Thursday, May 28, 2009

Capital Is Dead Labor, That, Vampire-Like, Only Lives By Sucking Living Labor

1.0 Introduction
Stupidity about Marx is never-ending. So I thought I would put up a post about Marx as a mathematical economist. This is exposition of unoriginal ideas. To amuse myself, I didn't review Sraffa or any other author when writing this.

2.0 The Technology
Consider an economy in which n commodities are produced, each in a separate industry. The technique in use is represented by the nxn Leontief matrix A and the n-element row vector a0 of labor inputs. A column, a.,j, in A and an element of a0,j represent an industry. The ith, jth element of A is the quantity of the ith commodity input per unit output of the jth industry. Quantities are here measured in physical units (e.g., bushels corn per ton steel). The jth element of the row vector of labor inputs, a0,j, is the person-years of labor services hired in the jth industry per unit output.

By assumption, all industries require at least some positive amount of labor to produce their outputs. All industries produce their outputs in a year, and they consume all their inputs in producing their output. This is a model of circulating capital alone; no fixed capital (e.g., long-lasting machines) appears in the model.

Assume that the economy is viable, that is, some levels of operation exist for the industries such that there is a surplus product available, after replacing used-up means of production, when industries are operated at that level. For simplicity, assume all commodities are basic. In other words, every commodity enters either directly or indirectly into the production of every other industry. Presumably, steel enters directly into the production of automobiles. Iron would enter indirectly into the production of automobiles through its use in the production of steel.

No choice of technique occurs in the model.

3.0 Quantity Flows

3.1 Labor Values
Let q be an n-element column vector, where each entry is the gross output of that industry. Each entry is measured in the corresponding physical units (tons, bushels, kilograms, etc.). Let y be an n-element column vector of net outputs. Gross and net outputs are related like so:
y = q - A q
q = (I - A)-1 y
where I is the identity matrix. The existence of the inverse follows from viability. If industries were operated at levels to produce the gross outputs, the net output would be available for consumption or accumulation after replacing exactly the inputs consumed in production.

The amount of labor hired to produce the net output y is:
L = a0 q = a0 (I - A)-1 y
Suppose net output consisted of only one unit of the commodity produced in the jth industry:
y = ej
where ej is the jth column in the identity matrix. The labor value of the jth commodity, that is, the amount of labor hired to produce one more unit of the jth commodity net, is:
vj = a0 (I - A)-1 ej
Labor values are expressed as an n-element row vector:
v = a0 (I - A)-1
Labor values then have a sensible meaning; nothing radical is involved in defining them.

3.2 The Standard Commodity
One might expect any arbitrary basket with a large number of commodities to have both some labor-intensive and some capital-intensive commodities, in some sense. On average, these will approximately cancel in an arbitrary basket. Accordingly, let's assume that gross outputs, net outputs, and the commodity in which wages are paid are all of average capital-intensity, in some sense.

Since both gross and net outputs are of average capital intensity, it seems sensible to assume they are composed of the same proportions, just different in amount:
q* = (1 + 1/R) y*
where the asterisks indicate a basket in standard proportions. R is a strictly positive constant. As a normalization condition, the standard system is assumed to employ a unit amount of labor:
a0 q* = 1
It follows that the gross output of the standard system is a right-hand eigenvector of the Leontief input-output matrix:
A q*= [1/(1 + R)] q*
The outputs of the standard system are guaranteed to be positive by setting [1/(1 + R)] to the maximum eigenvalue of the Leontief matrix, also known as the Perron-Frobenius root of the Leontief matrix.

The net output of the standard system, y*, is the standard commodity.

3.3 The Rate of Exploitation
The labor embodied in the gross output of the standard system is easily found. One has:
1 = a0 q* = v(I - A)q* = v q* - v A q*
Or, taking advantage of the fact that the gross output of the standard system is an eigenvector of the Leontief matrix:
1 = [R/(1 + R)] v q*
Hence, the labor value of gross output of the standard system is found as:
v q* = 1 + 1/R
Marx expressed the labor value of gross output as C + V + S. Constant capital C is vAq*, the labor value of the means of production used up in producing the net output. Variable capital V is the labor value of the value added by labor paid out in wages. Surplus value S is the labor value of the remaining net output, which is obtained by the capitalists. Since one person-year is employed, the sum of variable capital and surplus value in the standard system is unity:
V + S = 1

Let w denote the proportion of the net output of the standard system (that is, the standard commodity) that is paid out in wages. Hence:
0 ≤ w ≤ 1
And variable capital is given as:
V = w
It follows that surplus value is now defined:
S = 1 - w
Marx denoted the ratio of surplus value to variable capital as the rate of exploitation:
e = S/V = (1 - w)/w = (1/w) - 1
where e is the rate of exploitation.When the whole value of the net product is paid out to workers as wages, workers are not exploited and the rate of exploitation is zero. The rate of exploitation is otherwise positive, and increases without bound as the wage becomes a lesser proportion of the value of the net product.

