[PEN-L:5179] Re: Profit-rate equalisation

[Alan Freeman <100042.617@xxxxxxxxxxxxxx>]

>From Alan Freeman

Gil Skillman [PEN-L 5133] writes:
========================================================================
As I recall, Kliman and McGlone's approach (or if you'd prefer,  their
clarification of what Marx was about in Volume III, Ch. 9)  presupposes a series
of iterations in real time which begin with a set of input prices or values
taken as primitives and "ends" (or  approaches an end) with a system of
equations which approximates the Sraffian prices of production solution for a
given set of production  conditions.

...all thats being demonstrated in the Kliman/McGlone approach is that this
equation system is stable, in the sense of converging to a certain fixed point
from a given starting point.
========================================================================

It is genuinely to be welcomed that the carefully-considered approach being
developed by a growing number of writers who reject the equilibrium
interpretation of Marxs economics is receiving serious consideration,
especially by people who actually read what was written. However, I want to
argue that Gils polemic is misplaced and symptomatic of a widespread
misunderstanding of what is being attempted.

It is possible to illustrate the consistency and coherency of Marxs
transformation procedure using a system of difference equations with no
technical change (constant technical matrix) where in each period profits
equalize. This illustration requires the use of Marxs own transformation of
inputs, as outlined in Volume I and clarified in many other places, which makes
it clear that the value transmitted to the product by consumed constant capital
is equal to the value of the money paid for the inputs. (p317 Penguin edition,
full citation at end of this post, all citations Penguin edition)

The succession of prices generated by such equation systems converges, it is
true, to the simultaneous solution. Incidentally, it also converges under a very
wide range of assumptions concerning the distribution of profits, to an
equilibrium which supports this distribution of profits (for example if one
sector is assumed to make twice as much as all others). It can be proven that
what brings this about is not, as Gil suggests, the assumption of equal profit
rates but the assumption of a constant technology, as I hope he will agree if he
thinks about it.

But this assumption, as I am confident K/McG would be the first to explain and
which Andrew does not cleave to elsewhere, is not necessary to establish the
consistency of Marxs transformation procedure and their paper is not therefore
vulnerable to Gils criticism, since if the assumption is dropped their core
results still hold but his criticism does not. I will shortly give a theoretical
proof of this. The criticism breaks down because if the technical matrix is not
in general constant, there is no necessary convergence and no end.

So why do they make the assumptions, you might ask? Well, I wont presume to
answer on their behalf because Im sure theyll do a better job of that than I,
but I will guarantee that if they had not assumed equal profit rates, their
article would have been condemned just as roundly because it failed to address
profit-rate equalization. After all thats how this discussion began. Some
fights you cant win.

Before I show how the assumptions can be dropped, let me make a mild polemical
point, which is to ask whether Gil would dismiss dynamical systems derived from
neoclassical assumptions so quickly.

The convergence and the stability properties he describes are a property of a
very wide class of iterative systems. To cut a long story short, if you impose
some simultaneous condition on the final solution, and then apply the same
condition to generate a difference equation, then starting from almost any
arbitrary starting point, you will iterate towards the simultaneous
(=fixed-point, =equilibrium) solution. [Mathematical digression: technically,
the norm of the difference operator must be uniformly bounded above by a number
whose absolute value is less than one. I think thats right but welcome
discussion]

For example given the demand and supply curves

	D=6-3p
	S=1+2p

there is an equilibrium at D=S, hence p=1. Cobweb theory suggests we assume
lagged supply giving

	S(t+1)=1+2p(t)
	D(t)=6-3p(t)

If we further assume that in period t+1 a price is arrived at which equalizes
demand supply in that period we get p(t+1)=5/3-2/3p(t). Then if p is shocked and
starts at say 2, in successive years p(0)=2; p(1)=1/3; p(2)=13/9; p(3)=19/27 and
so it goes.

Cobweb theory doesnt treat this as just another tatonnement theory because it
claims [and, dont get me wrong, I dont support this claim] that the
intermediate prices in the iteration are at least an approximation to real,
observed prices.

