[OPE-L:3031] Okishio 5 of 4

Alan Freeman (A.Freeman@greenwich.ac.uk)
Mon, 16 Sep 1996 04:06:01 -0700 (PDT)

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[This continues the belated Okishio 4 of 4]

The story so far:

I tried to establish that two possible postulates form the
basis of two possible alternative world-views or paradigms
in economics. I call these

(E) the 'equilibrium is real' hypothesis
(D) the 'reality is dynamic' conjecture

I assert, and I think proved, that (E) cannot be true; that
is equilibrium cannot be real. I conclude:

Corollary 1 of (E): the ideality of Okishio
The results of the Okishio theorem, which we agree is a
deduction from (E), cannot apply to any real economy for
this reason.

There is a dispute about whether this constitutes a
refutation or not but there doesn't seem to be a dispute
about the corollary itself except possibly for Gil.

(D) is more accurately termed a conjecture than a theorem;
reality is infinite and we cannot enumerate all its
possibilities. I do think I proved that (D) applies to
Marx's dynamics - in chapter 11 of "Marx and non-equilibrium
Economics". I've presented quite a large number of models,
some highly specific, based on this over the years. They
are are all in the public domain, including a number of
illustrations on OPE which no-one responded to. So far I
have not observed the behaviour which Duncan describes in
[2915], namely:

"Much could be said against this position, but positive
results in simple models tend to hold the field against
general negative criticism in the history of thought."

I'm afraid I differ on the basis of practical empirical
experience and I fear that Duncan may be confusing his own
generosity with the practice of the profession as a whole.
I'm afraid most economists I've come across wouldn't
recognise a positive result in a simple model if it were
served up in the Ritz with applesauce. Marx's own work is
jam full of very positive results expressed very simply and
it is not as if the academic world has taken these to its
bosom. Not even the marxists endorse them.

I think, bluntly, what holds the field in the history of
economic thought are theories and models that confirm the
dominant prejudice of the people who pay the economists,
and that (E) is the primary form of this dominant prejudice.
(E) expresses in pure form the notion that a free
competitive market automatically and without external
interference arranges for a set of prices and quantities at
which all goods are sold.

The persistence with which this dogma has been refuted by
experience is matched only by the doggedness with which the
economists cling to it in the face of all available logic
and evidence. This calls for more drastic measures than the
mere presentation of models. Fundamentally I think the
method to be followed is that of a critique of the theory on
which the dogma is based, that is to uncover the logical
process that underlies its deductions and bring the
presuppositions of this process clearly to the light of day.

Why does the Okishio theorem highlight these issues more
effectively than almost any other? In my view, because
structural reasoning breaks down when dealing with technical

The differences between the results obtained from Okishio's
procedure, and the results obtained by virtually any dynamic
procedure, arise when the parameters change in a certain
systematic manner; essentially, when the fixed-point state
changes in the *same* direction instead of fluctuating up
and down. As long as prices merely oscillate around some
centre of gravity, as Allin discusses, the contradiction
between the comparative static and the dynamic solutions
does not emerge, though it is always present implicitly and
Marx is fully cognisant of this in his presentation.

But in the study of technical change we confront just such a
unilateral change, and the contradiction becomes apparent.
Speaking very informally, the technical coefficient matrix
becomes monotonically smaller over time according to a more
or less exponential rule. Such systems arise for example
under the conditions specified by Duncan in [3014].

Under this type of change, a discrepancy must emerge between
two alternative approaches to the variations in the
parameters A, L; between the fixed-point and the dynamic
solutions. I want to spend the rest of this post explaining
where this discrepancy in general comes from.

Duncan asks me for a model. I would like to discuss this
request in more detail at some point because my persistent
unease about model-making in economics has been strengthened
into a permanent objection by this debate. So I think I will
probably end up by giving a reasoned refusal to provide a
model, though I am not yet categorical about this (if this
disappoints can I hasten to point out that I've put plenty
of models in the public domain that I'm happy to resend to
anyone requesting).

