## Marx's Value Theory

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NP. I've said elsewhere, I don't think it's 'Marx's Marx', I think it's very much 'fdrake's Marx'.
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Would people be interested in me continuing through the book too? I have a couple of summary posts of what's happened so far planned, and some ideas for extra mathematical formalism to investigate. So there will be a delay due to the latter taking a decent amount of time.
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I'm reading and very much learning alot.
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Definitely!
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Mathematical note: the isomorphism of the measure of value and the standard of price.

I've been trying to understand the connection between value and price a bit better. If they're usually very similar but have the possibility of diverging in various circumstances that leaves some work in spelling out how the two relate quantitatively when things are going normally. In what sense is value as measured in the amount of abstract labour in the commodity the same as the price expression of that value?

I wrote before about labour times and values necessarily being calculable, and that the algebraic structure of value (and labour times) is that of an ordered field. There are a few things that characterise this relationship in terms of their socially necessary labour time and their direct price. Let commodity one in some amount x have direct price P1 and socially necessary labour time T1, and commodity two in some amount y have direct price P2 and socially necessary labour time T2.

(1) If we have two distinct commodities in two different amounts, x and y, the value of x and y together as measured in (socially necessary) labour time will be the sum of the labour times. T1+T2
(2) Under the same set up, if we take x and make it A times, it will have the same socially necessary labour time as A times x; A*T1
(3) Under the same set up, if we make y B times, the socially necessary labour time is B*T2.
(4) From (1), you can aggregate (2) and (3) to get the total amount of socially necessary labour time from the production of batches of commodities; giving the total socially necessary labour time as A*T1+B*T2.

Restating (1)->(4) in terms of prices just for explicitness:

(1) If we have two distinct commodities in two different amounts, x and y, the value of x and y together as measured in direct price will be the sum of their direct prices. P1+P2
(2) Under the same set up, if we take x and make it A times, it will have the same price as A times x; A*P1.
(3) Under the same set up, if we make y B times, the price is B*P2.
(4) From (1), you can aggregate (2) and (3) to get the total price from the production of batches of commodities; giving the price as A*P1+B*P2.

Let f be the function that maps a commodity's direct price to its socially necessary labour time. The first thing to notice is that this f is invertible; if things are going normally, any direct price is commensurate with a corresponding socially necessary labour time, and vice versa. For a given price, there is a certain amount of human labour in the abstract which that price expresses. This is to say that at a given time, there is a unique price for any amount of socially necessary labour time.

From (1) to (4) in both lists, you can see that the following relationships hold:

(1) f(P1+P2)=f(P1)+f(P2)=T1+T2
(2) f(A*P1)=A*f(P1)=A*T1

this means that the mapping from socially necessary labour times to direct prices is an isomorphism of fields. The two are so tightly coupled that socially necessary labour time and direct price are just different names for the same thing. However: this relationship between socially necessary labour times and their direct price expressions does not necessarily preserve the ordered structure of both of them. This is a useful distinction, as it is a substantially weaker form of the relationship between direct price and socially necessary labour time that Marx uses. You can find in Chapter (1):

Some people might think that if the value of a commodity is determined by the quantity of labour spent on it, the more idle and unskilful the labourer, the more valuable would his commodity be, because more time would be required in its production. The labour, however, that forms the substance of value, is homogeneous human labour, expenditure of one uniform labour power. The total labour power of society, which is embodied in the sum total of the values of all commodities produced by that society, counts here as one homogeneous mass of human labour power, composed though it be of innumerable individual units. Each of these units is the same as any other, so far as it has the character of the average labour power of society, and takes effect as such; that is, so far as it requires for producing a commodity, no more time than is needed on an average, no more than is socially necessary. The labour time socially necessary is that required to produce an article under the normal conditions of production, and with the average degree of skill and intensity prevalent at the time. The introduction of power-looms into England probably reduced by one-half the labour required to weave a given quantity of yarn into cloth. The hand-loom weavers, as a matter of fact, continued to require the same time as before; but for all that, the product of one hour of their labour represented after the change only half an hour’s social labour, and consequently fell to one-half its former value.

Marx uses direct proportion here (which still produces an isomorphism of fields). But direct proportion has one property that the above account does not; direct proportion is a monotonic increasing relationship between socially necessary labour time and direct price. This means it also produces an isomorphism of ordered fields ( it is an order preserving map ).

We can weaken Marx's assumptions (based on cotton loom prices) here and end up with much the same algebraic relationship; we don't need direct price and socially necessary labour time to be proportional to one another, we just need direct price to be a monotonic increasing function of socially necessary labour time.

One reason that such a weakening might be useful is in the analysis of the relationship between inflation and the minimum wage. If the minimum wage increases, we can expect an increase in the direct prices associated with any amount of social labour; labour becomes more valued. If inflation occurs, then the amount of each commodity that the same minimum wage as before buys is decreased; labour becomes less (relatively) valued. These two forces can (and usually do) operate at the same time, but I would be very surprised if the two had a linear relationship; this is what would occur when increases in the minimum wage were a constant multiple of inflation - that the graph between them is a straight line. This doesn't happen often (look at any real wage graph).

This means that f, the mapping from direct prices to socially necessary labour times, is a time varying function. The above relationship between the fields works at any specific time point, but the isomorphism breaks down for similar reasons to the ideal equivalence between real price and socially necessary labour time. That is, rapid fluctuations in the value of commodities or money undermining the stability of (equivalently valued) trade (exchange).
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Sketch for later: one feature of the relationship between direct and real price.

