In the Fall of 1963, I was living in Cambridge, MA, on leave
from an Assistant Professorship at the University of Chicago and doing a
visiting year at Wellesley College. For
four years, I had been deeply involved in the so-called Campaign for Nuclear
Disarmament, an effort to alert Americans to the dangers of nuclear war. I lobbied for an agreement with the Soviet
Union to put a halt to the nuclear arms race and even, perhaps, to reverse
it. I wrote, I lectured, I argued, I
marched, and I spent every day worrying about a catastrophic miscalculation by
American or Soviet forces leading to an “exchange” of nuclear weapons that
would kill hundreds of millions of people and make much of the earth’s surface
uninhabitable for countless millennia. I
grew more and more anxious as I studied such arcana as the number of feet of
concrete required to resist a thermonuclear blast and the probable wind
patterns determining radioactive fallout.
The previous year, in Chicago, I had spent the Cuban Missile Crisis
glued to my radio, with plane tickets to Canada and Mexico [depending on the
prevailing winds] and a Geiger Counter and food parcels in my VW bug.

Things came to a head for me personally one day that Fall as
I was having a heated argument in the Harvard Faculty Club [I think with
Zbigniev Breszinzki, though my memory may be wrong.] I must have snapped, because the next thing I
knew, I was running as fast as I could down Massachusetts Avenue toward Harvard
Square, having a full-blown anxiety attack.
When I got back to my apartment on Concord Avenue, I decided that I
could not go on as I had. My response
was to rise into what I would later learn to call the ideological superstructure
and start thinking about political philosophy – a cop-out, to be sure, but also
an act of emotional self-protection. Out
of that retreat from the battle came, among other things,

*In Defense of Anarchism*.
I recall this personal ancient history because I find myself
obsessively and unproductively fixated on the train wreck that we call the
Presidential election. The stakes are
lower, to be sure. The worst that can
happen does not compare with thermonuclear war, but it is bad enough to make me
lose sleep and what little composure I normally have. This morning, as I walked, I found myself thinking
not about Hillary Clinton or Donald Trump but rather, unaccountably, about the
precise definition of a neoclassical production function. This is something I have thought about
before, in part because it bears directly on the standard economists’ claims
about marginal productivity and the justification of profits in a capitalist economic
system, and in part because I do not think that even serious students of
economics understand exactly what a production function really is. This is one of those ideas – I have talked
about this before – that I think is a lovely intellectual object, one that I
can, with some time and effort, show to my readers in all its essential simplicity
and beauty, so that they can appreciate it intellectually as I do.

So, as Susie completes her packing for our trip to Paris tomorrow
[my much simpler packing is mostly done], I shall write a long blog post
explaining in excruciating detail the precise nature of a neo-classical
production function. I doubt that anyone
in the extended blogosphere will have much interest in this explication, but writing
it will give me some momentary peace.
Think of me, if you wish, as the violist in the string quartet playing
Haydn on the deck of the Titanic.

The first thing you must understand is that the classical
Political Economists and the neo-classical economists simplify the reality of
capitalism, for purposes of their formal analyses, in diametrically opposite
ways. The Classicals assume that there
is, for each commodity, one and only one dominant technique of production,
defined for purposes of their analysis by the list, or vector, of quantities of
inputs required for a unit of output.
Thus, to construct a hypothetical example, a bushel of corn may require
for its production one twentieth of a unit of corn [as seed], one half unit of
iron [as tools, etc.] and one fifth of a unit of labor. We would represent this as a vector of inputs
– 1/20c, ½i, 1/5l – per unit of output c.
As a consequence of this simplification, the appropriate modern
mathematical technique for analyzing Classical Political Economy arguments is
Linear Algebra. The neo-classicals
simplify the reality of capitalism by the opposite assumption of an infinite
number of techniques for the production of a unit of each commodity. For this reason, their mathematical technique
of choice is the Calculus. Now, neither
assumption is in fact true of capitalism.
Each is an ideal simplification.
For reasons that are not germane to this blog post, the alternative choices
have profound ideological implications.
The Classical choice highlights the fact of class conflict, making it
impossible to ignore. The neo-classical
choice makes it appear that capital and labor are engaged cooperatively in the
production of commodities, obscuring the fact of class conflict. [You can tell which side of this dispute I am
on.]

The central analytical concept of the neo-classical approach
is the

*production function*, which I shall now, at seemingly interminable length, explain.
The first thing you must understand is the concept of an
n-dimensional vector space. Think of
your everyday two-dimensional diagram with an x-axis running left to right and
a y-axis running top to bottom, the point where the two lines cross being
labeled the origin, or 0. Let us suppose
that each axis is marked off in units of the same sort and size – inches,
say. So the x-axis is marked off with
little cross lines at one inch, two inches, three inches, etc., [and also, to
the left of the origin, with minus one inch, minus two inches, and so forth],
and the y-axis is marked off the same way.
Suppose you draw a rectangle starting at the origin and going four
inches to the right and two inches up.
That is then, of course, a rectangle whose area is 8 square inches. It has the same area as a rectangle that is
one inch wide, on the x-axis, and eight inches high, on the y-axis. We can talk this way, and make these
comparisons, because the units marked off on the x and y axes are, as we say,
commensurable – inches all around. But
suppose we now draw a graph that charts the performance of NBA stars, plotting
height against average number of points scored per game. There will be a relationship, of course, but
since the units on the two axes are different and incommensurable, it would
make no sense to multiply one by the other [foot points or inches points is not
a measure with any meaning.]

