and of the consumer goods representing the variable capital.

k = c + v is the "cost-price" of producing some good; it is the capital actually expended; it does not include the persistent fixed capital.

pp = k + R(f + k) = (c + v) + R(f + c + v) = Rf + (1 + R)(c + v) is the price of production of some good. If one sets f = 0 for simplicity, it is just k + Rk = (1 + R)(c + v). But if one takes account of f, it has to be calculated over an entire production cycle. Or, alternatively, to apply the formula to an individual unit of a commodity, one uses f=(value of fixed capital)/N, where N is the number of units produced by the machine in the course of a production cycle. Finally, note that the formula for pp is only approximate, as the exact equation of this form would need to have every category     , including the rate of profit, calculated according to prices of production.

Category - When categories are underlined, it always indicates that they should calculated via prices of production, not values. For example, c represents the value of the constant capital, while c represents how much the constant capital would cost at prices of production.

pp = (1 + R )k = (1 + R)(c + v) This is the revised formula for the prices of production (when it is assumed there isn't any persistent fixed capital). Since prices of production appear on both sides of the equation, it expresses a relationship among prices of production rather than giving an explicit definition of how to obtain prices of production from values.

R is the rate of profit calculated via prices of production. Marx implicitly held that the rate of profit is the same whether calculated in value terms or prices of production. However, as is pointed out in the rticle, the rate of profit R does differ somewhat when calculated in value terms or in terms of prices of production.

Marx's helper formulas for the transformation process are given later in this list, just before the modified helper formulas

A and B are taken here to be two different commodities or products which have the same value, of five labor-hours, but B sells for twice the price as A, a single A selling for $100, and a single B for $200.

An economic category, such as c or v or s, may be measured in three-different ways in this article. When it is important to make such distinctions, they will be indicated as follows:

Category is the category measured in value terms, but the value is expressed in dollars, with one labor-hour represented by the average amount of money that a product with the value of one labor hour costs, averaged over the entire economy.

Categorylh is a category as measured in hours. It is the amount of socially-necessary labor-hours represented by the commodities in that category. It is a category measured not only in value, but with the value measured directly in labor-hours.

Category is, as mentioned above, the category measured in prices of production.

m is an amount of money, usually used for how much something costs. m usually is the price if things were priced at their value, and m if things are priced at the price of production.

L=Laverage=Lall, standing for labor-hours, is the average ratio between labor-hours and dollars; it is the amount of abstract labor-hours contained in a product worth one dollar, averaging over all the products in the economy. Alternatively, when everything is bought and sold at its value, it is the amount of abstract labor-hours contained in any product that costs one dollar.

D=Daverage=Dall, standing for dollars, is the average ratio between dollars and labor-hours; it is also the cost in money of the product of one labor-hour, in dollars per abstract labor-hour, when everything is bought and sold at its value.

L = 1/D, and D= 1/L.

vallh = L · m.

m = D · vallh .

When things are bought and sold at their prices of productions, these formulas with L and D break up into many formulas, each with its own separate L and D (such as LA or DB) since these ratios vary for different products. For example,

vallh = Lproduct · m or, more explicitly, valproductlh = Lproduct · mproduct where separate formulas have to be written for each product, thus:

vallh = LA · m or, more explicitly, valAlh = LA · mA

vallh = LB · m or, more explicitly, valBlh = LB · mB.

More generally, if one is considering the total or aggregate values and dollar sums for a basket of several products, A,B,C,etc., one has

vallh = LA mA + LB mB + LC mC + and so on,

or

vallh = Lshopping basket · m where Lshopping basket is an average L, averaged over A,B,C, etc.

Similarly,

val = Dshopping basket · vallh where Dshopping basket is averaged over the various products in the shopping basket.

U, or more explicitly, Uproduct, is the ratio between the value, measured in dollars, to the value of the product. Recall Lproduct is the amount of value, measured in labor-hours, represented by one dollar's worth of that product. The difference between the U's and the L's is that value is measured in dollars as far as U is concerned, not labor-hours. Wait, someone may say, wouldn't the amount of value, represented in dollars, of one dollar always be one dollar?! No! The point is that, once one switches to prices of production, the amount of value represented by a specific product that costs one dollar changes, depending on the organic composition of the capital producing that product. When one measures value in dollars, one represents a labor-hour by the average amount of dollars that a labor-hour represents, averaged over all products. By way of contrast, Uproduct represents the value, measured by the average amount of dollars a labor-hour represents, of a dollar's worth of a specific product. Thus, how far Uproduct differs from 1 represents the deviation between price and value introduced by prices of production, while Uentire output =1.

val = UX *m or, to be more explicit, valX = UX *mX .