However, I want to respond constructively by highlighting
some consequences of my 'reality is dynamic' theorem
(perhaps more accurately a 'conjecture') which have not been
understood and to do this using an illustration. Maybe this
will go halfway to satisfying the request.

The Reality constraint and the use of models

First off, my argument is not confined to saying there is
one particular set of assumptions, or one particular model,
which contradicts Okishio. This could be done, but I am not
going to do it.

My conjecture is much stronger.

I assert that any model and any reasonable dynamisation of
Okishio's equations will contain cases in which the profit
rate falls, where Okishio predicts it will rise. The claim
that 'reality is dynamic' is much stronger than finding one
particular model of reality with a falling profit rate; I
claim there is no coherent way to model reality without it.

Conversely I am pleased that at last we are starting to
discuss whether (E) is realistic. I am particularly pleased
that almost no-one [except possibly Bruce and Gil?] has tried
to claim it is realistic, one of those very big steps forward
that happen from time to time almost unnnoticed. However,
I think that there are consequences of this fact which have
not fully been absorbed.

The real difficulty is this: what kind of transition could
result in a move from one comparative static equilibrium to
another? My argument is: there is no such possible
transition which is compatible with (E). The process which
Duncan refers to, in which prices 'settle down' to
stationarity, *could* *not* *take* *place* without violating
(E). That is (E) *cannot happen*. The assumption of (E) is
self-violating. It is in fact valid only in the special
'case' where the economy is created ab initio with the
correct parameters. But this isn't a 'case' at all; it is
pure myth, pure religion, pure ideality. Moreover, and this
is particularly serious for Okishio's theorem, it
contradicts the assumption, clearly stated by Okishio, that
the economy moves from one state to another. My point is,
there is no means for this movement to occur within the
assumptions of the theorem.

I just don't think the full force of this argument has been
absorbed yet though I think some inkling of it is beginning
to surface, and so I want to return to it by explaining more
carefully the reasons it happens. These reasons do not arise
from any assumption about the process of technical change or
the conduct of the workers. They arise from the nature of
exchange and production, and in particular the movement of
prices. To simplify matters we will begin with values.

Value movements subject to the constraints of reality

Suppose under (E) the following holds: values are defined by


The state of this system is given by the three quantities
(v,a,l). In order to dynamise it, a temporal relation is
required between (v,a,l) at one time and at another.

The sequential/successive/dynamic approach dynamises the
system thus:


As such this is not a model. It is not determinate. It would
only become determinate if we added a further equation
specifying the movement of a(t) and l(t) over time; that is,
saying how the capitalists invest. That is what, on
reflection, I decline to do. Thus the equation above
defines, not one model but an uncountably infinite family of
models. Slot in your own assumptions of choice. Class
struggle, technical determination, phases of the moon,
sunspots: it doesn't matter; no matter *what* assumption you
put in, subject to a couple of very minimal constraints
corresponding to the assumption of reality, the profit rate
will fall when Okishio says it rises.

The (E)-based comparative static approach (which is
Okishio's) abstracts the coefficient matrix a as a parameter
and then 'dynamises' by writing (v, a, l) as a function of
time thus:


This *is* a model. It is determinate. There is no 'family of
solutions'. There is only one true religion. That's another
reason I don't like models.

First off, let's observe that these two procedures give rise
to *different* solutions for v, even when the time-evolution
of a and l are identical. Try it on the model of your

Already we see that a fundamental assertion of the Sraffians
is violated. They claim that values are *technically*
determined in each period by a and l. But in a dynamic
model, *any* dynamic model, this is not true. Instead,
values (and hence prices) are determined by the entire
history of the economy and its evolution.

Many problems with the simultaneous approach have been
discussed intensively on this list. Not least is this: if it
is meant to represent a relation of exchange (as Sweezy,
Meek, Laibman and most academic Marxist writers maintain)
then it is very hard to see how this exchange could take
place if values have changed from one period to the next.

If values in period 1 differ from values in period 2 the
awkward problem is that the capitalists must start
production in period 2 by buying their inputs from the
capitalists who produced them. But that means they must
exchange at the same values, not at different values.