If Gil wanted to apply his criticism consistently he should object that this is
only a method of reaching the market-clearing price. He should reject any claim
either from this theory, or from anything claiming to introduce time by means of
lags [adaptive expectations, rational expectations, neoclassical business cycle
theory, much Post-Keynesianism, probably Keynes himself, etc. etc. etc.] to
represent what happens in the real world, on the basis that if left to itself,
any such system will simply end (or approach an end) in a system of equations
which approximates the Walrasian General /full-employment equilibrium solution
for a given set of initial endowments. In short, the criticism removes the
dimension of time - if strictly applied - in the study of a market economy. If
Gil thinks this is unfair, then a good place to start might be to point to a
body of dynamic theory which he considers legitimate so that we can see how it
compares with Marxs. This is not a polemical point but a real proposal; I have
an ongoing discussion with Fred Lee on this exact issue.

Even a neoclassical can construct an answer to Gils argument by pointing to
situations in which the solution diverges, the presence of nonlinearities
leading to limit cycles or chaotic attractors, and so on.

For example try

	D(t)=17-0.9p(t)
	S(t+1)=-2+p(t)-0.01(p(t)-10)^3
	D(t)=S(t)

There is a fixed point at 10. Try and get there.

Marxist equivalents can be constructed but they are essentially toys. The
underlying problem is far more serious; the difference operator itself is in
general a function of time. The technical coefficient matrix is not constant (I
sincerely hope no-one launches a polemic in defence of simple reproduction) but
is itself a function of investment, innovation, class struggle, and so on.
Convergence is by no means a necessary property of such systems and Gils
criticism inapposite. The real issue is what laws if any, apply to a system
with no stationary state and no convergence properties? Such systems are not
unfamiliar; they are called real life and everyone except economists and
theologians has been studying them for the last four hundred years. I think
Marxs decisive theoretical instruments for studying their laws, or law, the
law of motion of capitalist society, are the concepts of Relative Surplus Value
and Expanded Reproduction, which - I have argued elsewhere - are inseparable.
Incessant technical change is at the foundation of his analysis of capitalism
and IMHO a challenge we still face is to produce an mathematical formalization
adequate to this concept.

The most substantive remaining element of Gils criticism is thus the following
remark, in my opinion much the most serious, and I think the core of the matter:

=======================================================================
1)  On what authority does one consider that the primitive input  prices or
values have any intrinsic connection to *labor* values?
=======================================================================

The answer is straightforward but if you try to understand it in a simultaneist
framework, you wont. If by authority you mean Talmudic authority I am happy
with citations at dawn - make my day - but I prefer the authority of reasoned
argument.

The relation is chronological. Marx considers reproduction as a succession of
periods of production and circulation. In a given period (say 1990) goods are
produced which have a value of (say) $5000bn. These are purchased. Some are
consumed, and some re-enter production in 1991. The connection between values
and prices is this: no matter how you diddle with the price system, you wont
create new value [or destroy it]. When you divide up this $5000bn between
current private consumption and the value which passes into the product, it
still adds up to $5,000bn at 1990 prices (or, as I prefer to put it, at labour
values represented in 1990 money) The value transmitted by the outputs of this
period to the product of the succeeding period is a definite proportion of the
accumulated social labour of the previous period, and the proportion of this
accumulated social labour which passes into the value of the product of any one
particular sector or capital is given by the value of the money paid for it.
[strictly, the value of the money in which its price is expressed at the time it
enters production].

The size of the pot from which this accumulated value is taken is determined by
the results of the previous period of production. But its distribution to the
capitalists is a result of the circulation process which follows, and is part of
the competitive struggle between them. Hence any attempt to reduce the
determination of prices to a set of production coefficients is simply
misconceived. But circulation cannot escape the law that new value is created
only in production, that is, through the expenditure of living labour, nor that
the new value added is proportional to the labour expended. The value in society
will not, for example rise because the earth is fruitful, because money earns
interest or because the currency inflates. When robots rule the earth they will
not create value unless they have become the owners of their own capacity to
metamorphose nature and sell this capacity on the marketplace. This is not a
small matter, so readers, think before you belittle it.