TX is the ratio between the price of production of a product, and its value, measured in dollars. Once again, this might at first blush seem to always be 1 by definition, but read the comment on why the U's aren't always equal to 1. T (for transformation) is used for this ratio because the transformation problem was first formulated as finding the price of production of a product of a certain value. The price of a product, when things are bought or sought at their value, is mX=valX. And the price of production is mX = TX ·valX =TX · mX. So the traditional transformation problem corresponds to calculating the T's.

m = TX · val.

TX = 1/UX.

UX = 1/TX.

Tall=Taverage =1 as this article sets the total prices of production equal to the total values. But TA varies depending on the organic composition of the capital used to produce A's. TA is less than one when A is produced in a labor-intensive sphere of production, and greater than one in the capital-intensive situation.

Similarly, Uall = Utotal product = Uaverage =1 but the UA's vary depending on the organic composition of the capital used to produce A's. However, UA is greater than one when A is produced in a labor-intensive sphere of production, and less than one in the capital-intensive situation.

The simple three-sector model of an economy undergoing simple reproduction (i.e., a omy) involves means of production, means of consumption, and the luxury sector, with capitalist profits, and only capitalist profits, used to buy the luxury goods.

Ttotal profits does not necessarily equal 1 in the three-sector model, unless the organic composition of the luxury sector is the same as the overall organic composition of the economy.

Capitalized categories -- in general indicate categories that that refer to the entire economy or to large branches of it: for example, c is the constant circulating capital used in producing a product, or a collection of products, while C represents the entire constant circulating capital of the economy.

S is the total surplus value generated in one economic cycle of the entire economy, measured in dollars.

P is the total profits produced in the entire economy. In the economic models used in discussing the transformation problem, the total surplus value and the total profit refers to the same physical amount of goods (this is not true for the surplus value and profits obtained by any one capitalist); however, the surplus value represents the total value of these goods, while the profits refer to the total of the prices of production of these goods. So P and S, the total profits and the prices of production of the goods representing the surplus value, are the exact same thing: P = S.

P = Ttotal profits · P = Ttotal profits · S.

C is the total constant capital for the entire economy.

V is the total variable capital of the entire economy.

E is the total size of the output of one production cycle, and equals C + V + S. Since the total of the prices of production equals the total value, E = E.

Marx's view was that the equalization of the rate of profit resulted in the total surplus value remaining the same (but being redistributed among individual capitalists in a different way). That is so, as expressed in both value and physical terms. It is not exactly so when measured by prices of production (except in the special case when Utotal profits = 1).

Marx's helper formulas:

P = S

V = V

C = C

E = E

R = R = S/(C + V) = P/(E - S)

The modified helper formulas:

P = Ttotal profits · S

V = Ttotal variable capital · V

C = Ttotal constant capital · C

E = E

R = S/(C+V) = P /(E - S) = TSS / (E - TSS), and thus does not generally equal R = S/(E-S)

Method of averaging (the weighted average):

Tcollection = Tprod1 (m1/(m1 + m2 + m3)) + Tprod2 (m2/(m1 + m2 + m3)) + Tprod3(m3/(m1 + m2 + m3)) where one is considering a collection of three products, with prices of production.

Revenue is the part of the total production of the economy that goes into consumption, rather than replacing or expanding the means of production.

Capitalist revenue is the part of the surplus value that goes for the capitalists' personal consumption rather than being reinvested in expanding the means of production.

REV stand for the capitalist revenue for the entire economy

SMPC stands for the part of the surplus value that is realized as means of production and consumption and can be used for expanding the scale of production.

REV - REV is the deviation of the prices of production of the total capitalist revenue from the value.

SMPC - SMPC is the deviation from its value of the price of production of the part of the surplus value that is realized as means of production and consumption.

g is the growth rate from one economic cycle to another in Anwar Shaikh's economic model of proportional growth referred to in the text.

Appendix 2: A counterexample to Shaikh's view of the transfer between two circuits of capital

Earlier, in the section "Anwar Shaikh and the transfer between two circuits of capital", I discussed his view that the discrepancy between total profits and total surplus value occurs because of transfers taking place between "the circuit of capital and the circuit of capitalist revenue". I showed that, despite his other contributions to the discussion of the transformation problem, this particular conclusion is mistaken. But it might also help those who are somewhat familiar with economic models to see a concrete example of how disproportion can result even without a "circuit of capitalist revenue".