As highlighted before, direct and real prices can (and usually do) diverge. This is pretty much a fact. A good requires a minimal cost to produce, but Bunnpris and Rema might sell it for different prices. This means that any supposed function mapping direct to real prices wouldn't actually be a function at all: it would be one-to-many. If Bunnpris sells Eldorado Tuna for 8.3NOK and Rema sells it for 8.2 NOK, it's the same commodity in the same amount, so both socially necessary labour times are equal. But the prices aren't. This means that direct price and price cannot have a direct mapping even when evaluating at precisely the same time.

A sketch of how to make a monotonic relationship between direct prices and prices might proceed as follows:
(1) Real prices are there to produce profit.
(2) It would be impossible to profit on the immediate sale of a commodity if the prices of production equalled the real price of the good in exchange.
(3) This means that the direct prices of production are a lower bound for the prices of sale.

So instead of associating the direct price of a commodity with a real price, we should associate the direct price of a commodity with a range of real prices. The range of sale prices for any commodity will have the direct price its greatest lower bound assuming sale is always sale for profit.

This means the function g that maps direct to real prices maps specific direct prices to intervals of real prices. EG, say the direct price of production for Eldorado Tuna is 3NOK, then any sale above 3 NOK is consistent with sale for profit. That means we can map:

direct price x->(direct price x, infinity)

And set up an ordering on the intervals (x, infinity) in terms of their greatest lower bound. EG, if direct price x < direct price y, then the corresponding order on real price intervals would be (direct price x, infinity) < (direct price y, infinity). This can be equipped with a distribution of real prices obtained from real sale data, which is the only way to represent all the different real prices that the same commodity can have.

One consequence of this is that while direct prices are essentially deterministic, real prices are essentially stochastic; the features of the distribution depending on where and when you are, and only easily summarised as an aggregate.
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Mathematical note: the algebra of commodities and the algebraic structure of value.

A tacit assumption in the mathematical model I've made so far is that commodity direct prices for a specific amount of that commodity are a real or rational multiple of the per unit price or per unit socially necessary labour time. Often, however, an exchange consists of multiple commodities for multiple commodities of equivalent value or of multiple commodities for their real price.

Given the previous sketch on the relationship of direct to real prices, I'm going to deal solely with the mathematical relationship between multiple commodities being traded for their direct price. This is because the manifest real prices of commodities are chaotic and impossible to map consistently, but the minimal price they could be sold for profit is more well behaved (at least in principle). I am making an assumption that the socially necessary labour time for the production of a given batch of commodities is constant here, and this means the analysis will not hold when the socially necessary labour time is changing due to innovation in production.

Say someone buys an iPhone with an iPhone case. The direct price of the sum is the sum of the direct prices. Similarly with the iPhone, the direct price of the iPhone is the sum of the direct prices of its components. So by writing:

iPhone + iPhone case = (component 1 of iPhone + component 2 of iPhone + ... + final component of iphone) + price of assembly of iPhone + iPhone case

I mean to say that the two sides are of equivalent direct price. There are two principles here:

(1) The direct price of a commodity is the sum of the direct price of its components plus the direct price of that commodity's assembly (as determined from the socially necessary labour time of assembly).
(2) The direct price of a set of commodities is equal to the sum of the direct prices of each commodity.

I think it's appropriate, when talking about the direct price or socially necessary labour time of a single commodity, to fold the price of assembly into the direct price. However when looking to compose commodities into other commodities the assembly price must be added on; adding it on keeps track of the money expression of the socially necessary labour time. The direct price must reflect the total conditions of production for that commodity.

How does this tie in with the algebraic structure of value? This is mostly book-keeping as the mathematical machinery developed with the ordered field isomorphism does the work of translating labour times to direct prices. The only thing to construct is a mapping from collections of commodities to their socially necessary labour times - the above construction maps this to direct price (and then the sketch maps it to real price). I'll deal with point (2) first then come back to point (1)

The meat of this is translating the meaning of addition of commodities to a trade to the meaning of adding socially necessary labour times. So when Apple produces an iPhone and an iPhone case, writing this as:

iPhone+iPhone case

means that Apple has produced one iPhone and iPhone case.*

Writing it in terms of direct prices we have the same thing:

Value of (iPhone) + Value of (iPhone case)

We also have that if there were 2 iPhones and 2 iPhone cases produced, we have:

2 iPhones + 2 iPhone cases

becoming

(A) Value of (2 iPhones) + Value of (2 iPhone cases)

which is equal to

(B) 2*Value of (iPhone) + 2*Value of (iPhone case)

there is a subtlety here. The algebraic structure of commodities differs from that of values - we can't make -1 apples, but we absolutely can subtract the value of 1 apple from a trade to reduce the value in accordance with removing a commodity from the trade. This means that the above way of composing commodities into multi-commodity trades through addition is not immediately an ordered field, since there's no sense of subtraction of raw commodities.

If we have a list of all commodities C={C1, C2, ... , Cn}, the set of multi-commodity trades which can be made from that list is just the collection of all sums of commodities in the sense of *. If C1=iPhone and C2=iPhone case this just gives C1+C2 as the items available in the trade. This makes the set of all trades the set of all (finite) such sums.