4.0 Price Equations
Let p denote a row vector of prices of production. Prices of production permit smooth reproduction in a competitive capitalist economy. They are defined by the condition that the same rate of profits is obtained in each industry:
p A(1 + r) + a0 w = p
where r is the rate of profits. Since profits are not earned on wages, the workers are paid at the end of the year. Wages are not advanced in this model. Since the same symbol for wages is used in calculating the labor value of quantities in the standard system, the standard commodity is the numeraire. Thus, the price of the standard commodity is unity:
p y* = 1

Recall that the net and gross outputs of the standard system are in proportion. One can thus calculate the price of the gross output of the standard system:
p q* = 1 + 1/R
Postmultiply the price equations by the gross output of the standard system:
p A q* (1 + r) + a0 q* w = p q*
p q* [1/(1 + R)] (1 + r) + w = (1 + 1/R)
(1 + 1/R)[1/(1 + R)] (1 + r) + w = (1 + 1/R)
The rate of profits is an affine function of the wage:
r = R(1 - w)
The above equation can also be expressed as:
w = 1 - r/R
The rate of profits ranges from zero to the maximum R. The maximum rate of profits is obtained when workers live on air, with a wage of zero. A higher wage is associated with a lower rate of profits, with a very simple relationship with this numeraire.

Total wages are a0 q* w. But, since one person-year is employed in the standard system, total wages are simply w.

Total profits are p A q* r, that is:
p A q* r = [1/(1 + R)] p q* R(1 - w)
p A q* r = [1/(1 + R)](1 + 1/R)R(1 - w)
p A q* r = 1 - w
The above is hardly surprising. The ratio of the rate of profits to the maximum rate is an increasing function of the rate of exploitation:
r/R = e/(1 + e)
When the rate of exploitation is zero, the rate of profits in the system of prices of production is also zero. As the rate of exploitation increases without bound, the ratio of the rate of profits to the maximum rate monotonically increases to unity.

5.0 Invariants
The following statements hold, whether the quantities mentioned are evaluated in labor values or in prices of production:
  • The gross output of the standard system is valued at 1 + 1/R
  • The net output of the standard system is unity
  • Variable capital is valued at w
  • Surplus value, that is, profits are (1 - w)
Furthermore, the rate of profits is positive if and only if workers are exploited.

This model certainly suggests that market phenomena are a veil over the exploitation inherent in capitalism. And calculations with labor values exhibit that exploitation.


Anonymous said...

I posted this on DeLong's blog...

"Along with the constantly diminishing number of the magnates of capital, who usurp and monopolize all advantages of this process of transformation"And this hasn't happened? "For the top one percent, that’s a pretty impressive period [from 1979 to 2005]. For the next 19 percent, there’s something happening. But for the bottom 80 percent, there’s just very little going on in terms of real income growth" From:

Boom, Bust, and Inequality"grows the mass of misery, oppression, slavery, degradation, exploitation;"Well, US workers have the longest hours and are unlikely to be in unions, so I guess that counts for oppression and exploitation rising... That income for the top 1% has to come from somewhere...

and best not to mention the pathetic official poverty definition nor the rising unemployment and homelessness just now...

"but with this too grows the revolt of the working-class"Ah, yes, he seems to have got that wrong... unfortunately...

"Centralization of the means of production and socialization of labor at last reach a point where they become incompatible with their capitalist integument. Thus integument is burst asunder. The knell of capitalist private property sounds. The expropriators are expropriated..."Hasn't happened yet, that is sure. But labour has become socialised (very few companies are run by their owners) and centralisation has continued to grow (compare the modern transnational corporations to what existed in Marx's time).

So, well, Marx made a prediction, most of which has come true. As for the "integument [being] burst asunder", well, who can made the prophecy that this will never happen?

