Saturday, 12 February 2011

THE THOUGHT OF KARL MARX LAST PART

Here are the price equations of our corn/iron/tools model, revised so as to represent workers as petty commodity producers. In order to indicate that capital cannot be transferred from the labor producing "industry" to other lines of production, I introduce a new variable, r, to stand for the rate of "profit," if there is any, in that "industry.

( 0Pl + 30Pc + 15Pi + 0Pt )( 1 + r ) = 150Pl

( 60Pl + 128Pc + 2Pi + 3Pt )( 1 + R) = 240Pc

( 20Pl + 16Pc + 1Pi + 5Pt )( 1 + R) = 60Pi

( 50Pl + 6Pc + 27Pi + 4Pt )( 1 + R) = 16Pt

It is not necessary to specify the real wage because that is implied by the input quantities of the first line of production. This is now a system of four equations in six unknowns: Pl, Pc, Pi, Pt, r, and R. We reduce the number of unknowns to five by stipulating that corn is the money commodity, as before, setting Pc = 1. That still leaves one degree of freedom.

It is not difficult to demonstrate [see the essay referenced above] that r and R vary inversely to one another. That, note, is different, formally speaking, from showing that the wage and the profit rate in the old system vary inversely to one another, although the underlying facts are much the same. A non-zero r means that the workers have somehow managed to extract from the capitalists some portion of the surplus. But this way of representing things, besides making no reference to the Labor Theory of Value, also opens the way to economists like Gary Becker to introduce the notion of "human capital." Workers who, in effect, drive the wage above bare subsistence can either consume the extra in the form of a somewhat less miserable standard of living, or they can -- as Becker might say -- invest in the improvement of their capital stock [themselves] by paying for education, which will allow them to produce a new and higher priced commodity -- namely, skilled labor. This way of thinking, crazy as it is [Marx is brilliant on this], leads naturally to the countless discussions of the "return on investment in education' in the form of higher lifetime earnings.

Notice, by the way, that if we wish to continue with this charade of labor as a commodity produced by a petty bourgeois entrepreneur, we must allow for several labor sectors, each with its own particular "product" [some form of unskilled or skilled labor or educated labor] and with its own rate of return. We can even analyze a failure of some children of the educated workforce to take advantage of the opportunities provided by their parents as their choosing to consume the additional portion of the surplus made available to them instead of investing it, thus reproducing the fairytale about prudent and imprudent capitalists [ants and grasshoppers.] And, of course, the variation in the rates of return in the several labor sectors permits us to talk about the relative exploitation of unskilled workers by much better compensated skilled workers.

The point of all this is to elaborate on the fundamental verruckheit, or craziness, of the manner in which bourgeois ideology [law, economic theory, philosophy, sociology, etc.] mystifies and conceals the exploitative nature of capitalism, making it appear that the failure of the workers to improve themselves is attributable to their imprudence rather than to the disadvantageous position in which they find themselves vis-a-vis capital.

[A propos, Susie and I saw "Company Men" last night, an interesting movie about the effects of the economic crisis on three privileged and advantaged corporate executives -- Ben Affleck, Chris Cooper, and Tommy Lee Jones, with a nice turn by Kevin Costner. It is worth seeing, I think.]

It is not difficult to show, formally, that the total profit appropriated by the corn, iron, and tools sectors is equal to what is lost to the labor sector by its inability to shift its capital to a more profitable line investment.

I have now come to the end of my story, even though there is a very great deal more to be said about Marx's theories, some of which I have said at length in my two books and several articles on him. Let me explain why I have taken so very much time and expended so much effort expounding and analyzing the Labor Theory of Value, only in the end to show [with arguments original to Vegara and myself] that the theory is incorrect.

Part of the reason, to be totally honest, is that I simply find the analytics of the theory fascinating and elegant. My entire life has been spent taking impossibly difficult theories [such as those in Kant's First Critique] and clarifying them in my own mind until I can present them to others as clear, simple, elegant, and beautiful. My second reason is to rescue Marx from the unfounded criticism that he is -- in the infamous words of Paul Samuelson -- a "minor post-Ricardian autodidact." I was so offended by those words when I first read them that I felt a need to show, clearly and simply, that they were just not true. My third reason is that I believe one cannot really come to grips with the profound truth at the heart of Marx's critique of capitalism without first working through this theoretical story and becoming completely clear on its precise status and limits. Speaking simply for myself, I could not have been led to an understanding of the ironic structure of Marx's critique, and the core irrationality of treating workers as though they are participants in a free market of exchange among producers and consumers, had I not first made my peace, intellectually, mathematically, and theoretically with Marx's own attempts to capture that craziness.

