Non-transformation problem math supplement

This is the mathematical/theoretical supplement to my video on the non-transformation problem in which I attempt to present the basic arguments of the TSSI (temporal single-system interpretation) of the transformation problem. I begin by reviewing 11 main points from that video in slightly more theoretical terms. If you feel you don’t need the review, skip them.

1. First of all, as the TSSI guys argue, this isn’t a new solution to the transformation problem. It is, they argue (and I agree with them), the way Marx understood the notion of prices of production. In fact, the idea of a transformation problem is a misnomer- there is no contradiction or puzzle to be solved. That is why they often refer to it as a non-transformation problem.

2. Labor creates value. We measure it in labor time. When we are simplifying an analysis, leaving out the ways in which individual prices deviate from values, we often say that if a commodity takes 10 hours to make its value is 10. But if we want to be more sophisticated we have to ask, “10 what?” Commodities only express their values in relation to another commodity. When one commodity becomes the universal commodity that all other commodities express their value in we call this commodity “money”. Thus value can be expressed as a money price. But money prices can and do deviate from labor values.

Value is always expressed through price. One does not buy commodities sometimes as values and sometimes at prices. There are many different transformations that take place between the concrete labor an individual worker does and the money price of a commodity. Through these transformations this concrete labor becomes more and more abstract. The process of exchange essentially mystifies us, rendering this concrete labor invisible. All we see are money prices.

3. Value can only be created in production, not in exchange. Exchange only allows us to redistribute value. In the case of monopoly, prices are artificially inflated above values allowing capitalists to obtain extra value in exchange. But this means that other capitalists in the economy are receiving less value. Given an amount of value already created in production, value is a zero-sum game: when one capitalist gets more than their share of value, another looses out. The same is true for deviations in supply and demand. When demand for a product shoots up there is a temporary rise in its price allowing these capitalists to receive extra profits at the expense of others. But this appearance of extra profit attracts other capitalists into this investment and soon profits return to normal. The case of prices of production is an even more general way in which value is redistributed among capitalists.

4. Though individual prices vary we hold three aggregate equalities to be true: total value equals total price; total surplus value equals total money profit; and the aggregate value rate of profit equals the aggregate price rate of profit. Rate of profit is the ratio of surplus value (or money profit) to total cost (c+v).

5.New value is constantly being added in production and so the total amount of value in the economy is constantly growing. But how much does it grow? Not all of this new value is carried over from one production period to the next. Some exists the circuit of capital through workers’ and capitalists’ consumption. So the total added to the social product is total labor added minus workers’ and capitalists’ consumption.

6. In Marx’s famous demonstration of the prices of production he creates an abstract model of an economy and shows how the value created in production is transformed into prices of production in exchange. Because he is trying to show how individual prices deviate from individual values he treats the inputs into this production process as if they were bought at their values. This is obviously a simplification. In reality, as Marx later points out, inputs are bought at prices of production as well.

7. Bortkiewicz seized upon this as proof of an internal contradiction in Marx’s argument. He took these prices of production and plugged them back into the transformation procedure as input prices. When he did this he discovered that it was impossible to maintain the three aggregate equalities Marx had held. Because he was plugging the new prices of production back into the input side of the procedure he also created a “dual system” in which there were two different measurements of values: values and prices.

8. What was the problem with Botkiewicz’s argument? Bortkiewicz was using mathematical modeling techniques created by neoclassical economists to build a general equilibrium model of an economy. General equilibrium models attempt to model an economy in which there is no change over time- they are static, stationary models. This is why it was essential for Bortkiewicz to maintain identical input and output prices. Essentially Bortkiewicz is saying that the price of a commodity at the beginning of the production period is identical to the price at the end of the production period. If, at the end of a production period, the output prices have changed, then we must go back and retroactively change the input prices. This then creates a bizarre feedback effect that severs value and price from having any relevant relationship.This is anathema to Marx’s whole concept of capitalism which he sees as a process of continuous expansion of value over time. Capitalism for Marx is not stationary. In the production process many things happen which can effect the value of commodities. The analysis of this production process is a large part of Marx’s value theory. Bortkiewicz, by insisting that input and output prices remain the same, was sweeping this entire dimension of Marx’s value theory under the carpet. Andrew Kliman has dubbed this logic “simultaneism”- the idea that input and output prices are simultaneously determined.

