[OPEL:6215] consistent periodisation

Alan Freeman (a.freeman@greenwich.ac.uk)
Tue, 24 Feb 1998 13:05:59 +0000

A reply to:

Re: [OPE-L]a consistent, temporal, national accounting system
Date: Sun, 22 Feb 1998 21:58:57 -0500 (EST)
From: "Duncan K. Foley" <dkf2@columbia.edu>

Duncan raises a number of interesting points in response to my argument
that a consistent periodisation must involve the alternation of consumption
and production.

"In real life, of course, circulation and production are all intertwined
and all going on at once. In most lines of production there are a lot of
circuits of capital in process at any moment, and they are not all neatly
synchronised... It is possible to construct period models that will give
the correct intuition as to what will happen in continuous time, but it is
also possible to construct period models that are completely misleading.
The key issue is whether the period model in question has a well-defined
limiting continuous time model as we allow the period length to go to

I absolutely agree that this is the key issue and it was the main question
that exercised me in writing the last chapter of "Marx and Equilibrium
economics" where I endeavoured to show that the stock-based approach
achieves this rigorously.

In fact it was through thinking about the question of how to approach a
limit that I came to the conclusion it was impossible to do this using
flow-coefficients for the MELT. Singularities would always arise, as a
result of the fact that the same coefficient appears in both stock
relations and flow relations (see my last post). I found that the only
circumstances which made the flow-coefficient work were those in which the
MELT was constant.

I also agree that many period models are completely misleading, but I would
modify Duncan's statement slightly. I would say that the models that
appear to give the correct intuition are also the most misleading ones. I
think a rigorous enquiry calls on us to be suspicious of what we find
intuitively reasonable, when this intuition itself derives from a
conceptual framework that we have come to suspect. I know it sounds
pompous, but I think this kind of sceptical activity is what generally
leads to revolutions in thought, so it's worth considering.

The entire framework for periodisation was set by Bortkiewicz and we have
been dogged by it ever since, because he adopted the assumption that all
capital turns over precisely in one period. That is, he abolished the
contradictions to which Duncan refers. He makes everything have the same
turnover. It doesn't help that he also insists prices cannot change, so
that the issue of changing prices is ruled out a priori and indeed
Bortkiewicz decrees that the introduction of changes in price leads to

Since that time, all period models that I know of have followed
Bortkiewicz's conceptualisation, not Duncan's; it is Duncan's that I think
is correct, but the problem is that we are trying to investigate a possible
the source of errors in models that have never integrated these insights or
even, to my knowledge, made the attempt or considered the difficulty.
Certainly I have found nothing in the Sraffian literature that deals with
it, discussions of turnover are in general few and far between.

My argument is this: if one is going to adopt a period model, one cannot
avoid making a decision about when the changes in that period actually
happen. One cannot just wave one's hands in the air and say that
they all happen all the time, because by adopting a period model one has
already excluded that possibility. Moreover, the minute one supposes
that something is turned over in less than a whole period, one has
actually introduced a shorter period and it is this shorter period
that must be considered before we can arrive at any valid identities.

The period model is the economists' equivalent to the physicists weightless
rods, perfectly smooth bowls, dimensionless particles, and so on. We know
that these assumptions are inadequate to reality but unless we replace them
by a better alternative, we must apply them doggedly throughout. We cannot
suddenly, halfway through an analysis that has hitherto assumed a
weightless rod, suddenly interject that real rods actually have weight. We
cannot, halfway through an analysis which has hitherto assumed a uniform
turnover equal to the length of the period, suddenly interject that this is
not the case. If we do want to make this interjection we must start again
at the beginning.

But in the present discussion we are not starting again at the beginning;
we are adopting without question the assumptions of our forbears who
devised the period model, and assessing the conclusions that follow from
these assumptions; such as, whether they reveal inconsistencies in various
arguments. I am only saying that if we do that, we must apply these
assumptions consistently. So, what are the issues involved in that?

I don't think there is any real problem about whether changes are
deemed to happen at the beginning or the end. There are only two real

(a) if changes happen at the end of one period, then they must happen at
the end of every period. Conversely, if they happen at the beginning of one
period, then they must happen at the beginning of every period. The
periodisation has to be consistent.

(b) time has to proceed in the model as it proceeds in the world: forwards.
You can't introduce assumptions that require people to sell goods they have
not produced, or use goods that do not exist; in a nutshell you cannot
date the purchase of a good from before the sale of the same good. This
is my essential point in the periodisation discussion. In any exchange,
people sell the goods that they have produced up to the exchange but not
after it, and people purchase the goods that they will use in the period
after the exchange but not before it. After all, if we don't conserve
matter there is very little prospect of conserving value.