I hope this tutorial has been useful to some of you. as I feared, it went on for a very long time, so long, I imagine, that I lost a number of readers along the way. But I have enjoyed writing it, and it will live forever in cyberspace, in the weird way things have of surviving in that realm.

Friday, 11 February 2011

THE THOUGHT OF KARL MARX PART EIGHTEEN

OK. I have solved the JStor problem. Here is the next part of the tutorial.

Let us step back for a moment from the equations to remind ourselves of the core of Marx's understanding of capitalism. Every social formation, every stage of history, he claims, is determined fundamentally by who owns or controls the means of production -- the material factors of production without which human beings cannot live and reproduce their conditions of being. In a feudal society, the land is the premier means of production, and it is controlled by the landed aristocracy, who compel others to work as peasant farmers on that land, and to yield up a portion of the annual product as a surplus, which the lords then use as they wish. Land continues to be essential to life in a capitalist economy, but it is joined by other factors of production, including machinery, raw materials, factories, technical knowledge, and also the money capital that entrepreneurs use to buy the factors of production.

The central fact of capitalism is the dispossession of the vast majority of working men and women, their exclusion from access to and control over any means of production. Propertyless, the great mass of people are forced to sell their labor to those who control those means, on terms dictated by the owners of the means of production. Despite the surface appearance of bargaining between legal equals in a market free of all coercion, the sale of labor for wages is in reality an unequal bargain, forced on the workers by their need for food, clothing, and shelter.

The distinction between labor and labor power was Marx's attempt to capture a fundamental structural feature of capitalism, in such a manner as to solve the Ricardian problem of price determination while also identifying the precise quantitative and qualitative origin of the profits that, year by year, swell the wealth of the entrepreneurs and cement their control over the eternally propertyless working class. Thus it was that Marx sought to prove, rigorously and quantitatively, the fundamental proposition of his critique of capitalism, which is that CAPITALISM RESTS ON THE EXPLOITATION OF THE WORKING CLASS.

But Marx's idea, ingenious as it is, proves not to be satisfactory, as we have seen, because it does not succeed in building his insight unambiguously into the formal structure of his theory. Every one of the propositions he advances concerning labor and labor value, many of which, as it turns out, are formally correct, can be replicated for any other input into production that is directly or indirectly required in all lines of production -- what Sraffa, in a different analytic context, calls Basic Commodities.

Now I am convinced that Marx's fundamental insight about capitalism is completely correct, so quite naturally, I am led to ask this question: Is there some other way of capturing Marx's insight in the formal structure of the equations that represent the determination of prices in a competitive capitalist market equilibrated by a system-wide rate of profit and a consistent system-wide set of relative prices? Because I was so powerfully impressed and persuaded by Marx's complex literary and sociological interpretation of what he calls the mystifications of capitalism [an interpretation that I analyzed at length in MONEYBAGS], I wondered whether the formal device by which we capture Marx's economic insight might also, somehow, build the irony and mystification into the equations, if indeed that is even imaginable, so that instead of two interpretations of CAPITAL, one economic and mathematical, the other literary and sociological, we would have a single integrated representation of the full complexity of Marx's thought.

Reflecting on this question for a while, I was led to the following train of thought. According to bourgeois ideology and economic theory, the market is a site of voluntary, uncoerced, mutually advantageous exchanges. The cobbler makes more pairs of shoes than he can wear, the farmer grows more wheat than he can eat. They meet in the market and agree to exchange shoes for wheat, to each one's benefit. Once money has replaced barter, the circulation of commodities is a bit more indirect, but no less free and uncoerced. The cobbler sells his shoes for money, and with his money buys wheat from a retail merchant, who in turn has bought the wheat from the farmer, who in her turn buys the cobbler's shoes from a stall in the marketplace. All of these purchases and sales are motivated solely by self-interest, and the only requirement, imposed fairly and universally, is that contracts for the purchase or sale of commodities be honored fully.

