PRELIMINARIES
=============
Duncan writes:
"the TSS MELT has an offsetting formal disadvantage, which is
that it can be measured only by stipulating an initial period
MELT, u0. We know from Andrew's examples in other contexts that
the choice of this initial value can lead to quite different time
series for the TSS MELT, and it is not clear how we can make the
measurement of the crucial u0 operational."
I think the second sentence is mostly incorrect. As Alejandro
Ramos noted in OPE-L:289, Alan Freeman has shown that any error
in the MELT due to mismeasurement of the initial condition decays
(and rather quickly), so that the path of the estimated MELT time
series quickly converges to the true one. So, after a bit of
time, the time series for the TSS MELT is always the same, no
matter what the initial condition. I'm not sure what examples
Duncan is referring to, but given the convergence of the
estimated MELT to the true one, I must have miscalculated.
Actually, Alan's proof refers to the measurement of consumed
constant capital in labor-time terms, but any error in that is
traceable to an error in estimating the initial value of the
MELT. Here I will refer directly to the MELT, and provide a
simpler proof. I also want to discuss how to estimate the
initial value of the MELT and explain why even the initial error
will be quite small.
But first, let me note that *nothing* of theoretical importance
is at issue here. I say this because, in prior discussion with
Duncan about this matter, we agreed that it is possible to
operationalize the TSS MELT, especially if the initial condition
is estimated wisely, given the convergence of the estimated MELT
to the true one. Yet, Duncan suggested, theoretical precision is
important. I agree. But there's no imprecision in the
theoretical conception of the TSS MELT. The only imprecision
that arises is imprecision in *measurement*. But that is true
for every single theoretical variable, including the simultaneist
MELT. Are the GDP figures wrong? By how much? How much of the
labor reported by the government is productive and how much is
unproductive? Etc.
I also note that all dynamical systems require initial
conditions. So if the need for initial conditions is a mark
against the TSS MELT, it is also a mark against a huge chunk of
existing science.
THE MELT
========
Duncan's derivation of the TSS MELT is correct. But, because it
may prove to be important, I'll quibble with his use of the term
"current period MELT." The TSS MELT refers to a moment in time,
not a span of time (period).
To simplify things, I'll use different notation. P is the
aggregate money price of gross output at the end of some period.
C is the aggregate money price of inputs entering production at
the start of the period. L is the total productive labor
performed during the period. I'll retain Duncan's "u" to denote
the TSS MELT. It is the ratio of total value in money terms to
total value in labor-time terms:
P
(1) u(t+1) = ----------.
C/u(t) + L
The deflation of C, a money term, by u(t), gives consumed
constant capital in labor-time, so the denominator is total value
in labor-time terms. (I have omitted time-subscripts for P, C,
and L, because their position in time is already contained in
their definitions.)
ESTIMATION ERROR
================
(1) is the *true* value of the MELT. Using u' for the
*estimated* value, we have, similarly,
P
(2) u'(t+1) = -----------.
C/u'(t) + L
Using (1) and (2), we can obtain the estimation error in
"percentage" terms. A bit of manipulation lets us write it as:
u'(t+1) - u(t+1) C u'(t) - u(t)
(3) ---------------- = ---------- * ------------.
u(t+1) C + u'(t)L u(t)
DECAY OF ESTIMATION ERROR
=========================
Examining (3), we see that the percentage estimation error at any
time equals C/[C + u'(t)L] multiplied by the estimation error at
the preceding time. No matter how C, L, and u' vary over time,
C/[C + u'(t)L] MUST be less than 1, since u'(t)L > 0. Hence,
each successive percentage estimation error MUST be less than the
preceding one. The errors decay over time. (The absolute error
must also decay unless the MELT is rising extremely rapidly.)
How rapid is the decay? It depends on the value of C/[C +
u'(t)L]. Multiplying (2) by C/[u'(t)P], we obtain
C/[C + u'(t)L] = (C/P)[u'(t+1)/u'(t)].
u'(t+1)/u'(t) is an estimate of "1 + growth rate of MELT." The
growth rate of the MELT is essentially determined by inflation
(of use-value) and productivity growth. From the definition of
the MELT, it follows that
1 + growth rate of MELT = (1 + inflation rate)(1 + productivity
growth rate)
or, using m, i, and q for the growth rates of the MELT, the price
level, and productivity,
(4) 1 + m = (1 + i)(1 + q).
So, in an economy such as the US, with low inflation and slow
productivity growth, u'(t+1)/u'(t) will not be much larger than
1. Measuring C as purchases of intermediate goods plus
consumption of fixed capital, and P as the sum of C and GDP, the
figures for the U.S. economy in 1996 indicate that C/P is right
around 0.5 (C is very close to GDP). So C/[C + u'(t)L] is
undoubtedly less than 0.6, even in times of double-digit
inflation.
This indicates that the decay in the estimation error is quite
rapid. If C/[C + u'(t)L] = 0.6, and the estimate of the initial
value of the MELT is off by 6.25%, the subsequent percentage
errors are 3.75%, 2.25%, 1.35%, 0.81%, .... Thus, as I've
remarked before, if you can sacrifice a few data points, five
periods or so at the beginning of the series, there's really
nothing to worry about. Certainly, given the magnitude of other
estimation problems (by how much is the GDP underestimated: 5?
10?, 15? No one knows), this is small potatoes indeed.
ESTIMATION OF INITIAL CONDITION
===============================
The larger the error in the initial condition, the larger will be
each subsequent error in the MELT series. So it may be wondered
whether my example of an initial error of 6.25% is too small. I
want to suggest that it is probably larger than the error one is
likely to make, and not very likely to be smaller.
The reason is that one is not just picking the initial value out
of the hat. Were that the case, the initial error could be quite
large, but some independent knowledge of the MELT's growth rate
is available, and can be used to minimize the error. (4)
implicitly gives the relation between u(t) and u(t+1). Since 1 +
m = u(t+1)/u(t), it follows that
(5) u(t+1) = u(t)*(1 + m) = u(t)*(1 + i)(1 + q).
Using (5), (1) can be rewritten, for t = 0, as
P
(1') u(0)*(1 + i)(1 + q) = ----------.
C/u(0) + L
Thus, if one has perfect knowledge of i and q, they, together
with the known figures for P, C, and L will give an exact value
for u(0):
P/[(1 + i)(1 + q)] - C
u(0) = ----------------------.
L
This isn't possible, of course. But information on i and q is
readily available. It is not the exact information one wants,
because both i and q are generally measured for GDP, not gross
product, and we also want, ideally, a measure of labor
productivity growth based on total labor, not living labor, but
information is also available to make adjustments.
In any case, even if the estimates of i and q are wrong by a good
bit, the initial percentage error will not be affected that much.
Given C/P = 0.5, and the true i and q both equal to 0.03, if one
estimates them as equal to 0.02, then the estimated u(0) is 4.2%
greater than the true u(0). If one estimates them as equal to
0.04, the estimated u(0) is 4.1% smaller than the true one.
I hope to address the theoretical issues in the very near future.
Ciao
Andrew
Andrew ("Drewk") Kliman Home:
Dept. of Social Sciences 60 W. 76th St., #4E
Pace University New York, NY 10023
Pleasantville, NY 10570
(914) 773-3951 Andrew_Kliman@msn.com
"... the *practice* of philosophy is itself *theoretical.* It is
the *critique* that measures the individual existence by the
essence, the particular reality by the Idea." -- K.M.