Transformation?- Math supplement

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+$s). The same calculation is made for all departments. Notice that now all three of Marx’s aggregate equalities exist: Total value (W) is equal to total money prices (P); total surplus value (S) equal total money profits ($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 (W) 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.

32 Responses to Transformation?- Math supplement

  1. Pingback: What Transformation Problem? « Kapitalism101

  2. One problem with the model (that I’m as yet unable to solve) is that the surplus value of D1 (creating constant capital) doesn’t get consumed; it gets “congealed” into the constant capital of the next phase. If we’re putting in 10 labor-units of surplus value from one phase to the next, it would see we have an expanding, not steady-state, economy.

  3. Or might be a problem… I’m still playing with the model in Excel, and I have money appearing out of nowhere.

  4. I think this is only a problem if one takes value to be physically determined. Obviously, if new value is added to the production process we can’t take these new prices of production and plug them back into the input side and expect to still have aggregate equalities. We don’t even need fancy algebra to understand this- it doesn’t make sense logically. But simple reproduction doesn’t require value to be stationary- in fact it is impossible for value to be stationary if, as you point out, we are adding new value to the constant capital each production period. The TSSI argues, and demonstrates, that simple reproduction can happen in physical terms while value still changes. The same physical amount of commodities can embody differing amounts of value.

  5. Paul says:

    Thanks for the detailed exercise, it really helps to clarify what’s going on.

    If anyone’s interested, I put this in a Google spreadsheet as I worked on it:

    http://spreadsheets.google.com/ccc?key=pKk4oJUqWO9iui9n3shV_Ow

    I think that should make it easy to experiment with changing the inputs to see how they affect one another.

    One suggestion: I was really confused at first over where “Total Labor” came from, and not unrelatedly my first guess for period 2’s S was totally wrong. I think it would help if the post explicitly reminded the reader that Total Labor is V+S, and why that’s so.

  6. Paul- this spreadsheet is really cool! Thanks a lot for posting this! I will try to address your concern about total labor equaling V+S at some point when I get a chance to do some editing. I know what it’s like when there’s just one thing that you can’t figure out- that the author hasn’t explained it because it must seem obvious to them. There were many times in preparing to write this post that I came across similar obstacles and had to stare at the problem for awhile before I figured out where a number came from. -Brendan

  7. If V stands for variable capital, another symbol is needed for total value. In German, value is Wert, hence the W. So in addition to helping one be pretentious and a pain in the ass, that’s a third reason for the use of W :)

    Bortkiewicz’s math is particularly confusing because he wanted to make it look like he was transforming values into prices. His procedures is basically that of taking value figures, manipulating them so as to get the underlying physical quantities (though that’s not easy to detect), and then taking the physical figures and computing prices on the basis of them. What’s really going on is that the same set of physical quantities give rise to one set of simultaneist value figures and another set of simultaneist price figures.

    So the simpler way to get Bortkiewicz’s figures is to start with the physical quantities. Imagine that the C, V, and W figures above are actually physical quantities. So the used up means of production are 30 macines in D1, 10 machines in D2, and 10 machines in D3. D1 produces 50 machines. The simultaneously determined value figures are 30*v1, 10*v1, 10*v1, and 50*v1, where v1 is the per-unit value of machines (these are simultaneously determined because v1 is both the per-unit value of machines as an input and of machines as an output.

    Likewise, the workers wages are 10, 20, and 20 loaves of bread (V), D2 produces 50 loaves of bread, and the value figures are 10*v2, 20*v2, 20*v2, and 50*v2.

    D3’s output, gold in Bortkiewicz’s example/model, isn’t an input, only an output, so there are 50 ozs. of gold, having a value of 50*v3.

    Now, there’s a distinct price system, also simultaneously determined, so we have 30*p1, 10*p1, 10*p1, and 50*p1, where p1 is the per-unit value of machines. Etc.