9. In order to find theoretical space for imposing a general equilibrium model on Marx’s price of production argument Bortkiewicz used Marx’s idea of simple reproduction. Simple reproduction is an abstract concept of an economy where there is no growth; where the entire social product is bought back and then reproduced on the same scale from one production period to the next. Of course, Marx didn’t believe that capitalism ever maintained simple reproduction. He used simple reproduction as a theoretical way of modeling a capitalist economy on the way to his theory of expanded reproduction. Never-the-less, the theoretical possibility of simple reproduction is still plausible and thus Marx’s prices of production theory must be able to work in the case of simple reproduction in order to be true.

10. The problem with Bortiewicz understanding of simple reproduction is that it is guilty of “physicalism”. Physicalism is the mistaken idea that value is physically determined. If I produce 2 widgets an hour, each widget is worth a half hour of labor time. If my physical productivity increases and I produce 4 widgets an hour, then it seems obvious that each widget is now worth half as much- only 15 minutes of labor time. But under physicalist assumptions the opposite is true: The value of the widgets remain the same and the total value I have produced goes up. Thus Bortkiewicz, Okishio and all the other neoclassical economists guilty of physicalism sweep under the carpet Marx’s entire body of theory that deals with the way in which changes to productivity alter value. For them an increase in physical output means a corresponding increase in value. In terms of the problem of simple reproduction, the physicalist would argue that in an economy with no growth the amount of value in the economy would stay the same.

11. The Temporal Single System Interpretation (TSSI) of the prices of production is anti-simultaneist and anti-physicalist. It is anti-simultaneist in that is argues that input and output prices, by definition, cannot be equal. New surplus value is added to inputs and this changes their value. To go and retroactively re-calculate the value of inputs at the end of a production period is absurdist. Instead, prices of production become the input prices for the NEXT period of production, just as they would in the real world. It is anti-physicalist in that it argues that simple reproduction can occur in physical terms without requiring value to stay stationary. In other words, an economy can still continue to grow in value terms, it’s prices can continue to fluctuate from one period to the next while still maintaining physical simple reproduction. To force values to correspond to physical outputs is anathema to the entire marxist project.

Now that I have reviewed the basic arguments, let’s take a look at some simple tables that illustrate the transformation procedure as envisioned by Marx and the TSSI school.

We begin by imagining an economy with three different departments. A department is a grouping of capitalist industries according to the function they serve in reproducing the basic class relations of capitalism. Department One produces means of production- all the machines and raw materials used as inputs into the production process like steel, screws, oil, robots, etc. Department Two produces means of subsistence for the working class like toilet paper, cars and bread. Department Three produces consumption goods for the capitalist class like yachts, caviar, and fancy wine. Often, similar models combine Departments Two and Three together into one department producing consumption goods for both workers and capitalists. For present purposes, it doesn’t really matter which we use. The point is just that someone has to make all the stuff workers and capitalists consume.

Staying true to the standard terminology we call the means of production made by Department 1 constant capital (c). We call the wages paid to workers variable capital (v). And we call the surplus value created by workers (s). So here’s our first table, without any numbers yet:

______________________Constant capital (c) Variable capital (v) Surplus value (s)

Department 1 (D1)

Department 2 (D2)

Department 3 (D3)

Department 1 produces means of production for all three departments. Department 2 produces consumptions goods which are bought by the wages paid to workers in all three departments. And Department 3 produces luxury goods for capitalists which are bought with the surplus value created in all three departments. In order for simple reproduction to take place:

____C V S Total value c+v+s

D1 C1+C2_C3

D2 V1+V2+V3

D3 S1+S2+S3

In other words, in order for simple reproduction to take place the total value of department 1 must equal the total demand for constant capital in departments 1,2 and 3; the total value of department 2 must equal the demand for wages (v) in all three departments; etc.

