aa r X i v : . [ q -f i n . GN ] N ov Marxism, logic, and the law of the tendency forthe rate of profit to fall
Robin Hirsch18th November 2020
I am not on a campaign to demolish Marxism. I am writing this note partlybecause I published a couple of articles [Hir04, Hir96] relating to Marxism and,decades later, I now think I was wrong on a number of points, so I would liketo clarify my own argument, mainly for my own benefit. But also, for politicalorganisations aiming for the overthrow of Capitalism and particularly for suchorganisations where the importance of political theory is emphasised, it seemsto me that the adoption of a mistaken theory could be a serious handicap totheir success.One worry about Marxism, as Martin Hyland once pointed out to me, isthat it is a theory of everything. Marxism is a theory of history, politics, class-struggle, natural science, philosophy, religion and economics, but the underlyingtheory, dialectics , claims to be a logic of change and interaction of processes infull generality. The track record of theories of everything is not so good andwhile that in itself is not enough to rule Marxism out, it does place a greaterburden of proof to provide a clear justification — either empirical or logical –for all its claims. Because its claims are so ambitious, we should require a higherlevel of proof than we do for, say, the licensing of a new drug.
Dialectics and Logic
My main point here, is that the underlying logic ofMarxism — dialiectics — is unscientific. By a scientific theory I mean a theorybased on evidence, using logic. It is widely accepted that science must bebased on evidence, but of course science is not simply the process of collectingevidence, rather such theory will be based on evidence and deduces a generalunderstanding of a topic based on logical reasoning from the evidence. Bothparts of this definition (evidence and logic) are problematic, the question ofevidence is † tied up with the notion of interpretation, and what is meant bylogic needs to be clarified. Nevertheless, I take this as a useful definition ofScience. What counts as a logical argument may depend on the field. But itshould include a clear definition of any non-standard terms, any assumptionsthat are being made, and it should be clear how each statement or propositionfollows from either evidence, from assumptions, or from what has gone before inthe argument. In formal logic, simple rules are given that allow you to deducenew statements from old (e.g. Socrates is a man, all men are mortal, therefore p and not p is true, then at least part of what we saymust be wrong. For those who do not accept this view, that by this definitionof ‘not’ it is impossible for both p and not p to hold, a more anthropologicalanalysis suggests that arguments based on contradictions are not convincing. Ineveryday conversation, a contradiction in an argument is seen as a good reasonto reject that argument (whereas, in Marxism contradictions are not only per-mitted, they are an essential part of an argument ). An argument containing acontradiction is not persuasive. When Matt Hancock argued that the UK wasavoiding testing as a strategy for COVID in March 2020 and argued that thiswas because Science showed that testing would not save lives, and anyway theGovernment were testing, the contradiction does not strengthen the argument,rather it undermines it. The two parts of his argument cannot both be true toreality, so something he said is clearly false.Of course, if we consider everyday conversational language, we also observe There are formal logics that deal with contradictions, e.g. paraconsistent logic . The in-tention here is to deal with inconsistent, contradictory information and try to resolve thoseinconsistency. The idea is not that contradictions exist in reality, rather they consider data-bases of information which may not be reliably true. hat contradiction frequently do appear. If I ask “is it hot in America?” youmay reply “well, yes and no”. But the meaning I glean from your answer is notthat there is a well-defined property that is true and false, instead I concludethat my original question is not clear, whether America is hot depends on manythings – the time and place most obviously, also, what you compare Americato, etc. — so that a simple yes or no answer would fail to describe the situationcorrectly. What is being suggested is not a logical contradiction but a variationin the property, and a difficulty in defining what counts as hot . It seems tome that while apparent contradiction in an argument provide a useful way ofidentifying inconstencies in someone’s beliefs and also may identify statementswhose truth varies accoding to the situation or is not properly defined, buta logical contradiction in an argument always weakens it. According to verysimple laws of propositional logic, if we take both p and NOT p as assumptionswe may draw any conclusions