For this reason I think the search to show how to determine input values from
the coefficients of production alone is doomed and if, as I suspect, this is
what you want us to do, then I understand why you misunderstand, but you should
direct your request elsewhere because neither Andrew, Ted nor myself think it
can be done. I think the idea that the structure of production determines prices
was demonstrably not Marxs conception. If we go to the marketplace, I with
cotton and you with candy, and by dickering drop the price of cotton and raise
the price of candy, the result is that value passes from me to you and I dont
understand by what magic the purchasers of my cotton are supposed to retrieve
this value by using the cotton to make cloth. Its been eaten. We had a
competitive struggle and you won, Punkt. See the p317 citation at the end. But
this means there is a determinant of price entirely external to production.
Price in any given period is jointly determined by the outcome of the antecedent
periods of production and circulation, and equally so is the value transferred,
via the intermediary of money, to the products of the succeeding period. People
seem to find this very simple idea shocking, but I think this is only because
they have been trained by academics to read Marx as if he was John the Baptist
for the Coming of Walras.

[In general IMHO any attempt to determine prices is vain precisely because
price is the disguised form of human (i.e. conscious) social labour. No
conscious individuals can predict a future they dont want, because if they
don't like the prediction, they will use their foreknowledge to frustrate it.
Gil cites futures arbitrage but this is a special case of the preceding
proposition. Thesis: suppose, deploying the latest Sraffa-Heisenberg Integrated
Technology on my Laplace Inference Engine, I can determine 1996 prices
perfectly. Then all I have to do is sell this information - or even act on it -
to falsify my predictions. The idea of information which has no effect on any
human actions is meaningless, so the thesis is false. QED]

Gil may respond - many do - by [pro]claiming this approach renders value
analysis redundant. But the explanation in no way asserts that production, or
human labour, determines nothing. The fact that circulation begins from, and
cannot modify, the results of the previous period of production is for me the
starting point of all dynamic analysis, with immense ramifications. There are
much more important questions at stake than whether individual prices are
directly regulated by individual values, either stochastically, tendentially or
on average [and again, dont get me wrong, they may very well be, but this is a
concrete and empirical question and has no bearing on the substance of Marxs
contribution]. For example, why do slumps happen, why does the profit rate fall,
why is there unequal exchange, why are there so many poor people, and why does
hardly anyone benefit from technical progress. Not to mention commodity
fetishism, the class struggle, the origins of racism and why wars happen. If the
consensus is that price determination is more important than these things, Im
on the wrong list. If the consensus is that neoclassical theory has answers to
them, Im on the wrong planet. Or you are.

Thus, for example, the account above provides a fully coherent and
understandable account of the tendency of the rate of profit to fall. The
capital on the basis of which the profit rate is actually determined is the
money capital which capitalists have had to pay in order to carry out
production. To cut a long story short this is the historic, not the current or
replacement cost of their assets. Someone who has bought a computer at its
historical cost of $4000 has to finance this cost; it doesnt help them if in
the meantime the computer has fallen in price to $2000. Their capital is $4000,
not $2000. Accountants understand this, bankers teach it and the petty
bourgeoisie has it drummed into them; it is part of the apprenticeship of
becoming a capitalist. As far as I can ascertain the only people who deny it are
Marxist economists. If Gil studies Andrews work on the Okishio theorem (in
which the assumption of a constant technology is not maintained) he will see
that it is the basis for a coherent defence of everything Marx says about the
rate of profit.

Second example: if circulation reallocates some of the labour of the previous
period to, for example, holders of financial assets, then less of this labour
will pass into the product of the next period. Thus, for example, if the general
climate is one of falling prices (so that the value of money is rising relative
to the value of other commodities), then the money stock of society will
accumulate in value as Marx says on p183 of the Contribution to a critique
(citation below)

This provides a perfectly rational explanation, for example, for liquidity
preference. In a generally deflationary period such as 1996-97 (ascertainable
if we denominate all prices in Yen) people who hold monetary assets which are
rising in value will appropriate a substantial share of the social labour of the
current period without throwing the value they own into production. Thats a
slump.

Third example: unequal exchange. Why do poor countries get poorer? Because
circulation transfers values from the direct producers or owners of value into
the accumulation funds of those in a position to appropriate it, either through
their possession of superior technology - which perpetuates itself since the
deployment of superior technology is itself enabled by accumulation - or through
control of the financial and commercial system - which also perpetuates itself
since the same accumulation fund is concentrated in the hands of the banking and
commercial sectors of the rich countries - or a combination of the two backed up
by military force.