This is so central that an illustration is useful. I simply
repeat the illustration given in chapter 11 of our book with
a minor change to avoid the difficulty of unsold stocks - in
the present context an unnecessary complication. For the
full details the chapter concerned can be referred to (I'm
happy to forward it by E-Mail). Consider a two-sector system
in which producers PI, PII make commodities CI, CII [to see
the tables properly you may have to change font. Use a fixed-
width font such as Courier]. Throughout we assume 1 hour is
represented in exchange by 1 dollar. The analysis is not
altered by varying this assumption, made only for the
purpose of illustration.

PI 40 0 400 produces 50
PII 10 0 300 produces 100

The simultaneous equation values are given by

50v1 = 40v1 + 400 (1)
100v2 = 10v1 + 300 (2)
giving v1 = 40, v2 = 7 (3)

Suppose in the next period productivity increases to give

PI 40 0 300 50
PII 10 0 200 100

The new simultaneous solution yields

50v1 = 40v1 + 300 (4)
100v2 = 10v1 + 200 (5)

giving v1 = 30 , v2 = 5 (6)

This all seems very reasonable if we just wave our hands
about as the economists do and say 'it's something like
that'. But let's look further. *Exactly* how do we get from
period 1 to period 2? How did the capitalists start off
period 2? What did they buy and who did they buy it from?

CI was worth 40 per unit at the end of period 1. That means
that the capitalists PI received $40 for each unit of CI
that they sold, according to any normal definition of
exchange. So they received a total of $50*40 = $2000.

But according to our second calculation, CI was also worth
$30 at the beginning of period 2. That means the capitalists
PII, for example, paid $30 for each of the 10 units they
bought; a total of $300 whilst the capitalists PI paid a
total of $1200. $500 has gone missing.

Where is it?

Please explain to me what 'assumption' allows A to sell
something to B in such a way that A pays less than B

This is more than just a wrong assumption; there is actually
no possible real mechanism by means of which the economy
could pass from its first state to its second state.

This is not altered and in fact it is made worse, if we
decide that we are going to allow exchange at prices
different from values. The transition of which Okishio
speaks, could not actually take place in a market economy. I
don't know if that counts as a refutation but it strikes me
as pretty damn awkward.

The Consequences for Profits

Now it is perfectly obvious that we are going to get a
different profit rate in the second period, depending on
which set of values we use. But since the changes have
raised productivity, the values have fallen. Ergo, period 2
costs will be higher, as will period 2 advanced capital.
Ergo, the profit rate predicted by the properly dynamised
system will be lower than the profit rate predicted by the
comparative static system.

This result doesn't depend on any particular behavioural
assumption at all. It doesn't depend on any expectations,
any real wage, any s/v, anything that anyone has thrown into
the discussion as relevant to the determination of the
profit rate.

That's why, though I respect the positive manner in which
Duncan has responded to the debate, I think his request for
more precise behavioural assumptions is misplaced. If I were
to supply more precise behavioural assumptions, what would be
discussed are the assumptions, not the differences of method.

Also, it would add to the general misconception that TSS
constitutes another alternative dogma, another alternative
Story of Everything. It isn't. It is a different method of
conceptualising the assumptions you already have.

Between those who support the TSS approach there are as many
different sets of ideas about how capitalists actually behave
as there are people. Is the real wage rising or falling? Is
the rate of exploitation rising, falling or constant? I don't
know. This is an entirely empirical question. But I can tell
you that regardless of whether they are rising, falling or
static, the dynamically-calculated profit rate will be lower
than the comparative static profit rate.

So I would turn it round. You tell me *your* assumptions
and I'll prove, on the basis of these assumptions, that the
dynamic solution yields a lower profit rate than the
fixed-point solution. This is the real nub of the matter.

Does that mean that I have failed to provide such
assumptions or that I regard this debate as irrelevant? Not
at all. It has been a totally fascinating and very focussed
discussion and actually I have all sorts of very clear ideas
on the behaviour of capitalists. I just don't consider them
relevant to this debate. I'll introduce them some other time.