This can seen by using a model of a very simple two-sector economy that has only means of production (the material form of constant capital) and consumer goods (the material form of variable capital); there are no capitalist luxury goods at all, and all profit is ploughed back into increasing production. Let's also assume that the rate of exploitation is 100%, so that the v = s (i.e., there is as much surplus value as variable capital expended on wages). In the sector devoted to means of production, let's say that it uses 3 units of means of production for every unit of variable capital. Let's measure in units of millions of dollars to make it be a respectable production cycle of a small economy. Then we might find that the value of the means of production that are produced in the first production cycle is 500:

500 = 300 (means of production) + 100 (consumer goods) + 100 (surplus value).

With respect to variable capital, let's assume that the consumer goods which the variable capital is spent on are produced by a process that uses 1 unit of means of production for every unit of variable capital. Then we might find that the value of the means of consumption produced in one production cycle is 300:

300 = 100 (means of production) + 100 (consumer goods) + 100 (surplus value).

This works out quite well, as 500 units of means of production are produced in a production cycle, 400 of which replace the used up means of production (300 units of means of production used up in producing means of production, and 100 units used up in producing consumption good), leaving 100 units of surplus product (which is the material form of the surplus value which has been produced in this sector). Similarly 300 units of consumption goods are produced in a year, 200 units of which go to replace the used up consumer goods (100 used up in producing means of production, and 100 used up in producing consumer goods), and 100 are left as surplus product.

However, the organic composition of these sectors differs dramatically, with the sector producing means of production having an organic composition of 300/100 = 3, while the sector producing means of consumption has an organic composition of 100/100 = 1. And when everything in priced according to value, the rate of profit differs in these two sector, with the sector producing means of production having a profit rate of 100/(300+100)=1/4 or 25% (assuming that there is no fixed capital to worry about, so that the rate of profit is just S/(C + V)), and the profit rate for the other sector being 100/(100+100) = ½ or 50%. The overall rate of profit for this simple economy is 200/(400 + 200) = 1/3, or approximately 33.3%

Putting this in a chart, we have



Used up C
Used up V
S
VAL
rate of profit
leftover product(37)
C(38)
300
100
100
500
1/4
100
V
100
100
100
300
1/2
100

In the next production cycle, something has to be done with the left-over product. I'll specify a particular way of doing this. Let production of consumer goods be expanded, using 200 units of means of production and 200 units of variable capital, instead of 100 units of each. But let the production of means of production stay the same.

So there is the following chart for the second cycle of production:


Used up C
Used up V
S
VAL
rate of profit
leftover product
C
300
100
100
500
1/4
0
V
200
200
200
600
1/2
300

This works, as in the means of production sector, the 300 units of means of production needed for carrying on production have been created in the last cycle (which produced a total of 500 units of means of production). And the 100 units of consumer goods needed to carry on production are available out of the 300 units of consumer goods created in the last cycle. Similarly, with regard to the sector producing consumption goods, it requires 200 units of means of production to in order to carry on, and that is available from the means of production produced in the last cycle since 500 means of production were produced, and only 300 units are needed by the means of production sector. And 200 units of consumer goods are needed, and this is available because of the 300 units of consumer goods produced in the last cycle, only 100 are needed in the means of production sector.

But production can't be expanded in the same way in the next, third cycle, as there are no surplus means of production created in the second cycle, and such a surplus would be needed to support further growth. I'll come back to this point later on. But first let's check what happens when one switches over from calculating in value terms and goes over to calculations in prices of production. I calculated the prices of production using the iterative method which I hope to discuss in a future continuation of this article. This method has been championed by Shaikh and others; it gives proper prices of production; and, for simple models such as the above one, it is easy to set up on a spreadsheet. I won't describe the method or my spreadsheet here, but simply give my results, which are verified by the fact that redoing the charts with these prices show that the rate of profit is indeed equalized in both sectors.

The results for the first cycle of production are as follows: the price of production for a quantity of the means of production is approximately 1.0871 times its value (TC=1.0871), while the price of production of a quantity of consumption goods would be approximately .8548 times the value (TV =0.8548). Thus the price of production of goods in the sector with the higher organic composition goes up, and the price of production in the sector with the lower organic composition goes down, as expected.