    For the value system, one needs the amounts of direct labor (living labor) performed. Let’s assume these equal V+S, so they are 10 + 10 = 20 in D1, and 20 + 20 = 40 in both D2 and D3. (Other numbers can be used, as long as the D2 and D3 figures are equal, and double the D1 figures.)

    Then the value equations (constant capital used up + value added by living labor = value of the product) are

    30*v1 + 20 = 50*v1
    10*v1 + 40 = 50*v2
    10*v1 + 40 = 50*v3

    So at this point, you use the top equation to find the numerical value of v1, and plug in the answer into the other equations to then find out the numerical values of v2 and v3.

    As for the price system, the idea is that

    expenditures on machines + expenditures on labor + profit = price of product

    and profit = (uniform) rate of profit times the expenditures

    and this is written as

    (30*p1 + 10*p2)*(1 + r) = 50*p1
    (10*p1 + 20*p2)*(1 + r) = 50*p2
    (10*p1 + 20*p2)*(1 + r) = 50*p3

    where r is the uniform rate of profit.

    At this point the math becomes tedious. What you do is divide the first equation by the second. The (1 + r) cancels out. And you can divide everything by p2 so as to get the ratio Y = p1/p2 and then solve the equation to find the numerical value of Y:

    30*p1 + 10*p2 50*p1
    ————- = —–
    10*p1 + 20*p2 50*p2

    or

    30*Y + 10 50*Y
    ——— = —-
    10*Y + 20 50

    (To solve this, cancel the 50’s on the right, and then multiply through by 10*Y + 20, then rearrange terms, and then use the quadratic formula to solve for Y.)

    At this point you can use Y and the top 2 equations to find the numerical value of r, then then plug in these things into the final equation to find Z = p2/p3. (The final equation can be rewritten as (10*Y*p2 + 20*p2)*(1 + r) = 50*Z*p2) or (10*Y + 20)*(1 + r) = 50*Z .)

    Finally, you want to get prices, not just price ratios Y and Z, so you fake some relationship between price and value aggregates. What Bortkiewicz chose was that the total price of gold = the total value of gold:

    50*p3 = 50*v3

    or, since p3 = Z*p2,

    50*Z*p2 = 50*v3. Since Z and v3 are now known, you can use this last equation to find the numerical value of p2. Then you can find p1 = Y*p2 and p3 = Z*p2.

    Maybe I should have left well enough alone?

  8. I can think of nothing more rewarding than to see Andrew Kliman’s above post. Thanks so much Andrew for writing such a thorough, clear explanation. And, of course, thanks for all your work on this topic. It is of utmost importance.

  9. Anonymous says:

    Interesting blog. I’m glad I found it. Thanks again, Corbin

  10. Murray says:

    Hey Brendan,
    Just noticed this but in D1 you put the C+V as 20, when it should be 40 (for Period 1). This doesnt affect anything else though, so I assume it’s just a mistype.

    Fantastic work though, I’ve always been having problems grasping the “transformation problem” and this is far more clear and simple that a lot of the other models suggested. Cheers!

  11. Pingback: Das Kapital vol. 3 Part 2- opening thoughts « Kapitalism101

  12. Hi
    I am Vietnamese.
    Thank you very much for this article. It help me. This article helped me understand more about Transformation problem.
    I am going to reseach marxist economic. I have the addition of the labor theory of value Marx. In there, value not only creat in production. And service olso creat value, if you look at the capitalist economy from the owner that a comprehensive. In there, concepts such as exchange value, the value of commodities, goods, labor, surplus value, profits, and tax issues also have a more comprehensive look.
    Because my english is not good, I can not be translated from Vietnamese into English.
    Now I study of transformation problem, from which I can try to build theories growth based on theory of Marx, so that we can better understand the nature of growth, which would be neoclassical economics with models of Solow can not understand. My model is the same with Dorma model.
    email: damvanvi@gmail.com
    Best wish for you, olso thank Andrew Kliman,

  13. sorry
    labour in service olso creat value

  14. Daniel de França says:

    Dear K101,

    One thing is bothering me. The average rate of profit increased from 50% to 52%. According to Marx, shouldn’t it had decreased?

