## Marx's Value Theory

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No worries!
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Going through section 2 again, I got some details about the elementary form of value wrong - symmetry of the trade relation is baked in from the start. The elementary form of value consists of otherwise unstructured symmetric pairs of pairs, so instead of just being the minimal reflexive relation, it's the minimal reflexive+symmetric relation of trades seen from either side - another feature of it is that the relation is not consistent with a strict ordering on trades. There's also a discussion to be had over whether stipulating that the T relation is reflexive is a harmless embellishment, a decent exegesis or occludes part of Marx's account, which I will do. The expanded form still appears to consist in the transitivity condition and the money commodity still appears to function like the canonical representative of a value equivalence class, though the formal equality of these ideas with Marx's turns on whether it's OK to treat the trade relation as reflexive.

So yeah, there's some work to do - clearing up previous misconceptions and discussing the introduction of orders that respect the value relation and the commutative monoid of commodities. What I wrote previously - especially the bits after the elementary form - I believe is largely fine.

note to self: sum like vs mean like orders in prospect theory
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Marx recaps the 'spectral objectivity' of value in commodities and sets up what is at stake in providing a theory of value.

Commodities come into the world in the shape of use values, articles, or goods, such as iron, linen, corn, &c. This is their plain, homely, bodily form. They are, however, commodities, only because they are something two-fold, both objects of utility, and, at the same time, depositories of value. They manifest themselves therefore as commodities, or have the form of commodities, only in so far as they have two forms, a physical or natural form, and a value form.

This paragraph is an elaboration of the idea that commodities are composites of use and exchange value; the elaboration being that exchange value has some formal characteristics apart from the aforementioned relationship with socially necessary labour times. This formal structure of value in commodities is the ground upon which the magnitudes of value make sense...

The reality of the value of commodities differs in this respect from Dame Quickly, that we don’t know “where to have it.” The value of commodities is the very opposite of the coarse materiality of their substance, not an atom of matter enters into its composition. Turn and examine a single commodity, by itself, as we will, yet in so far as it remains an object of value, it seems impossible to grasp it. If, however, we bear in mind that the value of commodities has a purely social reality, and that they acquire this reality only in so far as they are expressions or embodiments of one identical social substance, viz., human labour, it follows as a matter of course, that value can only manifest itself in the social relation of commodity to commodity. In fact we started from exchange value, or the exchange relation of commodities, in order to get at the value that lies hidden behind it. We must now return to this form under which value first appeared to us.

and situates the structural analysis of value, initially, within an account of exchange as the exchange of commodities...

Every one knows, if he knows nothing else, that commodities have a value form common to them all, and presenting a marked contrast with the varied bodily forms of their use values. I mean their money form. Here, however, a task is set us, the performance of which has never yet even been attempted by bourgeois economy, the task of tracing the genesis of this money form, of developing the expression of value implied in the value relation of commodities, from its simplest, almost imperceptible outline, to the dazzling money-form. By doing this we shall, at the same time, solve the riddle presented by money.

which comes to underpin the application of prices to them. Thus...

The simplest value-relation is evidently that of one commodity to some one other commodity of a different kind. Hence the relation between the values of two commodities supplies us with the simplest expression of the value of a single commodity.

we begin with the simplest example of exchange; exchange of one commodity for another.
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exchange of one commodity for another.

Transfer of property rights? That's how gift economies are described, but there's worry that this is an ethnocentric description.
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This is 'old ground' for the thread, but it is presented in a lot more detail (and with one of my misconceptions cleared up).

SUBSECTION A: The Elementary or Accidental Form Of Value

{

Marx's pictogram of this constituent of the value form

x commodity A = y commodity B, or
x commodity A is worth y commodity B.

20 yards of linen = 1 coat, or
20 Yards of linen are worth 1 coat.

