# [OPE-L:7052] [OPE-L:547] Re: Linearity, basket decomposition, and

Gil Skillman (gskillman@mail.wesleyan.edu)
Sun, 28 Feb 1999 13:03:54 -0500

Alan writes:

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

I won't speak for Steve or Brendan, but I don't understand the sense in
which Alan's post "continues the discussion" with me, since none of his
raise some interesting new questions, so I react to some of them below, and
assess implications at the end.

>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)

Actually, as (4) is stated, the two *aren't* the same. Axiom 4 only
indicates that you can go from components to aggregate baskets, not
vice-versa. And as someone has already noted, the converse is problematic:
saying that you can exchange a pair of gloves for 4 apples does not say
you can exchange, say, the left glove alone for an apple.

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)

That's fine, but virtually no one else defines it this way. Palgrave's
does not support you on this, nor is there any textual evidence I can see
that Marx associated LOOP with commodity exchange as a matter of *definition.*

> 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,

(assuming unlimited arbitrage opportunities, which would not arise in the
presence of transaction and mobility costs or sufficiently incomplete
information)

> and I've shown textually that Marx did not
>drop it.

As I've said earlier, I don't see the grounds for this claim.

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.

This assessment rather puts the cart before the horse. First, Alan assumes
it is obvious that Marx presupposed LOOP in his analysis of exchange.
Perhaps he did, and his argument would certainly come closer to making
sense (subject to important caveats) if he did, but I wouldn't call this
assessment obvious. But second, nothing in my critique of Marx's Chapter 1
argument depends on the presence or absence of linearity. Thus there is no
"press-ganging" or "throwing overboard" or other nautical shenanigans on my
part, and in no coherent sense do Alan's comments serve to "continue the
discussion" with me.

>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.

It's not the *definition of the word* that's at issue in Alan's comments,
but rather the conditions of exchange under which linearity might or might
not be expected to hold. As I mentioned previously, basket
decomposition--which, again, Axiom 4 as written does not assert--is not
assured given any number of exchange conditions. This fact does not warp
the definition of "price" as "exchange ratio".

Gil