Review of Farjoun & Machover’s Laws of Chaos (1983)

Emmanuel Farjoun & Moshé Machover’s Laws of Chaos: A Probabilistic Approach to Political Economy (1983) (henceforth LoC)^[FM20] was republished in 2020; this republished version is reviewed below.

In this review, we learn exactly how LoC (& much of the Anglophone literature in general) fundamentally misunderstands categories of Karl Marx’s Capital^[1] such as price, production‐price, value, capital, labour‐power, etc.; learn a bit about these categories themselves; locate the particular source of the neoricardians’ uncomprehension; and ultimately indicate which aspects of LoC can nonetheless be salvaged.

Readers of LoC interested in the critique thereof may skip ahead to the section on ch. III. Those solely interested in the heart of the matter may skip ahead to the crux of ch. VI. Many (but not all) findings are briefly summarised in the conclusion.

The basic thesis of LoC is that the particular quantities that capitalist competition unconsciously produces — prices, profit‐rates, etc. — cannot be understood, even in principle, as being mutually strictly convergent upon a single “ideal” point. Instead, the facts that two commodities of the same type can be concurrently sold at distinct prices, that two firms can concurrently have distinct profit‐rates, etc. are entirely necessary — again, even purely theoretically — for the operation of competition itself; we therefore have no choice but to model any such quantity at equilibrium as a random variable, instead of as a number.

When, for instance, profit‐rates are at equilibrium, the profit‐rates aren’t all equal; instead, they conform to a particular distribution that enables competition to continue. Assuming that this equilibrium lasts, each individual firm at various times finds itself at various locations within the distribution, all without disturbing the equilibrium itself, because these movements balance one another out.

This chapter makes four essentially mathematical/empiric observations:

• That “a mathematical relation that holds amongst variable quantities does not, in general, hold between their respective averages”.^[2] In particular, the property that we’re tacitly assuming when we assume that the “mathematical relation” also applies to averages is: linearity.

• That the stability of an equilibrium cannot be taken for granted (i.e. many equilibria are actually unstable).

• That a capitalist system under “perfect competition” — even purely in principle — must cause its own profit‐rates to spread out, because many competitive tactics (internal influences, in this context) push the profit‐rate away from the average: temporarily selling cheaply so as to drive out competitors, introducing new techniques, etc.. A perfectly uniform profit‐rate therefore cannot be an equilibrium at all, because it’s disturbed even in the absence of external influences.

• And of course, that real‐life profit‐rates are never even close to uniform anyway.

These points are well‐taken, and correctly warn the reader against simplified, mathematised readings of Capital.

However, LoC was written by mathematicians, so they interpret the fact that Capital manages to go without introducing any explicitly mathematical nonlinearity into its complex theory (it’s all linear combinations, arithmetic & weighted means, and simple ratios thereof)^[3] to mean that Capital’s theory really is both a purely mathematical one, and also naïvely so. This will have various implications, such as LoC’s use of input–output matrices; but the deeper implications will be addressed below.

This chapter puts forth the classic ideal gas model as a vivid analogy for firms (≘ gas molecules) competing (≘ elastically colliding with one another) for profit‐rates (≘ speeds).

The problem with taking all profit‐rates to be equal, then, is apparently that it violates the second law of thermodynamics. The caveät is that this ends up explaining both too much & too little. It’s unclear why, in particular, the molecules in a gas have anything to do with capitalist firms, except that we’ve defined a correspondence between speed & profit‐rate. We’re forced to instead give the less particular explanation that the real world of competition clearly consists of innumerable actors & their interactions, and thus the corresponding microstates cannot be specified in detail. Therefore, a statistical‐mechanical approach is appropriate, and this brings with it thermodynamic concepts like entropy.

However, entropy doesn’t really mean anything in particular. In this classical statistical‐mechanical context specifically, entropy is only well‐defined for a system that’s already at equilibrium. Worse still, translating this concept onto capitalist competition poses further conceptual difficulties.

If we take the ideal gas analogy as more‐or‐less valid, then LoC’s key insight is that any state in which all gas molecules have the same speed will be immediately scrambled by the molecules continuing to collide and thus to exchange parts of their momenta. Interestingly, LoC never considers the reverse situation, which is prima facie at least as compelling: if equilibrium always manifests as all molecules having equal speeds, then how is this state arrived at in the first place? It would seem to be a massive coïncidence.

But framing the issue in this way would draw attention to the deep limitations of this statistical‐mechanical paradigm. In particular, this paradigm only functions conceptually (to say nothing of empirics) at equilibrium. This may be enough to satisfy the economists, but obscures the more crucial problems in this context: how can equilibrium be established, and how is it disturbed to produce crisis? Ch. VII at least begins to address these more dynamic questions.

This is tied to the fact that the statistical‐mechanical analogy is too abstract. Whereas classical statistical mechanics at least had classical mechanics upon which to draw for an underlying explanation, LoC never provides more than hints as to how to understand what it concretely means for two gas molecules (≘ firms) to collide (≘ compete).^[4] Granted, LoC admits passim that it’s offered only as a preliminary sketch.

With respect to crisis, one hint — albeit a still overly abstract one — is already given in ch. II: LoC makes use of an ergodic hypothesis. It’s not at all clear that a real capitalist system is ergodic in this way, and indeed, informal arguments can be made against ergodicity.

LoC’s working hypothesis is that the profit‐rates of the various firms in the economy (each firm weighted by its capital advanced) at equilibrium conform to a particular gamma distribution, because this distribution maximises entropy in the ideal gas analogy.^[5] This hypothesis is at least enticing, if only for its mathematical specificity.

This is followed by the introduction of a new random variable:

$Z$ [is] the rate of wage‐bill or the rate of labour‐costs.^[2] It has the following meaning. For each $i$, the positive number $Z\left(i\right)$ at time $t$ equals the current annual wage‐bill (or labour costs) of the $i$‐th firm, divided by its capital $K\left(i\right)$. […]

Note that the reciprocal of $Z$ (that is, $1∕Z$) is similar to what Marx calls organic composition, the only difference being that he measures both labour costs and invested capital in terms of their labour‐value, whereas we measure them in money terms.^[6]

Perplexingly, this equation of $1∕Z$ with organic composition implies that $K$ — LoC’s definition of “capital” — coïncides not with the price‐magnitude of Marx’s notion of capital,^[7] but rather with that of Marx’s constant capital. This reveals shortcomings of the definitions of LoC’s random variables, as well as numerous conceptual misunderstandings that would be remedied by reading Capital:

• When the capitalist advances capital in the hopes of turning a profit, her dream will evidently not be fulfilled if that capital cannot pay the wages of the necessary workers. The variable capital must therefore be part of the capital, advanced^[8] just like any other part.

• My best guess as to why $K$ is understood in this way is that many mainstream economists use this terminology: capital meaning “material elements of constant capital”. This vulgarisation^[9] stems from the fetishistic notions that capital is a thing with occult powers of profit‐generation, that it’s responsible for advancements in the social productivity of labour, and that it directly commands living labour‐power.

  Within the framework of capitalist production this ability of objectified labour to transform itself into capital, i.e. to transform the means of production into the means of controlling and exploiting living labour, appears as something utterly appropriate to them […], as inseparable from them and hence as a quality attributable to them as things, as use‐values, as means of production. These appear, therefore, intrinsically as capital and hence as capital which expresses a specific relationship of production, a specific social relationship in which the owners of the conditions of production treat living labour‐power as a thing, just as value had appeared to be the attribute of a thing and the economic definition of the thing as a commodity appeared to be an aspect of its thinghood^[2]

• LoC’s definition of “rate of profit” is quite a bit different from the usual one. We’d expect that, ceteris paribus, the profit‐rate of a firm would decrease when its wage‐bill (or rather, variable capital) grew, assuming that the rate of surplus‐value decreased such that the mass of surplus‐value remained unchanged. This is because the firm’s “costs” (or rather, capital) has enlarged, while the absolute amount of surplus‐value that it produces has stayed the same. On LoC’s definition, however, the profit‐rate would be unaffected.