}

SUBSUBSECTION 1: The two poles of the expression of value. Relative form and Equivalent form

This part of the account locates two components of the value form; the relative form of value and the equivalent form of value; and exhibits a structural symmetry between them. The equivalent and relative forms of value are a necessary constituent of the form of value. The relative and equivalent form together specify the behaviour of the value form whenever we say that x has the same value as y. Whenever we can say that x has the same value as y; ie whenever the relative and equivalent form are jointly in play; we can say that the elementary form of value is at play too. Additional structure will be applied to the elementary form of value, yielding more constrained (yet algebraically more rich) forms of value.

Note that underpinning this account is the commensurability of commodities under the analysis of socially necessary labour time (and its direct price, a term which has its conceptual origin here, though it is not used explicitly by Marx).

The whole mystery of the form of value lies hidden in this elementary form. Its analysis, therefore, is our real difficulty.

In a similar sense to which the analysis of capital pivots upon the analysis of the commodity, the analysis of the exchange value of that commodity pivots on the understanding of the elementary form of value - which then leads to the exegesis of richer forms of value that can subsume the elementary form.

Here two different kinds of commodities (in our example the linen and the coat), evidently play two different parts. The linen expresses its value in the coat; the coat serves as the material in which that value is expressed. The former plays an active, the latter a passive, part. The value of the linen is represented as relative value, or appears in relative form. The coat officiates as equivalent, or appears in equivalent form.

At this point, I will introduce the relation T. T is a relation on a set of commodities C. Whenever x is related in way T to y, I will write xTy, Formally, a relation on a set X is a set of ordered pairs of objects from X. In this way, T will be a relation on C. Read 'T' as 'is worth the same as', so xTy is read 'x is worth y'.

Marx begins with an example of the linen and the coat; say 2 square yards of linen is worth 1 coat. Then we have, abbreviating, (2 linen)T(1 coat). This is the relative form of value for 2 linen, and 1 coat is serving as the equivalent.

The next paragraph asks the question of whether (2 linen)T(2 linen) gives the value of linen in linen.

The relative form and the equivalent form are two intimately connected, mutually dependent and inseparable elements of the expression of value; but, at the same time, are mutually exclusive, antagonistic extremes – i.e., poles of the same expression. They are allotted respectively to the two different commodities brought into relation by that expression. It is not possible to express the value of linen in linen. 20 yards of linen = 20 yards of linen is no expression of value. On the contrary, such an equation merely says that 20 yards of linen are nothing else than 20 yards of linen, a definite quantity of the use value linen. The value of the linen can therefore be expressed only relatively – i.e., in some other commodity. The relative form of the value of the linen presupposes, therefore, the presence of some other commodity – here the coat – under the form of an equivalent. On the other hand, the commodity that figures as the equivalent cannot at the same time assume the relative form. That second commodity is not the one whose value is expressed. Its function is merely to serve as the material in which the value of the first commodity is expressed.

Marx answers 'no'; "It is not possible to express the value of linen in linen. 20 yards of linen = 20 yards of linen is no expression of value. On the contrary, such an equation merely says that 20 yards of linen are nothing else than 20 yards of linen, a definite quantity of the use value linen.".

It is certainly true that the value of linen will always be the value of linen, irrelevant of what that value is, just as it is true that the value of a coat will always be the value of a coat. Marx suggests that because '2 linen is worth 2 linen' does nothing to express the value of linen relative to anything else, that it cannot be an expression of the relative value of linen; simultaneously, x cannot serve as the equivalent form for x. I grant that this is true; it is completely uninformative about the value of linen to say that it is worth linen.

When two distinct commodities were brought into the exchange relation in section 1, the qualitative particularity of each could not serve as the quantitative measure for both as equal; all concrete particularities of labour were agglomerated into human labour in the abstract. Marx suggests that this account; by which two distinct commodities were exhibited in their equivalence; would not work for the same commodity to exhibit it in equivalence with itself; the commodity is itself in every way. The meeting of a commodity with itself in exchange may be seen as a degenerate, impossible case that undermines Marx's argument of the relation of time and value. See just below to ground this point in the text:

It is not possible to express the value of linen in linen. 20 yards of linen = 20 yards of linen is no expression of value. On the contrary, such an equation merely says that 20 yards of linen are nothing else than 20 yards of linen, a definite quantity of the USE VALUE linen.

