## September 08, 2018

Attention conservation notice: Over 14,000 words about an archaic monograph on political economy. Either you already care about it, in which case this adds nothing original, or you don't, in which it's vanishingly unlikely to spark an interest. Contains an extra 1920 words of quotations, a fair bit of linear algebra, and claims of being able to say what one of the great thinkers of western civilization meant better than he could. Finally, while I come from the sort of family where a great-uncle's middle name was, in fact, "Karl Marx", I am a mere squishy social democrat with liberal tendencies and a weakness for markets as tools for coordination and feedback, so I am at once too close for objectivity and too remote for involvement. Finally, I am, of course, neither an economist nor a historian of 19th century thought nor an expert on Marxism.

This was largely written over December 2015 and January 2016, and then left to the gnawing criticism of the mice. I post it now, because evidently it's time to clear out my drafts folder of ruminations on dead economists.

The other day, one of the occasional used-book dealers who comes by campus had, in addition to the usual collection of novels from the 1970s and Dover books on mathematics, a stout little ex-library hardback of Capital, volume I1. It was (perhaps appropriately) virtually free, and so I found myself moved to buy it, and then to re-read it, for the first time since I was a teenager. I will not pretend this is anything like a serious contribution to scholarship on Marx, something which already fills libraries.

Before plunging in, I should perhaps say that while I couldn't resist my title, what follows fortunately owes nothing, consciously, to Althusser.

## Part I: Commodities and Money

### Chapter 1: Commodities

#### Section 1: The Two Factors of a Commodity: Use Value and Value

Like everyone else, Marx recognizes that a commodity can have value or worth in two ways: it can be directly useful to human beings in some way, or it can be exchanged for something else. Marx wants to call the latter just "value"; this seems a bad idea, but I'll go along with it.

Marx does not explain why "these two commodities are equally valuable" must mean "these two commodities share some other common property", any more than "these two books contain the same number of words" does. Marx also does not explain why that common property must be "equal amounts of abstract, homogeneous labor are, at the present time, socially necessary to make them". The closest he comes is a footnote about that being the only logical way for neither party in an exchange to come out behind, in a way which suggests an opportunity-cost notion lying just out of reach.

Marx tries to clarify the "socially necessary" part of "socially-necessary labor time" by example of a new power loom reducing the time needed to make a given quantity of cloth, hence reducing the value of existing cloth. There is an interesting subtlety here, which is actually related to the problem that led Kantorovich to linear programming. If initially society has only one power loom, and more cloth is needed than that loom can supply, isn't the more laborious process also socially necessary? Would the value of cloth be set by some sort of average, or what? (I return to this problem below.)

The rest of the book does nothing to provide a better defense of the labor theory of value.

#### Section 3, The Form of Value, or Exchange Value

Marx's logical tools for reasoning about relations were totally inadequate. In this he was like absolutely everybody else until the very end of the 19th century, when those tools began to be developed by Peirce, Schroeder, etc. (In fairness, in sub-section B, "total or expanded form of value", Marx comes close to grasping what we'd now call an equivalence class. In further fairness, he doesn't actually grasp it.) To try to clear this up: "is just valuable as", or "is fairly exchangeable with", is a symmetric and transitive relation on bundles of commodities. If we extend it to also be reflexive, we obtain an equivalence relation, which, like any equivalence relation, partitions the space of bundles of commodities into equivalence classes. Any member of an equivalence class can of course be picked out and used as a representative of this class2. As a matter of fact, not logic, every one of those equivalence classes contains a representative which contains only a certain amount of a single commodity --- so many coats, or so many yards of linen, etc. (Strictly speaking, this pre-supposes that there is at least one commodity available in continuous amounts, unlike coats.) Moreover, these equivalence classes can be total-ordered: any two bundles of commodities can be compared in value, and if one set isn't worth more than the other, then the two sets belong to the same equivalence class. This "more valuable than" relation can be represented by the ordinary ordering on the numerical amount of any one pure commodity. Pretty much everything in sections 2, 3 and 4 that isn't wrong could be summed up in a page or two of math --- now; not in Marx's day.

Marx realizes that labor differs in regards to (for instance) skill, intensity, strength, etc., and that if he wants to have a labor theory of value, those all have to be equated somehow, reduced to some sort of common-denominator abstract, homogeneous labor. He recognizes, that is to say, that there is an issue here, but his solution is, in almost so many words, that the market takes care of that3.

#### Section 4: The Fetishism4 of Commodities

People act as though the (exchange) value of commodities was a relationship between commodities, when it's really a social relationship between their producers, or between their producers and the total labor-power of society. If production and distribution were organized differently, there would still be labor and labor time, but not commodities with exchange values proportional to socially-necessary labor time. As examples, he gives Robinson Crusoe, "patriarchal" (self-sufficient) families, payment of feudal dues in kind, and the future society of freely-associated producers, where everyone contributes labor and receives a sort of coupon proportional to their labor contribution, for a share of the total product not devoted to reproducing and expanding the productive apparatus.

The last bit is one of the handful of places in Capital where Marx says anything about what the post-capitalist economy is supposed to be like. (Admittedly, the point of the book is to explain the political economy of capitalism, not post-capitalism.) The difference between those coupons and money eludes me, unless the idea is that everyone gets an exactly pro-rata share of every good, in which case I predict tears, and the emergence of barter, or rather a de facto money, before bedtime. (My neighbors with the newborn needs more diapers than I do, they have no use for books on stochastics and political economy, neither they nor I can put away as much beer as Uncle Karl, etc.5) If, on the other hand, the coupons are basically money, then there need to be prices, denominated in coupons, on the goods. And so I fail to see how we won't still have commodity production and, to anticipate some later material, how there won't still be a gap between the length of the working day, i.e., value provided by a worker's labor power, and the value of the worker's coupon wages. The gap might be shrunk to the minimum necessary for the reproduction and expansion of society, without skimming off extra to support capitalists and their parasites, but Marx seems to be aiming at a bigger qualitative difference, and not getting there.

I see why this section on fetishism has been so influential, partly because Marx is getting at something important, and partly because the attempted de-mystification is itself pretty mysterious. A certain amount of mystery helps any ideology; it provides the moment of "click". Further commentary on this point is out-sourced to Ernest Gellner's "Notes Towards a Theory of Ideology", and to Raymond Boudon.

— If you want to play "people think $X$ is a property of things, but really it's a social relationship in disguise" games, neo-classical economics has got at least as much to offer as Marx. Prices confront each individual price-taker as objective facts, but --- so goes the Walrasian myth --- they are in fact the outcome of a social process, reflecting at once the possibilities of production, the social facts of initial endowments, and the purely subjective wants of everyone. There isn't even, in the neo-classical view, something like "socially-necessary labor time" to ground prices in. Indeed, in neo-classical models, prices can be incredibly different, with exactly the same productive possibilities and desires, merely as a function of initial endowments with goods and money. One might say that Debreu's Theory of Value was the first mathematically rigorous contribution to theories of the social construction of reality.

### Chapter 2: Exchange

People make things they don't want, in order to sell them for money, which they use to buy things they do want; since they start with a commodity, get money, and end with a commodity, Marx writes this C-M-C. He depicts this as a "metamorphosis" of the original commodity into a new commodity. This illuminates nothing, but serves the rhetorical function of writing (I paraphrase) "no one makes a profit on trading in a competitive market" in big block letters and underlining over and over. ("Profit" of course in terms of exchange value; Marx is quite clear that people only exchange at all because they prefer the use-value they get to the use-value they give up.)

### Chapter 3: Money

Marx re-capitulates the usual origin myth of money, from barter to a commodity serving as numeraire to money. (The "double coincidence of wants" doesn't appear, but the idea that barter is very inconvenient certainly does.) It is important for Marx that money itself have value, and so be something which is itself the product of labor; metals, like gold and silver, are extra handy because they are durable, divisible, and (can be made) homogeneous, thereby representing abstract, homogeneous labor. Indeed, the value of the metal is, precisely, the quantity of socially-necessary labor required to produce it.

Marx states a basic form of the quantity theory of money: the amount of circulating money $M$ in a country is determined by $M = \frac{\sum_{i}{p_i Y_i}}{v}$ where $p_i$ is the money price of the ith commodity, $Y_i$ is the amount of that commodity bought and sold per unit time, and $v$ is the velocity of money, the number of transactions each coin goes through per unit time. Several paragraphs are given over to verbal descriptions of what happens when each of these terms moves up or down. (Here I do have to fault him, algebra was a thing in 1867, and there could have been few readers who found those paragraphs easier going than a few symbols.) He correctly notes that it's not necessary for all prices to go up in order for the quantity of circulating money to have to increase.

I deliberately wrote this equation with $M$ by itself on one side, since Marx is quite insistent that changing the volume of circulating currency in the country cannot alter prices. (He explicitly takes issue with Hume on this point.) One might even go so far as to write it with an arrow for causal dependence, thus: $M \leftarrow \frac{\sum_{i}{p_i Y_i}}{v}$ I did not catch his argument for why this must be the case, especially since in a footnote he says that "bungling legislative interference" with $M$ can induce stagnation of trade. I kind of get why he wants this to be true --- it goes along with the general labor-theory-of-value idea, if sheer quantity of money could drive prices that would be bad --- but I didn't see an actual argument. It's true that money could be hoarded rather than circulating, so, even with a metallic currency, $M$ can't be identified with the sheer amount of gold in the country, but denying that changing $M$ could change the $p$'s is just weird.

(Later on, Marx modifies this quantity theory of money to allow for credit and repayment of credit, but it doesn't alter the points above.)