  On an assumption that the rate of surplus‐value is an immutable global constant, this definitional discrepancy has no mathematical implications. Even in this case, however, the conceptual discrepancy very much remains. Moreover, such an assumption is clearly nonsense within LoC’s explicitly nondeterministic framework.

• Bizarrely, as unlikely as it might be, it’s possible under LoC’s definitions for a firm to advance zero capital — and therefore to not have a well‐defined profit‐rate at all. All that’s needed is negligible constant capital, which can be achieved by a very low‐tech (read: virtually no meaningful tools/machinery required) extractive industry paired with landownership that gives the capitalist–landowner exclusive access to the natural resource to be extracted. A hypothetic scenario, to be sure, but still worth mentioning — especially considering LoC’s almost exclusive focus on mathematics.

In any case, in addition to the profit‐rates (represented by the random variable $R$) being gamma‐distributed, there’s now a desire to give $Z$ a distribution as well:

The same heuristic argument which has led us to the hypothesis that $R$ has a gamma distribution can be applied, mutatis mutandis, also to $Z$. Here, too, the hypothesis that $Z$ has a gamma distribution tends to be confirmed by empirical evidence. […]

For our own purpose it will be sufficient to assume tentatively, as a working hypothesis, that $R$ and $Z$ have gamma distributions $𝔊\left(\alpha ,\beta \right)$ and $𝔊\left(\mathrm{𝛼\prime },\beta \right)$ respectively, with the same second parameter $\beta$.

As discussed above, the ideal gas heuristic used for $R$ is already questionable, even if enticing. This “mutatis mutandis”, then, is doing a bit too much heavy lifting, considering that the necessary changes are — to say the least — nonobvious. What does it mean for two capitals’ organic compositions to “collide”? LoC gives the beginnings of an explanation, but only later on, in ch. VII.

Giving both gamma distributions the same second parameter $\beta$ is motivated by a bit of black mathemagic that uses this assumption to show that the expected rate of surplus‐value would be 100%, which seems to accord startlingly well with the empirics of advanced capitalist countries over long periods of time. An even more enticing hypothesis! Yet, LoC admits that there’s no apparent reason why this assumption would be satisfied. I certainly have no idea what the conceptual content of this mathematical assumption might be, and it seems the authors of LoC are similarly at a loss. I guess capitalism is just unanalysable!

Of course, the random variable approach lends itself at least as well to prices as it does to profit‐rates. For this purpose, LoC wants to separate labour‐power from the other commodities: it’s not produced by capitalist firms for profit, and so its price^[10] must be determined at least somewhat differently. Unfortunately, LoC makes this separation in a way that does considerably more harm than good:

However, in order to attribute a definite labour‐content [read: value] to a unit of labour‐power, we would have to assume the existence of a “standard real‐wage basket”, consisting of definite amounts of each type of consumer good, per each unit of labour‐power. This assumption seems quite doubtful, since in reality consumption patterns can vary widely, both quantitatively and qualitatively, within the working population of a country at a given time.

To be sure, consumption patterns do vary widely. But to conclude from this that labour‐power cannot have a value is, at best, fallacious. The value of labour‐power is determined just like that of any other commodity: by the socially‐necessary abstract labour time necessary to reproduce it. Indeed, LoC all but admits this in the preceding paragraph: “the price [labour‐power] fetches (wages)^[10] must normally just about cover the costs of its reproduction”.

The self‐furnished obstacle over which LoC has here stumbled is a partial collapsing of the use‐value vs. exchange‐value distinction,^[11] in turn due to their use of neoricardian input–output matrices. The exchange‐value of a commodity is in no way determined by the use‐values commanded by its production‐process;^[12] rather, the production‐process per se is (in general) not entirely observable, and exchange‐value is a social relation (more concretely: a social fact about the division of labour within society) that supervenes on the constellation of exchange moments (whence the exchange‐ in exchange‐value) in which the commodity‐type participates.^[13] This point will be dealt with in more detail below. There’s therefore no contradiction whatever between the value of labour‐power & the absence — or simply, nonsense — of a so‐called “standard real‐wage basket”.

Indeed, in this context, labour‐power must be a commodity (with, therefore, its associated value), because this is an historically‐defining feature of capitalism, without which the latter is inconceivable. Having disintegrated this conceptual apparatus, LoC can no longer tell the difference between capitalism & other socioëconomic formations.^[14] If the commodification of labour‐power seems to present a conceptual difficulty (one already resolved by Vol. 1 of Capital), then we should note that capitalism has no obligation to make itself conceptually simplistic enough to be amenable to the humble mathematicians who piously wish for nothing more than to make a couple of matrices out of it.

All this might seem like an abstruse philosophic objection, but the consequences of this miscomprehension are dire. By defining the wage random variable $W$ in terms of the “average unit wage” (a.u.w.) so that $𝖤\left[W\right]\stackrel{\scriptscriptstyle\mathrm{def}}{=}1$, the insights enabled by the concept value of labour‐power are obliterated:

• As mentioned above, capitalism becomes indistinguishable from other socioëconomic formations. (This is, evidently, the fundamental problem responsible for many of the problems listed below.)

• LoC’s model is powerless to suggest even the most basic explanations already offered by Marx — & even by some of his predecessors. For instance, a cheapening of corn lowers the value of labour‐power, thus producing what Marx calls “relative surplus‐value”. Not only is LoC oblivious to this sort of knowledge, but it cannot even begin to pose the relevant questions. In this example, the real domination of labour becomes meaningless.

• Any trace of workers’ action, the necessary minimum rate of accumulation — or really, anything — and these processes’ role in shifting the rate of surplus‐value, social standards of living, etc. is effaced.

• LoC suggests that testing for multimodality in $W$’s empiric distribution could potentially be useful for finding wage gaps along e.g. racial or gendered lines. But even putting aside that sociologists already have considerably less crude methods at their disposal, what this really amounts to is replacing the value theory with a very blunt empiric instrument that conspicuously lacks an underlying theory upon which to rest.

  For example, if a worker is paid some (arbitrary) wage below the a.u.w., is her labour‐power priced below its value? — is she able to reproduce her labour‐power in its normal, uncrippled quality? To what degree might she be implicated in the generation of superprofits via some sort of superexploitation? In what specific ways might her labour‐power be devalued or revalued as a result of racial, gendered, national, etc. structures? How is her actual financial position affected by work that’s not both stable & full‐time: part‐time, gig, etc. work? And so on…

• Where does the reserve army of labour fit into the $W$ picture? Not only do workers restricted to unemployment or underemployment (part‐time, subminimum‐wage, gig, etc. work) get paid wages in many cases, but the pressure that they exert on the market (& their partial displacement of some jobs) presumably has an effect on the wages of the fully‐employed. $W$ misrepresents or doesn’t represent the former case, and gives no insight into the latter.

• Taking wage‐prices at face value leaves LoC with no way to analyse labour‐power by skilledness, i.e. by cost of producing &/or reproducing it. This point is revisited in more detail below.

• Because $W$ is essentially purely empiric (again, “face value” only), it’s unclear how it might relate to social wages; so LoC simply explicitly disregards the socialised portions of wages.

• ⋮

The existence of commodity capital outside the factory or workshop compels LoC to incorporate this into its neoricardian framework:

we assume that each commodity is produced by the firm that sells it (or, more precisely, by the workers of that firm). At first sight this seems incorrect; for example, a supermarket sells sugar but does not produce it. However, economically speaking one must regard sugar‐at‐the‐supermarket, sugar‐at‐the‐wholesaler’s and sugar‐at‐the‐sugar‐mill as three different types of commodity. The wholesaler produces sugar‐at‐the‐wholesaler’s from inputs that include sugar‐at‐the‐sugar‐mill, as well as labour, etc.. Similarly, the supermarket produces sugar‐at‐the‐supermarket from sugar‐at‐the‐wholesaler’s and other inputs (including the labour of shop assistants etc.). Thus merchandizing is subsumed under production, and the distribution industry is regarded as part of industry.