My claims:
(1) Marx suggests that this account; by which two distinct commodities were exhibited in their equivalence; would not work for the same commodity to exhibit it in equivalence with itself; the commodity is itself in every way.
(2) the qualitative particularity of each could not serve as the quantitative measure for both as equal; all concrete particularities of labour were agglomerated into human labour in the abstract.

(1) is rooted in the use of 'are', bolded and italicised, in the above quote.
(2) is rooted in the last clause of the bolded sentence, with reference to the 'USE VALUE' of linen which I capitalised. Linen meets itself, so it still meets itself as an amount of a concrete particular.

This comes down to the question of whether T is reflexive; whether for all x: xTx. Perhaps Marx would answer no, but I would like to argue that this additional assumption does no damage Marx's account, and is worthwhile on the basis of its utility in formalising his value theory.

While it is true that linen meeting linen would not allow the derivation of the relationship of human labour in the abstract to value, in terms of the socially necessary labour time of linen, if a meeting of linen with another commodity had already been posited, the concrete particularities of linen as a use value would still be excluded from the relation expressing its value. This is to say, we may say xTx and 'x is the same as x' can mean different things. Specifically, xTy says 'x is worth y'; a numerical equality of something in each in some unit of measure; whereas 'x is the same as y' is a full identity of all properties.

So yes, I am in agreement with Marx on both points; that xTx tells you nothing about the specific value of x, and that xTx could not be analysed in the same way to argue for abstract labour's magnitudinal relation to time. But I believe that if we consider xTx, xTy, yTx and yTy together, we still exhibit the relative form of x in y; in which y is the equivalent (xTy). And the relative form of y in x; in which x is the equivalent (yTx), so in (xTx) and (yTy), the commodities can already be understood not to meet as identical use values, but to meet as numerically identical exchange values. The instances of the relation xTx and yTy are still value uninformative; and this property aligns well with the reflexivity of T. Just as with usual mathematical equality, x=x, we don't immediately derive a conceptual contradiction regarding relative and equivalent forms because x=x for all x, we treat x=x as an uninformative tautology about the value of x.

So from now on, I will assume that T is a reflexive relation, and that this does no damage to Marx's account.

Marx continues:

No doubt, the expression 20 yards of linen = 1 coat, or 20 yards of linen are worth 1 coat, implies the opposite relation. 1 coat = 20 yards of linen, or 1 coat is worth 20 yards of linen. But, in that case, I must reverse the equation, in order to express the value of the coat relatively; and so soon as I do that the linen becomes the equivalent instead of the coat. A single commodity cannot, therefore, simultaneously assume, in the same expression of value, both forms. The very polarity of these forms makes them mutually exclusive.

The two sentences are important for the mathematical structure of T:

No doubt, the expression 20 yards of linen = 1 coat, or 20 yards of linen are worth 1 coat, implies the opposite relation. 1 coat = 20 yards of linen, or 1 coat is worth 20 yards of linen.

This is literally the meaning of the symmetry of a relation: if xTy then yTx. If x is worth y then y is also worth x. This establishes that T is symmetric. It is now reflexive and symmetric. But it is worth lingering a little here to highlight some extra mathematical structure that Marx is aware of.

But, in that case, I must reverse the equation, in order to express the value of the coat relatively; and so soon as I do that the linen becomes the equivalent instead of the coat.

