Sorry for the earlier confusion. I return to the questions
raised in Andrew's [3412] in response to my [3402]
"What you seem to disagree with me about is this: I
think that, in Marx's theory, the total value of the
product, in LABOR-TIME terms, is the value of the
consumed constant capital (circulating + depreciation of
fixed) and the new value added by living labor. You seem
to say that the change in the value of the economy's
assets during the accounting period --- again in labor-
time terms --- is the value added by living labor.
These two concepts give different answers."
Yes, they do give different answers, though we need to
assess carefully where these are qualitative and where they
are merely quantitative. So far I have not found an instance
where they make a qualitative difference except for one: the
conservation of value in the case where fixed capital is
involved.
A couple of preliminary clarifications. Andrew writes
"My examples concerning the Okishio theorem --- the one
in the book and the earlier one in the RRPE --- assume
a constant MEV or value of money. They do not derive it
from anything. "
Andrew is right. I am wrong. I elided two questions, one the
determination of value and second the determination of the
MEV. In the example under discussion Andrew does not discuss
a change in the MEV.
Andrew cites his 3307
"Thus, I understand the MEV at any moment to be the ratio
of the money value to the labor-time value of the
product. (Actually, however, since money is spent on
and used to value non-produced assets, I think that, in
a *real* economy, the numerator should be the money
value of *all* alienable assets and the denominator
their labor-time value.)"
I missed this. I think this is an important agreement.
However there is a connection between determination of value
and the determination of the MEV which was at the back of my
mind when I posted [3402] and I jumped a couple of stages.
To return to the main point.
My end result is a different formula for value which gives
the same qualitative behaviour of value and the rate of
profit, which is why we get the same graphs. My formula
differs from yours in the *speed* of change of value and of
the rate of profit. This makes a particularly important
difference in a case relevant to the question raised by
Duncan: does our interpretation predict that the rate of
profit declines asymptotically to zero? In both our
interpretations, the rate of profit rises if both
accumulation and technical change cease, for example due to
a slowdown in accumulation. However the change is slower in
my case than yours, and the larger fixed capital is relative
to output, the more marked the difference.
Let me first talk through how my value calculation would
differ from yours in the case of your illustration. To recap
its basic elements. I've made one change to your original
notation which is to use N for labour time instead of L. I'm
also going to use Qo for Q(0), etc. Your equations are
Q=Qo.b^t
F=Fo.b^t
A=Ao.b^t
L = Lo.c^t
[Q = output, A = input, F = fixed capital, L= Labour
time]
A preliminary question does not directly bear on the final
answer but sheds interesting light on the issues at stake.
What is the relation between the growth of F and the
production of Q?
Any change in capital stock is simultaneously a change in
its use-value and in its exchange-value. Thus there is a
physical as well as an exchange-value condition on the
relations between Q, F and A. Three notational
simplifications:
(1)I'm going to use D instead of Delta (the change in)
throughout;
(2)if I use any letter on its own, the time subscript t is
presupposed so F means F(t) etc
(3)For F(t+1) I will use F(+1), etc
There is a use-value constraint on the magnitudes involved:
DF < Q-A
This follows because F is augmented from Q and cannot grow
in one period by more than the surplus. We can be even more
precise in the case of your example where F=Fo.b^t;
D[Fo.b^t] = F(+1)-F = Fo.b^(t+1) - Fo.b^t = F(b-1)
Therefore
F(b-1) < Q-A
Therefore since F, A and Q are all growing at the same
exponential rate b
Fo(b-1) < Qo - Ao (1)
In the case of maximum expanded reproduction, the whole
surplus is invested and the proportions would be
Fo(b-1) = Qo - Ao (2)
We don't have to assume this however; it is just interesting
to note because it gives physical meaning to the
bifurcations and leads to a nice simple formula for the rate
of profit in the special case of maximum expanded
reproduction[See Note 1].
Assuming condition (1) is met, there will be a physical
surplus of goods which is divided between workers'
consumption and capitalist consumption.
My usual procedure is to suppose the fully general case in
which there are stocks of consumer durables which I'll call
W (wages or workers, as you will), B (bourgeoisie bosses or
_*s, as you will). The definition of F is ambiguous since we
have to decide whether it includes W, B and the circulating
capital A in order to discuss the total stocks of society.