whatsoever. Bertrand Russell may have said once(although I cannot track down a reference) that the weaker the logic, the moresurprising the conclusions.Although there are three laws of dialectics, these laws turn out to be more liketopic headings (the law of unity, the negation of the negation, the transformationof quantity into quality) and they do not offer a way to deduce one thing fromanother, nor do they tell you when one argument follows logically from another.There have been quite a few cases where deductions have been claimed, basedon the laws of dialectics — for example, Hegel’s proof that the number of planetscould be only seven, or Marx’s demonstration that the ellipse is a form of motionthat realizes and resolves the contradiction between centrifugal and centripetalurges — but these arguments cannot be scientific. Value
Marxism distinguishes itself from the wider political left by an interest-ing analysis of exploitation and profit. The general anti-capitalist view is thatcapitalism is based on exploitation and is therefore sinful, being contrary to anatural concept of fairness. Marx shares this view, but adds that exploitationis also the root cause of a problem for the Capitalist system that dooms it.Exploitation leads to a rise in the organic composition of capital, which leadsto a fall in the rate of profit. This is the central content of the thesis elaboratedby Marx in the three volumes of Capital [Mar54]. According to this thesis, theCapitalist system is a transitory system which cannot last long. The reasonthat it cannot last long is that a mechanism built-in to the nature of Capitalismleads to a long term tendency for the rate of profit to fall. This mechanismworks as follows. The rate of profit is defined as the ratio SC + V , surplus valuedivided by total costs. In order to improve productivity, over time the organiccomposition CV will rise and consequently (assuming some bound on the rate ofexploitation SV ) the rate of profit will fall.Here I will argue that the argument stated above is logical, given certain Quantum mechanics provides a more difficult challenge. A photon may ‘be’ in two placessimultaneously when unobserved, but after observation it may be in a single place. However,there is no logical contradiction here, just a puzzle. ssumptions, but at least one of the assumptions (a bound on the rate of ex-ploitation) is unjustified. A consequence is that I do not know how long theCapitalist system will last.But first, lets examine some issues in this formulation of the theory. In thestatement “The Capitalist system is transitory and cannot last long” and in thephrase “long-term tendency”, we might ask, what counts as long? All previouseconomic systems have turned out to be transitory, so we might have guessed,without reference to the three volumes, that Capitalism will end eventually, butMarx’s thesis suggests more: that Capitalism will last significantly less timethan previous systems, in particular significantly less than the thousand yearsof Feudalism. After two centuries of Capitalism it can certainly be argued thatinsufficient time has elapsed for us to see this, on the other hand, the moretime that passes without an end of Capitalism, the more problematic this is forMarx’s thesis.A second preliminary comment is that the theory makes use of quantities,like
C, V, S , the rate of profit etc., and these quantities require an explanationin the theory — an explanation that provides numerical values for these quant-ities. Without such an explanation we cannot infer that a certain ratio of thesequantities will fall. In Marx’s writing there appear to be a number of types ofvalue: exchange value, necessary labour time and underlying value.The exchange value of a particular commodity tells you the ratio that youexpect to exchange this commodity for other commodities, under average con-ditions. Given certain assumptions about fluidity in the market, the exchangevalue of a commodity may be represented as a certain number in such a way thattwo commodities with the same number as their exchange value would representa fair exchange, i.e. the sort of exchange you would expect to be able to make,in average conditions. More generally, the ratio between exchange values of twocommodities tells you (inversely) the ratio that you expect to exchange quant-ities of each commodity. The monetary price of a commodity is an indicationof its exchange value . On the face of it, the exchange value of a commodityshould be observable and measurable, but as a result of enormous complexitiesrelating to currency exchange rates, hidden costs and subsidies, it is very hardto establish reliable exchange values in practice.The necessary labour time for a commodity is what it says, the duration ofall the work needed to produce the commodity, measured in hours. It has tobe modified to take into account the fact that some types of labour are morevaluable than others. A