Like the cobbler, the farmer, the coal monger, and the clothier, bourgeois law and ideology would have it that the worker too is an independent entrepreneur who buys his raw materials in the market [food, clothing, shelter] in order to bring a commodity back into that market for sale. The worker's commodity is a service, not a physical object, but in that respect he is no different from the plumber or the electrician. Like them, and all other commodity producers, the worker enters freely into a contract for the sale of that service, a contract that is enforced impartially by the law.

By a long historical process of development leading to the full instantiation of capitalism, the worker has been freed from all legal and traditional constraints on the sale of his or her labor. The worker can go to whichever buyer offers the best price for the labor, anywhere that the law of the land reaches, be it London, Manchester, Liverpool, or Bath. Nor are there any constraints on what sort of labor the worker contracts to provide. She is free to work in a blacking factory, a cloth factory, a coal mine, or on a wheat farm. And she is entirely free to walk away from any place of employment for a higher wage or an easier job. It is this complete freedom of entry and exit that creates what we come to call a "labor market," precisely analogous in its functions to the market for coal, iron, or wheat.

The buyer too, the employer, is bound by no traditional obligations to a particular group of workers. He can hire them at will, and fire them the moment it is in his interest to do so, just as no law requires him to continue to buy coal from a coal monger if his furnaces have been banked or even allowed to cool because of a shortfall in demand in the market for his goods. If the worker, or the coal monger, or the tool and dye maker protests that he needs to sell his goods or services, the buyer, the entrepreneur, replies that that is none of his concern. They were not compelled to enter into a bargain with him, and inasmuch as he has paid for the goods and services for which he contracted, at the price he contracted to pay, his obligations have been met.

Now, the free movement of buyers and sellers throughout the entire national or international market over time produces a stable system of prices, so that a quarter of wheat or a ton of coal sells for no more, and no less, in one place than in another [taking account of transportation costs.] What prompts this movement is the ceaseless search on the part of all the producers for the highest possible rate of return on their invested capital. There are rigidities, of course -- once a cloth manufacturer has purchased his machinery and built his factory, he has some economic interest in continuing to make cloth, at least until he can extract his capital [by selling the factory to another entrepreneur and the machinery on the second-hand market, perhaps] and transfer it to a more profitable line of production. Capitalism, as a twentieth century economist would eventually observe, is a process of "creative destruction." [Joseph Schumpeter, for those who are wondering.]

Precisely at this point, as Marx is at great pains to make clear, the whole superstructure of liberal bourgeois philosophy and political theory and law is introduced to justify the treatment of labor power as a commodity. Workers are treated in law, in ideology, and in philosophy as small producers, petty entrepreneurs who bring their product, like any other capitalists, to the market and exchange it for the products of other capitalists' enterprises. Their fixed capital is their bodies, which, according to classical liberal philosophy and jurisprudence, they own. Their circulating capital is the fund they spend for food and clothing. Assuming that they live at the level of bare subsistence, the worker-capitalists are not likely to hire labor services (although they may be forced to go to a doctor from time to time). Hence, all their capital will be constant capital, none of it will be variable capital, to use Marx's terminology. Why are the workers unable to move their capital freely to sectors paying a higher rate of return? The simple answer, once more, is that their fixed capital is their bodies and their circulating capital is the food they eat to stay alive. A steel producer who finds the return in steel declining can, given a long enough period of time, cash in his investment and shift his capital to clothing, rental housing, or luxury appliances. The worker who notices the absence of any significant rate of return on her capital investment, and who, like any prudent capitalist, wishes to shift to a more profitable line, will find it necessary to separate herself ("alienate herself," to use the technical legal term) from her body. And by a quite unfortunate metaphysical accident -- which, however, can scarcely be blamed on capitalism itself -- she is unable to survive that particular liquidation of her investment.

It will be objected that workers are not really petty capitalists. Just so. But the objection entirely misses the point of Marx's analysis. The workers must be made to appear as petty capitalists, in law, in political philosophy, and in the formal theory of political economy. A political economy that fails to model the essential mystification and ideological self-deception of capitalist economic, political, and legal relationships will be an inadequate theory of capitalism. An adequate political economy must capture that feature as false, in order to be true to the reality and to the appearance of capitalism. We have here a very strange requirement indeed. We need a formal model of an ironic, dialectical relationship between appearance and reality. The trouble with other attempts to capture Marx's meaning is that they are either literary renderings, which preserve the irony and the intricate interrelation between appearance and reality, but without the formal structure that will allow us to calculate the magnitudes of the relevant variables; or else they are formal models, like the Sraffa model, that lose entirely the element of mystification and self-deception. If we agree with Marx that capitalism has its own mad logic, then we will search for a model that embodies both the logic and the madness of capitalism. I suggest that the correct way to begin this process is to treat the workers as though they were petty entrepreneurs, producers, producing a commodity -- labor power -- for the market, and then capture the inner madness of this way of thinking of them by stipulating that they, alone among all capitalists, are unable to shift their capital about from sector to sector.

