Fred,

Let's see if we can pin down the disagreements here.

>  I don’t get your distinction between a quantitative theory and a
> qualitative theory.  How can marginal productivity theory (or any theory)
> be a quantitative theory without also being a qualitative theory.

Are you saying that we cannot abstract quality from quantity?  I think
it is entirely possible to say that the "substance" of profit is x
and, then, to say that the magnitude of profit is determined by y,
where x and y are two different propositions.  Cf. Marx on relative
value in the very first chapter of Capital.  First, he deals with its
qualitative nature and then with its "quantitative determination."
What's wrong with that?

> Marginal
> productivity theory concludes that profit = MPK.  The MPK is a quantity of *
> output*, so the quality (or substance) of profit in this theory is *output*,
> as opposed to labor or surplus labor in Marx’s theory.

Why opposed?  If you make output=labor, the opposition dissolves.
(Here I'm glossing over the distinction between use value and value,
because deep down -- as Marx shows -- they are the same thing, or --
to be a bit more "precise" -- one is the social form of the other.)

> This argument assumes that means of production can be added and output
> increased *without adding more labor*.  I think there are many production
> processes for which this is not possible; e.g. another shovel without
> another worker to dig another ditch.

I think you're making too much of this.  Of course you can add another
shovel without adding another worker and, thereby, alter the number of
ditches.  Let the period of production be five years, L=1 worker, K=1
shovel, and Q=100 miles of ditches.  It's entirely possible that just
by increasing the number of shovels to K=2, you wind up with an
altered Q (e.g. Q=300 miles), as the worker can now safely discard an
old, worn, and blunt shovel, replace it with a new, sharper one, and
go on digging longer.  Etc.

> This argument also assumes that means of production can be added and output
> increased *without adding more raw materials *(or intermediate goods in
> general).  This is impossible in all goods-producing industries which
> require more raw materials in order to produce more output; e.g. in order
> to produce another car, one needs more wheels, brakes, glass, etc.

I think this is another failure of abstraction.  Of course you can add
means of production other than raw materials, keep the amount of raw
materials fixed, and alter (e.g. increase) output.  Suppose you are
producing cars and using metal and glass as your raw material, and
also robots with sensors that monitor the amount of metal and glass
wasted in the production of your cars.  Increase the number of robots,
and the amount of raw material saved may increase.  (Similar results
could be attained by increasing L, human monitors, without increasing
raw materials.)  As a result, you can produce more cars with the same
amount of metal or glass. QED.

> This was my main critique of Krugman, and with respect to raw materials
> specifically.  If the quantity of machines (or labor) increases in
> good-producing industries, *raw materials must also increase *in order to
> produce additional output (e.g. cars and wheels).  So the marginal product
> of capital (or labor) is *meaningless *in these production processes,
> because the concept of the MPK requires that *all other inputs be held
> constant*.  You have not yet answered this key criticism, either in this
> post or in the post that you linked to on the Cambridge capital critique.

I guess what you're trying to say is that, under certain assumptions
about the granularity of input and output units, in some processes
more than in others, the ability of producers to substitute one means
of production for another is next to zero, because production
possibilities are such that you have virtual Leontief isoquants, which
lead to undefined marginal rates of input substitution (and hence
marginal rates of input-output transformation) at given points or
segments.  (I'm alluding to isoquants here as you yourself decided to
put aside the aggregation problem in your post, although I think we're
dealing with different versions of the same old argument.)  Well, in
that case, your decision rule (optimal choice of input and output
levels) has discontinuities and you have to plug the holes in your
decision rule with intuition or seat-of-your-pants decision making,
but that doesn't mean that the decision rule derived under the
assumption of continuity is stupid.  It just means that it's
insufficient to answer your every question.  If you want a rule that
is more fine-grained, then you have to subdivide your inputs and
outputs into finer units, which may entail additional costs.  Such is
life.

The fact is that the distinction between "raw materials" and "fixed"
means of production (in a firm's accounting or in the national
accounts) is much more fluid than it appears at first sight.  It
hinges entirely on our conventions about the period and scale of
production.  I'm not saying these conventions are entirely arbitrary
-- they make sense (are economical) for certain human purposes.  But
they are conventions nonetheless.

Hope this helps.
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