    • Daniel,

      In the TSSI simple reproduction is not a matter of value magnitudes needing not to change but of physical quantities of inputs and outputs staying exactly the same. I think this interpretation of simple reproduction (an abstraction in the first place) is valid because: 1. when reading Marx’s description of simple reproduction in Vol. 2 this is a totally valid interpretation of his position; 2. if value is added to the social product each period then total value has to increase regardless of whether physical inputs/outputs stay the same or increase (see the above debate I had with another viewer); and 3. it makes sense of Marx’s value theory while the other interpretation renders his theory inconsistent.

      • Daniel de França MTd2 says:

        But I am not referring to inputs and outputs, values or prices. I am referring simply to the rate of profit, which should fall. Why didn’t it fall, but instead increased to 52% from 50%?

      • The Profit Rate falls in Expanded Reproduction where an increase in the organic composition of capital causes its fall. In Simple Reproduction we do not assume an increase in organic composition.

      • Daniel de França MTd2 says:

        A growth of 4% in the rate of profit, from 50% to 52%, for 1 iteration, is pretty big. So, it seems to me, that even in expanded reproduction, I’d expect to see some increase of profits for some at least for small values of rate of expansion. Do you have a counter example for this? Maybe like long term tendency for null rate of expansion.

      • I think this is merely an artifact of the arbitrary rules of Simple Reproduction in which physical quantities remain the same while value is added to the social product each year through labor. As soon as we introduce expanded reproduction we see a growth in constant capital relative to variable capital and a fall in the rate of profit. In fact, as Marx shows in Vol 2, the only way to move from simple to expanded reproduction is to increase the size of D1 and the expense of D2.

      • Daniel de França MTd2 says:

        But, isn’t there not even a simplistic mathematical model showing the rate of profit falling?

  15. Ed George says:

    In Brendan’s example, between periods 1 and 2, the aggregate rate of profit increases because the aggregate rate of surplus-value increases. The aggregate rate of surplus-value increases because the price of wage goods falls after the imposition of the general rate of profit (the production price of wage goods is lower than its value), while, because labour productivity is held constant, surplus-value rises (s + v are the same if no more new value is created in production). The rise in the rate of profit is less than the rise in the rate of surplus-value, however, because the value composition of capital ( c/v or c/(v +c) according to taste) has risen, and we see in chapter 3 of volume 3 that the rate of profit is dependent on both the rate of surplus-value and on the value composition of capital. This rise in the rate of profit is inevitable in a three sector simple reproduction model between the period at the beginning of which factors of production are bought at value and that at the beginning of which factors of production are bought at prices of production if the sector producing means of consumption has, like in the real world, the higher value composition to start with (its new price of production after the imposition of the general rate of profit will be higher than the value of the commodity product, while the price of production of the commodity product of the other two sectors combined will be lower than its value). However, if you continue the iterations, the rate of profit immediately begins to fall (in my reworking of Kliman’s model on pp. 148-52 of Reclaiming Marx’s ‘Capital’ a new aggregate profit rate lower than the original emerged and stabilised to two decimal places by the 16th iteration). What is pertinent here is that the fall in the rate of profit envisaged by Marx in volume 3 is a consequence of a rise in the organic composition, but for Marx the organic composition is a rise in the value composition because of a rise in the ‘technical composition’, i.e. because of a rise in the productivity of labour, and that is not what is happening here.

    My reproduction of Kliman’s model, with my own commentary, is at the end of my notes on volume 3 chapter 9, here: http://readingmarx.wordpress.com/2012/08/03/capital-volume-3-chapter-9-formation-of-a-general-rate-of-profit-average-rate-of-profit-and-transformation-of-commodity-values-into-prices-of-production/

  16. Ed George says:

    Oops, About a third of the way in I said ‘means of consumption’ where I should have said ‘means of production’. I’ll just post in again, corrected.