C1+C2 is clearly the same as C2+C1, and C1+(C2+C3) is clearly the same as (C1+C2)+C3, only which goods are on the table matter at all, not how they are presented. This means that the + operation in sense * is associative and commutative. This means that the set of all possible trades is (isomorphic to) the free commutative monoid on C. It also comes equipped with the multiplicative group action of the whole numbers on it. EG C1+C1=2*C1 even though 2 is not a commodity. IE If N of the same commodity P are produced, this can be written as P*N (like 2 iPhones + 2 iPhone cases = iPhone + iPhone + iPhone case + iPhone case).

(2) The direct price of a set of commodities is equal to the sum of the direct prices of each commodity.

This states that the direct price of a multicommodity trade is just the sum of the direct price of each commodity. Now there's some extra formalism developed, principle 2 is equivalent to this:

the mapping from a set of commodities in a trade to its socially necessary labour time is homomorphism of commutative monoids (with 0 as the identity element), which are fields that have forgotten the idea of additive and multiplicative inversion (there's no such commodity as 1/iPhone or -iPhone). (A) and (B) as labelled above demonstrate the required properties of a commutative monoid homomorphism. Specifically, this mapping takes the above algebra of commodities and maps it to the additive part of the ordered field of values.

However, in this case it is not an isomorphism (and moreover one cannot exist). This is because the value of two distinct amounts of different commodities can be equal (so the mapping is not injective).

This then allows elements of (C,+) to be ordered by mapping them to their values. EG, if C1+C2 costs less (direct price) than C3+C4, C1+C2<C3+C4). Also, the equivalence classes of value take their form here as the preimage of a direct price. EG, if the value of C1 = C2 and both have price 2, C1 and C2 would be in the set of all elements that have price 2 - the inverse image of 2 under the homomorphism.

Edit: regarding the action of the whole numbers on the commodities. More generally, multiplication by fractions and reals should be allowed. EG, if a 1 g of diamond had a direct price of 1 unit, 1/sqrt(2) g of diamond has a direct price of 1/sqrt(2) units. It may be more appropriate to model a multicommodity trade as a vector space over the reals with a certain mapping to the reals that corresponds to the evaluation of direct price. I'll sketch this out later.

Edit2: the vector space idea doesn't work.
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Principle (1), the ability to decompose commodities into constituent commodities and assembly prices is tricky. Might take some time to model it.
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Have a sketch of how to decompose a commodity into its constituents while still satisfying the above algebraic relationships, but it's more based on things from chapters 1 and 2.

If you take an iPhone and split it into the commodities which are used to assemble it, the socially necessary labour time of the iPhone will be equal to the sum of the socially necessary labour times for the composite of commodities which make it plus the assembly time. If we define a function D that takes a commodity and splits it into the sources of socially necessary labour time that together constitute the total for that commodity we'd end up with something like:

D(iPhone) = SNC(C1 + C2 + C3 + ... +Cn)+SNC(assembly of iPhone)

where SNC is the socially necessary labour time. But we know from how direct prices and labour times work - they're additive/associative/commutative, this results in:

D(iPhone) = D(C1 + C2 + C3 + ... + Cn) + SNC(assembly of iPhone)
implies
D(iPhone)=D(C1)+D(C2)+D(C3)+...+D(Cn)+SNC(assembly of iPhone)

similarly, C1 through Cn will consist of commodities which have a socially necessary labour time for production and an overall assembly time for each commodity from its constituents. The trick here is to notice that this procedure of the iterative application of D bottoms out in raw goods, which have not been subject to labour and therefore have no exchange value. This makes D partition the commodity into the raw goods which constitute it - which do not contribute to the value - and the socially necessary labour times for transforming raw goods into the commodities. The consequence of this is that just as value consists essentially in (abstract) labour times, the socially necessary labour time consists of sums of abstract labour alone; transformative activities on raw goods, and transformative activities on bought commodities. So long as commodity labels remain in the the expression for D(x), the procedure iterates down to labour times, which are already isomorphic to direct prices.

What remains to be shown, though I think it's quite obvious, is that D as defined above agrees with the direct price decomposition as defined previously.
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Since there was interest, I'm going to continue giving my exegesis of the book, and take a break from the mathematical modelling above. I'll return to it once I've done more exegesis. I'll be going through chapter/section 1, which stretches from the opening page to the end of the section on the fetishism of commodities.

The wealth of those societies in which the capitalist mode of production prevails, presents itself as “an immense accumulation of commodities,”[1] its unit being a single commodity. Our investigation must therefore begin with the analysis of a commodity.

This is pretty straightforward. Capitalist production makes a lot of stuff, and people have bought that stuff; making capitalist production appear as the accumulation of stuff people have bought. Note that the analysis begins from something worldly; Marx has noticed that capital accumulates a lot of stuff that people have bought; it isn't a purely philosophical assumption.

A commodity is, in the first place, an object outside us, a thing that by its properties satisfies human wants of some sort or another. The nature of such wants, whether, for instance, they spring from the stomach or from fancy, makes no difference.[2] Neither are we here concerned to know how the object satisfies these wants, whether directly as means of subsistence, or indirectly as means of production.

Commodities satisfy human wants and needs. The only thing which matters for analysis is that they satisfy human wants and needs, not how. This is an important methodological technique as it abstracts from the particular nature of commodities; how commodity x satisfies needs y; to their general structure; that commodities always satisfy wants and needs.

Every useful thing, as iron, paper, &c., may be looked at from the two points of view of quality and quantity. It is an assemblage of many properties, and may therefore be of use in various ways. To discover the various uses of things is the work of history.[3] So also is the establishment of socially-recognized standards of measure for the quantities of these useful objects. The diversity of these measures has its origin partly in the diverse nature of the objects to be measured, partly in convention.