— What Marx would have made of our current system, where our money has been completely cut free from any sort of commodity backing, and no more labor is required to make a \$100 bill than a \$ 1 bill (or to record "\$100" in an electronic account than "\$ 1"), is an interesting question. It is, indeed, something I am ashamed of not having taxed my Marxist friends with before. (Obviously having non-commodity money doesn't meant that the exchange ratios of commodities couldn't equal the ratio of their labor-contents, but it does create a problem when you want to talk about exchanging a commodity for money or money for commodities, the C-M and M-C parts of a "circuit".) The best answer my inner Marxist can come up with, absent any actual research, is this: a dollar bill, a credit card, etc., only function as money because a large set of inter-locking organizations and less formal institutions make them effective: it's not just a matter of printing the bill or the card, but the whole system of banking and finance which makes sure everyone recognizes them as means of payment, that transactions balance, that keeps track of who has which claims on money, that banks don't (mostly) just go belly up, etc., etc. The value of our current money (my inner Marxist continues) derives from the very considerable amount of labor required to keep these institutions operating and effective. (You could even use this to explain inflation, by arguing that the "productiveness" [see below] of the monetary system rises faster than the productiveness of the rest of the economy, hence money prices must rise.) That still doesn't answer why a \$100 bill represents a bigger share of the socially-necessary labor of the monetary system than does a \$ 1 bill, though.

## Part II: The Transformation of Money into Capital

### Chapter 6: The Buying and Selling of Labour-Power

Money becomes capital, rather than merely being hoarded, when it is used to make more money, i.e., when someone uses it to buy a commodity, and then sells that commodity again, which he writes M-C-M'. Marx follows Aristotle in regarding this as somewhat unnatural, and money-lending at interest as even more unnatural, as opposed to merely using money to lubricate exchange. (The Aristotlean inspiration is quite explicit.)

Problem: You can't make any profit in a fair exchange (that was the whole point of chapter 2), and you can't generally make profits by coming out ahead in unfair exchanges. (As we'd now say, that's at most zero-sum.) So where does "surplus value" come from?

Solution: what a laborer sells is their labor power, the use of their productive abilities, typically for a certain period of time; the capitalist can come out ahead if the value of labor power is less than value added by the labor. E.g., if the normal working day is 12 hours but the value of the labor power for that day is only 6 hours, the capitalist can get 6 hours of surplus value out of this.

The value of labor power is set, like that of any other commodity, by the quantity of average labor time socially necessary for its production, or rather reproduction, i.e., whatever's required to produce what workers typically require, given the climate and customs of their country, by way of food, clothing, fuel, housing, training, etc., to maintain themselves in work and to raise the next generation of workers.

## Part III: The Production of Absolute Surplus-Value

### Chapter 7: The Labour Process and the Process of producing Surplus-Value

Let's temporarily fix the length of the working day (or week or other period) at so many hours. The value of the labor power for that period is either greater than, equal to, or less than that number of hours. (Remember, value has units of labor time.) If the value of labor power is greater than the working period, everyone's in trouble: average workers don't produce enough to reproduce themselves during the period they work. If the length of the working day exactly equals the value of labor power, then the workers can, just, reproduce themselves. This means, however, that there is no point in hiring workers to make money. (Or so says Marx; but I wonder if it mightn't make sense to pay to hire the above-average workers.) Finally, if the length of the working day is longer than the value of a day's labor power, there is, potentially, a surplus. That difference in time is the surplus value produced by the worker. (Again, exchange value is measured in units of necessary labor time.)

The problem of equating different sorts of labor is raised again at the end of this chapter, and again Marx basically says that the market takes care of this. Indeed, the footnote pointing out that what counts as "skilled" and "unskilled" is often really just the distinction between what is and isn't in short supply, and hence able to command high wages, while it has merit, would seem to undermine the case for taking quantity-of-average-labor as fundamental. (I note in passing that, e.g. David Autor's recent paper, purportedly tracking how wages have changed for occupations with different levels of skill, just re-defines "skill" as the wage rates prevailing in 1979. This is done without any comment, or even the cross-check of seeing if the same results hold if a different year is used as the reference point.)

### Chapter 8: Constant Capital and Variable Capital

It is actually more convenient for me to give the definitions of these terms below, under chapter 9.

Towards the end of this chapter: after a bad harvest, a given weight of cotton represents more labor than after a good one. How on Earth does this follow? If the harvest is bad because (e.g.) of drought or flood or disease, which affect only some regions but not others, the total harvest might be cut in two, but no more work expended on the surviving part than on the part that didn't make it. The best I can do is something like "well, we wouldn't even have that half of the harvest unless we had gone through all the work of the complete harvest", which might sometimes be true but is hardly a universal rule.

### Chapter 9: The Rate of Surplus-Value

The absolute amount of surplus value which results from production is the difference $s$ between the value of the output, and the value $C$ of the inputs. The value of inputs consists of (1) "variable capital" $v$, i.e., the value of the labor power hired, and (2) "constant capital" $c$, so $C=v+c$. Constant capital $c$ in turn consists of (a) the value of the raw materials and supplies consumed, and (b) wear and tear, or amortization, on what we'd call capital goods (tools, machinery, buildings for carrying on work, fuel, etc.). Relative surplus value is defined as $s/v$, the ratio of surplus value to variable capital alone. I note that these are flows rather than stocks --- the total value of the capital goods, and so (e.g.) set-up costs, does not figure into Marx's calculation (except perhaps through amortization).

Marx thinks it is very wrong to consider (say) the ratio $s/C$ of surplus value to total capital charges. Only $s/v$ is legitimate. I honestly can't fathom why, intellectually, this should be the case; for propaganda purposes however it seems pretty clear that he wants to say that if a coat has a value of \$20, of which \$ 10 is constant capital, \$5 is labor and \$ 5 is surplus value, the "rate of exploitation" is 100%, not 25% or even 33%. But why one should care about this "rate of exploitation", so defined, is again unclear. It's true that $s/(s+v)$ gives (according to the labor theory of value) the fraction of the working day during which the laborer's efforts go purely to the benefit of the capitalist, and that, algebraically6, this can also be deduced from $s/v$, and Marx makes much of this, but, again, why? For capitalists, labor power is just another input commodity, and they should care about the ratios of profits to total outlays. (Incidentally, it is striking in these theoretical chapters just how much Marx leans on constant returns to scale.) For laborers, surely the important variable is just $v$ --- really it's wages, which is something else altogether (Part VI below).

#### Aside: Constant Capital and "Socially Necessary Labor Time"

There is an important, but largely un-heralded, interpretive point about the labor theory of value implicit in this chapter. This occurs at the very beginning, when it's asserted that the value of the goods produced is not $s+v$, which is the quantity of labor used in this stage of production, but $c+v+s$, which adds in the labor value of all of the material inputs, including the amortization and depreciation of capital goods7. This means that the phrase "social necessary labor time" has to be interpreted in a very special sense, as including all the labor time which went into making the input materials, some share of the labor time which went into making tools, machines and other means of production, and so on transitively for their production as well. The value of any commodity is thus the sum of a series which has, if not actually infinitely many terms, then at least has no real bound on their number.

Still, as I tell my students, one person's vicious circle is another's iterative approximation, and iterative approximations, when they work, converge on fixed points. The fixed point is actually something we can work out here, at least post-Leontieff8. Say that producing one unit of commodity $i$ takes $a_{ij}$ units of commodity $j$ as inputs. (This includes capital goods, which get treated on the same basis as everything else.) Collect these into a matrix $\mathbf{a}$, of dimension, say, $k \times k$. Producing one unit of commodity $i$ also directly requires $v_i$ hours of abstract homogeneous labor, which we collect into the $k \times 1$ matrix $\mathbf{v}$. Self-consistency demands that the $k \times 1$ matrix of socially-necessary labor times $\mathbf{l}$ satisfy $\mathbf{l} = \mathbf{v} + \mathbf{a}\mathbf{l}$ or $\mathbf{l} = (\mathbf{I} - \mathbf{a})^{-1} \mathbf{v}$

Expanding $(\mathbf{I}-\mathbf{a})^{-1}$ in a power series, $\mathbf{l} = \mathbf{v} + \mathbf{a}\mathbf{v} + \mathbf{a}^2\mathbf{v} + \ldots$ So the labor embodied in one unit of good $i$ is, indeed, equal to the direct labor needed to make it, plus the labor directly needed to make the raw materials and means of production for good $i$, plus the labor needed for their raw materials and means of production, etc. (If this doesn't exactly redeem my joke about iterative approximation, it at least makes it more literal.) We can thus extract the vector of constant capitals $\mathbf{c}$ as $\mathbf{c} = \sum_{n=1}^{\infty}{\mathbf{a}^n \mathbf{v}} = \left( (\mathbf{I} - \mathbf{a})^{-1} - \mathbf{I}\right) \mathbf{v}$ We'll come back to this formula later. For now, I just want to high-light that all of the implausible assumptions (constant returns to scale, no unproduced means of production such as land, etc.) involved in this derivation are ones explicitly made by Marx. It is implied by his assumptions, and if you don't like it, you need to assume something else.

— I want to make it very clear that I claim no originality for this little bit of linear algebra; I won't try to trace the history, but it goes back at least to the 1960s. I can't remember if I learned it from my father, or just from his books.

### Chapter 10: The Working Day

Being a worker in 19th century England was appalling. One of the reasons it was so appalling was the length of the working day, which employers did everything possible to extend. Every effort at limiting the length of the working day, or even to collect systematic information, was met with immense resistance, flouting of the laws, and proclamations of doom. This chapter lets Marx unleash his savage indignation, and equally savage sarcasm, on targets who fully deserved every particle of his venom.