Now to redo the chart for the first cycle in terms of prices of production, one has to make the following changes:

The result is as follows:



C
V
P
PP
rate of profit
leftover product
C
326.13
85.48
131.94
543.56
.3206 (32.06%)
108.71
V
108.71
85.48
62.25
256.44
.3206 (32.06%)
85.48


And the overall rate of profit is calculated by dividing the total profit (131.94+62.25) by the sum of the total constant (326.13+108.71) and total variable (85.48+85.48) capital. It comes out at 32.06%, which is not surprising, as the two sectors of production both have the same 32.06% rate of profit.

This chart represents the exact same amount of production in the first cycle as before, and the use of the exact same amount of means of production and consumer goods, but they are expressed in prices of production rather than value. With these prices, one sees that the rate of profit has been equalized at 32.06%. This verifies that these prices are indeed the correct prices of production.

But the total surplus value used to be 100 + 100 = 200 units, while the total profit is now 131.94 + 62.25 = 194.19 units. Thus there is now a discrepancy between total surplus value and total profits. It isn't very big, being merely 5.81 out of a total surplus value of 200 units. But that's not too surprising as these discrepancies usually aren't very big. Nevertheless this is indeed a real discrepancy; and it exists despite all the profits from the first cycle of production being used to expand production in the next cycle. This discrepancy thus has nothing whatsoever to do with the "cycle of capitalist revenue", which doesn't exist in this economy. So this is the promised counterexample.

Now let's look at some features of this example. Considering the tremendous difference in the organic composition of the two sectors, the prices of production don't differ that much from the values. The biggest deviation is for consumption goods, and that is only 15%. This would seem to be in line with prices of production being perturbations (small corrections) from values. Moreover, the overall rate of profit calculated in value terms and in prices of production is rather stable: it doesn't change that much, going from 33.3% to 32%. The rate of profit for each sector is adjusted, but the overall rate of profit stays pretty stable. Of course a single example such as this can only be suggestive of a general result, not a proof.

It's useful to also redo the chart for the second cycle in prices of production. It then looks like this:



C
V
P
PP
rate of profit
leftover product
C
339.58
89.00
137.38
565.97
.3206 (32.06%)
0
V
226.39
178.01
129.63
534.03
.3206 (32.06%)
267.01

So here again we see that the total profits (89.00+178.01 = 267.01) differs from the total surplus value (200+100=300, as taken from the chart above of the second cycle in value terms).

It's also notable that the prices of production change when they are calculated for the second production cycle. The price of production of the means of production is now 1.132 times the value (T = 1.132), and the price of production of consumer goods is .8900 times the value (T = .8900). This is different from the first cycle. Why do the prices of production change from cycle to cycle in this example? Is this surprising? Not really. This is because the relative sizes of the two sectors have changed, due to the expanded production. The change in the redistribution of surplus value from one sector to another comes from a difference in the organic composition of the two sectors, but the influence that the different organic compositions exercise is affected by the size of the sector with that organic composition.

However, as mentioned above, it turns out that in the second cycle there are no surplus means of production available for expanding production further in the next cycle. This means that one can't simply proceed to a third cycle by repeating the transition from the first to the second cycle, i.e. leaving everything unchanged except increasing the production of consumer goods again, as that would require more means of production. So the only way the third cycle could absorb the surplus consumer goods is if there is some additional change: a change in the organic composition of the various sectors (due perhaps to technical change); some reason to store the left-over consumer goods, such as building up needed stockpiles; an increase in wages; or some other change. Otherwise the left-over consumer goods mean that the second cycle results in an unbalanced situation, where the excess of consumer goods may cause price changes and a slow-down of production in the third cycle.

Does this mean that the example I have given of the first cycle is unrealistic? No, not at all. It is a general property of expanded growth that, unless this growth is exactly proportional, it will eventually give rise to an unbalanced situation -- unless these disproportions are counteracted by changes in the organic composition of the sectors or other factors. For that matter, proportional growth itself can be upset by changes in the organic composition of the sectors, running out of sufficient labor power or resources, etc. Growth and change -- technical change, change in markets, change in the availability of resources or labor, and so forth -- give rise repeatedly to disproportionalities. So it would be unreasonable to assume that proportional growth is the only case of expanded reproduction that has to be considered. And in fact, Marx does not assume proportional growth in his discussion of expanded reproduction in Capital(39), nor do various other studies of expanded reproduction. []

Notes

(1)The value due to the labor embodied in the raw materials goes fully into the value of the product, whereas the value of the "fixed capital" such as machinery and buildings only passes gradually into the product as the fixed capital is used up.