Certainly, some of these costs are indeed productive ones: transport,^[15] refrigeration, chemical treatments, etc.. These involve the productive incorporation of labour (living & dead) inasmuch as they’re directly necessary for the desired commodity‐type to be realised in its normal quality (i.e. its normal use‐value). But we might wonder why Marx is so adamant throughout both Vols. 2 & 3 (& to a lesser extent, 1) about the costs of circulation sensu stricto (≝ pure circulation costs) being unproductive & not adding any value to the commodity (see also: Vol. 2, part i, ch. 6):

So that he can classify commercial capital as production capital, Ramsay confuses it with the transport industry and calls commerce “the transport of commodities from one place to another” (An Essay on the Distribution of Wealth, p. 19). […] The use‐value of a commodity is greater in the hands of the consumer than it is in those of the producer, because it is only here that it is at all realized. […] But a commodity is not paid for twice over, first for its exchange‐value and then for its use‐value as something extra. By paying its exchange‐value, I appropriate its use‐value. And the exchange‐value does not increase in the slightest by the fact that the commodity passes from the hands of the producer or the middleman into those of the consumer.^[2]

Commercial capital thus creates neither value nor surplus‐value, at least not directly. In so far as it contributes towards shortening the circulation time, it can indirectly help the industrial capitalist to increase the surplus‐value he produces. In so far as it helps to extend the market and facilitates the division of labour between capitals, thus enabling capital to operate on a bigger scale, its functioning promotes the productivity of industrial capital and its accumulation. In so far as it cuts down the turnover time, it increases the ratio of surplus‐value to the capital advanced, i.e. the rate of profit. And in so far as a smaller part of capital is confined to the circulation sphere as money capital, it increases the portion of capital directly applied in production.

What allows Marx to make this important distinction is his historical perspective. One of the many historically‐contingent aspects inseparable from capitalism — specifically, in its capacity as the absolute generalisation of commodity‐production — is that it incurs many faux frais that do not (or rather, would not) exist for socioëconomic formations in which commodity‐production plays a small or nonexistent role.

By transmogrifying the entire demesne of the commodity into one giant factory — or rather, a giant matrix of horrible numbers in a factorylike trenchcoat — LoC obscures the specificity of the capitalist mode of production, while also moving closer to the vulgar equation (filtered through random variables or not) of price with value.

This chapter is dedicated to the real motivation for LoC’s existence. We can now get to the heart of why, for all its clever tricks, LoC seems to be missing something at its core.

The so‐called “transformation problem” concerns how values are “transformed” into production‐prices in Marx’s theory, as given in Capital (especially Vols. 3 & 4^[16]). The authors of LoC therefore make it clear that they wish to adhere strictly to Marx’s exposition:

5. Marx’s own explanations concerning these modifications (in Part II of the third volume of Capital) are not formulated in a mathematically precise way, and are therefore in need of interpretation. The explanation given by us is what we consider to be a fair and reasonable rendering, in our own words, of what Marx had in mind.

Unfortunately, we’re soon made aware of the relativity of “fair and reasonable”:

9. For each $i\ne 0$, one gets an equation for the unit price of ${𝐂}_i$: this price is equal to the total price of the inputs used up per unit of output (given by the coëfficients $a_j^i$), plus $r$ times the price of the invested capital employed during one week per unit of output (given by the coëfficients $f_j^i$).^[17]

We’ll get back to the conceptual confusion later. For now, we merely observe that LoC has made an egregious mathematical error that Marx, explicitly & at length, warns the reader against making, quite early on in his first introduction of production‐prices:

This seems contradicted by the fact that the elements of productive capital are generally bought on the market in capitalist production, so that their prices include an already realized profit and accordingly include the production price of one branch of industry together with the profit contained in it, so that the profit in one branch of industry goes into the cost price of another. But if the sum of the cost prices of all commodities in a country is put on one side and the sum of the profits or surplus‐values on the other, we can see that the calculation comes out right. […] Considering the calculation as a whole, to the same extent that the profits of one sphere of production go into the cost price of another, to that extent these profits have already been taken into account for the overall price of the final end‐product and cannot appear on the profit side twice. They appear on this side only because the commodity in question was itself an end‐product, so that its price of production does not go into the cost price of another commodity.

Again, Marx argues against LoC’s approach in several ways here, but we begin by focusing on the highlighted statement. LoC speaks of “the total price of the inputs used up” and “the price of the invested capital employed”, thus including the profits contained within the inputs’ prices in the calculation of the output’s price (including the profit portion thereof).

That this is a form of double‐counting (or triple‐counting, etc.) is clear to Marx due to his holistic reasoning that “if the sum of the cost prices of all commodities in a country is put on one side and the sum of the profits or surplus‐values on the other, we can see that the calculation comes out right”.

Another direction of approach is to observe that LoC’s method results in spurious profits. Consider a firm $\mathrm{S}$ (the “seller”) whose commodity outputs are wholly purchased & used up as the sole material component of constant capital for another firm $\mathrm{B}$ (the “buyer”). For simplicity, we assume that no fixed capital is involved, that all turnover times are extensionally equal to the time‐period in question, etc.. Then the situation might look like this, supposing a general profit‐rate of 50%:

Now consider instead a firm $\mathrm{M}$, the merger of $\mathrm{S}$ and $\mathrm{B}$. Because $\mathrm{M}$ doesn’t pay its own profits out of its own pockets, $\mathrm{S}$’s ${50}_{\mathrm{profit}}$ conspicuously disappears:

The effect is rather drastic here, partly due to the unrealistically high profit‐rate of 50%, but the effect is evidently significant in any case. More specifically, it would seem that the mass of profit increases exponentially with the number of discrete capitals through which their material components pass (i.e. the “length” of the branch of capital)!

Marx continues with this same point, showing how the double‐counting can be eliminated at the cost‐price level (<dfn>…</dfn> tags added by me):

If a certain sum $p$ goes into the cost price of a commodity for the profit of the producers of the means of production, and on this cost price a profit of $p_1$ is added, the total profit $P=p+p_1$. The total cost price of the commodity, discounting all portions of the price that count towards profit, is then its own cost price minus $P$. Using the symbol $k$ again for this cost price, it is evident that $k+P=k+p+p_1$.

LoC, too, concludes that something is mathematically wrong with their own interpretation. But the humour lies in that they need to bring linear algebra & matrices into the picture to arrive at this conclusion. I’ve my doubts that Marx knew quite what a determinant was, and yet he still managed to nip this arithmetic misstep in the bud, simply by conceptually understanding what he was actually doing.

We should, at least in passing, note the hyperfocus on industrial workplaces to the exclusion of all else. Much of Marx’s aim in introducing the general profit‐rate & production‐prices in the first place is so that merchant’s capital (commercial capital & money‐dealing capital), interest‐bearing capital (banking capital), & rent can be subsumed under the theory. This is a serious problem for LoC, given that merchant’s capital & rent both affect how profit‐rates are equalised and, more fundamentally, how surplus‐value is distributed amongst capitals.

But the mathematical uncomprehension observed in the “Double‐counting” section above is really an indicator of a deeper, far more fundamental uncomprehension.

In excavating this fatal error, I’ll attempt to rephrase the basics of Marx’s theory & the mistakes made by LoC in Anglophone‐friendly language.^[18] For better or worse, that language isn’t English so much as it is a narrow subset thereof. I’ll also sprinkle in some more Marx‐like language here & there, so that the reader might get some “comprehensible input”, as it’s called in language‐learning jargon.

Of course the method of presentation must differ in form from that of inquiry. The latter has to appropriate the material in detail, to analyse its different forms of development, to trace out their inner connexion. Only after this work is done, can the actual movement be adequately described. If this is done successfully, if the life of the subject‐matter is ideally reflected as in a mirror, then it may appear as if we had before us a mere a priori construction.

In the quotation that begins the “Double‐counting” section above — & indeed, throughout this topic generally — LoC consistently speaks of “price” in terms of other “prices”. Simply put, this is a category mistake. We need to maintain a tripartite distinction:^[19] value vs. production‐price vs. price. More to the point, these three categories exist as separate layers in an apparatus of abstraction; they cannot be arbitrarily mixed, because they each serve a unique purpose.