Our result is valid *no* *matter* *what* assumption is
adopted. All you have to do is introduce the same assumption
in both treatments. If, for example, you want to assume
workers eat the same identical thing for eternity fine; put
it into both systems and you will find a lower profit rate
in the dynamic solution than in the fixed-point solution. If
you want a constant rate of surplus value, fine, put that
assumption in. If you want to look at the maximum profit
rate, fine, look at that. [Actually I would agree with
Patrick on this: I do think this is a reasonable line of
enquiry]. But actually it doesn't affect my result. This is
an extremely robust result. In *every* case the profit rate
is lower for the dynamic solution than for the fixed-point

It doesn't depend on the assumptions. That's why I'm not
going to supply any assumptions. If I start supplying
assumptions, everyone will discuss the assumptions instead
of the result, which is exactly what's been happening to
Andrew for the last year. YOU supply the assumptions.
WHATEVER your assumptions, I'll prove the two paradigms lead
to different results and that the dynamic profit rate is
lower than the fixed-point rate.

Value, price, embodiment and exchange

Now let us see how this problem affects the literature.

Well, one way out is to deny that the system v=va+l
represents exchange at all. This is the course taken by Gil,
Paul and Allin, as far as I can see. I think John Roemer
takes a similar view, at least implicitly. In some manner
v=va+l is then said to explain to us the labour 'embodied'
in the product.

But then exactly the same issue arises then in another form,
which is the origin of the prolonged debate about historical
and reproduction costs. However in this case, there is no
fixed capital so we may confront the issue in its simplest

According to the first solution, the labour which is
embodied in CI as a result of the first period of production
is 50*40 = 2000 hours. Good. No dispute.

But what happens in the second period? The capitalists begin
this period with means of production that they got from
somewhere. Where did they get them from? From God? No, they
got them from the outputs of the previous period of

But we have established this embodied 2000 hours in these
inputs. These same goods, with this same labour embodied in
them, are now supposed to transfer to the final output of
the second period, not the 2000 hours that were embodied in
them but 1500 hours which have not yet been embodied in
them, but are instead about to be embodied in the output of
this second period.

Now, I respect the requirement that there should be some
conception of the labour that is objectively embodied in a
commodity that is independent of the price that it sells at.
And I don't think any TSSer dissents from this idea at all.
On the contrary, the problem is the opposite: simultaneous
valuation *cannot* provide a reasonable concept of
embodiment and it was indeed this which first set me on the
path of dynamising the value relations. [For the record, in
the process of doing this Paolo came across the concept
of 'direct price' in Anwar's work and concluded this was the
way to render a dynamic valuation process consistent with
price formation. Then we came across the work of Ramos and
Rodriguez, of Wolff, Callari and Roberts and of Fred's
collaborators and found this the missing link to assemble
a completely coherent dynamicisation of both price and value.
Hence our debt to these authors, whatever our differences].

Embodiment is precisely the requirement that is grossly
violated by the fixed-point solution. Is the labour in the
commodity, or isn't it? If it is in it, why doesn't it pass
into the product? 'Embodied' means precisely 'in the body'
of the thing, so it is a great mystery, practically a divine
manifestation, if something that is supposed to be in the
body of the thing, vanishes into nowhere or appears from
nowhere when the thing is used in production. Indeed when
Marx speaks of deviations in the value of fixed capital
because of variations in what is socially necessary, he is
at some pains to say that this is a source of variation in
value which *differs* from what is embodied.

However, let us set this aside and turn to Okishio. And at
this point, I think, we can appreciate why the Okishio
discussion focusses the full force of the contradictions of
the simultaneous method. Okishio does not speak of values
but of prices. Therefore, the systems he speaks of *must*
satisfy the requirements of exchange. He speaks *directly*
of prices. Again, exactly the same considerations apply.
Construct a model, any model you like, and write it down
statically. Then you can produce two alternative

In the (E) case you will have something like

p(t)=F{p(t), A(t), L(t) r(t)}

where F is a function incorporating your assumptions.

