# [OPE-L:7022] [OPE-L:516] Linearity, basket decomposition, and exchange-equivalence

Alan Freeman (a.freeman@greenwich.ac.uk)
Thu, 25 Feb 1999 17:02:42 +0000

Continuing the discussion on equivalence, equality and linearity with
Steve, Brendan and Gil.

I'm going to begin by introducing linearity in a different way, as an axiom
of basket de/composition. I hope this will make things clearer.

I define an exchange relation as a relation with the following axioms:

(1) aRb -> bRa
(2) aRa
(3) aRb and bRc -> aRc

(4 )if bRx and cRy then (b U c) R (x U y), where U is set union.

Axiom 4 says we can decompose two baskets, or make them up from components
(the two are the same) and if the parts of two baskets exchange with each
other, then the two baskets exchange with each other. This is stronger than
linearity but makes the point clearer. To see it implies linearity suppose
eg aRb. Then by axiom 4 {a,a}R{b,b}, that is, (2a)R(2b). Inductively it is
easy to show in general (ka)R(kb)

Now to the discussion. Brendan [OPE 493] writes:

>This claim, that the sets of exchangeable commodities are distinct, and
>that all the members of a set are mutually exchangeable, is all that is
>required for equivalence. Assuming this partition it's easy to prove
>Reflexivity, Symmetry and Transitivity, and from RST you can prove the fact
>of such a partition. The claim that exchangeability is an equivalence is
>slightly weaker than the law of one price since it doesn't require the
>existence of a money commodity.

Alan responds:

Brendan, I think you are applying this general algebraic result from set
theory without paying attention to the fact that the objects to which you
apply it have a structure of their own. You take no extra account of the
fact that they are quantified; there is a relation between 'sets' of
commodities prior to exchange and if you throw away this relation, then I
think you get results that really do contradict any meaningful concept of
'exchange'.

If there is a relation between 2 coats and 40 yards of linen, I can sell
the coats either separately or together. If the exchange relations that
govern 1 coat are not consistent with the exchange relations governing 2
coats, you get some remarkable oddities. You must ask what happens if
people sell the 2 coats separately, which they are perfectly entitled to
do; otherwise you have to impose some very peculiar restrictions on
exchange which stretch the concept of commodity well beyond the limits of
credulity. In particular I can't seriously believe that Marx conceived of
his agents as being constrained in this manner.

I begin with the 'law of one price'. I don't think this does require money,
since the same constraint can be expressed as a constraint on relative
prices. For me, commodity exchange (you can call it the LOOP if you want
but for me it's just the commodity relation) demands that exchange ratios
between diverse goods are consistent when different quantities of the same
commodity are traded. This does not presuppose money; actually, money can
be deduced from it, not the other way around.

The issue is whether, given x units of B are equivalent to 1 unit of A, 2x
units of B are also equivalent to 2 units of A. If not, an awful lot of
things go pear-shaped.

For example suppose

{1A} R {2B}
{2A} R {3B}

The exchange-value of A in terms of B given by the first relation is 2
units of B; the exchange-value of A given by the second relation is 1.5.
These numbers are not the same and there is therefore no 'consistent'
pricing of A; one may not attach a unique number to A which is invariant
with respect to the quantity of A concerned. It has no 'unit price', which
is what most people mean by 'price'.

This is so regardless of whether there is a money commodity. Commodities
cease to possess a 'price' at all, even a relative price.

Next: by applying RST cyclically, and decomposing baskets appropriately, in
general one can acquire an arbitrarily large amount of A in exchange for
itself. Or, alternatively, conceiving of exchange as a merely legal
relation of titles, possession of anything entitles you to possession of
everything.

Thus in the above case if we take a basket of 2A and split it into its two
separate A's, and trade each A for a B, we will acquire 6 Bs. Splitting the
basket of six Bs in turn into 3 baskets of two Bs we can acquire 3 As.

Thus {2A} R {3A}

and by repeated trading, you can get as much A as you want out of
circulation.

This is in a certain sense obvious because arbitrage will arise; what I
think hasn't been grasped is that arbitrage implies not 'many prices',
but the complete absence of price in any meaningful sense. In real life,
arbitrage can exist only because purchases and sales are separated in time;
the closer that the time of circulation approaches zero, the more arbitrage
will collapse, obviously since if the time of circulation is zero and
arbitrage persists, one may make an infinite profit in no time.

It doesn't violate RST but it is a very strange consequence of an attempt
to find a non-linear generalisation of the exchange relation. In general it
makes everything equivalent to everything. There isn't actually any
partitioning; just one giant equivalence class.

The theory of Competitive General Equilibrium escapes this conundrum by the
device of equilibrium. It is in this sense that I think this kind of
reasoning still bears the marks of the mental shackles which this imposes,
oddly enough since you are attempting to generalise.

CGE supposes an equilibrium not merely of aggregate demand and aggregate
supply but of the individual excess demand functions. Coupled with a
marginal valuation with diminishing returns and diminishing utility, this
means that trade only takes place in the completed baskets that satisfy the
equilibrium conditions for the individual demand functions. Thus it does
not confront trade in part baskets, even though this is what all normal

Marx certainly does suppose that people will sell their commodities in
whole or part and I think it borders on the ludicrous to demand anything
else of the commodity relation. Thus after an extensive discussion of the
relation 20 yards of linen = 1 coat, he writes:

"But whenever the coat assumes in the equation of value, the position of
equivalent, its value acquires no quantitative expression; on the contrary,
the commodity coat now figures only as a definite quantity of some article.
For instance, 40 yards of linen are worth - what? 2 coats." (p57 Lawrence
and Wishart edition)

The same reasoning appears over and over again and it would be very odd if
it didn't. The bizarre innovation of supposing that people trade only in
whole baskets, and never break them up into parts or add them together

Brendan continues:

>What does RST entail? Not market equilibrium, or even "linearity." If two
>apples exchange with 3 pears, but 4 apples exchange with 7 pears, that
>would be consistent with RST.

Absolutely, RST does not entail linearity (or more generally, basket
de/composition). But that's not the question. The issue is whether you can
theorise the *exchange-relation* without basket de-composition. That's why
it's an independent postulate. I've identified some pretty bizarre
consequences of dropping it, and I've shown textually that Marx did not
drop it. Therefore, if you want study exchange, you have to assume
de/composition, and if you want to interrogate Marx's argument, you have to
assume de/composition. Consequently, linearity.

You can study many interesting things without supposing linearity. However,
I don't think you can study exchange, at least as it is normally conceived,
and I don't think you can study Marx at all.

I think that it is fine for you, Gil and Steve to travel the route of
trying to generalise from the exchange-relation by dropping the axiom of
de/composition AKA linearity. I don't think you can legitimately retro-fit
Marx by supposing that he is accompanying you on your voyage. If you do,
you will of course find him uncooperative, but that's because you've
pressganged him into a crew he never intended to be part of. Don't
throw him overboard: you shouldn't have taken him in the first place.

Moreover, if you want to use words such as 'price' to describe what happens
without the axiom of composition, then I think it's incumbent on you to
provide a definition of this word that makes a little more sense.

Alan