    In Brendan’s example, between periods 1 and 2, the aggregate rate of profit increases because the aggregate rate of surplus-value increases. The aggregate rate of surplus-value increases because the price of wage goods falls after the imposition of the general rate of profit (the production price of wage goods is lower than its value), while, because labour productivity is held constant, surplus-value rises (s + v are the same if no more new value is created in production). The rise in the rate of profit is less than the rise in the rate of surplus-value, however, because the value composition of capital ( c/v or c/(v +c) according to taste) has risen, and we see in chapter 3 of volume 3 that the rate of profit is dependent on both the rate of surplus-value and on the value composition of capital. This rise in the rate of profit is inevitable in a three sector simple reproduction model between the period at the beginning of which factors of production are bought at value and that at the beginning of which factors of production are bought at prices of production if the sector producing means of production has, like in the real world, the higher value composition to start with (its new price of production after the imposition of the general rate of profit will be higher than the value of the commodity product, while the price of production of the commodity product of the other two sectors combined will be lower than its value). However, if you continue the iterations, the rate of profit immediately begins to fall (in my reworking of Kliman’s model on pp. 148-52 of Reclaiming Marx’s ‘Capital’ a new aggregate profit rate lower than the original emerged and stabilised to two decimal places by the 16th iteration). What is pertinent here is that the fall in the rate of profit envisaged by Marx in volume 3 is a consequence of a rise in the organic composition, but for Marx the organic composition is a rise in the value composition because of a rise in the ‘technical composition’, i.e. because of a rise in the productivity of labour, and that is not what is happening here.

    My reproduction of Kliman’s model, with my own commentary, is at the end of my notes on volume 3 chapter 9, here: http://readingmarx.wordpress.com/2012/08/03/capital-volume-3-chapter-9-formation-of-a-general-rate-of-profit-average-rate-of-profit-and-transformation-of-commodity-values-into-prices-of-production/

    • Ah, yes that makes sense re value of means of consumption and the rise in the rate of profit. George, your blog post on chapter 9 is quite an impressive feat. I’ve only managed to skim it. The demonstration at the end of the post of the stabilization of the profit rate after 16 iterations of Klimans schema…. this sounds vaguely familiar, as if I’ve heard others mention this before. Have you seen anyone else do this?

  17. Ed George says:

    Yes. I came across a paper by Anders Ekland (http://www.assoeconomiepolitique.org/political-economy-outlook-for-capitalism/?p=1168&aid=2247&sa=0) which noted the Kliman-McGlone model and David Laibman’s critique of it (‘Rhetoric and Substance in Value Theory: An Appraisal of the New Orthodox Marxism’, in Freeman, Kliman, Wells, The New Value Controversy – which I have a pdf of). Ekland reproduces Kliman’s model from Reclaiming, and notes Laibman’s crticism that the values generated do not converge towards a new equilibrium, and set out the 14th iteration, which suggested that they do. That’s when I decided to spreadsheet the Kliman model myself, to see; as I have said, by the 23rd iteration all values are stable to two decimal places (although the convergence is infinite). When I get 5 minutes I’ll upload my spreadsheet somewhere,

    We should note that ‘all’ the Kliman model seeks to prove is that von Bortkiewicz’s claim that, in a three department simple reproduction model, if a general rate of profit is introduced and input values are transformed into prices of production output goes unsold is false. Kliman shows that, because output prices of one period are the input prices of the *next*, reproduction on the same physical scale *does* occur. You wouldn’t get this – the sudden imposition of a general rate of profit and the convergence to a new state of balance – in the real world; but as a refutation of von Bortkiewicz’s refutation it is critically important.

    • Ed, I should have asked this a while ago: What do you think is the significance of the fact that the prices stabilize in the later iterations of Kliman’s model?