In the dynamic case this will on the contrary read

p(t) = F{p(t-1), A(t), L(t), r(t)}

If, further, you seek a closed or determined model then
there will be further equations like

A(t) = G{p(t-1), A(t-1), L(t-1), r(t-1)}

Now, there are various constraints that might be imposed
upon F; the principal constraint I insist upon is that goods
should be sold for the same amount of money for which they
are purchased. There is a similar constraint on use-values
that says goods should not be used unless they exist. I
don't know if this violates the existence of a futures
market but it does at least obey the known laws of physics.

In this case, if A changes over time so as to represent a
successively more productive economy, the 'Okishio' profit
rate will differ from the dynamic rate in a systematic
manner and perfectly reasonable circumstances arise in which
the Okishio rate rises, and the dynamic profit rate falls.
That's what it's all about.

How to dynamise?

All that I have done, and all that any 'successivist' or
'sequentialist' does, is to seek a means of dynamising such
equations in such a manner that the money paid by the people
who buy things is the same as the money received by the
people who sell the same things, that the labour embodied in
a commodity before it enters in production actually
coincides with the labour it transfers to the product, and
that goods do not get consumed before they exist.

There is no special secret to this. There is no 'model' that
uniquely represents the dynamic approach. On the contrary,
there are a wide variety of models and partial attempts to
dynamise in existence. Duncan's 1982 models, for example;
Paul's model presented to the 1996 EEA conference; a very
detailed computer model I presented at the CSE in 1991; the
models I gave in my C&C article and in the book; Andrew and
Ted's models; Paulo Giussani's examples presented in Paul
Mattick's journal, plus the further model he presented in
our book; several of John's models, and so on. I've placed
innumerable examples on OPE that can easily be considered as
models if people so desire.

I'm *tired* of presenting models. I've been presenting
models for eight years and unfortunately, economists do not
behave as Duncan asserts. They do not use these models to
test the coherency of generally held prejudices. What they
actually do is to reassure themselves that these generally
held prejudices are safe. They take issue with one minute
assumption in the model (pretty ironic in view of the way
the (E) assumption is used), heave a sigh of relief and set
the whole thing on one side. I don't think they ever will do
anything else. Something deeper is at stake. Models have
become a means of staving off facing the issues, not a way
of engaging in scientific discourse.

So I constructed, not a model but a general theorem that
applies to all models. To explain this let me observe that
some of these dynamic models are only partial. They dynamise
only one aspect of the state vector. As far as I can
ascertain, Duncan's models does not specify what happens to
use values, whereas Paul's model dynamises the stocks and
flows of use-value but has a comparative static measure of

My argument is not that there is some special model that
refutes Okishio; my argument is that every model refutes
Okishio. Every dynamic model contains within it a specific
case that refutes Okishio provided only that it dynamises
the process of exchange in such a manner that the money paid
for something is the same as the money received for it, and
matter isn't created out of nothing. Apart from that you can
adopt any assumption you like.

So I want to turn the whole thing round. Take a model you
agree with and some assumptions you agree with. I allow you
free choice of model, free choice of assumptions, free
choice of argument. On the house. The only constraint I
impose is this: the money paid must equal the money
received, and the usual laws of physics apply. *These* are
the constraints that Okishio violates. Then I'll prove to
you that your model produces different results if you
dynamise it coherently with reality, from if you dynamise it
in accordance with (E); and moreover, if productivity rises
monotonically over time, your profit rate will not only be
systematically lower than Okishio's, it will fall until and
unless the capitalists disaccumulate in money terms -
*exactly* as Marx says.

It seems to me highly unreasonable to treat this as a
'special assumption' or an 'incomplete market' or 'just one
way of looking at the world'. Pray tell me what kind of
complete market allows me to give the shopkeeper less than
she receives from me. Pray tell me what kind of futures
market lets me consume things that haven't yet been
produced. I'd very much like to go there and buy things. Or
sell things.