### Chapter 11: Rate and Mass of Surplus Value

Surplus value adds up across workers.

## Part IV: Production of Relative Surplus-Value

This part has a very odd structure. The first chapter in it, 12, is a continuation of the stuff about surplus value from the previous part. The rest of it is a historical survey of the development of the forces of production, and with them the social relations of production, from the most basic cooperation with minimal tools to "modern industry", i.e., powered automatic machinery. The connection between the first chapter and the rest is frankly weak. Marx claims that he'll show all these developments were drive by to increase the fraction of the work day going to surplus value, which he never really delivers on.

### Chapter 12: The Concept of Relative Surplus-Value

Recall that surplus value is the difference between the value produce by employing labour-power, and the value of the labour-power itself. Further recall that the value of labour-power is the quantity of labor, in the form of commodities, socially necessary for the worker to keep working and to bring up the next generation. If a capitalist wants to increase their surplus value, they can either lengthen the working day ("absolute surplus value"), or they can reduce the value of labor-power ("relative surplus value"). The latter requires increasing the productiveness of labor, so that less time is needed to produce the commodities which labor-power needs to reproduce itself.

### Chapter 13: Cooperation

The "productiveness"9 of labor increases when workers cooperate, even if they don't specialize. Think of many people lifting a large weight together which none of them individually could shift, or being able to bring in a harvest before anything spoils, etc. Sometimes the gains from cooperation arise because the laborers can share equipment, even if it's just a work-place. Additional increases to productiveness can follow from even elementary division of labor (think of a bucket brigade for moving water from a well), as well as more refined ones.

Cooperation often needs people to organize it; when cooperation is based on shared equipment, someone might own it. These are the most basic ways in which the capitalist can gain an entry into the process of production, as the equipment owner and organizer of cooperation. Reading between the lines here a little, Marx doesn't (or at least shouldn't) see a problem with what economists would now call "factor payments to capital", i.e., putting some of the product aside to pay for wear and tear, or even for improvements. He shouldn't even, by his own rights, object to paying someone for doing the work of organization, since that would be part of the socially-necessary labor process. That capitalists, because they own the means of production, should get to claim the whole of the surplus product, that is the fundamental sticking point.

### Chapter 14: Division of Labour and Manufacture

By "manufacture", Marx means the systems of refined division of manual labor within single trades --- the sort of thing famously celebrated in the opening of The Wealth of Nations. (Marx has some rather catty footnotes about how un-original Smith's passage was.) Something which might, before, have been done by a single artisan is instead broken up into a multitude of small steps. Each step gets its own crop of "detail-laborers", who do just that, often with specialized tools. This leads to savings of time (since the same laborer isn't switching tasks), and increase of skill, and hence increasing productiveness for the system as a whole. But, says Marx, it cripples the detail-laborers intellectual and often physically. (I wonder whether the detail laborers would have agreed about the former.)

Marx is at pains to draw out a couple of points here:

• While division of labor is ancient, the manufacturing system had a definite historical period, which he dates "from the middle of the 16th century to the last third of the 18th century";
• Manufacturing was not pursued because it increased productiveness or enhanced use values (as in the ancient arguments for the division of labor, going back to Plato if not before), but because it reduced costs and increased profits;
• Being a detail laborer was no fun at all;
• The manufacturing system had definite limits, precisely because it depended on manual labor.

### Chapter 15: Machinery and Modern Industry

This is a really magnificent historical analysis of the effects of introducing machinery into production, especially when that machinery is driven by in-organic power sources. Marx begins with distinguishing a "machine" from a "tool" . He quotes some definitions which say the difference is that a "tool" is powered by a human body, a "machine" by something else. This he rejects as unsatisfying (fairly enough; it makes an ox-drawn plow a machine, but the same plow drawn by the farmer a tool). Rather, he says, what makes a machine is that the actual effective instruments are not wielded by a human being, but by the contraption. (He quotes Babbage favorably on this point.) Whether the ultimate motor is a human body or something else is beside the point. Once you've built the machine, substituting one form of power for another might be a tricky technical feat, but it is just a technical feat, and the sort of thing which technologists get very good at.

The tools wielded by machines generally begin with the detail-labor tools developed by the manufacturing system, rather than the less specialized ones of handicrafts. But they then adapt to the conditions of mechanical use, often but not always by multiplying. The actual laborer's role, since it is not to wield the instruments, is to supervise the machine, correct its faults, and bridge gaps between mechanical processes. The progress of what we would now call automation comes in here, moving more and more processes into the ambit of the machine, and reducing the number of mistakes which the human operative must correct.

Every form of mechanized industry demands a specialized division of labor, but it is a transient division, one which is going to constanty be changed by the progress of technology. Thus it demands an education which suits its recipients to turn their hands to many different kinds of jobs over the course of their life10. Cf. Gellner's Nations and Nationalism, and for that matter every education writer for at least a century.

Recouping the fixed costs of machinery, especially set-up costs, pushes capitalists towards continuous production --- hence an additional reason for lengthening the working day, for night work, etc.11 It also pushes towards the lower possible labor costs, so employing women and children, especially because muscle power is less important; and it pushes towards intensifying the working day if lengthening it isn't possible.

Marx realizes that it's possible to mechanize and industrialize agriculture and the countryside; he thinks that if this is done by capitalists, it will have bad long-term consequences for the soil. It's a little hard to for me to grasp exactly what he's getting at, but I think he's worried about disrupting chemical cycles, e.g. of carbon or nitrogen12. Why there couldn't be capitalist businesses collecting "consumed ... food and clothing" from towns and re-processing them into fertilizer, he doesn't say.

## Part V: The Production of Absolute and Relative Surplus-Value

### Chapter 17: Changes of Magnitude in the Price of Labour-Power and in Surplus-Value

I realize I'm sounding like a broken record, but really, would it have killed Marx to have used a little algebra, and spared us all this prose? Let $t$ = length of the working day (units: hours), $g$ = the value of goods required for reproduction of average labor (units: hours), $\rho$ = productiveness of average labor at standard intensity of labor (unitless), with $i =$ that intensity, typically 1 (also unitless). Then the value of labor power $v = \frac{g}{\rho i}$, while $s = t-v$ is the absolute surplus value, and $s/v = \frac{t}{v} - 1$ is the relative surplus value (per Marx's definition). Thus $s = t - \frac{g}{\rho i}$ and $\frac{s}{v} = \frac{t}{g}\rho i - 1$ All of the "laws" deduced here follow immediately.

(In passing: surely Marx is right that the intensity of labor matters, but how does this fit with his stated position of reducing everything to average labor? For that matter, how, non-tautologously, might we measure "intensity of labor" $i$? It would seem to be hopelessly confounded with productiveness $\rho$.)

In these chapters, "rate of profit" is defined as the ratio of surplus value to working capital $s/(c+v)$. (It's possible he also introduced the concept earlier and I missed it.) To motivate this, remember that the labor value of the product includes the labor value $c$ of the raw materials and means of production used up in making it. So the total value of goods produced is $s+c+v$, while the value of the inputs is $c+v$, and so the profit is exactly the surplus value $s$. Since the numerator is a flow (surplus $s$), it makes some amount of sense for the denominator, too, to be a flow (variable capital or wages $v$ and constant capital or inputs and wear-and-tear $c$). It is thus not to be confused with our modern notion of "rate of return on investment", where the denominator is a stock of capital. Whether capitalists will care about rate of profit, in Marx's sense, or rate of return on investment, would seem to depend on how important start-up costs are.

## Part VI: Wages

Since reproducing labor power requires goods (and services) which are themselves products of labor, labor power has a value. Since money requires labor to produce, it has a value. Therefore labor power has a price in money. This price is wages. Whether it's expressed as a price per unit time, or a price per unit output, is secondary; piece rates will get adjusted to reflect the productiveness of labor.

There are a lot of gyrations in these chapters to get around the fact that the value of labor power doesn't tell us much about real wages, "i.e., the means of subsistence placed at the disposal of the labourer".

## Part VII: The Accumulation of Capital

### Chapter 24: Conversion of Surplus-Value into Capital

Any society needs to reproduce its means of production if it's to go on. This means that some fraction of production must not be consumed in the usual way, but go towards at least replacing old means of production. (Though I do not think Marx says it in so many words, this would correspond to the "constant capital" portion of the total [working] capital.) In simple reproduction, society just replaces its means of production, without expanding them. This would leave the surplus value available for the consumption of the capitalist. Part of reproduction is providing the labor-force with the means of subsistence, which are of course things produced by the laborers; under capitalism, this takes the form of the variable capital or wages, but it is always, under any society, about workers collectively producing what they (and their families) need. This takes other forms under other modes of production, and Marx hints at the later discussion of "primitive accumulation", i.e., how capitalism could have gotten started from other modes of production/reproduction. (How social reproduction will work after capitalism is not discussed.)

Marx says that if the surplus value is \$100 a year on a capital of \$ 1000, after ten years the capital consists entirely of surplus value, even if the surplus value is entirely consumed each year. (I have changed his pounds to dollars to suit my keyboard.)

If a capital of \$1000 beget yearly a surplus-value of \$ 200, and if this surplus-value be consumed every year, it is clear that at the end of 5 years the surplus-value consumed will amount to $5 \times \ 200$ or the \\$ 1,000 originally advanced. If only a part, say one half, were consumed, the same result would follow at the end of 10 years, since $10 \times \ 100= \ 1,000$. General Rule: The value of the capital advanced divided by the surplus-value annually consumed, gives the number of years, or reproduction periods, at the expiration of which the capital originally advanced has been consumed by the capitalist and has disappeared.