If C = {x,y}, and xTy, then T on y is:

$T=\{ \{x,x\}, \{y,y\}, \{x,y\},\{y,x\}\}$

from symmetry and reflexivity. Look at what happens when we reverse the order in which elements are brought under T, calling this new relation T*

$T^*=\{ \{x,x\}, \{y,y\}, \{y,x\},\{x,y\}\}$

it's exactly the same relation as T, just with the order of the last two pairs flipped. So:

$T=T^*$

those familiar with some order or category theory may notice that T is self dual. This means it is the same as before when the order of all pairs is flipped. Writing this out in words might make the (for me, squee inducing) conclusion clearer:

T is: (1) x is worth x
(2) y is worth y
(3) x is worth y (x relative, y equivalent)
(4) y is worth x (y relative, x equivalent)

T* is: (1) x is worth x
(2) y is worth y
(3) y is worth x (y relative, x equivalent)
(4) x is worth y (x relative, y equivalent)

the part of the relation dealing with the relative and equivalent forms of value inverse their order when the order of the pairs are reversed. This means that the notion of relative form is dual to the equivalent form in the category theoretic sense, and the relation of values T is self dual as a consequence of that (and of symmetry).

Whether, then, a commodity assumes the relative form, or the opposite equivalent form, depends entirely upon its accidental position in the expression of value – that is, upon whether it is the commodity whose value is being expressed or the commodity in which value is being expressed.

And it seems Marx was aware of this duality of structures - assuming the relative or equivalent form are just positional accidents which imply the other accident. The structure Marx diagnoses of the elementary form of value is unperturbed by the assumption of the reflexivity of the 'is worth' relation.
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The role property rights play in capital comes later. Just before the dolphins.
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Ok. But nobody would trade 20 yds of linen for 20 yds of linen (if the bolts were identical.)
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Yes.
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What would the exchange relations look like in a mode of production that satisfied the elementary form of value? This breaks down into two questions:

(1) What would the exchange relations look like in a mode of production that satisfied the elementary form of value alone with no additional constraints?
(2) If (1) does not look like the capitalist mode of production, what additional constraints are required to more accurately model the form of value in capitalist production?

The logical space of exchange relations which satisfy (1) is very broad, and does not even need to have a stable numerical equivalence of values over repeated exchanges. This can be shown thusly:

If we have 4 commodities, x=1 coat, y= 1 pair of trousers, z=1 yards of linen,2z=2 yards of linen. Assume they trade concretely like:
(1) 1 coat is worth 1 pair of trousers|xTy
(2) 1 pair of trousers is worth 1 yard of linen|yTz
(3) 2 yards of linen is worth 1 coat|2zTx

then make it satisfy the symmetry condition (flipping the relations around):

(4) 1 pair of trousers is worth 1 coat| yTx
(5) 1 yard of linen is worth 1 pair of trousers| zTy
(6) 1 coat is worth 2 yards of linen| xT2z

z->y->x->2z is a possible sequence of trades, using zTy, yTx, xT2z. This means we could start with 1 yard of linen and end up with 2 despite only going through trades of equivalent value. To be able to say that 1 yard of linen is worth 2 yards of linen through the medium of exchange should tell us that something is very different from how exchange works under capitalist production. IE that the elementary form of value alone and without additional constraints is not a good picture of how value works in capitalist societies, at least most of the time.

It is within the potential of the elementary form of value to amplify (or diminish) the value owned through trade of things held equivalent, which is to say that considerations of value magnitude are only done relationally through T and not through an equation of numerical magnitudes of the same unit. The ability to alter held values in this manner is unlikely to survive in a stable community which has common knowledge of how much of one thing trades for another. Thus the situations this raw elementary form of value could hold sway in are those that have sufficiently rapid fluctuations in the relative values of commodities (fluctuations more rapid than the actions of trade) or when some part of the system of exchanges occurs isolated from another.

Another thing to note is that the elementary form of value doesn't need to have anything which resembles money in any way; all commodities are as good as any other as the repositories of value. Under the conditions above, if there were a common currency for the system of exchange it could literally be used to buy more of itself through a series of favourable trades, supposedly of equal 'value'.

Such a mitosis of money has precedents in the real world. However, the mitosis in reality leverages temporal instabilities in the trade relation - which are juxtaposed without temporal indices in the elementary form. The elementary form above should hence be thought of as the elementary form at a given time and place because the determination of its features depends more on accidental irregularity than essential regularities.
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