1. Andrew is correct and I was wrong on the stability of the difference
equations defining the TSS MELT. (I append his discussion below for the
convenience of OPE-Lers who might have forgotten the details.)
2. It's possible to strengthen Andrew's derivation a bit. In a circulating
capital one-sector model where a units of output combine with l units of
labor at the beginning of the period to produce 1 unit of output at the end
of the period, let x be gross output, p be the money price of output, and u
the TSS MELT. Then, writing p(-1), u(-1) for the previous period's values
of the variables, the TSS MELT is defined by:
(1) px = ulx + (up(-1)ax)/u(-1)
Here px represents the money value of the stocks of commodities in
existence at the end of the period, which, according to TSS reasoning,
consists of the money equivalent of the labor expended during the period,
ulx plus the money equivalent of the labor embodied in the stocks of
commodities available at the beginning of the period, p(-1)ax/u(-1).
Sometimes TSSers want to take prices as given and calculate the MELT, and
sometimes, as in Andrew's examples attacking the Okishio Theorem, to take
the MELT as given and calculate prices. The problem of the stability of the
difference equation is equivalent in these two formulations, as we can see
by defining the new variable z = p/u. (z has the dimensions of labor time
per unit of output, so it is like the inverse of labor productivity.) Then
we can see that the difference equation (1) defining the TSS MELT implies:
(2) z = l + z(-1)a
It is not difficult to show along the lines of Andrew's demonstration
below, that an initial deviation in the measurement of z, due either to the
MELT or to the price level, will shrink over time in general. It would be
nice to generalize this theorem to a fixed capital setting.
3. Two other theorems emphasize the close relation between the TSS MELT and
the NI MELT:
If prices are stationary, the TSS MELT converges to the NI MELT.
If relative prices are stationary, and the rate of change of prices is
constant, the TSS MELT converges to the NI MELT.
4. In my opinion the decisive problem with TSS MELT remains the fact that
it imputes changes in the valuation of existing stocks of commodities due
to price changes that result from technical change to living labor time.
Marx seems quite explicit in rejecting this way of thinking, as Fred
Moseley's quotes have documented. (It's also much harder to make
operational in a context of fixed capital of various vintages, because of
the need to keep track of the labor time equivalent of the money value of
each vintage of capital separately, but this is perhaps not a theoretically
decisive argument.)
Duncan
>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.
>
>
Duncan K. Foley
Department of Economics
Graduate Faculty
New School University
65 Fifth Avenue
New York, NY 10003
(212)-229-5906
messages: (212)-229-5717
fax: (212)-229-5724
e-mail: foleyd@newschool.edu