skilled worker may be able to produce more value inone hour than an unskilled worker, so an adjustment is needed to take this intoaccount. It gets complicated because the skill is itself a commodity that may bepurchased and the value of that skill would then be determined by the amountof labour needed for the necessary instruction of the worker. There is a danger Its an interesting problem to find conditions which are necessary, or sufficient, for ameaningful exchange value to be defined, and to construct other models which summarize thepatterns of exchange of commodities in situations where a global exchange value cannot bedefined. I might look at that further some other time, here we may assume that an exchangevalue is well-defined. hat we are already losing track of exactly what we mean by “necessary labourtime”, clarifying that would be an interesting project, and we will return to thislater, however it is not our main concern for this note.The underlying value is the true value of a commodity, when distortions andillusions created by the system are removed. It is not directly observable ormeasurable, but it is needed to understand the behaviour of exchange value,labour time and everything else. It is the holy ghost of this value trinity . The Law of Value is sometimes invoked to explain the connection between thesedifferent types of value, but the definition of this law of value varies considerablyand is nowhere stated precisely.One version of the Law of Value is the
Labour Theory of Value (LTV).According to this, the exchange value of a commodity is determined by theamount of labour that is required for its production. The problem, of course,is that LTV is empirically and systematically false. Because of the equalisationof profit rates between different sectors of the economy, the exchange valueof a commodity will differ from that predicted by LTV. Marx was aware ofthis problem and referred to it as the transformation problem . Marx arguesthat there is a transfer of underlying value from labour intensive industries tocapital intensive industries, caused by the equalisation of profits. This meansthat the exchange value of a commodity will differ from the necessary labourtime. Since LTV is false, Marx’s theory still requires an account of how valuesare determined .One weak version of the law of value states merely that all value originateswith labour. The problem with such a qualitative weakening of the law is thatMarx’s thesis concerns numerically valued quantities, like the rate of profit, sothe theory must explain and provide numerical values, and the weakened law of Unobservable, hidden parameters may well be required in order to explain how Capit-alism works, but any theory which adopts such parameters needs particular justification —either empirical or logical — of how these hidden parameters can be used to obtain a correctunderstanding of observable things, like the collapse of capitalism. To illustrate a problemwith the notion of a discrepancy between observed values and true underlying values, considerthe well-known Newtonian equation f = ma . Imagine a physicist postulated that the truelaw was f = mv but because of certain peculiarities in the way we observe these quantities,we should multiply the right hand side of the equation by av to account for the illusion thatis observable to us. The true law is f = mv but after adjustment it seems (superficially)that f = ma . The argument would be discounted, if for no other reason that it falls foul ofOccam’s razor by introducing unnecessary and unobservable properties. All the same, for much of volume 1 of Capital, Marx gives the strong impression that heis using the LTV. In his laboriously worked examples he estimates that values produced bya worker over the ten hours of a working day, e.g. 4 hours for capital costs and depreciationcosts, 3 hours for worker’s own wage, 3 hours for the boss. It is true that Marx and marxists have written extensively about the relation betweenunderlying value and exchange value. Explanations are given as to why the rate of profit rosein particular periods of history. It might be that in a certain period, Capitalist were ableto raise the rate of exploitation, or the organic composition did not rise, or perhaps that achunk of value was stolen from one sector and deployed in another, somehow causing the rateof profit to rise. Such arguments are based on hindsight. What they do not give is a wayof working out, given data about an economy, how these quantities (exchange value, rate ofexploitation, organic composition, etc.) will interact and how to work out what the rate ofprofit will be. alue says nothing about these. Without a quantitative account of underlyingvalue it is hard to see how it is possible to prove that the rate of profit will fall.A stronger version of the law of value states that underlying value is de-termined by necessary labour time, but makes no