Production‐price serves as a link connecting value with price. This abolishes neither value nor price in the process, as all three categories must be concurrent for the abstraction to be coherent:

• At the value level, one portion is attributable to dead labour (which will, for sociohistoric reasons, come to exist almost exclusively in the form of constant capital — but we’re not there yet), and the rest is attributable to freshly‐added living labour.

• At the production‐price level, the notion of capital becomes necessary.^[20] Here, the line is no longer drawn between dead & living labour; instead, it’s drawn between the value “advanced” qua capital (this value is the cost‐price in this context) & the profit that accrues to the capital.

  By redrawing the line, the specific nature of value (which we’ll get to later) is prima facie obliterated; what was surplus‐value at the value level is now mere profit at the production‐price level. This surplus‐value qua profit is allotted to an individual capital in proportion to its share in the total social capital; for example, a capital making up 5% of the total capital captures 5% of the entire society’s surplus‐value as its profits.

  Note that this definition relies on the existence of separate capitals; thus the categories production‐price & profit cease to be meaningful at the level of the whole society, because they become coëxtensive with value & surplus‐value, respectively.

• The price level is the phenomenal level; that is, it’s superficially observable. Prices are responsible for concretely mediating the categories that exist at more abstract levels, because prices are what confront the consciousnesses of everyday individual persons.

  However, this mediating category gives absolutely no direct insight into the underlying substrates of value etc. that govern it but are simultaneously mediated by it.^[21] This is because capitalism proceeds anarchically and through ceaseless accidental fluctuations (as in LoC’s ideal gas analogy), wherein prices can be entirely arbitrarily assigned. Even totally value‐free things & ideas can have prices placed on them — and, in many cases, paid.^[22]

To get from value to production‐price, two conceptual leaps had to be made:

• We needed a notion of capital.^[20]
• Capital had to be split up into multiple discrete capitals that enter into competition^[23] with one another.

It follows from the above that if capital (to anticipate this point once again) is indeed value in the process of self‐augmentation, then production‐price must simply be value in a reorganised form.^[24] The value level already conceptually exists — it’s the level at which the notion of capital makes any sense — and then the action of competition merely redistributes that value, thus justifying the additional category production‐price.

We can already see that LoC’s collapse of this abstraction hierarchy is incoherent: treating production‐prices (or worse, prices) as if they entered into the calculation of production‐price implies that the value level — including capital itself — conceptually doesn’t exist, or at least that it has no particular significance.^[25] This is, however, consistent with LoC’s treatment of the random variable $K$ (“Kapital”) as criticised in “The magnitude of ‘capital’” above.

In more Anglophone‐friendly terms, the category error being committed by LoC here is somewhat analogous to a failure of quasiquotation. For example, if I’ve defined a way to write natural numbers — say, by our familiar system of Hindu–Arabic numerals, or by Roman numerals, or… — then I might want to extend it to the integers by introducing the minus sign. I might attempt this by declaring: If n is a written number, then the string “−n” is also a written number.^[26] Because I failed to use quasiquotation properly, it would seem that — interpreting the declaration a bit too literally — the string “−n” per se is itself a writing of a particular number, where we interpret n as no more than the lowercase Latin letter n.

Similarly, for LoC: a value is always already a production‐price. This, in turn, tends — at least, typically & on average — to be quantitatively the same as the price (at the moment of sale). Furthermore, to be quantitatively the same is just to be the same. In the analogy, everything is just a string; for LoC, everything is just a price. Abstraction is thus collapsed, and much like writing negative integers becomes impossible in the analogy, defining any of Marx’s categories becomes impossible too.

It might be argued that this is too many levels of abstraction; and more to the point, that at least some of these levels seem to be worryingly unobservable. But, if anything, the authors of LoC should know better than anyone else, given their background in physics, that theoretic abstractions that aren’t directly observable are the cornerstones of the sciences.

Another analogy is already given by LoC: it doesn’t make sense to assign a temperature to an individual particle. Kinetic energy is not the same as temperature, and to collapse this abstraction (presumably because individual particles are worryingly lacking in direct observability) is to regress to caloric theory — analogously: to regress to Ricardo. Moreover, we have to start adding more unobservable abstractions very quickly as soon as we want to take into account real gases, chemical reactions, high energies, etc., etc..

This is not to say that Marx’s exact method and his exact categories & terminology are the only correct approach possible. But the way to improve upon a theory, regardless of whether or not that theory is ultimately discarded, is not to regress to the dead theories that it was destined to lay to rest.

But I digress. We now turn to answering some common questions or objections, and to filling in some of the substance missing from the above exposition.

For it is characteristic of “scientific” thought to accept the immediate fact as the real object of knowledge without perceiving and conceiving the mediation that underlies it. It is on the basis of such gnoseology that in capitalist society social appearance becomes reality and vice‐versa.

The most prosaïc reality of price is, to be sure, lost on noöne — least of all Marx. Yet Marx also repeatedly insists that competition has a distorting effect on the capitalist’s ability to perceive economic phenomena; for example: “everything is expressed upside down in competition, and hence in the consciousness of its agents” (Vol. 3, part iii, ch. 13).

If we adopt the standpoint of the capitalist anyway, then we regress to Adam Smith, whose lapses into the habit of taking appearances at face value prevented him from working out the implications of his (what Marx calls “esoteric”) theories. If we jump immediately to the phenomenal level without investigating what might underlie it, science is thereby short‐circuited; the result is either incoherent, or an overly verbose articulation of common sense.

As mentioned above, it is still true that price must be the “concrete mediator” of other categories like e.g. value, for exactly the reason that it’s the level at which the economy’s agents operate. But what exactly is it “mediating”?

[E]ven if there were no chapter on “value” at all in my book, the analysis I give of the real relations would contain the proof and demonstration of the real value relation. The chatter about the need to prove the concept of value arises only from complete ignorance both of the subject under discussion and of the method of science. Every child knows that any nation that stopped working, not for a year, but let us say, just for a few weeks, would perish. And every child knows, too, that the amounts of products corresponding to the differing amounts of needs demand differing and quantitatively determined amounts of society’s aggregate labour. It is self‐evident that this necessity of the distribution of social labour in specific proportions is certainly not abolished by the specific form of social production; it can only change its form of manifestation. Natural laws cannot be abolished at all. The only thing that can change, under historically differing conditions, is the form in which those laws assert themselves. And the form in which this proportional distribution of labour asserts itself in a state of society in which the interconnection of social labour expresses itself as the private exchange of the individual products of labour, is precisely the exchange value of these products.

Where science comes in is to show how the law of value asserts itself. So, if one wanted to “explain” from the outset all phenomena that apparently contradict the law, one would have to provide the science before the science. It is precisely Ricardo’s mistake that in his first chapter, on value, all sorts of categories that still have to be arrived at are assumed as given, in order to prove their harmony with the law of value.^[2]

The problematic is clear: the society has to allocate its labour‐time somehow. Because the commodity — by definition — is produced not for its direct usefulness (i.e. use‐value) but for its ability to sell (i.e. exchange‐value) for money with which other commodities can in turn be purchased, a society whose production is virtually entirely commodity production (i.e. capitalism) cannot allocate its labour‐time unless something within each exchange moment contributes to this allocation. Without this “something”, commodity production can never gain a nonnegligible foothold.

This is the basic topic of part i (and, as we’ll see, part ii) of Vol. 1 of Capital, so you should read that. Nonetheless, some have evidently had trouble reading it, so I’ll attempt to summarise the bits presently relevant.

For commodity production of any kind to proceed, each commodity must be exchangeable for (roughly) an equivalent. This implies that there’s some common factor even between commodities with incommensurable use‐values. If we (rather suggestively) call this deepest economic substrate the commodity’s value, then this “value” is what makes exchange possible, and is therefore also the aforementioned “something” having to do with labour‐time that allows commodity production to displace other modes of production without subsequently collapsing within the week.

Exchange‐value is the form of value that supervenes on the web of exchange‐relations (hence the name) that connects the various otherwise‐isolated commodity‐producers. Because value’s raison d’être is to allow for exchange, exchange‐value is the only adequate form of value. We can therefore usually ignore the difference between value & exchange‐value.