Qualities of iron/paper etc are what wants and needs they satisfy and how they satisfy them; this is why studying human behaviour as it regards those commodities reveals the uses, and is a historical endeavour. How much of a given thing there is is also a function of the raw physical amount and the systems of measurement which have been developed. EG, cows come in whole number lots, iron comes in grams, paper can come in notepads or pieces etc.

The utility of a thing makes it a use value.[4] But this utility is not a thing of air. Being limited by the physical properties of the commodity, it has no existence apart from that commodity. A commodity, such as iron, corn, or a diamond, is therefore, so far as it is a material thing, a use value, something useful.[/b]

This is simply to state that how an item satisfies certain wants and needs depends on the material properties of that commodity (and how those properties are made use of). Corn satisfies hunger by being a food, specifically a hardy, insect resistant crop full of carbohydrates and nutrients.

This property of a commodity is independent of the amount of labour required to appropriate its useful qualities. When treating of use value, we always assume to be dealing with definite quantities, such as dozens of watches, yards of linen, or tons of iron.

This is to say that iron, as a construction material for munitions and boats, does not have the ability to be used for that construction if it comes in the quantity of a microgram. So Marx takes the dependence of use values on the amount of a commodity as something largely irrelevant; so long as we ascribe a use value to a thing, that can only be done insofar as there is enough of that thing for the use value. This ties in neatly with:

The use values of commodities furnish the material for a special study, that of the commercial knowledge of commodities.[5] Use values become a reality only by use or consumption: they also constitute the substance of all wealth, whatever may be the social form of that wealth. ||| In the form of society we are about to consider, they are, in addition, the material depositories of exchange value.

the bolded part. Use values are only use values insofar as they are used. Unprocessed iron in the ground, or an iron repository in a closed factory, do not strictly speaking have use values since they are not being utilised. There is still the []potential of use[/i] in those items afforded to them by their material properties. To put it another way, the formal character of a use value is to be expended through use.

Another feature of use values is that their accumulation constitutes wealth. Someone would be wealthy in virtue of having all they want and need forever. Wealth, as related to use values alone, is constituted by the fulfilment of a wants and needs; and having access to sufficient resources for it requires a certain availability and accumulation/storage of use values.

The sense of wealth in the previous paragraph doubtlessly seems very twee compared to the one we have now; but this is because the sense of wealth as the accumulation and availability of useful resources does not reflect the structure of wealth now; wealth under capitalist societies. In those societies, the family home of the commodity, goods also have exchange value and serve as its material depositories. Similar to previous discussion, commodities in certain amounts serve as a material expression of a social structure.

Exchange value, at first sight, presents itself as a quantitative relation, as the proportion in which values in use of one sort are exchanged for those of another sort,[6] a relation constantly changing with time and place. Hence exchange value appears to be something accidental and purely relative, and consequently an intrinsic value, i.e., an exchange value that is inseparably connected with, inherent in commodities, seems a contradiction in terms.[7] Let us consider the matter a little more closely.

bolded bit is the first reference to the 'lawless irregularities' that are the concrete acts of trade; exchange. Nevertheless Marx insists that there is, regardless, a social form of exchange which embeds itself in each exchange; constraining them, mediating them and ultimately reproducing itself as a social form in those concrete acts; which nevertheless is not a purely material property of the commodities themselves. From the previous vocabulary, Marx's reference to exchange value as intrinsic to the commodities sets up the value form as a real abstraction. It is intrinsic to the social significance of the commodity and the social form which uses them as such, not to the good stripped of all context and left as a barren composite of matter.

A given commodity, e.g., a quarter of wheat is exchanged for x blacking, y silk, or z gold, &c. – in short, for other commodities in the most different proportions. Instead of one exchange value, the wheat has, therefore, a great many. But since x blacking, y silk, or z gold &c., each represents the exchange value of one quarter of wheat, x blacking, y silk, z gold, &c., must, as exchange values, be replaceable by each other, or equal to each other. Therefore, first: the valid exchange values of a given commodity express something equal; secondly, exchange value, generally, is only the mode of expression, the phenomenal form, of something contained in it, yet distinguishable from it.

Marx points out that in order for exchange to occur as it usually does, the commodities which are exchanged must be equivalent in some manner. This is not necessarily to say that they are equal in all respects, like a numerical identity or both being rigidly designated by the same name; but to say that they count as one another in a sortal sense of equivalence. And just like such sortals, counting as one another is ultimately a consequence of the social organisation of trade.

I will highlight it again because it's important; exchange value is embedded in the commodities, but is distinct from every commodity. The mathematical structure of this embedding is what I've been wrestling with in a lot of these posts.

Let us take two commodities, e.g., corn and iron. The proportions in which they are exchangeable, whatever those proportions may be, can always be represented by an equation in which a given quantity of corn is equated to some quantity of iron: e.g., 1 quarter corn = x cwt. iron. What does this equation tell us? It tells us that in two different things – in 1 quarter of corn and x cwt. of iron, there exists in equal quantities something common to both. The two things must therefore be equal to a third, which in itself is neither the one nor the other. Each of them, so far as it is exchange value, must therefore be reducible to this third.

This is an interesting argument; a kind of transcendental deduction. If all the properties of x and all the properties of y are held equal in some sense, what renders that equality must be external to the properties of both; and instead is a feature of the relation of the two. Because changing the quantity of one commodity changes the quantity of the other that may exchange with it, the commensurability of x and y must be expressed in terms of an abstract quantity with which they are both equal.