To repeat, since this is a strange way of thinking, all of the entrepreneurs are ceaselessly searching for more profitable lines of production in which to invest their capital -- SAVE FOR ONE GROUP, THE WORKERS. The workers have what might, with full and deliberate ironic intent, be called a metaphysical problem. Their capital is not factories or mines or farms or money but their bodies. The inputs they must obtain to enable them to produce their labor services are the food, clothing, and shelter they need to continue to live. If the return on labor services sinks so low that the labor producer, the worker, barely makes enough to pay for her inputs, offering virtually no rate of profit whatsoever, the only way she can cash in her capital, so as to shift it to a more profitable investment, is -- to cash it in -- to die. She is trapped in her condition as provider of labor, and has no choice, if she would live, but to sell that labor in the market at whatever price she can get.

This is the bitter reality that lurks beneath the bourgeois ideological surface of equal uncoerced exchange. The status of the cloth manufacturer is precisely equal to that of the machine tools manufacturer. That is why they find it easy and natural to frequent the same clubs, buy homes in the same gated community, and send their children to the same public [i.e., private] schools. It is the structure of the underlying reality that explains the ever-increasing wealth of the entrepreneurs and the perpetual penury of the workers.

So I asked myself, how can I capture this complex situation, this contrast between surface appearance of equal exchange and underlying reality of relentless exploitation, in a system of equations that will express formally the bitter irony of construing workers as free and equal small commodity producers and at the same time yield a satisfactory account of the determination of equilibrium prices in an economy exhibiting a uniform rate of return on invested capital? The result of my ruminations was a revision of the corn/iron/tools model, which is readily generalizable to an economy with any finite number of sectors of production.

Tomorrow, I will set the revised model before you and discuss some of the conclusions we can draw from it [and from its generalization to any number of sectors of production].

A BRIEF PAUSE

I had already written most of the next blog post, when I took another look at my 1981 article. There I found that I had said certain things much more clearly and forcefully [I was thirty years younger.]. I would like simply to reproduce portions of that article in this blog, although without including the rather complicated mathematics that accompany my exposition there, but I am at the moment having trouble accessing the article online [JStor, or such like], and I am loathe to sit at my computer typing into a blog post what I ought to be able to cut and paste. So be patient as I attempt to solve this problem. Perhaps later today I shall be able to put up a lengthy post that will bring my tutorial pretty much to completion [though there is a vast amount remaining to be said, of course.]

Wednesday, 9 February 2011

THE THOUGHT OF KARL MARX PART SEVENTEEN

I write these next words with tears in my eyes -- not because it turns out that Marx is wrong. No, I am a grown-up, I have lived through twenty years of Reagan, Bush, and Bush without breaking. I have shown that I can take it. The tears are for my one claim to economic fame. In 1981, I published an essay entitled "A Critique and Reinterpretation of Marx's Labor Theory of Value," in a journal called PHILOSOPHY AND PUBLIC AFFAIRS. [I believe it is available on-line.] In that essay, I proved an extremely important theorem that shows that Marx was wrong to impute the exploitative capacity of capitalism to the labor/labor power distinction. I was, I firmly believed, the first person ever to realize the underlying logical flaw in Marx's argument, and to demonstrate it mathematically. The proof was not much from a mathematical point of view. Indeed, when I had first actually proved the theorem several years earlier, I was ignorant of linear algebra, and had used nothing but elementary algebra and some ingenious labeling moves. After the essay appeared [since I made the mistake of publishing it in a philosophy journal, almost no one read it who was capable of appreciating it], the brilliant, mathematically extremely sophisticated Marxist John Roemer published a reply and criticism in the journal in which, in passing, he pointed out that the same theorem had been published two years earlier by Josep M. Vegara in a monograph entitled ECONOMIA POLITICA Y MODELOS MULTISECTORIALES. [For those who are interested, the proof appears in section 3.5 of that book.] As I am linguistically challenged, and do not even read Spanish, let alone Hungarian or Japanese, I had of course not noticed Vegara's book. In the essay, I go on to develop an entirely original alternative analysis of exploitation, which in due course I shall repeat here [Roemer criticized that too]. But since this was my one shot at a genuine formal proof to which my name could be attached, I still weep, all these years later, for the lost opportunity. I am not the only person to whom this has happened, needless to say. At some point in the 70's, Samuel Bowles presented to his class at UMass an elegant formal refutation of Marx's famous claim about the falling rate of profit. When he finished, one of the graduate students said, "Sam, Okishio proved that in 1960." Bowles was reduced to publishing a note in 1981 in the CAMBRIDGE JOURNAL OF ECONOMICS with the rather modest title "Technical Change and the Profit Rate: A Simple Proof of the Okishio Theorem." I think I will stick to philosophy. In that field, if someone publishes an idea before you do, all you need to do is put brackets around your essay and a negation sign in front of the brackets, and then publish. You will get just as much respect.