I frankly do not follow his argument at all:

The capitalist thinks, that he is consuming the produce of the unpaid labour of others, i.e., the surplus-value, and is keeping intact his original capital; but what he thinks cannot alter facts. After the lapse of a certain number of years, the capital value he then possesses is equal to the sum total of the surplus-value appropriated by him during those years, and the total value he has consumed is equal to that of his original capital. It is true, he has in hand a capital whose amount has not changed, and of which a part, viz., the buildings, machinery, &c., were already there when the work of his business began. But what we have to do with here, is not the material elements, but the value, of that capital. When a person gets through all his property, by taking upon himself debts equal to the value of that property, it is clear that his property represents nothing but the sum total of his debts. And so it is with the capitalist; when he has consumed the equivalent of his original capital, the value of his present capital represents nothing but the total amount of the surplus-value appropriated by him without payment. Not a single atom of the value of his old capital continues to exist.

This seems, at best, like moralizing about what workers are owed, and not value accounting.

What makes rather more sense to me is the point that with compound interest, any initial capital that might have been accumulated by personal savings and effort is swiftly overwhelmed by capitalized surplus value.

Also more sensible are the two following points:

1. Taken as a whole, the working class produces, through its labor, all its own means of subsistence. What this looks like, however, is that the laborers of any one capitalist exchange their wages for (for the most part) the goods sold by other capitalists.
2. If the market works and laborers receive the value of their labor power, they are just able to reproduce themselves. To stay alive, they must therefore sell that labor power again.

The last point is worth emphasizing. Once someone becomes a laborer, for Marx, they have to pretty much stay one: they enter the process with nothing to sell except their labor power, and they get just the value which lets them keep on selling their labor power. They are trapped. There might, of course, be accidents which let some few individuals save enough to become capitalists, but they cannot be general.

Chapter 24 specifically focuses on the accumulation of capital, i.e., capitalists not consuming all of the surplus value, but converting (some of) it back into more capital used for production, both as constant capital (means of production) and variable (wages). Marx, quite correctly in my view, sees this as the fundamental thing which both distinguishes capital from earlier modes of production, and makes it (temporarily) progressive:

Except as personified capital, the capitalist has no historical value... And so far only is the necessity for his own transitory existence implied in the transitory necessity for the capitalist mode of production. But, so far as he is personified capital, it is not values in use and the enjoyment of them, but exchange-value and its augmentation, that spur him into action. Fanatically bent on making value expand itself, he ruthlessly forces the human race to produce for production's sake; he thus forces the development of the productive powers of society, and creates those material conditions, which alone can form the real basis of a higher form of society, a society in which the full and free development of every individual forms the ruling principle. Only as personified capital is the capitalist respectable. As such, he shares with the miser the passion for wealth as wealth. But that which in the miser is a mere idiosyncrasy, is, in the capitalist, the effect of the social mechanism, of which he is but one of the wheels. Moreover, the development of capitalist production makes it constantly necessary to keep increasing the amount of the capital laid out in a given industrial undertaking, and competition makes the immanent laws of capitalist production to be felt by each individual capitalist, as external coercive laws. It compels him to keep constantly extending his capital, in order to preserve it, but extend it he cannot, except by means of progressive accumulation.

And again, famously:

Accumulate, accumulate! That is Moses and the prophets!

This is followed by some rather acid remarks on the understandable tendency of individual capitalists to not want to subordinate everything to accumulation, but rather to enjoy themselves, and the idea that savings is "abstience" from consumption.

Going back just a little, however, I do not think Marx gives an adequate theoretical explanation for why accumulation is so vital to capitalism. Or, rather, he does have one, in two parts, but it fits awkwardly with the rest of his theory. Part 1 of the explanation is that

the development of capitalist production makes it constantly necessary to keep increasing the amount of capital laid out in a given industrial undertaking
and part 2 is that
competition makes the immanent laws of capitalist production to be felt by each individual capitalist, as external coercive laws

I think it's fairly easy to make sense of this, if one allows for substantial economies of scale. The argument would go like this:

1. The larger the scale of production, the lower the cost at which it can be profitably sold.
2. Firms which cannot match the lowest selling-price currently offered will tend to exit the market, either because they lose money, or lose market share, or both.
3. Therefore, the surviving firms will have the same costs of production, which will all be the lowest currently feasible.
4. A firm which re-invests some of its surplus can expand scale and lower its cost of production, so every other firm must match it or exit the market.

Of course, this presumes that the process of selection-by-competition invoked in (2) and (4) is quite fast, since otherwise it's an idle in-the-long-run-the-inefficient-are-dead observation. But leave aside Marx's reliance of the efficiency of competition. (In his own way, he really believed in capitalism.) This is yet another place where Marx makes sense, if we assume increasing returns to scale, but his value theory presumes constant returns to scale.

Marx may also have had in mind an argument about technical change:

1. A change in technique which increases the productiveness of labor reduces the value of a commodity, because it reduces the labor time socially necessary to produce it. (Cf. the bit about the new power loom back in chapter 1.)
2. Firms which can only produce a commodity by using more than the socially necessary amount of labor will exit the market.
3. Generally, adopting the new, productiveness-enhancing technique will have costs, which will have to paid from accumulation.

However, I like this second line of interpretation less than the first, because I feel the balance of Marx's text is against thinking of the imperative of accumulation as something relying on technical change.

What I want to emphasize here is that, yet again, Marx has a pretty sound point, which fits very poorly with his theoretical apparatus, since the latter relies crucially on constant returns. In a constant-returns world, a capitalist who is content to just consume his surplus value every year suffers no disadvantage at all, vis-a-vis one who re-invests and accumulates. The latter may, ultimately, enjoy a larger stream of personal income, but that's neither here nor there. Under constant returns, a single large capital enjoys no advantages over a multitude of small capitals.

### Chapter 25: The General Law of Capitalist Accumulation

The condition of laborers will just get worse and worse as capital accumulates. When capital accumulates very rapidly, wages might rise above the value of labor power, but if it does, capital accumulation will slow, and wages will come back down. As accumulation proceeds, Marx claims, the ratio of organic capital $c$ to variable capital $v$, $c/v$ or the "organic composition of capital", will tend to grow. (I will avoid the phrase "organic composition of capital".)

I do not follow the argument for this proposition at all. The supposed cause is the "development of the productiveness of labor". By reducing the quantity of labor necessary for producing the goods with which labor-power is reproduced, it lowers the value of labor-power $v$. But the value of constant capital is that of the socially-necessary labor time for replacing the goods making up constant capital, so increases in the productiveness of labor can also reduce $c$. There is no argument given, that I can detect, that $v$ should generally be reduced by more (in proportion) than $c$ is reduced. The best I can do, attempting to make sense of Marx here, saves his conclusion only for a very special case. Since this is a bit involved, I will push this to an addendum at the end of this chapter, and just say, in conclusion, it doesn't work generally.

This idea, about the rising ratio of constant to variable capital, is linked in Marx's mind to the rising industrial reserve army. He has, I think, two valid points here:

1. If the productiveness of a certain branch of industry rises, that will ceteris paribus reduce employment in it; those laborers are "made free".
2. When production expands (at constant productiveness), the laborers have to come from somewhere. The first process throws people into the industrial reserve army, the second process draws them out. (Even if, like a modern economist, one thinks of the labor supply as being a curve, of so many hours offered at so many dollars per hour, there are still adding-up constraints.)

If we add that the productiveness of labor generally rises under modern industry, it still does not follow that the industrial reserve army must grow either absolutely or in proportion to the total population. Still less does it follow that the real income of the working class, measured in the quantity, range, and quality of commodities they can command for their wages, must fall. After all, increasing the productiveness of labor will reduce the exchange value (in labor units) of those commodities!

There are also hints, in this chapter, of an even-more-famous idea, the tendency of the rate of profit to fall, though IIRC that is only fully developed in Volume III. This is related to the point about the supposedly rising ratio $c/v$. Recall that for Marx the rate of profit is $s/(c+v)$. If $v$ and $s$ stay the same but $c/v$ rises, then $c$ must increase and the rate of profit will of course go down. Increasing the productiveness of labor in a given industry will tend to increase $s$, so in order to make the rate of profit fall, it's especially important that the ratio $c/v$ increase.

Unfortunately for Marx, as far back as 1961 Okishio Nobuo showed that a technological change which increases profits when introduced by one capitalist is necessarily one which increases the profit rate when generally adopted. (Okishio's paper doesn't seem to be online, but Sam Bowles has a lovely little paper, "prompted by" "his students' critical scepticism", re-proving the theorem in two pages.) This makes it very hard to see how "the general law of capitalist accumulation" could possibly lead to a falling rate of profit, unless capitalists are not generally driven by competition to increase their own profits.

Of course the profit rate could still tend to fall for other reasons, and whether it has fallen or not is a tricky empirical question. The claim, after all, is not about the volume of profits, nor about return on investment. The numerator isn't even, strictly speaking, about "profits" in the ordinary accounting sense, but about surplus value, which gets divided into ordinary profits and many other things (debt servicing, taxes, land rent, embezzlement, protection money to criminals, etc.). Moreover, the denominator is the flow of outlays for the production process (wages $v$ plus raw materials plus amortization/depreciation of means of production $c$), not the stock of capital values (which is the denominator for return on investment). A very rough approximation, based on the usual system of national accounts, would be the share of "net operating surplus" in the national income. At least for the US, this actually did trend downwards from the beginning of the data in 1929 to about 1970, but has been, if anything, rising since about 1990 (source):

Enough, for now, about the rate of profit.