(optional c:) If the question of turnover arises, I would tend to say that
the period should be no greater than the smallest turnover period under
consideration. There might be another way to do it but I haven't seen
anyone achieve it.

Provided one sticks with these rules, I think none of the contradictions in
the simultaneous calculation that have been pointed out by Andrew, myself,
and others, can be abolished. Of course, it may be possible to abolish them
by softening assumption (c) but I haven't seen it done and I have very good
reasons for thinking it can't be done.


Duncan raises a second issue that I think is also important, namely, how
can we operationalise a periodisation. But I think the very posing of the
question, in the way that Duncan does, goes to the heart of the
methodological difficulty. The difficulty is that, whatever their intuitive
reasonableness, period models are wrong: that is, they are insufficiently
concrete, they have insufficient determinations, to describe reality
without error. Therefore, there *is* no way to operationalise a period
There will always be errors if one does so. One must in fact begin from the
continuous formalization in order to operationalise the accounting.

This does not at all rule out drawing up period accounts. One does not need
a period model in order to draw up a set of accounts showing all purchases
and sales during the period, and the changes in the values of all stocks,
as the accountants actually do. The errors arise only if one attempts to
take this as a fully accurate representation of the data in a period model.

Nor does it rule out a conceptual analysis of period models using a system
of accounts. After all, a period model is only a hypothetical special case
in which everything turns over in at least a year, and in which all
transactions take place at once. Since it is a special case, the same
accounting identities must apply. So we can address questions such as
'where does unproductive labour fit in the accounts' by these means, and
ascertain whether there is a consistent way to assign this to surplus value
in a period model. All I was asserting is that there is such a consistent
way to do it.

It does rule out treating a period model as an actual description of
reality and, to rule out any misunderstanding, that was not at all my

One cannot in fact solve either the accounting problem, or the
periodisation problem, by proceeding to map the one onto the other as if
this difficulty did not exist, and I think a great deal of the problems
arise from the attempt to do this. In my opinion there is a quite distinct
problem in actually calculating the magnitudes of which we speak,
independent of how the periodisation is conceptualised; this is that prices
are actually changing during the measurement period.

To solve this I would make the following suggestion: I think one must first
proceed to the continuous case in order to derive the fundamental
accounting equations, and then integrate these to obtain the aggregators
of annual expenditure. This is because only the continuous formulation is
sufficiently concrete to deal with the difficulties that Duncan describes,
namely, the intertwining of many turnovers and the fact that prices (and
values) change during the period under consideration.

In that case, I would say that what we actually observe in the accounts is
the path-integral, over the year, of the variable in question. For example
what is actually recorded as output is the integral of the price paid for
everything sold in the year, and this includes the variation due to price
changes and due to the output itself. If one wishes to make the
simplification (and it is a simplification) that the price is in fact
constant during the period, then one is working with an average price,
neither the price at the beginning nor at the end. In consequence, if one
uses actually-observed output as a means of estimating either the
start-of-period price, or the end-of-period price, there will always be an

If one then wishes to conceptualise this, I think the most intuitively
helpful way to do so is to consider these end-year accounts as the summary
of a large number of very small period accounts, say, each a week long.
It is always possible to aggregate accounts for successive periods, once
one has solved the problem of producing them in the first place. One simply
carries over the figures from each period into the next.

>From Duncan's remarks about this question, I think we would probably agree
on that.

Perhaps another way of looking at this is to say one begins from the
component of capital with the shortest possible turnover (generally labour
power) and treats everything with a longer period as fixed capital.



There is a possible fourth point in the differences about periodisation -
I'm not sure - which arises from
the following remark of Duncan's in response to Andrew:

"As I've pointed out above, this isn't actually consistent with the
treatment of the year as a period, in which prices and wages have to remain
constant by definition."

But if prices remain constant by definition during a period, then I hope
this means that they must change in between periods. Otherwise they can't
change at all. It is conceivable that Duncan might be treating this change
as taking place at a different time to the actual exchanges, as if the
Walrasian auctioneer sneaked into the room at midnight and switched the
tickets around.

For me the simplest idea is that the prices are formed when the goods are
exchanged. However, if this is not the case, and if prices are constant
through the period, then the accounting identities are going to be exactly
the same as if we move the acts of sale and purchase to the point where the
prices change. Then my principles (a) and (b) applies; if you move the
sales back, they all go back and if you move them forward, they all go
forward; you can't move period 1 forward and period 2 back. And finally
since every purchase is also a sale, we have the following:

(1) the sales constitute the revenues up to, but not after, the point
where the exchanges take place.

(2) the purchases constitute the costs after, but not before, the point
where the exchanges take place.

which is my basic point about periodisation. What I think is illegitimate
is to treat the purchase of a good as if it happened before the sale of the
same good.