So, herewith an informal, very simple, and elegant exposition of Vegara's Theorem. [God, how I hate writing those words.] The purpose of formal representations of arguments is to exhibit certain very general structural features of the arguments without the distraction of the content. Suppose I present you with the following argument: All Martians are vegans. All vegans are politically correct pompous asses. Therefore, all Martians are politically correct pompous asses. If you have never had an elementary course in logic, you may protest that there are no Martians, and that you are offended by the statement about vegans. These natural reactions are likely to blind you to the fact that the argument is a perfectly valid instance of what, in the old days, was called a syllogism in Barbara, namely All A are B, All B are C, therefore All A are C. [As a mnemonic device to help them remember which syllogisms are valid, the medievals gave all the various possibilities names in which in which a sequence of three vowels stood for the major and minor premises and the conclusion. Since "All A are B" was labeled a proposition in the "a" form, the name Barbara, with syllables aaa, stands for a syllogism in which each of the three propositions is of the form "All x are y."]

Marx, as we have seen, makes a number of formal claims about the relationship between the physical surplus, the quantity of labor directly indirectly required to produce a commodity, the prices at which goods exchange in a market ruled by competition, and the profit rate. Some of these claims are true, others are true under certain interesting circumstances, and still others are false. Marx also makes the very strong claim that all of these theses rest on the distinction, unique to the labor input in production, between labor and labor power. Taken all together, he claims to have shown rigorously that capitalism and its profits rest on the exploitation of the working class. What is more, he argues, on the basis of all of these claims, that exploitation is completely compatible with bourgeois morality, whose fundamental command is to give equals for equals.

We have translated many of Marx's claims into formal mathematical assertions, and have been able by this device to establish exactly when, and under what circumstances, they are true. Suppose now that we look back at the equations we formulated for the purpose of calculating labor values and prices and see whether, in those equations, there is some formal representation of Marx's signature innovation, the distinction between labor and labor power.

On first inspection, the answer would seem to be yes. Here, for example, is the equation we formulated, in the corn/iron/books model, for the labor values in the corn sector:

100 + 2Lc + 16Li + 0Lb = 300Lc

The labor inputs are distinguished by the fact that in the process of production, each unit contributes one full unit of labor value to the corn output, whereas the corn and iron inputs contribute only an amount equal to their labor value. The amount of labor required to produce a unit of labor is, we saw, necessarily LESS than one unit of labor [this follows necessarily from the fact that the system as a whole produces a physical surplus], and yet it contributes as full unit of labor to the output. The difference between those two quantities -- the amount of labor required to produce a unit of labor and a unit of labor -- is precisely the surplus labor that is transformed, in the sphere of circulation, into profit.

But suppose now we ask a question that it never occurred to Smith or Ricardo or Marx to ask, a question [I cannot let this go] which, for a brief moment, I thought I was the first person ever to ask: How much corn, directly and indirectly, does it take to produce one unit of iron, one unit of books, one unit of corn itself, AND ONE UNIT OF LABOR? Why on earth ask that question? There is no plausible distinction between corn and corn power, after all. But that is just the point. If we look at the equations, the only thing that tells us which input is labor, which is, corn, which is iron is the label. Suppose we keep the same physical characteristics of the model, but rewrite the equations so that they are set up to discover how much corn it takes, directly and indirectly, to produce one unit of each input. Following our labeling practice, let is call Ci the amount of corn, directly and indirectly required to produce one unit of iron, Cl the amount of corn directly and indirectly required to produce one unit of labor, Cb the "corn value" of books, and Cc the "corn value" of corn itself.