An Anarchist FAQ

YouNotSneaky! said...

Two things;
1. Robert, what in the world does this have to do with Brad Delong's post?
2. Ian, I replied over at Brad's (with a typo 2055=2005). More intellectually honest arguments please.

BenP said...

Sneaky but didn't Marx also say;

"The bourgeoisie, during its rule of scarce one hundred years, has created more massive and more colossal productive forces than have all preceding generations together. Subjection of Nature’s forces to man, machinery, application of chemistry to industry and agriculture, steam-navigation, railways, electric telegraphs, clearing of whole continents for cultivation, canalisation of rivers, whole populations conjured out of the ground — what earlier century had even a presentiment that such productive forces slumbered in the lap of social labour?"

Of course we have also had the holocaust, industrial world wars, not to mention poverty in the here and now, economic crisis...

As for the transformation of society, as someone once put it;

"Who built Thebes of the seven gates?
In the books you will find the names of kings.
Did the kings haul up the lumps of rock?
And Babylon, many times demolished
Who raised it up so many times? In what houses
of gold-glittering Lima did the builders live?
Where, the evening that the Wall of China was finished
Did the masons go? Great Rome
Is full of triumphal arches. Who erected them? Over whom
Did the Caesars triumph? Had Byzantium, much praised in song
Only palaces for its inhabitants? Even in fabled Atlantis
The night the ocean engulfed it
The drowning still bawled for their slaves."

DeLong's Marx "lectures" are much worse, charges of intellectual honesty might be better pointed elsewhere.

Any disinterested observer can only be impressed with Marx's insights of the dynamics of bourgoise politicla economy. Which of course they are.

YouNotSneaky! said...

What does this have to do with anything, or with Brad in particular? Here Marx is talking about the "past" hundred of years. But Brad is talking of Mars as a prophet - of the future not the past. And there Marx changed his tune (how else do you get the crises necessary for the revolution?)

Most disinterested observers think that Marx is sufficiently bunkum so that's it's not even worth bothering having an opinion on the subject. I think by "disinterested observer" you mean "myself and those who agree with me".

BenP said...

So you have an opinion so don't think its bunkum or are not disinterested?

What this has to do with DeLong is he does not do Marx justice, though this is not surprising.

You brought up intellectual honesty.

YouNotSneaky! said...

He's talking about Marx as a prophet. You're talking about some other aspect of Marx. The two things have nothing to do with each other. You're basically saying that DeLong should've genuflected at the altar of Marx, like all these other "disinterested observers" but instead he had the nerve to blaspheme.

The intellectual dishonesty was in reference to Ian cherry picking irrelevant dates. Marx wrote a century and a half ago, give or take. During this century and a half, the things that Marx said were going to happen, didn't happen. But Ian comes along, picks like a fifteen year period out of that century and a half, which has some phenomenon which very vaguely resemble some stuff Marx talked about and voila! Marx was a prophet after all and criticisms of him are "stupidities".

And then folks get together and pat each other on the back because they are all so smart to recognize that Friedman's Positivism was "unscientific".

YouNotSneaky! said...

And that still doesn't explain what the model presented in the post has to do with what Brad said. It's a total Non sequitur. Much like all those "so much for the ..." conclusions.

BenP said...

No, let’s be clear, Marx was not a prophet and would have been horrified at the thought of it. It's a deliberate and condescending attempt to impute religious mumbo-jumbo to Marx's ideas.

Marx sought to apply science to the study of history and political economy. You may disagree with the findings but with comments like "it's not even worth bothering having an opinion on the subject." you seem to have ruled yourself out from offering an - informed - opinion.

Marx's insights from a relatively early period of industrial capitalism still show dynamics at work easy to see demonstrated today. Look at the misery of Chinese workers in "free enterprise zones", banks and auto companies too big to fail, globalisation. These are all widely acknowledged insights for those not ideologically challenged.

Somebody else beat you pronouncing the end of history and that didn't work out to well, did it.

YouNotSneaky! said...

Ok, if Marx was no prophet then make that argument (which entails more than just asserting it). But don't bring up Marx's analysis of pre 19th century history to prove that point as obviously one has nothing to do with the other.

"These are all widely acknowledged insights for those not ideologically challenged."