To simplify matters I assume W and B are all consumed
entirely in one period (no consumer durables). Second, I'm
going to assume no unsold goods (stocks of Q=0 at the start
of production). Third, I'm going treat F as representing the
non-circulating component of capital so that the total
productive capital at the beginning of the circuit of
reproduction is F+A. Finally I assume W and B are already in
existence at the start of the circuit [See Note 2]. Then at
the start of the circuit of reproduction we have a total
stock of alienable goods of all types K where
K = F + A + W + B
During production A, W and B are consumed and replaced by Q
and we have
K(+1) = F + Q
This is before investment has taken place, that is before
the sales goods Q have been allocated to augmentation of F
and to the next circuit's W and B.
My argument is that fixed capital participates in the
formation of social value from individual value on the same
basis as newly-produced commodities. In this way, no new
value can be created in exchange other than by living labour
and any value destroyed in consumption appears either as the
value of W or the value of B. My procedure is therefore as
follows
Step 1: calculate the total value in the economy at the
beginning of the period
Step 2: calculate the total value in the economy at the
end of the period
Step 3: deduct the value consumed in any way from the
results of step 1, valued at v(t)
Step 4: add to the results of step 4 the value
transferred in the form of circulating constant capital
Step 5: add to the results of step 4 the new value
created by living labour
Step 6: the results of step 5 must equal the results of
step 2.
In the case where there is no fixed capital, this reduces to
the standard calculation. For if w represents the use-values
in the wage
Value at beginning of period = vA+W assuming wage goods are
in existence at this time
Value at end of period = v(+1)Q
Value consumed = vA+W
Value transferred = vA
Value added = L
Therefore
v(+1)Q = vA + W + vA -(vA+W) + L = vA+L
So it can be seen that without fixed capital, our results
are the same. With fixed capital the calculation is as
follows:
Value at beginning of period = vF + W (same assumption
re wagegoods)
Value at end of period = v(+1)F(+1)
Value consumed = vA+W
Value transferred = vA
Value added = L
Therefore
v(+1)(F+Q) =
Initial stock of value v(F+W+B+A)
LESS value used up -v(W+B+A)
PLUS value transferred + vA
PLUS living labour +L
v(+1)(F+Q) = v(F+A) + L
We can write this as
(v(+1)-v)F + v(+1)Q = vA + L or more simply
F DV + v(+1)Q = vA + L
which differs from your equation
v(+1)Q = vA + L
in the inclusion of the term I have referred to as the stock
revaluation term F DV. This term is equal to the change in
value of the stock F, due to changes in the social value of
the commodity concerned or, as we have been calling it,
moral depreciation. This is the term I was discussing with
Duncan under the heading of the IVA, though for reasons I
gave recently it may be a bit more complex than that: as can
be seen, the term in the equation above was derived, as you
say, on the assumption of a constant value of money and
cannot therefore from my point of view represent a monetary
effect.
Now when we come to the solution we have to reorganise it as
v(+1)(F+Q) = v(F+A) + L
and again it can be seen that if fixed capital is zero, this
reduces to the normal sequential equation.
In the case of your example F, Q and A are all growing at
the same rate. The solution behaves the same as yours,
therefore, except that we must replace your coefficient a
=A/Q by the coefficient (F+A)/(F+Q). Generally this will be
larger (->1 as F->oo) and the rate of change of value is
therefore slower. This corresponds to the fact that a large
stock of a commodity acts as a 'buffer' absorbing sudden
rapid changes in productivity so that the value falls only
slowly as the existing stock of this commodity gets used up.
That's the formal difference.
This becomes a bit sharper, though not much so, accumulation
and/or mechanisation stops. Suppose, for example, as a
result of the profit rate falling, the capitalists cease
accumulating and also technical progress halts and the
economy reverts to simple reproduction of use-values so that
the capitalists merely replace used-up fixed capital. All
magnitudes Q, F A and L are frozen and we go back to 'simple
reproduction'. This is perhaps equivalent to your 'one-time-
only mechanisation' situation.
The equation still reads
v(+1)(F+Q)=v(F+A)+ L
but now F, Q and A are constants. The solution is simpler:
let (F+A)/(F+Q)=f and note that 1>f>a. Let ve be the static
equlibrium value. Let vo now be the starting value of this
new phase of the process. Recall that because sequential
prices fall slower than equilibrium prices, ve<vo. For
notational convenience let k = (vo-ve)/ve
Simultaneous solution is ve=L/[Q-A]
Andrew's solution is v = ve+(vo-ve)a^t = ve(1+ka^t)
Alan's solution is v = ve + (vo-ve)f^t = ve(1+kf^t)
So both solutions converge on the equilibrium price from
above, bFut Alan's converges slower. The bigger the fixed
capital, the slower the convergence
This also gives us a quite simple form for the rate of
profit since the denominator is now Av [Recall A, Q, F are
the magnitudes reached when the music died]
We have for max. profit rate, letting vo = kve
Simultaneous: re = L/A.ve = Q(1-a)/A = (1-a)/a
Andrew: r = L/A.v = re/(1+ka^t)
Alan: r = L/A.v = re/(1+kf^t)
So both solutions converge from below on the new equilibrium
profit rate, but again at different speeds.