Let’s plug some numbers into this table and see if we can find some figures that allow for simple reproduction. I labored hard to find some numbers that would be nice and round so as to make the math easy.

_______________C V S Total value

__________D1 30 10 10 50

__________D2 10 20 20 50

__________D3 10 20 20 50

Total demand: 50 50 50 150

This example is really simple: all three departments produce the same amount of value. Notice also that the rate of exploitation is the same in each department- for each unit paid to workers in wages, one unit of surplus value is created. What is different in each department is the ratio of C to V (called the organic composition of capital.)

This is where the puzzle comes in. Let’s make another version of this table in which we compare the total cost of production in each department to the total amount of value created:

_____C+V (cost of production) V+S (new value added)

D1 40 20

D2 30 40

D3 30 40

Departments two and three produce more value than department one, yet we know that under conditions of perfect competition an average rate of profit will be established. An average rate of profit means an average rate of return on total cost. So even though department one creates less value, it gets the same rate of return on its investment in C+V as the other two departments.

Let’s now look at a final version of the table:

_______C V C+V S W $s P S/C+V $/C+V

D1 30 10 20 10 50 25%

D2 10 20 30 20 50 66%

D3 10 20 30 20 50 66%

Total: 50 50 100 50 150 50%

In order to be pretentious, we’ll follow tradition and use W for “total value”. As far as I can tell, there is no reason for this other than to be a pain in the ass. S/C+V (surplus divided by cost) is the rate of profit for each department. You’ve probably noticed that each department has a different rate of profit. Our goal is for each industry to have the same rate of profit, or more specifically, the average rate of profit which, as you can see from the bottom of the column, is a healthy 50%.

To do this we’ll need three more columns, the empty columns from the above table. $s is the amount of money profit each firm would receive under the average rate of profit. P is the price of production- W reconfigured now with $s instead of S. And $/C+V is the money rate of profit in contrast to the value rate of profit. Making those calculations we get:

_______C V C+V S W $s P S/C+V $/C+V

D1 30 10 40 10 50 20 60 25% 50%

D2 10 20 30 20 50 15 45 66% 50%

D3 10 20 30 20 50 15 45 66% 50%

Total: 50 50 100 50 150 50 150 50% 50%

To get the money profit for D1 we multiply D1’s cost of production by the average rate of profit (50%) to get 20. Adding his 20 to the total cost we get 60 (C+V+$s). The same calculation is made for all departments. Notice that now all three of Marx’s aggregate equalities exist: Total value is equal to total money prices (P); total surplus value (S) equal total money profits ($s) and the value rate of profit (S/C+V) equals the money rate of profit ($/C+V).

If your brain needs recharging now, step away from the computer, take a short walk, drink some tea and then come back for more when you are refreshed.

Remember that the point of this example is to show the way prices systematically deviate from values. Even though prices deviate from values they are still derived from values and deviate from them in a predictable, measurable way. The downside to this example is the fact that the inputs into each department (C and V) are measured in values and not prices of production. This is useful for explaining our above purpose, but not useful for more sophisticated modeling of an economy because in the real world C and V are bought at their prices of production and not their values. This is what caused Bortkiewicz to get his panties in a bunch. He showed that if you plug the prices of production (P) back into the beginning of this table, you can’t get all three of these equalities to hold.

His demonstration involves a lot of really confusing math. I was delighted that after a week of originally posting this math supplement Andrew Kliman himself (the author whose work most of this argument is based upon) posted a comment explaining exactly how to derive a Bortkiewicz transformation table. If you are interested in seeing how such a table is constructed, scroll down to the comments section for this thorough explanation.

Regardless it should not be hard to see how such a procedure would wreak havoc on the above example. During the production period new value has been added to the total social product, so plugging these new values into the input side of the equation will not allow aggregate equalities to hold.