Oh so I guess it's the "disinterested observers" vs. the "ideologically challenged". Right...... speaking of intellectual dishonesty, a big part of winning (or trying to, anyway) an argument rhetorically, rather than meritocratically is to define the ad hominen terms which are used to the sides in a discussion in just the right way. Those whom one agrees with are "disinterested observers" (which implicitly claims victory in a debate before it is concluded or even conducted). Those whom one disagrees with are "ideologically challenged".

Real arguments please. More intellectually honest arguments please.

"Somebody else beat you pronouncing the end of history and that didn't work out to well, did it."

Not sure what that has to do with anything but I guess this is a conversation about Marxism so I guess the category "things which didn't work out too well" is appropriate though not in the sense you intended it.

BenP said...

Hold on, this started from DeLong denigrating Marx as a prophet, which you have defended, is that a "real honest argument". You doth protest too much.

Juxtaposing Marx with John of Patmos is not an argument that Marx was a prophet. It is a silly attempt to link Marx with the ramblings of a mystic. Portraying Marx as some sort of Millenarian has been tried before and failed; Liberals just don’t have a cogent response to Historical Materialism.

Why bother with Marx if it's bunk and why now. Could this have anything to do with the obvious fact that the 'free market' is not a viable economic system without massive interference by the state and people are looking for a better explanation outside the neoclassical 'parables' - talk about theological ramblings...

YouNotSneaky! said...

"Why bother with Marx if it's bunk"

Because it's influential bunk.

"this started from DeLong denigrating Marx as a prophet, which you have defended, is that a "real honest argument""

I can only repeat myself:
Ok, if Marx was no prophet then make that argument (which entails more than just asserting it).

"You doth protest too much."

Umm, this is another one of those utterings which are completely meaningless but masquerade as argument. Who cares how much I protest? What matter is how I protest. I could've said the exact same thing about you.

"Could this have anything to do with the obvious ..."

And here is where we start constructing that strawman. Again, what does this have to do with the subject under discussion.

"It is a silly attempt to link Marx with the ramblings of a mystic."

No it's drawing an analogy. Marx prophesied the impoverishment of the workers because of capitalism, in essentially, apocalyptic, prophetic terms. Instead workers' income exploded and about two generations after Marx had increased by a factor of 10 (more or less). Now, to be fair, Marx wasn't alone among the classical economists who thought that incomes would stagnate or return to "subsistence levels" (Malthus being the obvious example) - but nobody these days tries to argue that Malthus was right in his predictions of the future but some people - Ian for example - do try to argue that Marx was right in his prophecies. Weirdly enough, despite all the evidence to the contrary.

BenP said...

Right, so DeLong says something outlandish and we have to busy ourselves proving the opposite.

“Now, to be fair, Marx wasn't alone among the classical economists who thought that incomes would stagnate or return to "subsistence levels" (Malthus being the obvious example)...”

Oops. Marx consistently opposed Says Law, specifically in Lassales use of Ricardo’s ‘iron law of wages’ – Marx and the Iron Law of Wages

“I can only repeat myself”. Agree, I think we have exhausted this discussion.

Magpie said...

Dear Mr. Vienneau

Thanks for a thought-provoking and exemplary article.

For one, I find it refreshing that you have publicly argued your case in the same language the arrogant and pompous bourgeois "economists" employ.

Although the use of mathematics may seem intimidating to most, in my case, it motivated me to go back to linear algebra. If there was nothing else in your article, this would be more than enough to earn my gratitude.

As Deirdre McCloskey has made clear: mathematics has been long used and abused by mainstream "economists" as a rhetorical device to give the appearance of science and precision to what is essentially mumbo-jumbo.

That you are using mathematics against the snake-oil salesmen, is poetically just.

You have gained in me a new regular reader.


escaiguolquer said...

Another great article as usual.
But I disagree on one point, and it's not a minor one: labour value cannot be just the labour cost of production. In all classic political economics, and of course for Marx, the value of commodities is labour value, wich includes, besides direct "living labour", the indirect value inherited from the capital consumed in production, wich in turn was produced by work, That "dead labour" is not taken into account in the article, when you say "The labor value of the jth commodity, that is, the amount of labor hired to produce one more unit of the jth commodity".
When you say, later "Labor values then have a sensible meaning; nothing radical is involved in defining them" I can agree, but no doubt that it's not the same meaning than for the classics.
I think we wouldn't be afraid to be radical. In fact, that only means to look for the roots of the facts as they are.

Again, congratulations and thanks a lot for your blog.