In this case, even though the fixed capital is valued at its
historic costs, it is being replaced slowly over time by
cheaper capital and its value is therefore sinking
asymptotically towards its stationary level.
Finally, I want to look at the conservation properties of
the two systems. Though this can be done for Andrew's full
illustration, and the same type of difference emerges, I
want to study it in the context of the one-time-only
mechanisation, or as it is here studied, a simultaneous
suspension of accumulation and technical change.
By definition, the total increment to the value of the
stocks of society in my solution is always equal to living
labour, less private consumption of all types.
In Andrew's case, what will the value of stocks become
after, let us say, the first period following the suspension
of change? To study the problem in its pure form, let us
suppose that there are no consumption goods of any form in
existence at the start of this period, the system being in
crisis.
Suppose for simplicity that F=90 and A=10, that Q will be 20
when production has happened, and that L = 10. The
coefficient a is thus 10/20 = 0.5. The equilibrium value is
then 1. But suppose because of the previous phase of rapid
technical progress the actual current value is still 2. The
total value of stocks in society is hence 2F + 2A = 200.
We now find that in the first production period Andrew's
equations yield
v = ve + (vo-ve)*(0.5) = 1 + (2-1)*0.5 = 1.5
At the end of this period, however, there is in existence 90
units of F that were not used up, and 20 units of Q that
were produced. The result is 110 units valued at 1.5, being
165 units of value. The total value of the stocks of society
have declined by 35 even though living labour to the value
20 was discharged.
This might be thought not disconcerting. Value vanishes all
the time. But suppose, instead, that there was a temporary
*decline* in productivity, so that the starting value was
not greater but lower than equilibrium, so vo = not 2 but
0.5.
In that case, the initial value of the stock would be
100*0.5 = 50. Andrew's equations would yield
v = 1 + (0.5-1)*0.5 = 0.75
But now at the end of this same period, the stocks in
existence would as before be 110 and their value would be
82.5, so that they have *increased* in value by 32.5 even
though L was only 10. Value has been created out of nothing.
My solution would yield the following: f = (F+A)/(F+Q) =
10/11
Case 1 (vo=2):
v = ve + (vo-ve)*(10/11) = 1 + (2-1)*(10/11) = 21/11
New value of stock = 110 * 20/11 = 210, exactly the value of
the old stock plus living labour.
Case 2( vo=0.5)
v = ve + (vo-ve)*(10/11) = 1 -0.5(10/11) = 6/11
New value of stock = 110* 6/11 = 60 = value of old stock
Plus living labour.
These differences are connected to the phenomenon of the
release and tie-up of capital. What puts people off my
solution is the fact that the gross value of current
production can appear less than the value directly consumed
or added. Consider case 1 above. The capitalist buys A for
10*2 and adds living labour 10 so if we merely add up the
'individual' value of this output we find it should be worth
30. But there are 20 units of output and each unit is worth
21/11. So its value is in fact 420/11 ~= 38. Where has the
extra 38 in value come from? I argue it is not 'created' but
transferred from the pre-existing stock by a process of
averaging, which sets a social value that is above the
individual value of the new product but below the historical
value of the stocks.
Last point on the MEV: of course, in the last case, you
might still maintain that value conservation was not
violated, *if* you maintain that the MEV had meantime
fallen. My take on this is that differences in theories of
value lead directly to differences in theories of the MEV
which may help account for some of the discrepancies that
worry Duncan.
Notes
=====
[Note 1] For a circulating capital model F = A neglecting
wages the stock condition reduces to
Ab = Q
that is, since A/Q = a, the technical coefficient
b=1/a
If moreover labour inputs are constant (L=constant) this
gives an equation for maximum expanded reproduction with
fixed labour inputs
v(+1)b^t=vab^t+l, that is
v(+1)=va + la^t
This has the special case solution
v = voa^t+lta^(t-1)
The denominator in the profit rate then becomes
vAo/a^t = Ao{vo+lt/a} or Ao{vo+ltb}
and the profit rate becomes
L/[Ao{Vo + ltb}]
which declines indefinitely as Duncan suggests, I think.
[Note 2] I had a discussion ages ago with Costas because he
felt that the assumption that goods were advanced at the
beginning of production was responsible for the result of an
FRP but the exercise can be repeated under the assumption
that these goods do not appear until the end and the
qualitative results are not, I think, any different.