Contained in this argument are the germs of the relative form and equivalent form of value, and of the universal equivalent. That we exchange x for y doesn't tell us why we can exchange x for y; we say that we can exchange them when they are of equivalent value, but this equivalence as a numerical property must be abstracted from the material properties and relative uses of both in the trade.
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Marx continues his argument for the externality of value to the commodities in an exchange.

A simple geometrical illustration will make this clear. In order to calculate and compare the areas
of rectilinear figures, we decompose them into triangles. But the area of the triangle itself is expressed by something totally different from its visible figure, namely, by half the product of the base multiplied by the altitude. In the same way the exchange values of commodities must be capable of being expressed in terms of something common to them all, of which thing they represent a greater or less quantity.

This time through an analogy. Updating the analogy somewhat is useful. Say we have a square and a circle and the two are equal in terms of area. This means that the square of side h, of area $h^2$ and the circle of radius r, with area $\pi r^2$, are equal. We can say that the square and the circle are of equal area because $h=\sqrt{\pi}r$. The only reason we can make this equation is that the two are already posited as equal in area; and the idea of 'equal area' is no more contained in the square than the circle. That they are of equal area follows from the application of the abstraction of area; which nevertheless can be read as an inherent/intrinsic property of both the circle and the square. The analogy functions by mapping 'are of equal area' to 'are of equal value' and of mapping 'circle, square' to 'commodity 1, commodity 2'. 'This size' and 'that size' become commensurable through 'area'. This commodity and that commodity become commensurable through exchange value.

This common “something” cannot be either a geometrical, a chemical, or any other natural property of commodities. Such properties claim our attention only in so far as they affect the utility of those commodities, make them use values. But the exchange of commodities is evidently an act characterised by a total abstraction from use value. Then one use value is just as good as another, provided only it be present in sufficient quantity. Or, as old Barbon says,“one sort of wares are as good as another, if the values be equal. There is no difference or distinction in things of equal value ... An hundred pounds’ worth of lead or iron, is of as great value as one hundred pounds’ worth of silver or gold.”[8]

Again, the emphasis here is that value is a (real) abstraction that operates on commodities in an exchange and isn't essentially constituted by material properties of either commodity. Since the use values of commodities depend entirely on their material constitution, this means that exchange value is essentially different from use value. A more striking way to use Barbon's equivalence is to suggest that because, say, 1 mole of iron is worth 1 mole of copper, 1 atom of iron is worth 1 atom of copper - the raw proportion of the two is all that matters in determining their relative value, irrespective of every everyday use of both being destroyed by the amounts present.
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Did Marx think there is such a thing as value other than that which created by consumer demand?
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Marx concludes his derivation of the distinction between use and exchange values:

As use values, commodities are, above all, of different qualities, but as exchange values they are merely different quantities, and consequently do not contain an atom of use value.

Which prefigures his argument for the alliance of labour and exchange value.

If then we leave out of consideration the use value of commodities, they have only one common property left, that of being products of labour. But even the product of labour itself has undergone a change in our hands. If we make abstraction from its use value, we make abstraction at the same time from the material elements and shapes that make the product a use value; we see in it no longer a table, a house, yarn, or any other useful thing. Its existence as a material thing is put out of sight. Neither can it any longer be regarded as the product of the labour of the joiner, the mason, the spinner, or of any other definite kind of productive labour. Along with the useful qualities of the products themselves, we put out of sight both the useful character of the various kinds of labour embodied in them, and the concrete forms of that labour; there is nothing left but what is common to them all; all are reduced to one and the same sort of labour, human labour in the abstract.

Breaking it down, since it's an important logical step in his value theory.

If then we leave out of consideration the use value of commodities, they have only one common property left, that of being products of labour.

Marx believes the 'if' there is vindicated by the qualitative independence of use and exchange values. He then asks 'what are the common properties left of two commodities x and y which are exchanged for one another in a definite amount?', and concludes quite sensibly that the only share property left is that they are both products of human labour. I think, instead of treating this as an argument from exhaustion, IE that Marx has surveyed the entire manifold of shared properties barring material properties of both commodities x and y and found that the only non-material property is that they are products of labour, we treat this as an argument to best explanation with a few motivating features.

Firstly, we already have that exchange is a social relation, and so the equivalence of two commodities in value reflects a social property. Secondly, we have that both commodities were produced; they were made. Something which satisfies both of these conditions is labour. Labour is a socially organised phenomenon that all commodities are brought into the world by; it transforms raw goods into commodities. So it's not ruled out by the ruling out of use values, and it is a shared non-material property of both commodities; they are products of labour.

That commodities are products of labour and also that the material constitution of the commodities does little to influence their equivalence exchange value also constrains the notion of labour which is suitable to facilitate the equivalence of exchange values.

But even the product of labour itself has undergone a change in our hands. If we make abstraction from its use value, we make abstraction at the same time from the material elements and shapes that make the product a use value; we see in it no longer a table, a house, yarn, or any other useful thing.

Imagine you are a potter and you're tasked with making a salad bowl and a plate, the salad bowl has to be deeper than the plate to facilitate tossing and the overall bulk of food expected to be in a serving dish; the plate can be much thinner than the bowl because it will be a vessel for less food than the salad bowl, and the food manipulated on it will be done with finer tools like forks and knives. All that changes between clay-arranged-platewise and clay-arranged-bowlwise are the concrete circumstances of labour in their production. The concrete circumstances of production are what shapes resources into use values.

However, the concrete circumstances of production are precisely what have been abstracted away when considering the shared properties of the commodities that may facilitate their equivalence in exchange value.