But barter is inadequate to the self‐actualisation of commodity production. This is not just for the obvious practical reasons, but more importantly, because the exchange of commodities is not the exchange of things (nor of their associated use‐values), but rather, the movement of exchange‐value; it is the allocation of (at least some of) society’s labour‐time. The commodity producer — under capitalism, or not — sells to anyone willing & able to pay, irrespective of whether the buyer has anything of personal use to her to offer in return. (And, once again anticipating this point: accumulation — a necessary process for capital’s sustenance — cannot be realised through barter.) Money must therefore be precipitated out as the most suitable form of exchange‐value; it is exchange‐value qua exchange‐value. Money is initially(!) only possible because a particular commodity (Marx uses gold purely as a representative example^[27]) can become a central commodity against which all others are valuewise compared.^[28]

This brings us to capital, where the impossibility of mere barter becomes absolute, because capital is value made autonomous (this is the point that absolutely must be understood). This implies that capital is a social relation, not a collection of commodities, nor a thing. From the perspective of capital, value is pure quantity; therefore, the only thing that it can “do” is change its own quantity — capital is value in the process of self‐augmentation. Noncapitalist commodity‐production occasionally has capital (e.g. in usury), but capitalism is special in this context because it generalises commodity production so thoroughly that labour‐power is commodified, thus yielding a new (& ultimately, dominant) sort of capital that exploits this commodification.

Although the irreconcilable contrast between Marx’s & LoC’s notions of capital is clear enough already, it’s made sharper by how Marx uses his notion to criticise other analysts. When at a certain point in a crisis, there’s said to be a “lack of capital”, but Marx points out that this isn’t true:^[29] in fact there is, on the contrary, an overabundance of capital, and it just happens to be stuck^[30] in circulation — lying unsold at a certain point within capital’s cyclic process of production‐&‐circulation. The equivocation of so‐called “capital” with loan capital, means of production, etc. by these analysts causes their theories of crisis to be incoherent. LoC’s (& all ricardians’) equivocation lies in its defining capital inertly: as hoarded, dead labour, some of which just so happens to be “exchanged against” living labour.^[31] The specific nature of capital is thereby effaced, and equivocating this inert notion of “capital” with Marx’s notion causes LoC to erroneously conclude that a “transformation problem” exists.

At any rate, we appear to have a chicken‐or‐the‐egg problem. On the one hand, value seems to be doing something very concrete: the allocation of society’s labour‐time. Prices, then, insofar as they represent value, are merely imperfect representatives. But on the other hand, value is an abstract category that cannot be “found” anywhere in real life — so it must be an emergent phenomenon arising from the vast number of concrete prices paid every day, vaguely analogous to how temperature is an emergent phenomenon of kinetic energy.

The basic form of Marx’s solution to this apparent circularity is to observe that value cannot just be labour‐time.^[32] For values to be commensurable in the first place, they must correspond (always bearing in mind that value per se is born of exchange, not of labour processes) not with concrete labour (e.g. blacksmithing labour), but with abstract social labour — that is, a certain quantum of physiological, nervous, cognitive, etc. exertion. Moreover, that abstract labour must be the amount socially necessary on average to produce the commodity, given the current state of the society & its environment.

Thus, a price doesn’t have to “know”, as it were, anything about the private details of the way in which a particular commodity was produced in order to participate in transmitting value. Conversely, it’s now clear that value, in spite of it being the only allocator of society’s labour‐time under capitalism, can never do its job in any other way than approximately, through rough averages that are continually — & indeed, chaotically — approached even as they move, mediated by the “chaos” of prices.

Crucially, the only problem with this apparent conceptual loop — values regulating prices, while also seeming to emerge from the exchange‐moments in which said prices are realised — arises if we attempt to reduce price to value (or really, to any other category), or vice versa. This is the source of the neoricardian impulse: premature reductionism.

The instinct for unwarranted reduction is so strong, in fact, that it often results in eliminative reductionism. Take, for instance, Sraffa’s “corn models” & his revealingly‐titled Production of Commodities by Means of Commodities (1960): value is eliminated, and in its place are the use‐values & technical coëfficients — all matrix’d up & ready to go — also utilised by LoC.^[33] Marx’s crucial use‐value vs. exchange‐value distinction is thereby collapsed, and therefore so is any possibility of real insight.

LoC, then, manages to partially dig its way out of this hole by very partially reïnventing — in more scientific‐looking garb — Marx’s categories & links: price is (somewhat) un‐reduced from value^[34] by replacing a supposèdly “deterministic” relationship with explicit random variables. And this process is indeed merely partial: value has been hypostatised into “labour‐content”; circulation costs are conflated with production costs; in the context of labour‐power, value is conflated with price; etc.. Still, at this rate, the neoricardians might manage to reconstruct the entirety of Capital’s theory by the time capitalism is abolished. I’m rootin’ for ’em.

By now, the reader is surely fed up with the so‐called “transformation problem”. Unfortunately, the only actual problem here is the amount of baggage in need of unpacking.

Fortunately, now that we’ve clarified the actual concepts at work here, we can more deeply understand Marx’s remark (quoted at greater length above) that “its price of production does not go into the cost price of another commodity”.

Because production‐prices exist at the level of competition between capitals, they must build upon the level of capital: value‐in‐process. Profit is calculated based on the total capital advanced — which is, by definition, in value terms, because that’s what capital means — in combination with the general profit‐rate.

If $C$ is the magnitude of the total capital advanced, $P$ the total profit on that capital over a time‐period $T$, and $r_{\mathrm{G}}$ the general profit‐rate throughout $T$, then we can calculate the production‐price $B$ of the total product (not a unit product) produced over the entire course of $T$ like so:

For a unit product, things get a little trickier. Thankfully, we can simply use the holistic reasoning of the total product^[35] to obtain the production‐price $b$ of a unit product like so (let $n$ be the number of unit products in the total product):

Dimensionally:

• $C$ is in value terms;
• $B$ and $b$ are in production‐price terms;
• $r_{\mathrm{G}}$ relates values to production‐prices, and therefore can be considered to have the dimension $\frac{\mathrm{production‐price}}{\mathrm{value}}$;
• $n$ is dimensionless;
• and none of the quantities concerned are in price terms, because price in no way enters into this nonphenomenal level of abstraction.

Note that, because $C$ is a capital magnitude and therefore in value terms, the calculation of production‐price is insensitive to fluctuations in the production‐prices &/or prices of inputs insofar as these aren’t the results of changes in the underlying values. This is true transitively; for instance, if the price of corn increases without any change in its value — because the general profit‐rate increased, or the price fluctuated for any number of reasons, or what have you — the variable capital remains unaffected (and so does the constant capital, even if corn is a direct &/or indirect means of production).

In the case of constant capital, this is consistent with the fact that living labour transfers the value of the material inputs to the product — the worker doesn’t “transfer” any production‐prices nor prices. On the other hand, the variable capital isn’t added to the product at all; instead, living labour adds fresh value, and an amount of this extensionally equal to the value of labour‐power is what actually replaces the variable capital advanced. Fittingly enough, Marx emphasises this in his criticism of Ricardo:

The characteristic feature of variable capital is that a definite, given (i.e. in this sense constant) part of capital, a given sum of value (assumed to be equal to the value of the labour‐power, although it is immaterial here whether the wage is the same as, or more or less than, the value of the labour‐power), is exchanged for a force that valorizes itself and creates value — labour‐power […]. [T]he portion of value that the capital laid out on wages adds to the product is freshly produced (and thus actually reproduced)

This section tacitly assumes that the rate of surplus‐value is the same for all capitals.

But we can do even better: to simultaneously satisfy both the mathematicians and those who care about the actual concepts at work, we can make use of a mathematical consistency condition that’s graciously provided to us several times throughout Vol. 3 of Capital. This condition follows immediately from the actual meaning of production‐prices:

for commodities of average composition. The quantity of profit that falls to the share of these commodities is equal to the quantity of surplus‐value contained in them. For the above capital, with its composition of ${80}_c+{20}_v$, for example, the important thing as far as the determination of surplus‐value is concerned is not whether these figures are the expression of actual values, but rather what their mutual relationship is; i.e. that $v$ is one‐fifth of the total capital and $c$ is four‐fifths.