Finally, in this chapter Marx expresses well-deserved contempt towards some economists' views on wages and population, to the effect that rising wages leads to more baby laborers which pushes down wages. To paraphrase him a little, that process would take 15--18 years to take effect, while wage fluctuations are vastly faster.

#### Addendum: The effect of increasing the productiveness of labor on the ratio of constant to variable capital, a.k.a. the general tendency of the organic composition of capital to do whatever it feels like

Let me recall a result above, from the aside about embodied labor values. When $\mathbf{v}$ is the input-output matrix of production, and $\mathbf{v}$ is the vector of labor directly required to produce one unit of each output, the embodied labor values are $\mathbf{l} = (\mathbf{I} - \mathbf{a})^{-1} \mathbf{v}$ and the constant capital required per unit of each good is $\mathbf{c} = \left( (\mathbf{I} - \mathbf{a})^{-1} - \mathbf{a}\right) \mathbf{v} \equiv \mathbf{\alpha} \mathbf{v}$ introducing the abbreviation $\mathbf{\alpha}$ for the matrix which sums up direct inputs, plus direct inputs' direct inputs, plus direct inputs' direct inputs' direct inputs, and so on ad infinitum. I need a name for the goods showing up in the $i^{\mathrm{th}}$ row of $\mathbf{\alpha}$, so I will call them the "ultimate inputs" used to produce $i$, or just the "ultimate inputs" for short. It is these ultimate inputs which fix the ratio $c_i/v_i$: $\frac{c_i}{v_i} = \frac{\sum_{j}{\alpha_{ij} v_j}}{v_i} = \alpha_{ii} + \sum_{j\neq i}{\alpha_{ij} \frac{v_j}{v_i}}$

Now, a technical improvement which increases the productiveness of labor will have to reduce the elements of the vector $\mathbf{v}$. Suppose for simplicity that it just reduces one element, say that $v_i$ gets pushed down to $v^{\prime}_i < v_i$. Further suppose, and this is crucial, that the technical change doesn't otherwise alter the input-output matrix $\mathbf{a}$, and so $\mathbf{\alpha}$. Call the new labor requirement $v^{\prime}_i$. The old value of the ratio $c_i/v_i$ was, remember, $\alpha_{ii} + \sum_{j\neq i}{\alpha_{ij} \frac{v_j}{v_i}}$ while the new one will be $\frac{c^{\prime}_i}{v^{\prime}_i} = \alpha_{ii} + \sum_{j\neq i}{\alpha_{ij} \frac{v_j}{v^{\prime}_i}}$ which is necessarily larger than before, since $v^{\prime}_i < v_i$. So far so good for Marx.

But, and this is crucial, this last paragraph presumes a technical improvement which is only labor-saving, without using more of anything else. If labor becomes productive because it has more or better means at its disposal, those need to show up as larger elements in the input-output matrix $\mathbf{a}$. So we're really facing a new matrix $\mathbf{a}^{\prime}$ where $a^{\prime}_{ij} > a_{ij}$, at least for some $j$. (Of course some other entries could be smaller, e.g., now-obsolete tools might not be called for.) Thus we'd need to compare the old ratio $\alpha_{ii} + \sum_{j \neq i}{\alpha_{ij} \frac{v_j}{v_i}}$ to the new $\alpha^{\prime}_{ii} + \sum_{j \neq i}{\alpha^{\prime}_{ij} \frac{v_j}{v^{\prime}_i}}$

The denominators of the ratios in the sums have shrunk, $v^{\prime}_i < v_i$, but the new numerators $\alpha^{\prime}_{ij} v_j$ could be larger or smaller than the old ones, $\alpha_{ij} v_j$. The over-all effect will depend on whether the ultimate inputs used to produce $i$ have shifted towards goods which require a lot of direct labor or goods which require little. There is certainly nothing in the logic which says whether the sum of the ratios must have increased.

Labor-saving technical change can thus cause the ratio $c_i/v_i$ to fall, if it shifts the ultimate inputs towards ones which require little direct labor. Conversely, a labor-raising technical change could increase $c_i/v_i$, if it shifts the ultimate input mix towards goods with high direct labor requirements, or even just raises $\alpha_{ii}$ enough.

Finally, I want to consider one additional wrinkle, which is what happens to $c_i/v_i$ when there is technical progress in another industry, say $h$. In particular, let's go back to the case where it's only the direct labor requirements that get reduced, so that $v^{\prime}_h < v_h$. This, as we say, increases the ratio $c_h/v_h$, but what about industry $i$? We go from $\alpha_{ii} + \alpha_{ih}\frac{v_h}{v_i} + \sum_{j \neq i, h}{\alpha_{ij}\frac{v_j}{v_i}}$ to $\alpha_{ii} + \alpha_{ih}\frac{v^{\prime}_h}{v_i} + \sum_{j \neq i, h}{\alpha_{ij}\frac{v_j}{v_i}}$ which is evidently lower than before, unless $\alpha_{ih} = 0$. So even a change which drives up $c/v$ for one industry can lower this ratio for another, downstream industry which uses $h$ as an ultimate input. If we care about the "organic composition of capital" for the entire economy, which of these two effects would win out would depend on the sizes of the industries affected. Why should raising effect on industry $h$ should always overwhelm the lowering effect on all industries that ultimate use $h$ as an input?

To sum up, Marx's idea, to the extent I can make sense of it, only works for the case of a pure improvement of technique, where less labor is needed to get the same output with exactly the same inputs. Even then it only applies to the industry directly affected, with a contrary effect on all downstream industries. Beyond this, the effect of increasing the productiveness of labor on the ratio of constant to variable capital would seem to depend strongly on the details of the technical change.

There could, of course, be some other consideration Marx could advance here, which would say that only certain patterns of technical change are favored under capitalism, and these would imply restrictions on $\mathbf{\alpha}^{\prime}$ such that the ratio will rise. (Of course he couldn't have put it in just those terms.) But I don't see any such argument in this chapter, or elsewhere.

## Part VIII: Primitive Accumulation

### Chapter 26: The Secret of Primitive Accumulation

The official theory of "previous accumulation" through which capital arises is that the proto-capitalists are better at saving than other people are. Marx replies that the actual way primitive accumulation happened was simply cheating and theft. Many of Marx's earlier chapters were about how laborers will end up exploited even if the market proceeds exactly as it's supposed to; the following chapters are about how nothing happened the way it was supposed to.

### Chapter 30: Reaction of the Agricultural Revolution on Industry. Creation of the Home-Market for Industrial Capital

Starting around 1500 in England, the process of primitive accumulation kicks off. This had two prongs: one was appropriating rights to the land, commons, etc.; the other was actively forcing peasants off the land13. Before, the peasants owned, or at least leased for generations, their means of production, especially the land. Afterwards, their only means of supporting themselves were wage labor, beggary or crime. (Many did not support themselves, and died.) A side-effect of driving families off the land is that they became much less self-sufficient, guaranteeing a market for the output of capitalist manufacturing, even if the latter were no more productive than the old subsistence handicrafts.

Marx takes it for granted that all these processes worked together to reinforce the growth of capitalism. I do not, however, see anything in the text which makes the growth of capitalism a final cause at which these aimed. I also don't see anything which rules that out. A lot of really bad, only-nominally-materialist social and economic theorizing has ever since been resolutely on Team Final Cause, and I wish he'd been clearer about what he meant.

The life of agricultural laborers in 19th century Great Britain and Ireland was nasty, brutish, and short, but over-crowded rather than solitary.

Marx mentions merchant capital in these chapters, and finance capital, but offers no clear account of how either works; there are hints, though just hints, that both are mere trickery and cheating. But transporting goods from one place to another is something that takes labor and creates use values, and so with the modern device of "indexing" goods by location, transport at least could be brought within Marx's framework of production. The value of, e.g., American cotton in Lancashire includes the labor necessary to transport it from New Orleans to Liverpool, plus the value embodied in wear and tear on the ship. Even storage could be put in this framework, indexing goods by time. So Marx should have thought of merchant capital as (at least in part) just a special case of capital employed in production14. Explaining finance capital the same way is more than I feel up to.

### Chapter 31: Genesis of the Industrial Capitalist

Lots of rich British families got their money from things like the slave trade and the imperial exploitation of India. Some of them went on to become free-trade liberals.

### Chapter 32: Historical Tendency of Capitalist Accumulation

This is the one hopeful chapter in the book. It is also the next to last one, and just three pages. It begins by re-capitulating that capitalist private property begins by expropriating individual private property: the laborers must lose their own means of production before they can become proper proletarians, and capitalists can employ them. But this was historically necessary, if we were ever to get beyond the "mediocrity" of petty production, to develop really powerful and advanced means of production. These are means which can only be used in common, can only be social rather than individual production, and it is one of the ironies of history that they develop under the form of the individual, private property of capitalists. As capitalist accumulation goes on, as technology advances, as the market expands, the working class will become more and more miserable (along with the industrial reserve army), and production will, in its capitalist form, become more and more centralized, while becoming, in reality, more and more socialized.