One further reason such an abstraction makes sense is that, insofar as commodities are different use values, they represent qualitatively incommensurable uses and are the result of qualitatively incommensurable labours. With reference to later; the only thing the expanded form of value preserves is value; it destroys all specificities that are not value determinative.

This only leaves a ghostly image of labour; a trace of labour devoid of specificities; a generalised labour that appears as labour only insofar as labour has been done. This is human labour in the abstract, the texture of value. What can be said of such an abstract form of labour? Well, this could be facilitated by looking at labour phemomenologically.

What commonalities are there between the potter making a plate and the potter making a bowl? Well, the potter is using clay, but that is a property of the material constitution of the commodities they make. The potter shapes the bowl to be deeper than the plate; but the different spinning and moulding patterns are concrete features of labour and therefore cannot facilitate this abstract equivalence. The bowl and the plate are both made of clay; but the bowl and a knife would be equally exchangeable in certain quantities, of equal value, independent of the material constitution. The bowl and the plate, made through the variation in thickness and depth on the potter's wheel, are as alien from one another as the bowl and the knife insofar as exchange is concerned. Being of common material constitution didn't help, neither did being involved in the same household activity (eating). The only things which remain are quantitative properties under which various qualitatively different labours can be held equivalent; just as they are held equivalent in value.

What property of every concrete labour is immediately given a magnitude? We have a choice between that labour was expended tout court, and of a magnitude of that exertion. Such a magnitude, however, must be indifferent to the intensity of work; as work intensity is a concretely qualitative feature of labour. It is indexed to a particular productive processes, rather than a generalisation over productive processes. So we may be able to say that labour is 'light' or 'back breaking', but this only provides an ordinal measure of the exertion; it brings productive processes into qualitative contrast as much as it brings them into a rough ordinal agreement. How much of back-breaking commodity x is worth light commodity y? The analysis of exertion intensity does nothing to answer this question.

The same argument could be applied to the average expenditure of calories in different productive processes; but such an average over all productive processes renders each of expectation proportional to...

The labour's duration, and that duration signals exertion over the duration. We find in this spectre of labour simply that a person exerts themselves over time. Dealing with the intensity of labour results in dealing with the duration of labour by proxy. So we're now left to analyse how the amount of abstract labour congealed in the commodity is determinative of its exchange value.

Edit: I believe Marx would have folded the energy expenditure/work intensity away into the concrete specificities of labour. The important thing to note about such properties is that they allow the results of productive processes to be partitioned into classes of equivalent value - the ordinal measures do that minimally but do not allow intra or inter class comparisons of differing magnitudes. The caloric expenditure does allow bringing different productive processes into equivalence classes of intensity, but we end up with essentially the same model as Marx does following this path. Caloric expenditure becomes average caloric expenditure over all people and productive processes per unit time, then the amount for any particular process is proportional to the duration. And the duration is what Marx works with.

Literal proportion, as highlighted before, is an isomorphism of ordered fields, so it makes absolutely no difference to the mathematical developments I've expressed either.

Supply and demand fluctuations affect real prices but not direct prices. Direct prices are a reflection of the minimal amount of labour (or modal amount of labour depending on whether we're in productive disequilibrium and the two fail to coincide) required to make that commodity in the sphere of capitalist production. Real prices are whatever the bloomin' thing costs, and only reflect direct prices insofar as those direct prices provide the greatest lower bound for the sale price of that commodity assuming sale for profit.
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Direct price isn't necessarily the lower boundary. I'd sell you hardware for less than it costs to make it hoping I can sell you software downstream.

The hardware was made by a robot. Some algorithm that includes depreciation and phases of the moon is used to estimate the cost of manufacturing it.
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Direct price is the lower boundary assuming sale of that commodity for profit. This is exactly the assumption I made. If it's not sale for profit of that commodity the lower bound doesn't hold.
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Let us now consider the residue of each of these products; it consists of the same unsubstantial reality in each, a mere congelation of homogeneous human labour, of labour power expended without regard to the mode of its expenditure. All that these things now tell us is, that human labour power has been expended in their production, that human labour is embodied in them. When looked at as crystals of this social substance, common to them all, they are – Values.

Again, pay attention to the homogeneity of labour in the abstract; it is labour done by the everyperson and only insofar as the general properties of labour can it relate to and determine value. Also, Marx's reference to this labour as a 'social substance' reinforces that value is produced through the social organisation and facets of production and labour in general. Abstract labour and value are social stuff embedded in commodities intrinsically as a feature of capitalist production, and lay there intrinsically so long as production is capitalist production.
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Again, pay attention to the homogeneity of labour in the abstract; it is labour done by the everyperson and only insofar as the general properties of labour can it relate to and determine value. Also, Marx's reference to this labour as a 'social substance' reinforces that value is produced through the social organisation and facets of production and labour in general. Abstract labour and value are social stuff embedded in commodities intrinsically as a feature of capitalist production, and lay there intrinsically so long as production is capitalist production.

I don't mean to suggest that the causality is one way here - I think it's also part of the account that so long as commodities are exchanged as equivalents the mode of production is capitalist. The social structure of exchange is just as important as the social structure of production in capital; and the two reinforce the other socio-economically. Try to reorganise production, it will happily collapse back into capitalist production through wealth concentration so long as commodities are bought for money; try to do away with money, you will be isolated from the fruits of production.
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Direct price is the lower boundary assuming sale of that commodity for profit. This is exactly the assumption I made. If it's not sale for profit of that commodity the lower bound doesn't hold.