IMHO all these accounts - all directly obtainable from Marxs analysis - are
scientifically superior to anything else around. But you cant offer them on the
basis of an equilibrium interpretation of Marx, and you certainly cant arrive
at them from any branch of neoclassical economics I know of. And finally, of
course, I dont think anything in the above argument in any way invalidates the
usual arguments concerning exploitation, class struggle, anal retentiveness or
whatever you use Marx for. So why get so hung up about it?

The Math
========

How does this lead to a defence of Marxs transformation of values into prices?
To see the matter clearly, write the value output of society in any given period
as follows:

	v(t+1)X(t)=C(t)+L(t)

where v = unit values, X = output, C = constant capital and L = the value
product of labour. Now assume the price of labour power (the value of variable
capital) to be V(t). Then surplus value in the given period is L(t)-V(t). Call
this S(t).

For now (stocks of fixed capital is more complex but vindicates everything Marx
says) assume all output is consumed so X represents all commodities in
circulation. Sale at _any_ given set of prices p(t+1), whether profit is
equalized or not, results in a total price of output

	p(t+1)X(t)

What is the relation between the aggregate of prices p(t+1)X(t) and the
aggregate of values v(t+1)X(t)? As Marx explains in Volume I (p264 citation at
end) value cannot be created in circulation, so that the sum of the two is the
same. (total price=total value). The difference between the two is a vector
which I call E(t), the price-value difference arising from circulation at the
end of period (t).

	p(t+1)X(t)=v(t+1)X(t)+E(t)

Since the sum of values is equal to the sum of prices,

	sum(E)=0

Note that since E includes money, there may be a transfer of value to or from
societys stock of money, leading to a general nominal fall or rise in all money
prices. (citation from CCPE:183 below) This is why the relations concerned are
clearest if expressed in labour hours, the intrinsic measure of value. They can,
however, be expressed in the money of a definite time, since the money of a
definite time is the expression of a definite quantity of social labour. What is
illegitimate is to express v in the money of 1990 and p in the money of 1991,
the most usual confusion.

What is the relation between total profit and total surplus value? Surplus
value, we have seen, is L(t)-V(t), the difference between the value produced by
living labour and the value with which the capitalists must part in order to
employ this living labour (otherwise known as the money wage). Profit is the
difference between the sales of the capitalists and their costs:

	profit=p(t+1)X(t)-C(t)-V(t)
		=v(t+1)X(t)+E(t)-C(t)-V(t)
		=C(t)+L(t)+E(t)-C(t)-V(t)
		=L(t)-V(t)+E(t)
		=S(t)+E(t)

a redistribution of surplus value by the very same vector of price-value
differences E(t). The sum of profits is therefore equal to the sum of surplus
values plus sum(E), which is by definition zero. Thus the sum of profits is
equal to the sum of surplus value, Marxs second equality.

This is so simple, I am each morning amazed that anyone thinks Marx meant
anything else.

K/McG treats a special case of this in which the vector E gives rise to an equal
profit rate at every stage of the evolution of prices. But it is exactly that, a
special case. Other special cases are possible, for example, a vector E which
produces any given ratio of sectoral profit rates [Naples treats a distinct and
extremely interesting case in which profit rates equalize for all commodities
except money. I suspect this may not always converge, an important case to
study]. However my general result is true with or without the assumption of any
special relation between sectoral profit rates, and moreover this general result
is true whatever happens to the technical coefficient matrix A. The static,
equilibrium case emerges, just as Newtonian dynamics is a special case of
relativistic dynamics, as a special case of value dynamics. Neatly satisfying
Marshall:

the problem of normal value belongs to economic Dynamics; partly because
Statics is really but a branch of Dynamics, and partly because all suggestions
as to economic Rest, of which the hypothesis of a Stationary state is the chief,
are merely provisional, used only to illustrate particular steps in the
argument, and to be thrown aside when that is done. (Principles 1920 8th edn
p366 n2, cited in Currie and Steedman Wrestling with Time Manchester 1990,
p19)

Marshall in the fine old tradition of British empiricism latches onto the fact
that equilibrium is a special case because he is too practical to deny it. But
also in the fine old tradition of British empiricism he cannot provide the
general framework because he is too philistine to produce it. So we get a
provisional hypothesis (equilibrium) which is supposed to be thrown aside, but
which provides the exoteric theoretical foundation of the work. His epigones
have out of self-interest wisely refrained from sawing off the branch they sit
on. This is no reason for the rest of us to saw off our own legs for fear of
unseating him.