The upshot is famous, but enough of a rhetorical high point to deserve quotation:

As soon as this process of transformation has sufficiently decomposed the old society from top to bottom, as soon as the labourers are turned into proletarians, their means of labour into capital, as soon as the capitalist mode of production stands on its own feet, then the further socialisation of labour and further transformation of the land and other means of production into socially exploited and, therefore, common means of production, as well as the further expropriation of private proprietors, takes a new form. That which is now to be expropriated is no longer the labourer working for himself, but the capitalist exploiting many labourers. This expropriation is accomplished by the action of the immanent laws of capitalistic production itself, by the centralisation of capital. One capitalist always kills many. Hand in hand with this centralisation, or this expropriation of many capitalists by few, develops, on an ever-extending scale, the cooperative form of the labour process, the conscious technical application of science, the methodical cultivation of the soil, the transformation of the instruments of labour into instruments of labour only usable in common, the economising of all means of production by their use as means of production of combined, socialised labour, the entanglement of all peoples in the net of the world market, and with this, the international character of the capitalistic regime. Along with the constantly diminishing number of the magnates of capital, who usurp and monopolise all advantages of this process of transformation, grows the mass of misery, oppression, slavery, degradation, exploitation; but with this too grows the revolt of the working class, a class always increasing in numbers, and disciplined, united, organised by the very mechanism of the process of capitalist production itself. The monopoly of capital becomes a fetter upon the mode of production, which has sprung up and flourished along with, and under it. Centralisation of the means of production and socialisation of labour at last reach a point where they become incompatible with their capitalist integument. This integument is burst asunder. The knell of capitalist private property sounds. The expropriators are expropriated.

(A later, lesser paragraph re-assures us that overthrowing capitalism will be much less bloody and violent than the original primitive accumulation was.)

This is one of the great apocalyptic visions of modern times; indeed, over the last two centuries, it has rivaled in influence that of St. John the Divine. It is a real pity that it does not follow at all from the preceding analysis. There is nothing in that analysis which indicates that the capitalist mode of production cannot keep reproducing itself forever. Nor is there anything in that analysis which says that if capitalism does end, the replacement will be the socialization of the means of production in the hands of the organized working class.

## Chapter 33: The Modern Theory of Colonisation

In one of the great anti-climaxes of modern intellectual history, Marx ends volume I with an extended polemic against a now-obscure political projector. The unfortunate E. G. Wakefield15 wrote a book which was (in Marx's account) all about how hard it was to get proper capitalism going in settler colonies like Australia, because there too many of the settlers owned their own means of production, rather than being property-less laborers, hence such acquisition of productive property ought to be discouraged by the state. The point Marx is trying to drive home is, of course, that capitalist private property has got nothing to do with holding on to the fruits of your own labor. Still, to go from the rhetorical heights of chapter 32 to an extended fisking of a fourth-rate policy entrepreneur (as we'd now call him) shows that Marx, whatever his other virtues as a writer, had little sense of drama.

# Some Reflections

## On matters of style

#### On "critique"

The subtitle of capital is A Critique of Political Economy. Evidently, by "critique" Marx didn't meant just subverting the concepts used by political economists; instead he tried to provide a better theory of political economy. I wish more of his successors shared this laudable ambition.

#### On Marx's modes of expression

I make no apology for having been very free in translating Marx into modern mathematical terms. If Marxism, or even Marxian economics, were a living tradition of scientific inquiry, rather than of scriptural exegesis, it would be continually refining the way it expressed its theory, in particular adopting new mathematical and logical tools, and even building new tools adapted to its own needs. The adherents of the tradition would simplify, formalize, generalize, abstract, apply, reformulate, articulate, refine, and replace. Capital would be as absurd a starting point for modern Marxian political economy as Principia Mathematica Philosophiæ Naturalis is for modern Newtonian mechanics and astronomy. Even if there were no important new discoveries (and who gets everything right the first time?), a progressive tradition would have access to all the ideas and tools of thought developed since Marx's time, and so should be able to improve on him.

It is no coincidence, comrades, that many of those who set on this road end up at not-very-Marxist destinations. (Sam Bowles and Herb Gintis, whose works have been very important for my own intellectual development, might serve as Exhibits A and A'.)

#### On the voices

Marx's voice --- his very characteristic voice, running here over a range from detached exposition through righteous indignation to acid sarcasm and lacerating contempt --- is the reader's constant companion through the whole book. He quotes, generously, from his predecessors in political economy (bourgeois though they be); he quotes, mockingly, from hapless ideological foils (sometimes other passages from those very same economists); he quotes, approvingly, from factory inspectors, public health workers, parliamentary reports; he quotes the ancients (often in the original Greek). What he hardly ever does is give voice to a laborer, and even when he does, I think it is always mediated through some middle-class social reformer or parliamentary record.

## On the labor theory of value

This was not Marx's invention; he inherited it from classical political economy, it having been embraced whole-heartedly by Smith and Ricardo. (The latter was, in many ways, Marx's template.) It has remained a shibboleth of Marxism ever since. I can detect in this book nothing resembling a really solid argument on its behalf. The closest approaches are places where "quantity of socially necessary average homogenized labor" it is just begging to be replaced by "social opportunity cost". Among other advantages, this would let labor (or labor powers) be heterogeneous, like other factors of production.

As an explanation for prices, i.e., exchange values, the theory is a failure for reasons having to do entirely with its logical structure. I think this is important, but I don't know of any way of putting it non-mathematically. I will accordingly confine it to an excursus at the end of this section.

Much of what Marx has to say about technological change, the struggle over the length of the working day, etc., does not in fact depend on the labor theory of value, and can be explained in merely monetary or opportunity-cost terms. (I have tried to indicate this above.)

Marx thought his great advance over his predecessors was to distinguish between the value of labor power, and the value added by the exercise of that power. What remains of this? That there is a cost to creating and maintaining a person's capacity to do work, to producing or re-producing labor power, is obvious. That the opportunity cost of (re-)producing a day's labor power is not the same as the opportunity cost of withdrawing the use of that labor power for a day should also be clear, at least upon reflection16. That the cost of producing labor power had better be less than the cost of going without it, or the whole society is in trouble, once again should be obvious. That, finally, this leaves a surplus, which has to be divided up somehow, is also true. None of this changes if instead of saying "labor power" we say "human capital"; knock yourself out that way if it makes you feel better.

But equally nothing has to change if in place of labor power / human capital we substitute any other factor of production which is in limited supply and has alternative uses. It could be linen or coats or arable land or potable water or microprocessors or 6mm ball bearings. Even the air, as we know only too well these days, has a limited capacity for dealing with the by-products of our productive processes and our consumption, which goes into the social opportunity cost. Indeed, the analytical Marxist John Roemer once proved that while capitalist profit is impossible without exploiting labor, it is equally impossible without exploiting every other commodity17.

It is still the case that the surplus which results from production must be allocated somehow, and why should it go to the capitalist, or even go through the capitalist? Production is positive sum; that always means there's a struggle over the surplus. This is fundamentally political, it's about the social relations that exist around production and not just technical questions about who does what.

Capitalism pre-supposes a market economy, and production for the market, "commodity production". But it has, classically, three more core features: (1) the means of production are privately owned; (2) workers who do not own the (other) means of production sell their labor power (rent out their human capital) for wages; (3) the owners of the means of production are residual claimants on the surplus product; (4) the owners of the means of production, or their agents, claim the right to direct the work process. (In the corporate form of capitalism, the means of production are owned by corporations, which are the residual claimants; shareholders do not actually own corporations.) All four are logically distinct, and I think could even be dissociated institutionally. There doesn't seem to be anything impossible about a world where the owners of means of production can hire laborers, but are taxed at 100% on any profits over the value of their time put into helping organize production. Capital goods might not fetch very high prices in such a world, and private investment might be very low, but those are different issues.

#### Excursus: embodied labor content versus exchange values

Recall back to where we worked out how much (abstract, homogeneous) labor is embodied in any given commodity. To make one unit of good $i$ uses $v_i$ hours of labor directly, and $a_{ij}$ units of good $i$. So the total labor embodied must satisfy $\mathbf{l} = \mathbf{v} + \mathbf{a}\mathbf{l}$ or $\mathbf{l} = (\mathbf{I} - \mathbf{a})^{-1} \mathbf{v}$ where $\mathbf{a}$ is a $k\times k$ matrix, $\mathbf{v}$ is the vector ($=k \times 1$ matrix) of direct labor values, and $\mathbf{l}$ is the vector of total labor values, for all of the $k$ different commodities in the economy.

For the labor theory of value / "law of value" to hold, we need the vector of exchange values $\mathbf{p}$ to either equal $\mathbf{l}$, or at least be proportional to it. But exchange values, too, need to obey an adding-up constraint. Each hour of labor earns a wage, but since hours of abstract homogeneous labor are our units for measuring value, we know that the wage is always $=1$ numerically. The workers will exchange their wages for a bundle of goods they consume to reproduce their labor-power, say $b_i$ units of good $i$ per hour, or in vector form $\mathbf{b}$. Thus producing one unit of good $i$ will require $a_{ij} + v_i b_j$ units of good $j$, once wages are exchanged for commodities. Collect this into the matrix $\mathbf{m} = \mathbf{a} + \mathbf{v}\mathbf{b}^T$. Assuming (as Marx emphatically does) a uniform rate of profit $r$ across industries, can we find a self-consistent vector of exchange values $\mathbf{p}$? It would need to satisfy $\mathbf{p} = (1+r)(\mathbf{a}\mathbf{p} + \mathbf{v})$ since, again, labor time is the unit for exchange value. But, precisely because the wage is numerically 1, we know that $1 = \mathbf{b}^T\mathbf{p}$, so we can write $\mathbf{p} = (1+r)(\mathbf{a}\mathbf{p} + \mathbf{v}\mathbf{b}^T\mathbf{p})$ or $\mathbf{p} = (1+r) (\mathbf{a} + \mathbf{v}\mathbf{b}^T)\mathbf{p} = (1+r) \mathbf{m}\mathbf{p}$ Thus the exchange values are the eigenvector of $\mathbf{m}$ with eigenvalue $\frac{1}{1+r}$. Since all the entries in $\mathbf{m}$ are non-negative, the Perron-Frobenius theorem guarantees that it has a (dominant) positive eigenvalue, with all the entries in the corresponding eigenvector being non-negative. (That last is a good sanity-check for exchange values!)