(2)"Circulating" capital is that part of the capital which is used up in a production cycle, and so must be replaced in order to carry on the next production cycle. This includes both the capital represented by materials which are used up in the course of production, such as raw materials, and that used to pay workers' wages.

(3)More accurately, the surplus value is divided between that portion retained by the capitalists as their profit, and that part which is transferred to other exploiters, or to the use of the capitalist state, as rent, interest, and taxes.

(4)One usually sees another formula here, just s/(c+v), not s/(f+c+v). That's because, for many purposes, what Marx calls the "persistent fixed capital" is just a complication and can be left out in order to simplify the discussion. Moreover, when the persistent fixed capital is important, small c may be used to mean the full constant capital, which in this article is instead specified as c + f. In Capital, c is used both ways, depending on context. However, because in economic crises like the ongoing depression, the depreciation of fixed capital becomes a major issue, it's useful to make the point explicitly that f is not involved in the value of a product, but is involved in calculating the rate of profit.

(5)This doesn't mean that a capitalist can make more profit by hiring more workers than are needed to do the job. These formulas assume that the workers work at the average, or socially-necessary, rate of intensity, and only the necessary labor is employed. The value of a product doesn't go up because excessive amounts of labor are used, and yet that extra labor has to be paid.

(6)It might seem natural that a ratio involving the amount of living labor is called the "organic" composition of capital. But it might then be natural to think that a high organic composition should be "labor-intensive", when in fact it is "capital-intensive". I don't know why Marx defined it in this way. Well, "organic" can mean something fundamental, rather than something living. And a fundamental of capitalist domination of labor is capitalist ownership of the means of production and other constant capital. From that point of view, a high organic composition of capital would correspond to more and more constant capital per worker. I mention this sheer speculation on how the "organic composition" got named only in order to help the reader remember more easily what is a high organic composition, and what a low one.

(7)Ch. IX: "Formation of a General Rate of Profit (Average Rate of Profit) and Transformation of the Values of Commodities into Prices of production" in Part II: "Conversion of Profit into Average Profit", Capital, vol. III, pp. 164-5, Progress Publishers, emphasis as in the original.

(8)Marx didn't use separate symbols to indicate whether he was evaluating something in terms of value or price of production. Instead, in certain passages on the transformation problem, where he thought the distinction had to be borne in mind, he would raise it explicitly.

(9)It should be remembered that in such examples it is assumed for simplicity that products are selling at the prices of production. One leaves aside cheating and other ways of one buyer or seller getting the best of another, as well as such issues as scarcity or oversupply, that affect the price offered everyone.

(10)These models also specify that the technique of production remains the same, and a commodity still requires the same physical amount of raw materials, the same amount of labor, etc., to produce. As a result, in these models the value of a given physical amount of goods remains the same after the transformation to prices of production. So both the total physical amount of the goods bought by the profits, and the total value, remain the same.

(11)Similarly, the total surplus value is the same as the total profits when things are priced at their value. So S=P.

(12)Recall that the physical goods represented by the total surplus value and the total profits are the same, so Ttotal surplus value is the same as Ttotal profits.

(13)Recall that in my calculations in this article, I always take the total of the prices of production to be equal to the total values. This amounts to using this equation to set the standard of money. Sweezy, however, spends a good deal of attention on setting this or that standard for money.

After setting the total of prices to the total values, I then investigate whether the other helper formulas are satisfied, such as whether the total profits equals the total surplus value. Sweezy, by way of contrast, sets the total profits equal to the total surplus value, and then checks to see whether total prices end up equal to the total values.

(14)Sweezy, The Theory of Capitalist Development, p. 122.

(15)Ibid., p. 123, emphasis added.

(16)Ibid., emphasis added.

(17)E.K.Hunt and Mark Glick, "Transformation Problem", in The New Palgrave Marxian Economics, p. 358.

(18)See James N. Devine, "The Utility of Value: the 'New Solution,' Unequal Exchange, and Crisis", Research in Political Economy, vol. 12, pp. 21-39, available at http://myweb.lmu.edu/jdevine/JD-1990-UtilityofValue.pdf, for a brief, sympathetic account of the "new solution".

