[OPE-L:2432] Dynamical systems

Alan Freeman (100042.617@compuserve.com)
Thu, 30 May 1996 12:42:59 -0700

[ show plain text ]

Paul C (#2402) writes

"Clearly if you allow input and output prices to differ, your
equations acquire sufficient additional degrees of freedom
for you to be able to impose the constraints that you wish.
The question is whether this system of equations has greater
or lesser economic plausibility than the standard

I think this is the central issue of economic *theory* per se
that has to be addressed.

Paul is right to identify that it is not just ourselves but the
Kaleckians (and many other Post-Keynesians, and Steve on our
own list; also I think Riccardo?) who believe that the correct
foundation for a dynamic theory includes two not very unreasonable

(a)the money received by the seller of a good is the money
paid by the buyer of the same good;

(b)prices change.

Neither of us sees anything plausible about the assumption that
the prices which goods are sold for on December 31st 1996 must
be the same as the price which the same goods were bought for
on January 1st 1996. In fact, it's about the most implausible
idea to hit the ground running since the Immaculate Conception.

This leads to a problematic in which the most important thing
to establish are the transition rules governing the passage
from one set of values, prices and quantities at one point in
time, to another set of values, prices and quantities at
another point in time. In such a problematic, as with any
normal system of lagged or differential equations, there is a
temporal sequence defined by an axiom which asserts (this is
best specified in the language of mathematical State Theory)
that the state at time T depends, and only depends, on states
at times t<T. A simultaneous formulation on the contrary
asserts either that the state at time T is self-determined
without reference to previous times, or worse still that it
depends on states at times t>T.

But once this simple axiom is accepted this by no means settles
the vast range of possible transition rules that one can
envisage. As you remark, the degrees of freedom become
enormous. Instead of the two degrees of freedom of the Sraffa
model (profit rate plus normalisation condition), every
sectoral price and every sectoral quantity, as a part of the
state vector of time t, is subject to the axiom above and a
fully-specified model has to explain the transition rules that
give these prices and quantitites *purely* in terms of the
preceding prices and quantities.

The reason there must be this many degrees of freedom in these
models, is because there are that many degrees of freedom in
real life, as I am sure you will appreciate. Therefore, if any
model removes these degrees of freedom a priori, it necessarily
loses the capacity to portray real life.

This is why I prefer to speak of this approach as a paradigm
rather than a model. We do not specify what these transition
rules must be, only that they must satisfy the basic temporal
axiom that the present depends on the past.

Any *actual* model, being a simplification, will specify
certain interrelationships between state vectors at time T and
previous times t, and in so doing, begins to limit the degrees
of freedom. But the paradigm as a whole cannot be judged by the
properties of any one particular model within it, any more than
Newton's theory of gravitation is overturned by a model which
demonstrates there are green cheeses orbiting the moon. Newton
only tells us that if there are such green cheeses, there must
be certain relations between their velocity and their
trajectory; it cannot tells us whether the cheeses exist. That
is a matter for empirical verification. It is the same with the
temporal paradigm. A paradigm is falsified not by the empirical
failure of a single model expressed in that paradigm but by the
existence of phenomena which *no* model in the paradigm can

The questions which then begin to arise, in outline, are the
following two types of problem (there are more):

(a) we can specify general *categories* of model, just as
different general *categories* of differential equations can
be specified (and their topological features can be mapped
qualitatively by defining attractors, limit points, saddles
and so on). I think, for example, that one broad category
consists of all Post-Keynesian models in which there is no
operative concept of value, and another is Value-theoretic
models in which the concept of value appears not only as an
interpretation of price but also as as a limitation on the
possible state transitions.

(b) we can define what in dynamical theory is known as
'constants of motion' of the system for each such broad
category. Thus systems that have a potential function will
exhibit the dynamical conservation of value (analogous to the
Hamiltonian), which I consider a central and indispensible
requirement of a valid economic theory.

A very early and decisive difference between two very broad
categories of dynamical systems of this type is precisely the
issue of the mark-up, and that for me is why value is
indispensible. In a nutshell, I would strongly deny that the
capitalists can impose an arbitrary markup and that, I think,
is the significance of Marx's determination of the maximum rate
of profit from the quantity of new value added, and of the
actual rate of profit from this latter quantity less the money
paid in the *previous* period as wages.

Giussani's 1993 article in the International Journal of
Political Economy explains this particularly well. It
guarantees, for example, that there is a determinate and
positive general profit rate even for many economies in which
traditional static theory either throws up its hands in horror
or predicts a negative profit rate. This includes cases where
the Hawkins-Simons conditions (positive net product, alias
positive leontief inverse) are violated, which personally I
think is a norm rather than the exception it is portrayed in
the literature.

This requires that the labour value added in each period be
defined independent of the price level. It is this closure
condition that provides for and indeed guarantees the stability
of value-theoretic dynamical systems.

The reason that this, and not some monetary condition, is the
decisive and necessary closure rule is that only in this way
can we express the real productive resources which this profit
represents. It is of no use at all to know that the profit
might be 10,000 or 100,000 or 100,000,000 unless one also knows
what productive capacity this represents, and the only truly
universal productive resource at present around is us chickens.

The essential problem with all non-value-theoretic dynamical
systems of this type is that there is nothing to 'tie down' the
size of the markup the productive resources it commands.

Such models are in my view characterised by more or less
arbitrary external monetary constraints, or arbitrary
assumptions not rooted in the fundamental features of the
economy, to foreclose the possibility of an arbitrarily large
general price increase. Though many such systems are
nevertheless (I think) in practical terms streets ahead of
simultaneous so-called 'marxist' systems, and have also made
what is for me the indispensible theoretical step of re-
introducing normal time, this is nevertheless a decisive
weakness and that's why I think value is needed in a dynamical

However within the general rate of profit established by such
calculations, any number of sets of relative prices are
possible and corresponding to this (and in a 1-1 relationship
with it) any number of sets of sectoral profit rates. No
equalisation is implied by such a general condition.

I don't at all see this as a weakness but a virtue. The real
world really is that complicated. If in advance of analysing
the actual features of an actual economy, all the degrees of
freedom are already accounted for, we cannot model the
complexity of that real economy and cannot allow for the actual
observed diversity between economies.

This is why I also insist that the method we propose is not a
model but a paradigm. Within a paradigm, there are many
different possible models. Any particular such model *will*
specify the transition rules and *will* remove many degrees of
freedom. The test of each such model is of course empirical.

But the *paradigm* itself, the method of working, cannot impose
any particular set of transition rules. What it should specify
is the axiomatic foundation which any actual set of transition
rules should respect among which I would number

(1)true temporality (see above)
(2)a distinct category of value
(3)linear values and prices
(4)objective values; the value and price vectors are the same
for everyone
(5)value cannot be created or destroyed in exchange
(6)new value added in any period is directly proportional to
total hours worked
(7)physical stock conservation: total use value of each type
= previous total, less consumption, plus production.

Axiom (7) might seem trivial but there are perfectly serious
theories which violate it.

This is not exhaustive. I would be interested in a 'comparative
anatomy' of dynamical models in which we sit down and try to
honestly work out what are the underlying *axioms*, in the
sense above, of each general category of model.

I'd also be interested to know how far Mike W would consider
the above list of axioms drift from his sceptical 'wish list'
for the desirable characteristics of a model.