      • Ed George says:

        That the equalisation of prices happens, as well as the way in which it happens, is a function of the model, which is in turn a function of the terms of Marx’s own chapter nine. It has, this accepted, no special significance, other than that prices will tend to stabilise if the productivity of labour does not change.

        What Marx sets out in chapter nine is essentially: if there were to be a general rate of profit, what would it look like? He does not consider at this stage in his exposition what the mechanisms by which the profit rate either does, or tends to, equalise are.
        Why he does this, I think, is the way that he couches his whole argument, although not explicitly, as a critique of Ricardo. At the end of chapter eight, Marx had reached the conclusion that ‘at a given rate of surplus-value it is only for capitals of the same organic composition – assuming equal turnover times – that the law holds good, as a general tendency, that profits stand in direct proportion to the amount of capital, and that capitals of equal size yield equal profits in the same period of time. […]. There is no doubt, however, that in actual fact, ignoring inessential, accidental circumstances that cancel each other out, no such variation in the average rate of profit exists between different branches of industry, and it could not exist without abolishing the entire system of capitalist production. The theory of value thus appears incompatible with the actual movement, incompatible with the actual phenomena of production […].’ This is essentially where Ricardo, on Marx’s reading, had arrived. What chapter nine does, then, is assert the logical necessity of the equalisation of the rate of profit; it is only in chapter ten that Marx looks at how this equalisation might come about.
        Kliman’s model is a refutation of von Bortkiewicz’s argument (endorsed by Sweezy and, subsequently, many others) that, taking Marx’s argument in chapter nine at face value, unless input prices and output prices are the same in the same production period (i.e. unless simultaneous valuation occurs) then the reproduction of production cannot occur. This is the origin of the argument that ‘Marx forgot to transform the input prices’. Kliman, by observing that the input prices of one period of production are by definition the output prices of the previous one, conclusively and decisively demolishes von Bortkiewicz’s would-be refutation of Marx.

        But Kliman’s model, because of what it purports to do, is situated entirely within Marx’s terms of reference of chapter nine (I think it says a great deal about how little Marx’s detractors understand Marx that it is chapter nine that has attracted the weight of the criticism of Marx’s account of the equalisation of the rate of profit rather than chapter ten). In Kliman’s model, as in Marx’s chapter nine, a necessity of an average rate of profit is imposed, to the effect that each sector of production realises a rate of profit which is that of total social production. There is no account of how this deus ex machina imposition might come about: it is as though the capitalists sat down collectively at the end of the production period and decided it. Before the imposition the model, of simple reproduction with a constant productivity of labour, is evidently in equilibrium. After the introduction of an average rate of profit – which is in Kliman’s model imposed anew at the end of each production period – the system stabilises and moves, in an infinite egress, to a new (infinitely far away) equilibrium state.

        When the first imposition of an average profit occurs, the system passes from a ‘value’ system to a ‘price’ one, i.e. from one in which the commodity product of each sector is sold at cost price plus sectoral surplus-value to one in which it sells for cost-price plus average profit. At the end of each production period, the rate of profit is again averaged out and the total surplus-value distributed between sectors of production accordingly. In this distribution of the total surplus-value those parts of social production producing at below average value composition give up a part of ‘their’ surplus-value while those parts operating at above value composition receive ‘extra’ surplus-value. Within each period of production these flows of surplus-value cancel each other out, so that total ‘value’ always equals total price at the social level in each period.

        When surplus-value is distributed in this way, given that the sector producing means of production has a higher value composition of capital than the other sectors, the production price of the product of this sector rises with respect to its ‘value’, while that of the sector producing workers’ means of subsistence falls (as does that of the sector producing capitalists’ means of subsistence).