If such static equilibrium states make sense to you, then read no further. If you put your shoes on before your socks, eat after brushing your teeth and get out of bed before you wake up, then don’t bother to read the rest of this supplement. It is based on an understanding of time in which time only moves forward and the future can’t rewrite the past.

Instead of plugging these new prices of production into the input side of the same equation, the TSSI uses these figures as input prices for the NEXT period of production. The original inputs into period one are taken as given datum- the prices of production from some previous period of production. This generates the following results. I have posted both periods here so you can see the progression from one to the next.

Period one:

……………C V C+V S W $s P S/C+V $/C+V

…..D1 30 10 20 10 50 20 60 25% 50%

…..D2 10 20 30 20 50 15 45 66% 50%

…..D3 10 20 30 20 50 15 45 66% 50%

Total: 50 50 100 50 150 50 150 50% 50%

Period two:

________C V C+V S W $s P S/C+V $/C+V

D1 36 9 45 11 56 23.6 68.6 24% 52%

D2 12 18 30 22 52 15.7 45.7 73% 52%

D3 12 18 30 22 52 15.7 45.7 73% 52%

Total: 60 45 105 55 160 55 160 52% 52%

Now, if you are like me the first time you saw this sort of period two table you are probably saying, “Where the hell did those number come from? I wish I was at the bar right now…” Let me talk you through it. It’s actually quite fascinating.

Bortkiewicz would be horrified by the argument that I am about to make. In period two the economy has undergone simple reproduction. The entire social production has been reproduced in exactly the same physical quantities. But these physical quantities now have different values and money figures.

Here’s how the calculations were made. I simply used the proportions at which C and V were distributed between departments to create the following table. We also assume that in each period the same amount of total labor is produced in each period. This gives us the following physical reproduction table:

_____C V S Total Labor Total units

D1 3/5 1/5 ? 20 50

D2 1/5 2/5 ? 40 50

D3 1/5 2/5 ? 40 50

So Department one always buys 3/5ths of the constant capital in the economy while Departments two and three always buy 1/5th each. The workers in Department one always buy 1/5th of the wages goods while workers in Departments 2 and 3 always buy 2/5ths each. How much surplus is produced in each department? Surplus is found by subtracting the amount of wages from the total labor from each department.

Let’s be super nerdy and look at a second way of doing the same calculations. I assume that in period 1 each unit of a physical commodity was equal to one dollar. Over the course of period one those values change. But in order to have physical reproduction take place we just need to have all three departments produce 50 physical units and to have their inputs be in the same proportions in physical terms. At the end of period one 50 units of C are equal to 60 dollars. This means that the price per unit of C, at the end of period one, is $1.20. $1.20 times 30 gives us 36- the input value of department one’s C for period two. The same calculations can be made for all departments thus:

_____Units P at end of period one Price per unit

D1 50 60 1.20

D2 50 45 .90

D3 50 45 .90

There is one last mystery that may be bothering you: It still may not be apparent how simple reproduction has taken place from period one to period two. Perhaps you are smarter than me, but it took me awhile to figure it out, even after reading the explanation.

In period one it was easy to see how simple reproduction was accomplished. The total demand for C, V and S were equal to the total output of Departments 1, 2 and 3. But if you look at the table for period two this is not the case. Why not?

This is because now the transformation is set in motion. What is important for physical reproduction is for the total social product to be bought and sold at the same proportions in each period. So at the beginning of period two the total demand for C and V in all three departments are equal to the total output of departments 1 and 2 in period one. After the capitalist class has sold its C and V ($105) at the beginning of period two it has 45 left over (150-105) to purchase the remaining luxury goods produced by department 3 in period one. The new S produced in period 2 will go toward the purchase of luxury goods at the beginning of the next period.

You may also complain that the total W has changed from period to period. Remember that new surplus is being added each period. This means that while the same amount of physical units are being created their value is increasing.

If you are truly interested in understanding this concept I recommend that you try to work out some tables of your own using different numbers. It’s kind of like doing sudoku puzzles, but hipper. Then you should read Andrew Kliman’s “Reclaiming Marx’s Capital”, the book that this argument is based upon.