(19)E.K.Hunt and Mark Glick, "Transformation Problem", p. 361, in The New Palgrave: Marxian Economics, edited by John Eatwell, Murray Milgate, and Peter Newman. This article contains, among other things, a brief explanation and characterization of the "new solution".

(20)Revenue is the part of the total production of the economy that goes into consumption, whether workers' or capitalists' consumption, rather than replacing or expanding the means of production. Hence the "capitalist revenue" is the part of the profit that goes for the capitalists' personal consumption rather than being reinvested in expanding the means of production.

(21)Anwar Shaikh, "The Transformation from Marx to Sraffa" in Ricardo, Marx, Sraffa: The Langston Memorial Volume, edited by Ernest Mandel and Alan Freeman, p. 55. Shaikh's article is also available from a link at his homepage at http://homepage.newschool.edu/~AShaikh/.

(22)Ibid., pp. 56, 82.

(23)See the three-part series "Labor-money and socialist planning" in Communist Voice for a refutation of the idea of labor-money or of the labor-hour being a natural unit of socialist calculation. (www.communistvoice.org/00LaborHour.html).

(24)See Paul Cockshott, "Venezuela and New Socialism", Thursday, September 6, 2007, http://21stcenturysocialism.blogspot.com/2007/09/venezuela-and-new-socialism.html.

(25)The New Palgrave: a Dictionary of Economics, 1987, first edition, edited by John Eatwell, Murray Milgate, Peter Newman. The second edition appeared in 2008, but I don't have access to it. The New Palgrave: Marxian Economics, referred to earlier in this article, is one of a series of volumes that consisted of reprints of those of the articles that bore on a certain subject. The introduction of that volume grandly proclaimed that The New Palgrave "is the modern successor to the excellent Dictionary of Political Economy edited by R.H. Inglis Palgrave and published in three volumes in 1894, 1896 and 1899. A second and slightly modified version, edited by Henry Higgs, appeared during the mid-1920s." It also pointed out that the authors of the various entries had their own, differing views. Although it asked its contributors to be fair-minded, it didn't try to suppress the differences among them. Instead it sought a balanced (but bourgeois) viewpoint on economics through the total sum of all the entries, not through any one of them.

(26)Michael Parkin, "Inflation", entry in Vol. 2 of The New Palgrave, "E to J", p. 832.

(27)W.E. Diewert, "Index numbers", The New Palgrave, vol. 2, "E to J", pp. 767-780.

(28)Franklin M. Fisher, The New Palgrave. Volume I, A-D, pp. 53-5.

(29)He also seems blissfully unaware that bourgeois microeconomics itself uses a number of aggregated quantities, although it emphasizes different ones from macroeconomics.

(30)Heinz D. Kurz, The New Palgrave, volume I, A-D, "The most recent critique of neoclassical theory", emphasis as in the original.

(31)Capital, vol. I, Chapter I, section 1, pp. 44-45, Charles H. Kerr edition (reprinted by The Modern Library).

(32)Ibid., vol. I, Chapter I, section 3, Subsection 2.2.3.

(33)Marx, Grundrisse: Foundations of the Critique of Political Economy (Rough Draft), Translated and with a Foreword by Martin Nicolaus, "The Chapter on Money", pp. 172-3, emphasis as in the original.

(34)Peter Howitt, "Money Illusion", The New Palgrave, Volume 3, K to P, p. 581.

(35)For more on money illusion in establishment environmental economics and on the type calculations in Nordhaus' book A Question of Balance: Weighing the Options on Global Warming Policies, see "Market lunacy: the use of financial calculation to answer material questions" in "THE CARBON TAX: Another futile attempt at a free-market solution to global warming" in Communist Voice #42, August 2008 (www.communistvoice.org/42cCarbonTax.html).

(36)See "Marx and Engels on protecting the environment" in Communist Voice #40, August 2007 (www.communistvoice.org/40cMarx.html).

(37)The leftover or surplus product equals the entire production of means of production minus the amount needed to replace used up means of production in both sectors. This is not an equivalent for either the profits or surplus value obtained by this sphere of production: this leftover product is simply the surplus over the total use of the product in the economic cycle. Similarly for the leftover consumer goods.

(38)The C row represents the production of means of production, which is also the concrete form of the constant capital of both sectors of production. The V row represents the production of consumer goods, which is also the concrete form of the variable capital of both sectors of production. The VAL column gives the total value of the production of the various rows.

(39)See Capital, vol. II, Chapter XXI "Accumulation and Reproduction on an Extended Scale". []


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