        Why does the stabilisation of the system take the form of an infinite egress rather than that of an instaneous fact? By definition, prices are stable when the production prices of one period are equal to those of the next one. But when the average rate of profit is first imposed, prices rise, and they keep on rising, by progressively smaller amounts, in an infinite egression towards the new equilibrium point. Why do prices rise? Because, when the average rate of profit is imposed, the sector producing means or production ‘benefits’ from the redistribution of surplus-value and sees the price of production of its commodity product rise over its ‘value’ (because average profit in this sector is greater than secroral surpls-value).
        The form of reproduction considered in the model is simple reproduction in physical terms. The same product is produced, with the same inputs, period on period. Thus if, in one period, the production price of a given mass of means of production is higher at its end than at its beginning, i.e. higher at the beginning of one period than at the beginning of the previous one, to maintain production on the same physical scale – a requirement of the model – more money must be laid out as constant capital. Although no more physical constat capital is required, to buy the previous amount at its new price more money capital is needed: surplus in money form is thus accumulated. At the beginning of the process, just after the imposition of the ‘rule’ of an average rate of profit, the production price of workers’ means of production (for the same reason, the redistribution of surplus-value) falls compared to their ‘value’ – wages effectively fall – offsetting the amount of ‘extra’ capital, accumulated surplus in money form, necessary (although the rise in the production price of means of production soon works its way through into the cost price of means of workers’ subsistence and these too begin to rise).
        Because one of the assumptions of the model is constant productivity of labour, even though the (initial) fall in the price of workers’ means of subsistence means that the ‘extra’ capital that needs to be accumulated in order to maintain production on the same physical scale is less than the rise in the production price of the means of production it is the whole of the latter that is passed on in the price of the social commodity product: because labour productivity is constant, no new value is created in production – the sum of surplus-value plus variable capital in money form is constant at both the sectoral and aggregate levels, and should the latter fall the former rises. But the extra monetary surplus accumulated as constant capital at the beginning of the production period is passed on in the form of the price of production of the commodity product at its end, i.e. in the form of the constant capital component of cost price at the beginning of the next. This explains the constant, albeit progressively, and eventually effectvely infinitely, smaller, price inflation eperienced once the necessity of an average rate of profit is imposed.

        The profit that is averaged out in each production period is the aggregate surplus-value – the sum of the sectoral surplus-values – produced, and redistributed in function of the size of the total capital laid out in each sector. But in each sector in each period the surplus-value produced is what it is in function of the variable capital component of cost price (and rate of surplus-value, which is constant and unchanging), i.e. in function of the production price of workers’ means of subsistence in the previous period, already modified by the imposition of an average rate of profit in this period; but the production price of workers’ means of subsistence in any given period is in turn determined by the constant and variable capital components of cost price also modified in function of the average rate of profit, i.e. the redistribution of surplus-value, in previous periods. This progressive modification of commodity prices has the effect that the ‘extra’ capital that needs to be accumulated becomes progressively smaller: the distribution of social constant and variable capital across sectors of production means that the ‘value’ price of the product of the means of production sector is always higher than the ‘price of production’ price; but the progressive modification of production (and hence cost) prices through the repeated imposition of average profit means that this difference converges to a particular value, and, when this (effectively) happens, the price of means or production (effectively) ceases to rise, and prices of production in general are (effectively) stable between production periods (as I observed, to two decimal places, the system is effectively stable by the twenty-third iteration, although the convergence to the new equilibrium is effectvely infinite). The regression back is not infinite, because its starting point is the arbitrary imposition of the ‘rule’ of average profit at the point at which the system passes from a ‘value’ system to a ‘price’ one, i.e. from one in which the commodity product of each sector is sold at cost price plus sectoral surplus-value to one in which it sells for cost-price plus average profit. The egression forwards towards the new equilibrium, however, is.

        What is the value of this for the real world?

        First, as a refutation of von Bortkiewicz’s own refutation of Marx, Kliman’s model is definitive and decisive. Unless Kliman is in turn refuted, the ‘Marx forgot to transform the input prices’ argument in favour of simultaneist valuation can no longer be admitted as a valid criticism of the internal coherence of Marx’s value theory.