We can certainly do this. Nothing stops us from formulating such equations. But will they have any economic meaning? Do we have any assurance that we won't get absurd results, such as a negative value for the "corn value" of books? And, most important of all, will it turn out that it takes less than one unit of corn to produce a unit of corn, so that there is some "surplus corn value" generated each time we use corn as an input into production?

The answer to all of these questions, and any other similar questions we might ask, is a simple YES. EVERY SINGLE RESULT WE PROVED IN OUR VARIOUS SYSTEMS WITH REGARD TO LABOR, LABOR VALUES, AND SURPLUS LABOR VALUE, CAN BE PROVED IN EXACTLY A PARALLEL MANNER FOR CORN VALUES, OR FOR IRON VALUES OR TOOL VALUES. The only thing we cannot do is replicate these results for "book values," because books are not required as directly or indirectly as an input into every line of production. They are a luxury commodity, as I called them.

It is necessarily true that all of the corn values or iron values or tools values] will be positive, that the corn [or iron or tools] value of the physical surplus will exactly equal the surplus corn [or iron or tool] value extracted from the corn inputs, that in an economy embarked on a von Neumann balanced growth path:

(total money profits)/(total surplus corn value) = (total money prices)/(total corn values)

and so on and on.

Clearly, Marx and Smith and Ricardo are right that there is something special about labor that sets it apart from all the other inputs into production, but the distinction between labor and labor power is not it, because the revised equations we wrote [well, we did not actually rewrite all of them, but you get the idea] appear to demonstrate that in our system corn is exploited! What on earth is going on?

I have an answer [and this one IS in some sense original with me, I think], but I must prepare now to lecture this afternoon about Michael Oakeshott, so my answer will have to wait until tomorrow.

Tuesday, 8 February 2011

THE THOUGHT OF KARL MARX PART SIXTEEN

Now Marx makes his move. [This paragraph is actually quoted from page 127 of my book, UNDERSTANDING MARX. It seems to me somehow vaguely cheating to quote from a book rather than to write something new, but I did write it myself, after all.] The prices at which commodities exchange in the market are merely the surface of the capitalist market, the appearance. The underlying reality is the extraction of surplus value from the workers in the sphere of production. As Marx says in a letter to Engels in the year following the publication of volume one, "profit is for us first of all only another name or another category of surplus value ...[S]urplus value gets the form of profit, without any quantitative difference between the one and the other. This is only the illusory form in which surplus value appears."

The equality of profits with surplus value is an economy-wide fact, Marx claims. In any single industry, the particular capital intensity may cause prices to deviate from surplus value, so that the equality of the two, and the grounding of capitalism on exploitation, is hidden from view. But in the society as a whole, the total of profits, rent, interest and other unearned income must exactly equal the surplus labor value extracted from the workers. Marx adds the assertion of a second claim: That the total price of all the commodities sold in a time period must equal the total labor value of those commodities, even though the price of an individual commodity may lie above or below its labor value.

As it stands, Marx's solution to the problem posed by Ricardo is incoherent. The problem is one of units. Prices are measured in units of whatever is being used as money in the economy. Labor values are measured in units of labor time [hours, person-years, etc.] Strictly speaking, they cannot be equal unless one is deliberately stipulating the equality as a way of defining the money unit, in which case the equality is a tautology. But there is a simple way of capturing what Marx really intends to assert, namely in the following equality, which is definitely not a tautology:

(total profits/total surplus value) = (total prices/total labor values)

Since the numerator of each side of the equation is measured in money units, and the denominator is measured in labor time units, the units cancel out, leaving Marx asserting the equality of two pure fractions.

Marx also asserts one more equality, between the money profit rate and something he calls the value rate of profit. This latter quantity is simply the ratio between the surplus value extracted in the economy as a whole [ S] and the sum of the constant capital [C] and variable capital [V], measured in labor units. In other words:

The value rate of profit = S/(C+V)

So Marx claims that S/(C+V) = R

Is Marx right? Let us start by checking our little corn/iron/tools economy, the one that does not have equal organic composition of capital. We can do this, albeit a trifle tediously, by plugging into Marx's equation the values we got for prices, profits, labor values, and profit rate in that system. Here is what we get:

Total profits are simply the total money cost of all the capital inputs in the system, including labor [evaluated by means of the money wage], multiplied by the system wide profit rate, which is 1/3. When we carry out this calculation, we find that

Total profits ~ 110.28 money units [which, we will recall, was specified as units of corn, because we set Pc = 1.]