We can say a little bit more if $\mathbf{m}$ is irreducible, which is to say that tracing back the chain from any one good ultimately requires every other good (i.e., for every $i, j$, ${\left(\mathbf{m}^{n}\right)}_{ij} > 0$ for some finite $n$). Then the dominant eigenvalue $\frac{1}{1+r}$ is non-degenerate, with a unique, positive eigenvector $\mathbf{p}$, and there are no other positive eigenvectors. That last uniqueness means that there isn't any other way of assigning exchange-values to goods which satisfies the adding-up constraints imposed by technology (including the reproduction of labor power).

But now we have trouble for the labor theory of value. Remember that the vector of labor contents is $(\mathbf{I} - \mathbf{a})^{-1} \mathbf{v}$ while we've just established that the unique vector of prices is the dominant eigenvector of $\mathbf{a} + \mathbf{v} \mathbf{b}^T$ These two vectors will not, of course, agree in general. For that to happen, we'd need the vector of labor contents to also be an eigenvector, so it would have to satisfy the equation $\left(\mathbf{a} + \mathbf{v} \mathbf{b}^T - \frac{1}{1+r} \mathbf{I}\right) (\mathbf{I} - \mathbf{a})^{-1} \mathbf{v} = 0 ~ (LTV)$

Between $\mathbf{a}$, $\mathbf{v}$ and $\mathbf{b}$, there are $k^2+2k$ parameters here ($r$ doesn't count, because it's fixed by the others). This last matrix equation, (LTV), is equivalent to a system of $k$ equations those parameters must satisfy in order for the labor theory of value to hold (under all Marx's assumptions). There are more unknowns than equations, so one expects there will be some solutions. Naively, in fact, there'd be a $k^2 + k$ dimensional space of solutions. (I say "naively", because because the matrix inversion, and the presence of the eigenvalue $r$, make the equations non-linear in the parameters.) But it's certainly false that (LTV) will hold for all choices of parameters, nor will it hold for most, for any reasonable sense of "most". (Solutions are a measure-0 subset of parameter space, and, if we start from a solution, an arbitrarily small perturbation gives us a non-solution, while the reverse isn't true.) Generically, exchange values just can't be proportional to labor contents.

There are some potentially neat math-y questions here:

1. Can one give a nice characterization of the part of parameter space where (LTV) holds? What is its dimension? (Someone may have done this.)
2. Suppose instead of looking for exact solutions to (LTV), we looked for approximate ones, say ones where all exchange values were within $\pm \epsilon$ of labor contents. Would that occupy much of the parameter space, for not-too-large $\epsilon$? If so, one might award the labor theory of value a sort of consolation prize.
3. Suppose one started with an arbitrary vector of prices $\mathbf{p}^0$. Exchange at those prices would lead to supra-normal profits in some industries, and losses in others. Under what assumptions about how people update prices will the price vector converge to the unique $\mathbf{p}$? This will require somehow modeling the out-of-equilibrium dynamics. My first thought is to just have every capitalist add a fixed mark-up rate to the cost of their inputs, but that will only get us to $\mathbf{p}$ if the mark-up happens to be $r$... (Perhaps something where capitalists learn what mark-up to charge?) If it turned out that the only conditions under which $\mathbf{p}$ was a stable, rapidly-achieved equilibrium also entailed (LTV), I would be astonished and impressed.

(It will not, I think, do any good to show that $\mathbf{p}$ is an unstable equilibrium, or one which is reached only glacially slowly, because the whole point of the labor theory of value is that actual prices, which everyone admits fluctuate for all sorts of accidental and un-important reasons, are supposed to track labor contents. That means labor contents are equilibrium prices, and the only equilibrium prices are $\mathbf{p}$.)

Even I, however, will admit that these are only diversions, not important either practically or even big-picture theoretically. The assumptions of classical political economy, which Marx shares, determine both labor contents and exchange values uniquely, and those are equal only by a great coincidence. If you want to hold on to the labor theory of value, I guess you still could, but you'll need to replace those assumptions with others, and show that the whole thing works. For my part, I don't see why anyone should bother.

— To amplify a point I made above, when solving for labor-contents, the fact that Marx didn't do this pretty simple math is not to his dis-credit. (After all, none of the other classical political economists, e.g., Ricardo, did either.) This problem, like the one about labor contents, is only simple if you have modern linear-algebraic notation (and the concepts that go with it), and Leontief's idea of an input-output matrix from the 1930s. This problem also needs the Perron-Frobenius theorem, from c. 1900. No blame attaches to Marx (let alone his predecessors) for not anticipating all this in the 1850s and 1860s. Later Marx-ists have less excuse.

## "The" Cost of Production, or, Again with the Fixed Costs

It is central to Marx's value theory, and so to everything else here, that every commodity has a well-defined cost of production per unit, viz., the amount of (average, homogeneous) labor time socially necessary to replace that unit. There are two great problems here, about fixed costs and about multiple technologies. The first is an (unacknowledged) internal contradiction in Marx's economic thought; the second was simply a blind spot.

Whenever Marx tries to make his value theory rigorous, he presumes, more or less explicitly, that every unit of a commodity produced requires combining the same inputs in the same proportions, with labor time very much included as an input. With this assumption, there is indeed a well-defined value for the total amount of labor required to produce, or replace, a unit of a single commodity, as we've seen. (As we've also seen, this won't generally match exchange value, but that's another story.) But this assumption is empirically vulnerable in two ways.

The first vulnerability is that, even if we grant (as this assumes) that there is only one way of producing any given commodity at any one time, many technologies have fixed costs, say $c_0$. If, for example, you're going to produce coats in a factory with powered machinery, you need to erect the factory, build the machines, arrange a power-supply, etc., before you can produce even one coat. The marginal cost of one extra coat may be very low --- so much labor $v$, plus the labor embodied in the raw materials, the fuel, and the marginal wear and tear on the physical plant, totalling $c_1$ --- and might even be constant as we increase the number of coats produced. (At some point we'd run in to physical limits of the plant and its workers, but let's ignore that for now.) But the total cost of producing $x$ coats is going to be $c_0 + x(v+c_1)$, meaning the average cost per coat will be $v+c_1 + \frac{c_0}{x}$. So what is the cost of production, the quantity of socially-necessary labor time, for a coat? My inclination is that, because Marx is so concerned with the over-all process of production, he ought to go with the average cost, and that if someone could have expressed the point to him, he'd have said that a pro rata share of the fixed costs was included in the constant capital term $c$. But, if so, his value theory would have to be totally re-done, in a way which somehow incorporates the scale of production. If, on the other hand, he goes with marginal cost, the "socially necessary" part of "socially necessary labor time" is being let go, because those fixed costs aren't just fripperies.

And, in fact, Marx was very aware of the importance of fixed costs, and how they lead to economies of scale. As I've indicated, he has very astute things to say about them, but generally when he's considering economic change rather than value theory, so they're important in the treatment of technological development in Chapter 15, and (I think) in the account of how competition among capitalists forces them to accumulate in Chapter 24. It's just that this recognition contradicts his value theory!

The other empirical vulnerability of Marx's assumptions about production is that he assumes there is one way of producing each commodity at any one time. If we allow that multiple techniques can be economically viable at the same time, then the cost of production of is ill-defined. At the very beginning of the book, Marx gives the example of the introduction of a new power loom thereby reducing the socially-necessary labor time embodied in existing cloth. But, as I said back then, suppose society (currently) has only one such loom, and wants (or needs) more cloth than it can supply. The old looms will, then, still have to be used. But this means that there is no one cost of production for cloth. Again, it was just such a situation, where the necessary output could not all be achieved using the most efficient techniques, that led Kantorovich to linear programming.

Now, Marx might say in response that the firm(s) using the new power loom will earn above-average profits, accumulate capital in the form of more of those looms, and eventually drive the old ones out of business. Even granting this, however, we would only have well-defined costs of production in some sort of long-run equilibrium in between episodes of technological change. The relevance of this to actually existing capitalism is, to say the least, unclear.

## But what about power?

A logically distinct point that Marx runs together with extracting surplus value is that the capitalist, in hiring workers, gets to order them around --- that there is domination in the work-place, that bosses boss. This is emphatically true and not well-explained by either classical political economy or the Utopian-competition variety of neo-classical economics. After all, in most market transactions, the buyer doesn't care how the seller gets what they're selling, nor does the buyer claim the right to oversee the production of what's sold. There are resources within contemporary neo-classical economics for explaining the authority of bosses, and they suggest ways in which Marx was right about power, but not for the reasons he thought he was. This is consequential for where we go from here.

## Was it worth re-reading?

For me, yes, though my "is this book worth reading?" threshold is notoriously low. I am still happy to acknowledge Marx as one of my intellectual ancestors. (Indeed, dear reader, he was probably one of yours, whether you are aware of it or not.) But in general, though, I'd say no. Even if you really want an understanding of how the economy works which doesn't presuppose the benevolence of capitalism, why go here? Too much of the book is bound up in its least defensible parts, and too much of it is a brilliant mind wrestling with basically mathematical problems without using enough math. (In some cases because the math didn't exist yet, but still.) This has become a book of great historical importance, but of merely historical importance.