The argument just given provides a general mathematical proof of Marxs two
equalities and the basis for the economic dynamics which Marshall recognized as
necessary, but could not provide - just as neoclassical theory cannot provide it
even today, just as the traditional equilibrium interpretation of Marx cannot
provide it, and for the same reasons.

The crucial difference between Marxs account of the value-price relation and
all other accounts is that Marx derives the relation _prior_ to his treatment of
production; a point which many forget. Suzanne de Brunhoff has drawn attention
to it and others pick it up from time to time but somehow it keeps getting set
on one side (VI:196 citation below). The transformation of inputs is
therefore, far from being forgotten, already accomplished in Volume I. Also,
as distinct from neoclassical theory, the category of money is derived
analytically prior to the analysis of production. It does not have to be tacked
on at the end as an afterthought because it is the starting point of the
analysis.

The consequence is that Marx, to paraphrase Laplace has no need of the
hypothesis of equilibrium. This IMHO is why his theory of value provides a
vastly superior basis for economic dynamics to any other, and why I consider it
the only scientific basis for political economy.

The Citations
=============

I:196 cited by de Brunhoff Marx on Money Urizen 1976, p27:

The magnitude of the value of a commodity therefore expresses a necessary
relation to social labour-time which is inherent in the process by which its
value is created. With the transformation of the magnitude of value into the
price this necessary relation appears as the exchange-ratio between a single
commodity which exists outside it. This relation, however, may express both the
magnitude of the value of a commodity and the greater or lesser quantity of
money for which it can be sold under the given circumstances. The possibility,
therefore, of a  quantitative incongruity between price and magnitude of value,
i.e. the possibility that the price may diverge from the magnitude of value, is
inherent in the price-form itself. This is not a defect, but on the contrary, it
makes this form the adequate one for a mode of production whose laws can assert
themselves only as blindly operating averages between constant irregularities.

I:264

The consistent upholders of the mistaken theory that surplus-value has its
origin in a nominal rise of prices or in the privilege which the seller has of
selling too dear assume therefore that there exists a class of buyers who do not
sell...A may be clever enough to get the advantage of B and C without their
being able to take their revenge. A sells wine worth L40 to B, and obtains from
him in exchange corn to the value of L50. A has converted his L40 into L50, has
made more out of less, and has transformed his commodities into capital. Let us
examine this a little more closely. Before the exchange we had L40 worth of wine
in the hands of A, and L50 worth of corn in those of B, a total value of L90.
After the exchange we still have the same total value of L90. The value in
circulation has not increased by one iota: all that has changed is its
distribution between A and B. What appears on one side as a loss of value
appears on the other as surplus-value; what appears on one side as a minus
appears on the other side as a plus...The sum of the values in circulation can
clearly not be augmented by any change in their distribution...The capitalist
class of a given country, taken as a whole, cannot defraud itself.

I:317

The definition of constant capital given above by no means excludes the
possibility of a change of value in its elements. Suppose that the price of
cotton is one day sixpence a pound, and the next day, as a result of a failure
of the cotton crop, a shilling a pound. Each pound of the cotton bought at
sixpence, and worked up after the rise in value, transfers to the product a
value of one shilling; and the cotton already spun before the rise, and perhaps
circulating in the market as yarn, similarly transfers to the product twice its
original value.
Before the polemics note two phrases: price, and after the rise. Also note:
in several other places where Marx writes Preis the Penguin translation
wrongly substitutes Value. Alejandro Ramos, for ever diligent, deserves credit
for noting this.

A Contribution to the Critique of Political Economy p183:

The most common and conspicuous phenomenon accompanying commercial crises is a
sudden fall in the general level of commodity-prices occurring after a prolonged
general rise of prices. A general fall of commodity-prices may be expressed as a
rise in the value of money relative to all other commodities, and, on the other
hand, a general rise in prices may be defined as a fall in the relative value of
money.