Total surplus value is calculated by subtracting the total amount of necessary labor in the economy [which is the labor value of all the real wages earned by the workers] from the total amount of labor employed in the economy. This is equal to 150(1 - Lw).

Total surplus value ~ 103.71 units of labor.

Total prices are arrived at by simply adding up the money price of the total output:

Total prices ~ 441.12 money units

Total values are calculated in analogous fashion by adding up the labor value of all the outputs.

Total values ~ 415.20 units of labor

Now we can check Marx's claim.

(total profits)/(total surplus value) ~ 1.063 money units/unit of labor

(total prices)/(total values) ~ 1.062 money units/unit of labor.

Marx is right! [Indeed, the minor difference is actually a result of rounding. The actual result is a strict equality.]

But this is just for our little corn/iron/tools model. What about the general case? Well, there is bad news, and there is good news. The bad news is that if you add a theology books sector, the equality break down. The good news is that the equality holds for EVERY POSSIBLE LINEAR REPRODUCTION MODEL IN WHICH THERE ARE NO LUXURY GOODS. What does all of this mean?

Well, when there are no luxury goods, the entire physical surplus is being used to expand the magnitude of production. In other words, an economy without any of its surplus diverted to luxury goods is engaged in the fastest possible rate of growth. "Accumulate, Accumulate, that is Moses and the Prophets to the capitalists," says Marx. Now, to the more alert among you, the following question may occur: As capitalists attempt to reinvest all of their profits in expanded production, how do we know that they will not be frustrated by bottlenecks and gluts -- too little of one input, too much of another? Can we be certain that balanced growth is in fact possible, whatever the technical specifications of production may be?

The answer, I am happy to say, is yes. There may indeed be mismatches between supply and demand at first, but it is a mathematical fact [proved in the Appendix of my book] that for any linear single-product system of production, there is some vector of activity of levels of the different industries, arrived at over time by the forces of competition, that supports a process of balanced growth, year after year. The growth rate, needless to say, is equal to the profit rate.

Interestingly enough, the great twentieth century mathematician John von Neumann proved a theorem in growth theory exactly along these lines. As a consequence, an economy embarked on maximum growth is said to be on a von Neumann balanced growth path.

[There is one really hinky other case in which Marx's equation is true, but as it has no apparent economic meaning, and involves some hairy propositions about maximal eigenvalues, I will leave it alone. You can find it discussed in Abraham-Frois and Berrebi's fascinating book, THEORY OF VALUE, PRICES, AND ACCUMULATION.]

This is the theoretical high point for Marx, the outer limits of the success of his version of the Labor Theory of Value. The key to the entire development in Marx is the distinction between labor and labor-power, which opens up the possibility of surplus value. Tomorrow, I will show you that Marx is all wrong, but that nevertheless his underlying intuition about capitalism is correct. I will also suggest an alternative theoretical formulation of this underlying intuition. Stay tuned.

Monday, 7 February 2011

THE OLD DOPE PEDDLER

Some of you are undoubtedly old enough to remember Tom Lehrer, the satirical songwriter and singer who got his start, in the late 1940's, as a Teaching Fellow in Mathematics at Harvard [see Volume One of my Memoirs for a story about Lehrer in those days.] One of my favorites among his many tunes is The Old Dope Peddler, who, in Lehrer's felicitous phrase, was "doing well while doing good." The line came to mind as I was reading The NEW YORK TIMES at one a.m. this morning [old guys like me never sleep through the night.] My eye fell on a report that Arianna Huffington, of The Huffington Post, has sold her wildly successful left-wing blog to AOL for 315 million dollars [300 million of it in cash.]

According to the story, The Huffington Post is visited 25 million times each month, so the sale price works out to a tad more than $12 a visit a month. Hmm, I thought. Since The Philosopher's Stone is currently recording 16,000 visits a month, more or less, by that reckoning, it ought to be worth roughly $200,000. I could almost pay off my mortgage. Arianna will become the head of "content" at AOL, so maybe I could find a buyer willing to let me stay on as content provider.