1. Specifically, the Modern Library reprint (n.d.) of the 1906 American edition prepared by Ernest Untermann. When I quote long passages, though, I have sometimes merely taken them from the online edition at marxists.org, which is evidently a slightly different translation.^

2. Commentary on the role of this axiom of choice in a truly liberatory mathematics is referred to the appropriate literature.^

3. "Skilled labour counts only as simple labour intensified, or rather, as multiplied simple labour, a given quantity of skilled labour being considered equal to a greater quantity of simple labour. A commodity may be the product of the most skilled labour, but its value, by equating it to the product of simple unskilled labour, represents a definite quantity of the latter labour alone. The different proportions in which different sorts of labour are reduced to unskilled labour as their standard, are established by a social process that goes on behind the backs of the producers, and, consequently, appears to be fixed by custom."^

4. Marx means "fetishism" in the 19th century anthropological sense, not in that of a paraphilia, though of course generations of scholars since Freud have had fun with the obvious puns.^

5. Cf. Russell's Proposed Roads to Freedom (ch. VIII, p. 129): "There would still have to be money, or something analogous to money, in any community such as we are imagining. The Anarchist plan of a free distribution of the total produce of work in equal shares does not get rid of the need for some standard of exchange value, since one man will choose to take his share in one form and another in another. When the day comes for distributing luxuries, old ladies will not want their quota of cigars, nor young men their just proportion of lap-dogs; this will make it necessary to know how many cigars are the equivalent of one lap-dog. Much the simplest way is to pay an income, as at present, and allow relative values to be adjusted according to demand. But if actual coin were paid, a man might hoard it and in time become a capitalist. To prevent this, it would be best to pay notes available only during a certain period, say one year from the date of issue. This would enable a man to save up for his annual holiday, but not to save indefinitely." --- I would add that the advantages of giving people money to spend as they wish are only enhanced if some young men would like lap-dogs and some old ladies would prefer cigars. But here I come to close to all-too-familiar ground.^

6. If $s/v = r$ is given (Marx doesn't give it a letter), then clearly $s/(v+s) = 1/(1+v/s) = 1/(1+1/r)$ is fixed. But if we know $s/(v+s) = R$, then $r = 1/(1/R - 1)$.^

7. This makes me even more unhappy about how Marx, in his chapters on value, assumes constant returns to scale and ignores the problem of fixed costs. To the extent there are costs in setting up a productive enterprise at all, regardless of how much is produced, e.g., erecting a factory building or installing an assembly line, they ought to show up in $c$, since those acts of labor were socially necessary to production. But Marx turns this all into wear-and-tear, or amortization and depreciation, which is hardly adequate. (Never mind how we allocate the fixed cost for a factory that produces multiple commodities.) I am almost tempted, rather perversely I admit, to try to re-interpret everything as referring to marginal values.^

8. What follows looks trivial, because, with modern mathematical and conceptual tools, i.e., linear-algebra notation and Leontief's idea of an input-output matrix, it is trivial. I don't know enough about the history of linear algebra to say whether the notation and mathematical concepts (or some equivalent precursors) were available in Marx's day, though I suspect they were (and, if so, would almost certainly have been expressed in German!). One can't, of course, blame Marx for not anticipate Leontief's work in the 1920s and 1930s. But an interesting question, which I am not at all competent to resolve, is whether the tables of the 18th century Physiocrats, with whom Marx was familiar, were close enough to Leontief's input-output matrices that he could have at least posed the question in its linear-algebraic form. Since I am writing a commentary on Capital and not writing alternate-history story where Marx spends decades in the British Museum compiling The Structure of the British Economy in 1849, followed by a further decade of inverting the I-O matrix by hand, I shall bring this aside to a close.^

9. I am going to deliberately use Marx's word "productiveness", rather than the modern "productivity", because the latter refers to monetary output per unit of labor time, while Marx means the output of use-values, and I think that distinction is important here.^

10. A paragraph is, I think, worth quoting in full (omitting Marx's footnotes). "Modern industry never looks upon and treats the existing form of a process as final. The technical basis of that industry is therefore revolutionary, while all earlier modes of production were essentially conservative. By means of machinery, chemical processes and other methods, it is continually causing changes not only in the technical basis of production, but also in the functions of the labourer, and in the social combinations of the labour-process. At the same time, it thereby also revolutionises the division of labour within the society, and incessantly launches masses of capital and of workpeople from one branch of production to another. But if modern industry, by its very nature, therefore necessitates variation of labour, fluency of function, universal mobility of the labourer, on the other hand, in its capitalistic form, it reproduces the old division of labour with its ossified particularisations. We have seen how this absolute contradiction between the technical necessities of modern industry, and the social character inherent in its capitalistic form, dispels all fixity and security in the situation of the labourer; how it constantly threatens, by taking away the instruments of labour, to snatch from his hands his means of subsistence, and, by suppressing his detail-function, to make him superfluous, we have seen, too, how this antagonism vents its rage in the creation of that monstrosity, an industrial reserve army, kept in misery in order to be always at the disposal of capital; in the incessant human sacrifices from among the working-class, in the most reckless squandering of labour-power and in the devastation caused by a social anarchy which turns every economic progress into a social calamity. This is the negative side. But if, on the one hand, variation of work at present imposes itself after the manner of an overpowering natural law, and with the blindly destructive action of a natural law that meets with resistance at all points, modern industry, on the other hand, through its catastrophes imposes the necessity of recognising, as a fundamental law of production, variation of work, consequently fitness of the labourer for varied work, consequently the greatest possible development of his varied aptitudes. It becomes a question of life and death for society to adapt the mode of production to the normal functioning of this law. Modern Industry, indeed, compels society, under penalty of death, to replace the detail-worker of to-day, grappled by life-long repetition of one and the same trivial operation, and thus reduced to the mere fragment of a man, by the fully developed individual, fit for a variety of labours, ready to face any change of production, and to whom the different social functions he performs, are but so many modes of giving free scope to his own natural and acquired powers."^

11. Marx never, that I can tell, squares these (entirely correct) observations about fixed costs, which imply (initially) increasing returns to scale, with his reliance on constant returns to scale in his more theoretical chapters.^

12. "In the sphere of agriculture, modern industry has a more revolutionary effect than elsewhere, for this reason, that it annihilates the peasant, that bulwark of the old society, and replaces him by the wage-labourer. Thus the desire for social changes, and the class antagonisms are brought to the same level in the country as in the towns. The irrational, old-fashioned methods of agriculture are replaced by scientific ones. Capitalist production completely tears asunder the old bond of union which held together agriculture and manufacture in their infancy. But at the same time it creates the material conditions for a higher synthesis in the future, viz., the union of agriculture and industry on the basis of the more perfected forms they have each acquired during their temporary separation. Capitalist production, by collecting the population in great centres, and causing an ever-increasing preponderance of town population, on the one hand concentrates the historical motive power of society; on the other hand, it disturbs the circulation of matter between man and the soil, i.e., prevents the return to the soil of its elements consumed by man in the form of food and clothing; it therefore violates the conditions necessary to lasting fertility of the soil. By this action it destroys at the same time the health of the town labourer and the intellectual life of the rural labourer. But while upsetting the naturally grown conditions for the maintenance of that circulation of matter, it imperiously calls for its restoration as a system, as a regulating law of social production, and under a form appropriate to the full development of the human race. In agriculture as in manufacture, the transformation of production under the sway of capital, means, at the same time, the martyrdom of the producer; the instrument of labour becomes the means of enslaving, exploiting, and impoverishing the labourer; the social combination and organisation of labour-processes is turned into an organised mode of crushing out the workman's individual vitality, freedom, and independence. The dispersion of the rural labourers over larger areas breaks their power of resistance while concentration increases that of the town operatives. In modern agriculture, as in the urban industries, the increased productiveness and quantity of the labour set in motion are bought at the cost of laying waste and consuming by disease labour-power itself. Moreover, all progress in capitalistic agriculture is a progress in the art, not only of robbing the labourer, but of robbing the soil; all progress in increasing the fertility of the soil for a given time, is a progress towards ruining the lasting sources of that fertility. The more a country starts its development on the foundation of modern industry, like the United States, for example, the more rapid is this process of destruction. Capitalist production, therefore, develops technology, and the combining together of various processes into a social whole, only by sapping the original sources of all wealth --- the soil and the labourer."^

13. Note 12 in chapter 27 mocks William of Orange for awarding lands to a lady whose "endearing offices are supposed to have been --- foeda labiorum ministeria". Since royal land grants have usually recognized success in wholesale slaughter, using them to reward excellence in oral sex seems like a real advance for civilization and decency.^

14. And, indeed, it's been pointed out to me that he basically does this, in Volume II (ch. 6, sec. 3).^

15. Wakefield turns out to have been a remarkably colorful character, but that doesn't improve the quality of his ideas.^

16. That the opportunity cost of one less day's labor power is also the worth of expanding the available labor power by one day is less obvious. However, it drops out of basic optimization. Lagrange multipliers are shadow prices, and give the value, here in terms of opportunity costs, of an incremental loosening of the constraint. When the constraint is that there is only so much labor power available, the claim follows.^

17. You can find a very simple proof of the same proposition, along with some other interesting reflections on Marx's labor theory of value, in R. P. Wolff, "A Critique and Reinterpretation of Marx's Labor Theory of Value", Philosophy and Public Affairs 10 (1981): 89--120 [JSTOR].^

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