Another, more straightforward, formulation of the same condition:

We shall further assume, in order to avoid needless difficulties, that this capital is of average composition, so that its production price and its value coïncide; which is the case whenever the product of this individual capital can be treated as the product of a part of the total capital corresponding to it in size.

The average composition is a weighted average, where the capital magnitudes are the weights. A mathematically equivalent, & more conceptually apt, formulation is that the average composition is the composition of the total social capital: add up all the constant & variable capitals (keeping the two separate, of course), divide $\frac{c}{v}$, and this composition is obtained.^[36] The above “consistency condition” is now clear enough: the total product of the total social capital must have a production‐price equal to its value, because there’s only one (1) capital amongst which to share the surplus‐value! For a smaller capital of the same composition (i.e. of average composition), this proportionality is simply scaled down according to the capital’s relative magnitude.

Unfortunately for LoC & other neoricardians, this consistency condition is clearly violated by their interpretation. Said interpretation is thus invalidated even on purely mathematical, nonconceptual grounds (much like in “Double‐counting” above). In particular, the means of production used by a firm whose capital is of average composition might very well have (for instance) a production‐price larger than its value. Because the neoricardian interpretation uses production‐prices as inputs to production‐prices, the cost‐price for such a firm would, ceteris paribus, be larger than its capital value, and likewise would the production‐price of the total product be larger than its value — plainly violating the consistency condition.

At the end of the day, the total surplus‐value generated over a given time‐period by society is a definite quantity of value, and the (value) magnitudes of the capitals advanced by the various firms^[37] involved in the production of that surplus‐value are also given. All that needs to be done — assuming a purely hypothetic “perfect equilibrium” — is to modify the production‐prices (but importantly, not the values; and prices don’t enter the picture at all) such that the surplus‐value is redistributed, in the form of profits, according to each firm’s share in the total social capital. A firm whose capital makes up 1% of the total social capital gets 1% of the society’s surplus‐value as its profits, and so on.

Production‐prices in this context are theoretically unproblematic, and clearly pose no mathematical difficulties. It’s only if we were to assume that the difference between value & production‐price were relevant to the determination of production‐prices themselves — thereby catastrophically collapsing the entire abstraction, and thus defeating the entire purpose of production‐prices in the first place — that this equilibrium picture could possibly be seen as mathematically complicated, much less unworkable.

Fortunately for the authors of LoC, they have a readymade response to anyone who attempts to clarify the true nature of the so‐called “transformation problem”:

The blinkered faithful have managed to ignore the whole issue, and are satisfied by repeating the word of the Master, who surely knew best. The more sophisticated orthodox Marxists — a contradiction in terms, for how can one be both orthodox and Marxist? — have tried to fudge the issue by putting far‐fetched or scholastic interpretations upon what Marx actually said.

One gets the distinct idea that “far‐fetched or scholastic” is a phrase that would be applied similarly to Vol. 1, part i of Capital, and to any claims — made by Marx or anyone else — that capital is a social relation.

In reality, understanding Marx’s basic conceptual apparatus is of great import even if it turns out that the theory of production‐prices is totally unnecessary — something about which I personally remain agnostic. The bungling of concepts involved in the positing of a “transformation problem” implies a deep misunderstanding of the entirety of Capital — even the parts that LoC seems to approve of — and indeed almost all of Marx’s theoretic works. We can & should improve upon & move beyond Marx, but the way to do this is not to put so little effort into understanding him that we wind up regressing to his superseded predecessors.

If this seems like too many words, it’s because it’s necessary not only to explain the relevant theory, but also to simultaneously combat the inexhaustible preconceptions surrounding it. Many fall here & there into the various quicksands of (neo)ricardianism on innocent accident — I would know, because I once did too — and it is these readers, not so much the dedicated neoricardian authors, who I hope will benefit from this review of LoC.

But wait — there’s more! LoC has many grievances to air about production‐prices, both in this chapter and in others. We limit ourselves to only one more passage:

In short, this model is powerless to make any definite statement about the behaviour of the ratio $\bar{\mathrm{\pi }}\left(\mathrm{C}\right)∕\mathrm{\lambda }\left(\mathrm{C}\right)$, the ratio between price of production and labour‐content. Since specific price — the ratio between market price and labour content — presumably fluctuates around $\bar{\mathrm{\pi }}\left(\mathrm{C}\right)∕\mathrm{\lambda }\left(\mathrm{C}\right)$, the model cannot make any definite statement about specific price either. Unlike the original model of Capital Volume 1, this “improved” model fails to make what we would regard as even a roughly correct prediction about specific price.^[2]

Thus, even if the transformation problem could be solved mathematically, the resulting model would […] actually be inferior to the original unmodified model in respect of prices.

At first, the aim of this passage is a bit perplexing: isn’t part of Marx’s justification for introducing production‐prices in the first place to explain otherwise‐unexpected dispersion in what LoC calls specific price (roughly: the ratio of price to value)? The model’s being “powerless to make any definite statement” about how specific price is distributed over the entire economy taken as a whole isn’t a bug — it’s a feature.

Putting aside the strange fascination with specific price, the wording here is revealing: LoC wants Capital to make statements about the “behaviour” of certain ratios, and to make “roughly correct prediction[s]” about them. The goal is, then, not to gain insight into the inner structure of capitalism and what makes it historically unique & contingent, but rather, to make positivistic empirical predictions about its everyday trivia. For LoC, Capital is not, as its title seems to claim, a “critique of political economy”, but rather, just another work of political economy.

LoC provides an amended version of Vol. 3’s (in)famous argument for the law of the tendential fall of the average profit‐rate, which culminates in this equation:

The benefit is that we now see increases in $e_{\mathrm{M}}$ as being, by logical necessity, unable to indefinitely counteract an increase in $k∕N$ (note that this ratio is similar to, but not to be confused with, the value composition of the total social capital). If $k∕N$ rises without bound, then the general profit‐rate must tend toward zero. This is because $e_{\mathrm{M}}∕\left(1+e_{\mathrm{M}}\right)$ can never exceed 1, no matter how high $e_{\mathrm{M}}$ might be.

With this in mind, LoC’s focus on the profit‐rates being randomly (gamma‐)distributed rather than uniform allows them to present a model of accumulational crises. The average profit‐rate never manages to get “dangerously low” from a total‐social‐capital‐as‐one‐big‐firm perspective, because by the time the average gets to be even mildly unfavourable, the firms in the lowest segment of the profit‐rate distribution start to fail in significant numbers. In turn, this crisis (as it may be) devalues capital due to:

• Firms going to the wall, much of their capital now being effectively destroyed.
• Desperate optimisation of the production process, resulting in labour productivity increases.
• The firms that do fail being more likely to be the less productive ones in their respective spheres, thus further increasing the social productivity of labour. (LoC doesn’t mention this effect.)

This, by the way, implies that (as LoC is keen to stress) any tendential fall of the average profit‐rate cannot — at least, not inevitably — be secular. Any fall can be “corrected” by devaluation, and this devaluation may or may not take the form of crisis.

Increase of $N$ can play a role as well, but here we start to worry about technical limitations of the — now typically more automated — production process. More to the point, the size of the working population is conditioned by capital, not the other way around.

Although this basic imagery was in no way new to marxist crisis theory in 1983 (cf. Grossman, Mattick^[38]), LoC’s exposition at least has the merit of a more vivid illustration, and a (once again, not exactly novel) deëmphasis of “inevitable” secular trends.

Shifting our focus to below the level of the total social capital, it’s not immediately clear why the organic compositions of the capitals of various firms wouldn’t be totally arbitrary. Because LoC rejects production‐prices, such arbitrary compositions would, in combination with a given rate of surplus‐value, result in similarly arbitrary profit‐rates. For competition^[23] to be capable of producing an equilibrium distribution of profit‐rates (in Marx’s terms: a general profit‐rate), then, it needs to do so by affecting individual organic compositions.