        Second – and this is really the last made point expressed in other words – Kliman confirms that the price of an input at the beginning of one period is – by definition – the same price it has as an output at the end of the preceeding one. To calculate the value of a factor of production one does not need to engage with an infnite regress of labour inputs but rather to count the moey one has paid for it. Value does not exist alongside price as if in some parallel universe; value finds its own manifestation as price.

        But Marx’s treatment of the equalisation of the rate of profit in chapter nine is esentially a mathematical one, based on logical necessity; von Bortkiewicz’s would-be refutation is based on the same terms; as is Kliman’s refutation of von Bortkiewicz. But in the real world, the equalisation of the rate of profit is not a mathematical phenomenon, but a social one: surplus-value is not redistributed by a committee of capitalists; capital itself flows, in function of its own mobility, to where it gets the best return. This is how, in practice, rates of profit do tend to equalise; and this is what Marx deals with in chapter ten, the decisive theoretical part of his exposition.

        In this chapter, Marx shows how, at any given moment, there is a given level of social need for a given type of commodity. This social need, which is expressed as money, represents a determinate quantity of social labour time. This social labour time is the commodity-type’s market value, the market value of an individual commodity – in money form its market price – being an aliquot part of the total market value. Competition between capitals producing the same commodity within a given branch of production produces a tendency towards the equalisation of individual market prices independently of the specific conditions of production of the individual capitals. If, in a given branch of production, the total social labour time in which the social need for a given commodity consists coincides with that embodied in the commodity product (the latter taking the monetary form of prices of production) – i.e. if demand coincides with supply – then the total market prices of those commodities equals their total market value and an average profit is realised in the branch of production in question. If, however, supply outstrips demand – if more social labour time is embodied in production than that expressed as social need – then either the unit price of commodities will fall, or some of the commodity product will either go unsold, or some of it will remain longer in circulation than normal, or a combination of these factors will occur; but, in short, market price will fall below market value and a below than average profit will be reaped. If, contrariwise, more social need is expressed in the monetary form of social labour than that embodied in production – if, in other words, demand exceeds supply – then market price will rise above market value and a greater than average profit will be reaped in the branch of production in question. At the aggregate level, since the social need for a given commodity represents an aliquot part of the total aggregate labour time available to society, total market price is equal to total market value: differences between market price and market value in given branches of production cancel themselves out in the aggregate. Capital, which is disinterested in use-value other than insofar as its sale as a commodity allows capital’s self-valorisation, flows, in function of its own mobility, from those branches of lower profitability to those of higher profitability: production expands where demand outstrips supply and contracts where supply outstrips demand. This movement, the product of the competition between capitals of different branches in search of above average profits, is the mechanism whereby the rate of profit for the whole social capital tends to equalise. Were, on the one hand, the productivity of labour, and, on the other, the scale and distribution of social need fixed and unchanging, this movement of capital in function of the interplay of supply and demand would generate an equilibrium point where market price and market value would be equal in all branches of production and an average profit would be garnered everywhere. Yet both the productivity of labour and the shape of that social need expressible as money change constantly: pressure of competition and the need to realise ever more surplus-value impel capitalists to cheapen individual commodities, reducing market value (as well as increasing the organic composition of capital), while social need is determined by the relations between classes and the division of ‘revenue’ into wages and surplus-value, as well as the division of the surplus-value itself into its various forms. As such, instead of a fixed equilibrium towards which the process of equalisation of the rate of profit inexorably converges, alongside the process of equalisation there obtains an equally strong process of disequalisation of profit rates, such that permanent and shifting unevenness of profit rates and the consequent constant movement of capital towards surplus profit mark the essential dynamic of capitalist reproduction.

    • Uncle Marx says:

      Uncle Karl approves this. My head just exploded.

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