# [OPEL:6215] stock-based and flow-based MELT

Alan Freeman (a.freeman@greenwich.ac.uk)
Tue, 24 Feb 1998 13:05:32 +0000

In the discussion on the MELT, and whether it is best treated as a stock or
a flow measure, I think Duncan does us a service by focussing on the
relation between new value and living labour. I think homing in on this
question offers the best promise of clarity.

Duncan is convinced that the use of a stock measure violates the principle
that value created is given by new labour time worked.

Of course, I am equally convinced that the use of a flow measure for the
MELT violates this same principle.

To save space, I'm going to use the words 'value conservation' in this post
to mean this idea. To save diversionary debates, I'm not going to use it
for anything else. It's just a shorthand.

This, for me, is the surest indicator that we are applying a different
concept to one or all of the words new labour, new value, stock, flow, or
MELT. I think it is very important indeed for everyone to recognise that
our disputes arise out of these conceptual or paradigmatic differences and
not out of trivial mistakes or slips in either the temporal or the
simultaneous approach.

I want to see if we can reduce the difference to its very simplest element.
For me the best way to do this is discuss the following question: what
happens when the MELT is changing?

I want to say why I think the flow calculation violates value conservation
in the presence of stocks with a changing MELT. My argument can be put most
forcefully by proceeding directly to a continuous formulation, but I will
restate it discretely afterwards. I'll use differential rather than
integral calculus because I think most people are more familiar with it
[but see footnote].

Let us write down an expression for the money measure of a stock whose
labour-value is K and whose money value is \$K. Using m for the MELT we get

[1] \$K = m.K

That is, the money value of the stock, like every money value, is given by
its labour value, times the MELT. Now differentiate this. We get

[2] (\$K)' = m'K + K'm

Now let's enquire into the meaning of some of the terms. For simplicity
assume nothing is consumed during the period under consideration. If new
labour is being added at a rate L, then to me, this is what is meant by K',
so that

[3] K' = L

Equally the rate of growth of the monetary expression of K is what we all
term 'Monetary Value Added' or MVA. Thus

[4] (\$K)' = MVA

Now substitute these into [2] to give

[5] MVA = m'K + Lm

Now, Duncan and Fred both define m by

[6] m=MVA/L

Substituting this into [5] gives

[7] MVA = m'K + MVA

that is

[8] m'K = 0

Therefore, only two presuppositions really support this calculation
consistently; either capital stock is zero, or the MELT does not change. I
don't wish to be insulting but I think that if we examine most of the
discussion about periods carefully, we find that the outcome boils down, in
the last analysis, to slipping in one or other of these suppositions: a
constant MELT or zero fixed capital.

I have put the contradiction in this extreme form to focus maximum
attention on the issue that I suspect is at the heart of the debate. Taking
the discrete form, I'm going to use the delta notation because this
focusses the issue. Suppose that instead of an infinitesimal change in K,
\$K and m, we have finite changes dK, d\$K and d\$m (I can't get a delta out
of this verkokte e-mail system)

Take equation [1] (please) and substitute into it the expressions K+dK,
\$K+\$K' and m+dm, to get

[10] \$K + d\$K = (m + dm).(K + dK)

Now, dK or the increment to K is the new labour added and d\$K or the
increment to \$K is the money value added. So we can write this as

[11] \$K + MVA = (m + dm).(K + L)
= m.K + dm.K + mL + dm.L
= \$K + dm.K + m.L + dm.L

by equation [1], the definition of \$K. This gives

[12] MVA = dm.K + m.L + dm.L

Now substitute into this the definition m = MVA/L. This yields

[13] MVA = dm.K + MVA + dm.L

whence

[14] dm.K + dm.L = 0

which can work only if dm is zero, or the initial capital stock is
negative. As the interval of time reduces to zero we arrive at K=0,
consistent with our finding in the continuous case.

Now the many contradictions that arise in the present discussion, in my
opinion, all arise from equation [11], but are complicated by incomplete
theorisations of periodisation. In consequence, what happens is that by
substituting different partial assumptions into [11] we can derive one or
another contradiction, much in the same way as the mathematical puzzles
which prove 1=2 by substituting into the expression x = 0/0 and disguising
that this has been done.

I suspect, though I am not sure, that the origin of the conceptual
difference is that Duncan and Fred cannot (if they think about it) accept
the equation

[1] \$K = m.K

that is, they cannot accept that existing capital is revalued when the MELT
changes. Indeed, this is the essence of the difference about equation [1],
because all I do is turn this around to derive my MELT.

In short the essence of the discussion is a different concept of the MELT.

The fundamental issue at stake, as I see it, is whether the MELT is a
universal coefficient of the economy which yields the value of all monetary
magnitudes including stock magnitudes; or whether it is a coefficient that
only applies to flow magnitudes.

All I say is that if \$100 of new value counts as 100 hours, then \$100 of
old value must also count as 100 hours. If this principle is applied
consistently, no violation of the value conservation principle results that
I can see. Since I have defined K' to be L, the integral of K over any path
must necessarily be the sum of the labour-time worked along that path, so
that the increment to K must be exactly equal to the total living
labour-time of the period concerned.

Alan

Footnote on integral equations
==============================

Duncan rightly says that continuous formulations lead to integral
equations. However, to every integral equation there is a corresponding
differential equation and this form is a bit easier to handle, though the
integral equations which Duncan derived in his work in the early 80s are an
extremely useful way of expressing these differential relations, and a
mathematical complication arises from the treatment of delays.

The presentation above makes no reference to delays and these equations
would have to be true also in the presence of delays, since they are quite
general identities.

I think the integral equation form of

[2] (\$K)' = m'K + K'm

is

[2a] \$K[t,T] = Integral[t,T] (m'K).dt + Integral[t, T] (K'm).dt

To go back from 2a to 2, differentiate with respect to T, the upper limit
of integration.

I'd be very interested to study the consequence of reformulating Duncan's
1982 equation system as the corresponding differential system. I think
these equations were an extremely important contribution. The difficulty
lies in the role played by delays; mathematically I suspect what is
required is to find a way to proceed from a convolution to the
corresponding delay differential equation.