LoC gives the basic explanation for this mechanism like so:

[I]f a firm has a very high [organic composition], this means that its production process is very [constant‐]capital‐intensive, usually because it is very highly automated. But a highly automated system tends to become a new commodity in its own right, which — rather than serving as a basis for the operation of a separate firm — is bought directly by those firms or consumers who need the products of the given process, and is used by them as a part of their operating capital or as a consumer durable.^[2]

For example, in the past there used to be such things as ice factories, which used to sell ice to individuals or firms for the purpose of refrigeration. But once the process of production of ice (or of low temperatures) became sufficiently automated, it was embodied in refrigeration units and domestic electric refrigerators, which are bought by firms and individuals directly for their own use.

(In the other direction, a low organic composition implies either an uncompetitively low productivity, or no basis for the wage‐relation in the first place.)

This explanation is quite useful, and I haven’t encountered it elsewhere. Taking this process as the theoretic basis — instead of production‐prices — seems plausible to me, although it does come with at least two caveäts:

• Because this process is essentially technological, it’s not clear that it can always happen in the “right” way, so as to facilitate competition towards a hypothetic(!) equilibrium. This isn’t a theoretic problem, however, because it merely implies that things won’t always work out smoothly — which is evidently true, and also witnessed by the actual dispersion of profit‐rates.

• Again because this process is essentially technological, it doesn’t operate — at least, not directly — at the level of value; nor of production‐price, price, etc.. It’s therefore not incompatible with production‐price theories. This point is taken up in more detail below: “The heart of the matter”, “What we can do about it”.

Are equilibrium profit‐rates really gamma‐distributed? This chapter presents some nice‐looking charts that tentatively suggest an optimistic “maybe”. Not being an econometrician myself, I cannot comment any further on the matter. Nonetheless, the point that profit‐rates really are quite dispersed is well‐taken.

The outstanding empiric result, however, is that the rate of exploitation — at least, in manufacturing industry of the advanced capitalist countries within a century or so of LoC’s original publication — seems to pretty much just be… 1. Like, the number one.

In correlating wage‐bills with profits^[39] (both in price terms), LoC manages to produce an incredibly high diachronic correlation coëfficient of 0.997 for their U.S. manufacturing data. The synchronic correlations are less impressive — typically ≈0.95 or so — but still good. These correlations seem to imply that not only is the average rate of surplus‐value roughly 1, but that this average rate is very stable both over time, and between branches of industry.

This is what lends support to LoC’s hypothesis (given at the end of ch. III, as mentioned above) that profit‐rates and organic compositions are gamma‐distributed, and that their distributions have the same second parameter $\beta$ (the rate parameter). From this hypothesis, in combination with the additional (but more commonly postulated) hypothesis that the rate of exploitation is the same for all firms, it follows that this rate should be 1.

Some caveäts might be attached to this hypothesis:

• Although I was able to follow LoC’s mathematical reasoning, the reasoning is just that: a mathematical proof. The proof gives absolutely no insight into what its social implications might be, or by what means it might actually operate in practice. In LoC’s words: “one needs a probabilistic ‘bridging’ theory that connects the macroëconomic with the microëconomic, and shows how to aggregate correctly”.^[2]
• We’re left to wonder what the situation might be in poorer countries, on which LoC has no data. We can’t be too surprised if it turns out that imperialism, labour outsourcing, etc. have something to do with the picture here.
• Everything here is purely in price terms. In some cases, e.g. the calculation of rates of exploitation, this seems probably(?) fine. In others, we just have to pray that the category errors don’t distort the resulting image.
• The picture is apparently obfuscated by rent, the social wage, etc., all of which LoC ignores or eschews.

When it comes to values & prices — in particular, what LoC calls “specific price” (the ratio $\mathrm{price}:\mathrm{value}$) — LoC’s empirics have nothing to say. This is quite unfortunate, given that the claimed advancement over Capital boils down — once we extricate the conceptual confusion — to the ability to predict specific price (i.e. its distribution), which then has implications for how competition^[23] actually operates.

Nonetheless, LoC is able to provide empirics on something approximating organic compositions of capitals. Once again, everything is in price terms; rather than value composition, we have relative labour costs ≝ wage‐bill divided by total expenses. And once again, we get something that could plausibly be, or plausibly not be, a gamma distribution. It seems unimodal, widely dispersed enough to evidently not be a constant, but narrow enough to cast doubt upon the idea that organic compositions are totally arbitrary byproducts of the technical character of the production process.

In fact, it may be that the actual mechanics by which firms influence their rates of profit and bring them closer to the optimum is not only via an appropriate pricing policy, as tradition has it, but rather by taking care that their organic composition and labour costs will not be out of line.^[2]

But there’s a problem: price isn’t value.^[40]

If the theory of production‐prices is basically correct — or even a little bit correct — then we should expect the associated process of equalisation to percolate (as it were) its way through the various branches of capital by means of production‐prices — the latter being, as Marx makes clear, continually in the process of formation & re‐formation.

Value composition, and therefore also organic composition, exist exclusively at the value level. Thus, after the full double‐“transformation” from value to production‐price to price, we’d expect this process to surface (i.e. at the level of appearance) in a way that’s at least consistent with what LoC finds.

Regrettably, it’s already impossible — before they even get started — for the authors of LoC to make this connexion, both because they misunderstand what a production‐price is (by conflating it alternately with both value & price), and because they mistakenly interpret Marx’s theory as “deterministic”.

This consideration doesn’t necessarily point to a particular theory (Capital’s or LoC’s) as being the “correct” one. In fact, I would venture a hypothesis that perhaps both theories are simultaneously true, or at least have elements of truth to them. This is possible basically for two reasons:

• Vol. 3 is very clear that even though the general profit‐rate (and therefore also production‐prices etc.) is typically in effect, this effect is a global one that requires a lengthy & involved process before it can reässert itself locally. Local changes, then, tend to reflect simple changes in value that are relatively unmediated by production‐price:

  For all the great changes that constantly occur in the actual rates of profit in particular spheres of production (as we shall later show), a genuine change in the general rate of profit, one not simply brought about by exceptional economic events, is the final outcome of a whole series of protracted oscillations, which require a good deal of time before they are consolidated and balanced out to produce a change in the general rate. In all periods shorter than this, therefore, and even then leaving aside fluctuations in market prices, a change in prices of production is always to be explained prima facie by an actual change in commodity values, i.e. by a change in the total sum of labour‐time needed to produce the commodities.

  This leaves room for the mechanics that LoC has in mind. Thus, we have value‐level competition/optimisation locally & in the short run, vs. the competitive self‐allocation process of capital via production‐prices at the global, secular level.

• It would, in a way, be a bit strange if competition amongst the entire capitalist society proceeded exclusively via a single mechanism. In this sense, our equilibrium picture of smooth competition should perhaps be one in which LoC’s proposed mechanisms do some of “the work”, while production‐prices etc. do the rest.

Here, LoC asks about how concrete labour‐time relates to value (the latter of which has been hypostatised as “labour‐content”). Do all types of labour generate the same value, or are certain types of labour more “skilled” in a way that makes them more valuable? LoC refers to the former as the “egalitarian solution”:

Marx, however, rejects this simple egalitarian solution. He does so, we believe, because he is aware that such a solution would be blatantly incompatible with the deterministic relation that he posits between value and price.^[41]

[…]

In any case, Marx does not suggest in Capital any other method, independent of any prior knowledge of the prices of commodities, for determining the skill coëfficients, nor do we know of any plausible method of this kind. […] We fail to see any method, theoretically convincing or even plausible and empirically meaningful, for determining skill coëfficients. We must therefore do without them.

Here, the authors of LoC project their preoccupation with empirical prediction onto Marx. In reality, Marx rejects the egalitarian solution because it’s incompatible with the notion of value itself — even when considered in isolation from price.