I understand. It's a limited analysis.
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Eh, I still find it convincing for the most part. On the back of a commodity which is sold below its direct price, in order to generate a profit other commodities must be sold sufficiently above their direct price. This is just how to make a profit when using that strategy. That for a given owner of a profile of commodities, some of which are sold below their direct price, others must be sold more above their direct price to still create a surplus despite the loss. This still conforms to the analysis so far. Relabel commodity with commodity profile and you're done - the profit being the sum total of profits minus loss of those commodities which are being sold below their direct price.

In a world where every commodity, or even most commodities, were sold below their direct price I'd think your criticism was more convincing. Especially considering when you aggregate things as above (and as, say Nintendo probably did for the Wii, analysing in terms of yearly/quarterly etc total surplus or loss ) it works just the same as before.
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In a world where every commodity, or even most commodities, were sold below their direct price I'd think your criticism was more convincing.

My interest is in agendas and how they differ by culture and era. If you compare the agendas of a 19th Century British factory owner to that of an American raider in 1985, the word "capitalist" sort of fades in significance. The raider isn't trying to make a profit. He or she is not even interested in running a successful business.

The commodity that broke the system in 08 can't really be analyzed in terms of cost.

Much of what seemed carved in stone to Marx is gone now. What made it seem so solid back then? Psychology. Culture. That's what's fascinating to me.

Carry on.
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I don't really see the point in what you're saying, it doesn't function as a criticism. It's like saying that studying organic chemistry is useless because of its insufficient emphasis on the organic chemistry of dolphins. Regardless, Marx does write at length about business, private property, business cycles and speculation (which was a thing back then too!) elsewhere in the book.

Don't worry frank, the dolphins will come later.
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Anyone who thinks aggregate economic trends reduce entirely to psychology and culture must be completely baffled by the accumulation of wealth, redistributive measures, the entanglement of race, gender, poverty and crime. Even the most incompetent politician or business owner tries to influence the conditions of production and exchange systemically. This while acknowledging that all of these things come down to the objective features of social arrangements.

Being less informed about how things work than incompetent politicians and business owners is not a perspective that should be promoted. I think I'll start calling it Taylor Swiftism.
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I don't really see the point in what you're saying, it doesn't function as a criticism.

As I said, it's a limited analysis.

Anyone who thinks aggregate economic trends reduce entirely to psychology and culture must be completely baffled by the accumulation of wealth, redistributive measures, the entanglement of race, gender, poverty and crime. Even the most incompetent politician or business owner tries to influence the conditions of production and exchange systemically. This while acknowledging that all of these things come down to the objective features of social arrangements.

Being less informed about how things work than incompetent politicians and business owners is not a perspective that should be promoted. I think I'll start calling it Taylor Swiftism.

This is close to completely meaningless.
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Instead of my reconstruction of Marx's argument, he continues:

Let us now consider the residue of each of these products; it consists of the same unsubstantial reality in each, a mere congelation of homogeneous human labour, of labour power expended without regard to the mode of its expenditure. All that these things now tell us is, that human labour power has been expended in their production, that human labour is embodied in them. When looked at as crystals of this social substance, common to them all, they are – Values.

We have seen that when commodities are exchanged, their exchange value manifests itself as something totally independent of their use value. But if we abstract from their use value, there remains their Value as defined above. Therefore, the common substance that manifests itself in the exchange value of commodities, whenever they are exchanged, is their value. The progress of our investigation will show that exchange value is the only form in which the value of commodities can manifest itself or be expressed. For the present, however, we have to consider the nature of value independently of this, its form.

A use value, or useful article, therefore, has value only because human labour in the abstract has been embodied or materialised in it. How, then, is the magnitude of this value to be measured? Plainly, by the quantity of the value-creating substance, the labour, contained in the article. The quantity of labour, however, is measured by its duration, and labour time in its turn finds its standard in weeks, days, and hours.

I think I've highlighted the relevant points of it before, and explaining how 'x expresses y' or 'x is embodied in y' work through real abstractions is something I've done a lot too. So I'll skip it here, especially since I reconstructed the argument above (with some embellishments).

One really interesting part of this bit is that just as the structure of value is calculable, so is the structure of time conformable with that value structure. We split time periods into days, hours, minutes etc; and the division of time facilitates both the management of and ease of measurement of labour. Again; value is inherently calculable. We can keep track of it and do algebra with it.

Trouble, trouble, trouble!

I don't think I'll ever make a better pun than Taylor Swiftism. :(
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On the back of a commodity which is sold below its direct price, in order to generate a profit other commodities must be sold sufficiently above their direct price. This is just how to make a profit when using that strategy. That for a given owner of a profile of commodities, some of which are sold below their direct price, others must be sold more above their direct price to still create a surplus despite the loss. This still conforms to the analysis so far. Relabel commodity with commodity profile and you're done - the profit being the sum total of profits minus loss of those commodities which are being sold below their direct price.

While I agree that Frank's interjection is more or less entirely irrelevant, I do wonder - and this is a stray thought that I haven't fully thought all the way through - if this holds for so-called debt capitalism: when the commodity being sold simply becomes a vehicle for debt, and where the goal is to keep the debtor in debt for as long as possible in order to squeeze interest out of them in the long-term (allowing a commodity to then be sold below cost price in the short term so as to recoup the losses in interest payments in the long term); one can imagine a model where the commodity then is always sold below cost and it is time which becomes the source of value, as it were. Would this kind of thing - even as an idealized model - affect the analysis here?
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How, then, is the magnitude of this value to be measured? Plainly, by the quantity of the value-creating substance, the labour, contained in the article. The quantity of labour, however, is measured by its duration, and labour time in its turn finds its standard in weeks, days, and hours.Marx
Human labor. He doesn't realize he's developing industrial engineering.