What earthly use would The Philosopher's Stone be to a canny investor? Well, what earthly use will The Huffington Post be to AOL? As always, the answer is advertising. Apparently AOL is bewitched by the thought of those twenty-five million monthly visitors. Would this site be a good place to advertise? I don't see why not. Surely, anyone attracted by a 50,000 word tutorial on the thought of Karl Marx or a 240,000 word Memoir of an obscure philosopher would be on the lookout for Canadian prescription medications or online dating opportunities.

I am open to any reasonable offer. The Web is a wondrous thing, withal.

Sunday, 6 February 2011

THE THOUGHT OF KARL MARX PART FIFTEEN

The time has come to see how Marx solves Ricardo's problem. This is going to get a trifle gnarly, so you will have to follow along carefully. Our exposition will take us from Volume One into Volume Three [Chapter 10]. Bear with me. To save space and time, I will simply state the results of solving little equations and carrying out little calculations. Those who are serious about mastering this subject are encouraged to work the equations out for themselves on a piece of paper.

Recall the problem we face: Ricardo was aware that in the general case, when different lines of production exhibit differing degrees of capital intensity, prices are not proportional to labor values. He was convinced that the labor required directly or indirectly in production was still the key to understanding the determination of the distribution of the social product, but he simply could not figure out what happens when some lines of production are more labor intensive and others more capital intensive.

To focus our attention, it will be useful to introduce a new model for analysis. [I trust you understand that everything I say about these little three sector models can be proved quite generally for a system with any finite number of lines of production. The formal proofs are all in the Appendix to my book UNDERSTANDING MARX.] So, consider this new model, with three sectors: corn, iron, and tools. Notice that in this model there are no luxury goods [theology books]. Corn, iron, and tools are all required, directly or indirectly, in the production of all three sectors. This, we shall discover, is essential to Marx's analysis. I will discuss it at length a little later on. Notice also that I have specified the real wage, which is once again 0.2 units of corn and 0.1 units of iron per unit of labor.

.........................Labor...........Corn..........Iron........Tools.........Output

Labor.....................................30...............15..........0................150

Corn.................80................128................2...........3................240

Iron ..................20...............16...............1...............5...............60

Tools.................50.................6..............27..............4...............16

Using Lc to stand for the labor value of corn, Li to stand for the labor value of iron, and Lt to stand for the labor value of tools, we can form three labor value equations, as before, from the input and output data of those three sectors. When we solve the equations, we get the following result, using the symbol "~" to mean "approximately equal to."

Lc ~ 0.9344

Li ~ 1.2168

Lt ~ 7.3712

In Marx's terminology, the total amount of embodied labor required by a sector of production is called its Constant Capital, which he labels C. The total amount of direct labor required he calls its Variable Capital, which he labels V. He uses these terms because the capital inputs yield up a constant amount of embodied labor to the output in the production process, but the labor power inputs yield up an amount of labor that varies according to the capitalist's ability to wring more labor out of his workers.

Having solved the equations for the variables Lc, Li, and Lt, we can now plug these values back into the table and calculate the ratio of constant to variable capital in each sector, which Marx calls The Organic Composition of Capital. This is the same ratio that Ricardo would call the ratio of embodied or indirectly required labor to directly required labor. These are all different terms for the same mathematical quantities. When we do this, we find the following:

The organic composition of the corn sector is 0.1713

The organic composition of the iron sector is 0.1164

The organic composition of the tools sector is 0.2271

The organic composition of the entire system is 0.1746

Quite obviously, this is not at all a case of equal organic composition of capital. We can now set up the price equations, using Pc for the price of corn, Pi for the price of iron, Pt for the price of tools, W for the money wage, and R for the profit rate. Because we have specified the real wage, this system of three equations now has four unknowns: Pc, Pi, Pt, and R. When we set the price of corn equal to 1, we have a system that can be solved. The results are as follows:

Pc = 1

Pi ~ 1.364

Pt ~ 7.455

R ~ 1/3

At these prices, the real wage ~ .3364

As should be obvious, in this system prices are not proportional to labor values.

Pc/Lc ~ 1.0702

Pi/Li ~ 1.12097

Pt/Lt ~ 1.0014

Now Marx makes his move. Tomorrow I shall tell you about Marx's big idea. [Hey! This stuff is so deep in the weeds even my eyes glaze over. I have to do something to keep you coming back.]