Labour‐power has been commodified; and like any commodity, it has a value. As discussed above, LoC doesn’t see it this way. Nonetheless, irrespective of how one sees it, the fact remains that society must allocate some of its labour‐time to the production & reproduction of labour‐power. If some type of labour‐power requires more of this time for its (re)production, and this type makes possible a particular labour process — i.e. it has a sui generis use‐value from the standpoint of capital — then this additional abstract labour‐time socially necessary for the (re)production of this type of labour‐power will be reflected in its greater value.

Capital makes a distinction between the extensity of labour vs. the intensity of labour: the former is its duration, whereas the latter is the rate at which it generates value per unit time — Marx uses the term “specific gravity”.^[42] The category of labour intensity is applied to some more obvious cases like, for instance, labour becoming less intense (“more porous”^[43]) as a result of its overextension, thus placing another physical limitation on the production of absolute surplus‐value. But he also applies it to other cases; for instance, the comparison of labour‐power of different countries.^[44] And of course, he also applies it to skilledness of labour‐power:

All labour of a higher, or more complicated, character than average labour is expenditure of labour‐power of a more costly kind, labour‐power whose production has cost more time and labour than unskilled or simple labour‐power, and which therefore has a higher value. This power being of higher value, it expresses itself in labour of a higher sort, and therefore becomes objectified, during an equal amount of time, in proportionally higher values.

Note that Marx is very mathematically precise in this formulation: we get proportionally higher values during an equal amount of time. If the labour‐power has twice the value of average simple labour‐power, then its labour intensity is likewise double. We therefore, having read the first seven chapters of Capital, have obtained exactly the “method, theoretically convincing or even plausible and empirically meaningful, for determining skill coëfficients” that LoC failed to stumble across.

Inasmuch as labour‐power is a commodity like any other, this is effectively yet another application of the fluid vs. fixed distinction, but now applied to variable capital: the value fixed in the labour‐power — for instance, by a process of training & education — only enters the total product gradually over the worker’s lifetime.

Nonetheless, LoC defends skill coëfficients by demonstrating that knowing how much of the various types of labour go into the production of commodities actually overdetermines skill coëfficients. This is meant to address the charge of apparent (but illusory) circularity: skill coëfficients explain observed prices, but the coëfficients themselves supposèdly can only be observed via those same prices. In LoC’s words, skill coëfficients are “a non‐vacuous consistency condition”.

LoC then goes on to defend its choosing the “egalitarian solution” anyway. This amounts to the purely empiric observation that most labour‐power simply isn’t significantly more valuable than simple labour‐power. The value of simple labour‐power — i.e. that necessary to raise someone to adulthood and reproduce them from day to day, in accordance with the relevant social standards — is already so large that additional skilledness adds relatively little to this value. Wage differentials are then explained as the result of arbitrary sources of bargaining power, which are then taken to be independent of the value of the labour‐power in question.

Although this is a useful observation (assuming its validity) that justifies our often tacit assumption that all labour‐power is more‐or‐less simple, we’re left to wonder what exactly the point of the theory is. If we want insight into the inner mechanisms of capital, then we ought to choose the “solution” that follows from the category of value (or “labour‐content”) itself. We therefore once again get the idea that LoC is less concerned with inquiry than it is with matrices capable of spitting out plausible numbers, and histograms that seem like they almost fit a certain curve.

If we’re being honest with ourselves, it all starts here:

Marx’s own views on this question are not always unambiguous, and may even be inconsistent. A plain reading of his definition of value in the beginning of the first volume of Capital seems to be that values are determined in the sphere of production alone. But in the third volume he seems to say (or at least can be interpreted as saying) that a commodity does not possess a definite value independent of whether it actually gets sold, or of the price for which it gets sold.

Because value, for Marx, is a social fact^[45] (at this point, I am literally begging. pleading. this must be understood), and LoC, by inexplicable contrast, wants to find value as a thing or quantity in a particular location, Marx is accused of plain ol’ inconsistency. The fact that an entire body of Anglophone literature exists that’s based upon this abject uncomprehension is one that can only be described as migraine‐inducing.^[46]

The basic structure of Capital’s theory has already been clarified in the discussion of ch. VI above (see also: ^[13]). By clinging to empiric appearance, LoC collapses value with production‐price with price, obliterating this structure. Likewise, capital is then misunderstood as being hoarded, dead (= thingified) labour, rather than as value made autonomous and therefore a social relation.^[45] This, in turn, partially collapses the use‐value vs. exchange‐value distinction as well. The collapse of value with production‐price with price, and the reïfication of capital, are both exacerbated by the use of input–output matrices (whether augmented by random variables or not) that reduce values, capitals, use‐values, etc. to technical coëfficients.

Evidently, this robs Capital of virtually all its content, transmogrifying it into thousands of pages of complete nonsense. On the bright side, for those (such as, perhaps, LoC) only here to do economics, this isn’t necessarily a problem.

The problem arises when this uncomprehension makes LoC capable of seeing Marx’s value ↔ price relations, general profit‐rate, etc. as a “deterministic” theory with “random noise” added post hoc, instead of the rich theory full of conceptual layers & mediations that it actually is. Because the authors of LoC want to have their own ostensibly working theory, they are therefore forced to reïnvent — now in 20^th‐century mathematical garb — these same conceptual layers & mediations.

What separates LoC from the common neoricardian text, then, is that it actually manages — albeit very incompletely — to do this reïnventing.^[47] We should therefore attempt to give credit where credit’s due, and that means taking what can still be salvaged, so that it can be synthesised with the more mature theory found in Marx’s works.

LoC’s primary strategy for recovering the categories that it has catastrophically collapsed is, of course, mathematisation with random variables. The inspiration from statistical mechanics allows for a somewhat more concrete approach than the abstract categories used by Marx. Specifically: to the extent that the statistical‐mechanical analogy can be made sense of in terms of actual social facts & categories, the theory laid out in Capital is enrichened.

However, there is one invariant of utmost importance: this cannot be done unless we consistently respect the concepts & categories at work in Capital. This doesn’t necessarily mean keeping them as‐is, but it does mean preventing ourselves from regressing behind Marx (e.g. to Ricardo). It also means refraining from ascribing anaemic, economistic theories to him as a sneaky way of defending similar theories (note that this remains true irrespective of what Marx “actually meant” or “believed”).

Two caveäts apply:

• Although this process — if done well — can indeed enrich the theory, the simple fact of making it more mathematical is not, in & of itself, a virtue. There’s a fine line between genuinely deepening one’s insight into the nature of the capitalist mode of production and just doing economics.
• For better or worse, the people who understand statistical mechanics and the people who understand Capital tend to be nonoverlapping populations. This is most clearly evidenced by LoC.

In any case, the following aspects taken from LoC seem potentially useful:

• Thinking of the establishment of a central tendency (an “average”) in terms of the establishment of a unimodal probability distribution, when appropriate.

  This presumably requires knowing what the relevant maximum entropy probability distribution is, which in turn implies that entropy is somehow defined. Good luck with that. Maybe information‐theoretic entropy is of some assistance?

  Some special cases are easy — but perhaps not so helpful, for that reason. For instance, as LoC explains in detail, price distribution looks similar enough to the conditions for the central limit theorem that a normal distribution becomes the obvious choice.

• If an explanation with real conceptual content can be provided, then we have LoC’s magical “proof” of the rate of surplus‐value tending — as always, via ceaseless fluctuations & exceptions — towards 1: “Distributing $Z$”, “Rate of exploitation”.

• There’s some hint — albeit an extremely vague & faint one — at ergodicity having something to do with equilibrium. This is potentially useful if (big “if”) it can be grounded in a crisis theory with real conceptual content.

As discussed at greater length in “The heart of the matter” above, LoC presents its own theory as an alternative to that of production‐prices: firms use organisational/technical changes to regulate their organic compositions, thereby guiding their individual profit‐rates toward socially‐expected levels.

This theory is actually consistent with that of production‐prices; recall that phenomenal (i.e. in price terms) “compositions” don’t necessarily align with actual organic compositions, which exist at the capital (i.e. value) level. The main difference between the two theories is that LoC’s proposed mechanisms are local, whereas production‐prices are global. The two therefore aren’t opposed, and should be taken as (not necessarily exhaustive) parts of a larger picture.