assertion about a relation-ship between exchange value and underlying value. However, what counts forCapitalists when making investment decisions is the exchange value of the com-modities they (have their workers) produce. The true underlying value is of nointerest to them, what matters is what they can exchange their commodity for.Capitalists will happily invest if they expect that the rate of profit, measuredby exchange value, is high, it won’t trouble them one bit if the rate of profit,measured by underlying value, is low. There is no reason why Capitalism shouldnot continue happily in such conditions.Since it is exchange values that matter to capitalists, Marx’s theory mustexplain why the rate of profit, measured as a ratio of actual exchange values,tends to fall if the organic composition rises. Having adopted a definition ofvalue let us state and examine the argument for the declining rate of profit.Our assumptions are that (i) the organic composition CV rises without bound(ii) but the rate of exploitation is bound say SV ≤ b , (iii) V > SC + V > ǫ for some ǫ > SC + V ≤ bVC + V ≤ b VC The first inequality follows from assumption (ii), the second follows from as-sumption (iii). Since CV is getting bigger without bound (assumption (i)) itfollows that eventually CV > bǫ . At that point, SC + V ≤ b ǫb < ǫ . At this point,if not earlier, Captalism must end, by assumption (iv). It all works, its logical.But note that if we drop any one of our four assumptions, the conclusion doesnot follow.It may be of interest to note that the argument from the four assumptionsto the conclusion that the rate of profit must fall to arbitrarily small levels,does not depend on the use of underlying value, nor of necessary labour time,it works equally well using exchange value as a measure.The first assumption is that over time in order to raise productivity, invest-ment will tend to shift into capital intensive forms of production, hence theorganic composition of capital tends to rise. However, productivity can also beincreased by investing more intensely in the value of labour-power. The valueproduced by a highly skilled worker in one hour can greatly exceed the valueproduced by an unskilled worker. All investment in education and welfare canbe considered as a cost of production counted as variable capital V . Given that,it is not clear that there has to be a tendency for the organic composition ofcapital to rise. Empirical evidence may help with this question, although evenif historically the organic composition has risen, this would not prove that itwill rise without any bound.The second assumption, that there is an upper bound on the rate of exploit-tion, is also problematic. If we are assuming that the organic composition ofcapital rises without bound it is easy to conceive that the rate of exploitationalso rises without bound. With a very high organic composition of capital aworker will be highly productive. The cost V of his/her labour need not becorrespondingly high, it is only required that V covers the cost of keeping theworker at a basic level of health, strength, skill and competence. It is not clearwhy the rate of exploitation could not be arbitrarily high. The third and fourthassumptions are not problematic.The contrapositive of the argument above shows that assuming (i), (iii) and(iv) we may deduce that while Capitalism continues, the rate of exploitationwill increase without bound.In summary, I find the various notions of value in Marxism confusing, inparticular the so-called Law of Value is most unclear. The value that is well-defined (though hard or impossible to measure) and critical for the health ofthe economy is exchange-value. Using exchange-value as a measure of value,Marx’s argument for the declining rate of profit is a logical consequence of fourassumptions. However, two of those assumptions are doubtful and without allfour of them we have no proof of a long term tendency for the rate of profit tobecome arbitrarily small.I do not doubt that Capitalism, like all previous systems, will end. It isclearly a dynamic system prone to booms and destructive slumps. It is quiteconceivable that a slump more severe than previous slumps may have alreadystarted or will occur before very long, and that this would wreak enormous dam-age to our lives. Whether Capitalism could recover from such a slump, perhapsin a new form, or whether a new economic system would emerge, remains open.The nature of any new system that might arise from such a crisis is a matter ofspeculation.
References [Eng1883] F Engels. Introduction to dialectics of nature. In
Marx and Engles,Collected works , pages 338–353. Lawrence and Wishart, London, 1968.[Eng1877] F Engels. Anti D¨uring. See , 1877.[Hir96] R Hirsch. Mathematics — the pure science?
Science and Society ,60(1):58 – 79, Spring 1996.[Hir04] R Hirsch. Dialectics and logic.
Cultural Logic , 7, 2004. ISSN 1097-3087.[Mar54] K Marx.