That's not a criticism. It's an interesting historical thingy.

Not everybody is trying to kill you, fdrake. Some people are just staring at the clouds mumbling to you. Nothing threatening.
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Next Marx addresses a common confusion, reiterating the distinction between concrete labour and human labour in the abstract.

Some people might think that if the value of a commodity is determined by the quantity of labour spent on it, the more idle and unskilful the labourer, the more valuable would his commodity be, because more time would be required in its production. The labour, however, that forms the substance of value, is homogeneous human labour, expenditure of one uniform labour power.

It's homogenous because it's had all the things which make labour non-generic have been stripped away; and this was necessary because the specifics of the labours didn't matter to the magnitude of the value of commodities.

This does something methodologically too; because exchange value abstracts away from the specifics of labour, it characterises human labour in the abstract and its inherent calculability as a systemic property of capitalist production. So, methodologically, the analysis of value has to 'zoom out' to the total productive behaviour of, for Marx, society. Today, we should update this to the global sphere of production.

I don't think this update does too much to the account, as 'society' is largely a placeholder. 'Society' functions as the playground of the relevant social and productive relations. Now the entire world does.

The total labour power of society, which is embodied in the sum total of the values of all commodities produced by that society, counts here as one homogeneous mass of human labour power, composed though it be of innumerable individual units. Each of these units is the same as any other, so far as it has the character of the average labour power of society, and takes effect as such; that is, so far as it requires for producing a commodity, no more time than is needed on an average, no more than is socially necessary.

If we have a total of values for a society, the individual values which constitute it are a proportion of that value; and those values have a magnitude of value in accordance with their proportion. The specific magnitude of value that a commodity congeals is whatever labour is deemed socially necessary within it. Socially necessary labour time as an abstraction has two functions here:

(1) It provides the link between human labour in the abstract and the conditions of production; stuff in general is produced in whatever time it is required to be.
(2) It comes along with a specific value magnitude; the required time.

The labour time socially necessary is that required to produce an article under the normal conditions of production, and with the average degree of skill and intensity prevalent at the time. The introduction of power-looms into England probably reduced by one-half the labour required to weave a given quantity of yarn into cloth. The hand-loom weavers, as a matter of fact, continued to require the same time as before; but for all that, the product of one hour of their labour represented after the change only half an hour’s social labour, and consequently fell to one-half its former value.

It should be noted that 'the labour' underpinning 'the labour time' in the first sentence is human labour in the abstract; generic labour done by the generic person. Marx is highlighting that human labour in the abstract has a kind of average skill and intensity associated with it; individuals in a factory may differ in speed of work, regardless the production has a socially necessary required time to produce a given number of its goods.

What interests me here stylistically is the seamless transition between describing human labour in the abstract to something which could be well understood outside the context. If someone were to say 'Toyota halved the required time to produce their cars, they're doubling their income by reducing costs', it'd make sense. Regardless of the reliance on the agency of a company (Toyota) which only as an aggregate produces cars - which are individually produced by labourers and machines. I think going from concrete labour to human labour in the abstract is very similar to the move implicit in understanding the sentence 'Toyota halved the required time'; of labour devoid of its contextual richness, reduced to the exertion of an aggregate.

While I agree that Frank's interjection is more or less entirely irrelevant, I do wonder - and this is a stray thought that I haven't fully thought all the way through - if this holds for so-called debt capitalism: when the commodity being sold simply becomes a vehicle for debt, and where the goal is to keep the debtor in debt for as long as possible in order to squeeze interest out of them in the long-term (allowing a commodity to then be sold below cost price in the short term so as to recoup the losses in interest payments in the long term); one can imagine a model where the commodity then is always sold below cost and it is time which becomes the source of value, as it were. Would this kind of thing - even as an idealized model - affect the analysis here?

I think this is a good point, the production of commodities for sale is only one part of the way profit can be made under capitalist production. I think ultimately this comes down to the ability to commodify and size up the price of anything. That is to say, I believe most financial operations take place in the internal tension of value creation and price. Marx knows you can hang a price on anything. There's some material directly about speculation later too.

One interesting thing I've seen bandied about in this kind of discussion - of capital as debt capital - is that financial operations are highly destabilising; crashes, hyperinflation etc occur as part of the financial operations of capital. It's part of the usual economic narrative to attribute the capacity for financial transactions to destabilise an economy to the unsustainability of their relation to demand, supply, and the real profits of the companies (or real repayments of the debtors) the speculation is based on. I think a similar move would be available to Marx; just as finance capital is an inherent part of capital, so too is the destabilising tension between it, value production and wage labour.

I don't know what you're talking about at all, sorry.
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I was checking to see if the zombie apocalypse has made it to Scotland
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Another part of my response should have been; financial operations don't typically come along with the sale of commodities; they don't produce profit through the direct extraction of surplus value, rather they re-appropriate already created value through purchase. Things like land or oil extraction rights or shares or improvements in mental health from therapy are not commodities for Marx, regardless of the fact they're on sale.
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Lastly, I don't see an analysis of commodity production which characterises it as the sole producer of values as something completely at odds with an analysis of the re-appropriation of value through money repayments of debt.
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