Now think of a diagram that has three or four or one hundred
ninety three axes, each one perpendicular to all of the others and
incommensurable with all of the others.
Unless you have really good mathematical intuition [which I do not], you
cannot possible

*imagine*that, but you certainly can*understand*what it is. That is an n-dimensional vector space, where n is three or four or one hundred ninety three or any other finite positive integer [there are infinite dimension vector spaces, but we are not talking about those.]
All right. Got
that? Now, let us think about commodity production. Neither the Classicals nor the neo-classicals
ever talk about how stuff is actually made.
They leave that to engineers and design specialists and workers and
other inferior orders of society. [Marx

*does*actually talk about how stuff is made, at great length in*Capital*– it is one of the many ways in which he differs from just about everyone else who writes about economic theory.] Instead, they talk about how many units of each input, including labor, are required to produce one unit of output of a specified commodity [see my little example above.]
The neo-classicals, as I said, assume effectively an
infinite number of techniques of production for each commodity [or a very large
finite number of techniques with each input denominated in small units.] Let us suppose, to pluck a number out of the
air, that there are 25,984 different possible inputs available for the production of some commodity, say corn – two inch
nails, three inch nails, steel of nineteen different varieties, long fiber
wool, short fiber wool, a hundred and fifty seven varieties of paint, etc etc
etc. We can now construct a 25,984
dimensional vector space, with each dimension measured off in the appropriate
units for one of those possible inputs [inches, gallons, pounds, etc.]. If you think about it for a moment, you will
see that each point in that vector space is identified by where it is relative
to each of the 25,984 axes. It is a
point with 25,984 coordinates [just as a point in a two-dimensional graph has
two coordinates, the x coordinate and the y coordinate.]

Suppose, just to take a specific example, that we are
interested in corn production. Each
point in this vector space will represent one possible set of inputs into the
production of a unit of corn. Obviously,
most of those possible inputs will be just plain useless for growing corn. I mean, how on earth are you going to use a
dozen wingnuts to grow corn? Never
mind. That just means that the
coordinate along the wingnut axis will be zero.
No problem.

Are we clear? Now,
pick some point, representing one particular combination of inputs into the
production of corn. To make this a
little easier to understand, I am going to suppose that this point represents,
let us say, one acre of arable land, one shovel, ten ears of corn [potentially
for seed], and 728 hours of skilled agricultural labor, and zero for all the
other 25,980 inputs. In other words,
this point has coordinates of zero for 25,980 of the 25,984 axes. Can you sort of picture that? It is like a point in 2-dimensional space
that is at the origin along the x-axis but is some way up the y-axis. OK?

Now comes the point that you seem never to find explained in
an ordinary Economics textbook. There
are countlessly many different ways of combining this particular set of outputs
to produce corn. Here are just
three. You can invent lots of others
yourself.

Technique 1: On the
first day, a single worker arrives at the acre of land with the shovel and the
ten ears of corn. He sets the shovel
down next to the land, eats the ten ears of corn, and then spends four hours
praying to Ceres, the Roman goddess of agriculture. Each day for half a year he returns and
spends four hours praying [half a year is 182 days, or 728 hours]. This technique produces zero bushels of corn.

Technique 2: On the
first day, a single worker arrives at the acre of land. He removes the kernels of corn from the ears
of corn, scatters them evenly across the acre of land, and spends the remainder
of four hours walking around the borders of the acre of land, complaining
loudly about the difficulty of growing corn.
Each day for six months he returns and repeats his four hours of complaints,
until, on the last day, he harvests the corn that has grown. This technique produces one bushel of corn.

Technique 3: On the
first day, ten workers show up. One of
them, the foreperson, directs the others to take turns using the shovel as they
walk up and back on the acre turning the soil over and preparing it for
planting. On the second day, the workers
return and the boss tells them to use the seed corn for planting. She makes sure that the kernels are evenly
scattered in the rows turned over by the shovel, and are then covered
over. Periodically the team returns to
weed, using up most of the 728 hours of labor, but keeping in reserve just
enough to harvest the crop on the last day.
This technique yields ten bushels of corn.

Now comes the payoff for all of this nonsense.

**The production function of a commodity is a mapping of an n-dimensional input space onto a one-dimensional output space. For each point in the input space, representing one unique vector of inputs into the production of the specified commodity, it identifies the technique [or set of techniques, if there are several] that yields the largest amount of output, and maps that input point onto the point in the one-dimensional output space corresponding to that amount of output.**
In the example above, the production function for corn maps
the input point [0, 0, 0, …, 1 acre, 1 shovel, 10 ears of corn, 728 hours of
agricultural labor, 0, 0, 0, … ] onto the output point 10 bushels.

**For every single point in the 25,984 dimension input space, the production function maps that point onto some point along the one-dimensional output axis. I.e., it maps the input point onto the maximum amount of corn producible by any of the endlessly many different techniques using that combination of inputs.**[Just as in the example above.]
Each commodity has its own production function. Routinely, neo-classicals assume that the production
functions are continuous, which means that two points in the input space of a
production function that are arbitrarily close to one another will map onto
arbitrarily close points along the output axis.

Now comes an apparently minor point that I really think most
economists, even professional economists, do not really grasp ]this goes back
to my early observation that economists never talk about how stuff is actually
made.]

**The fact that two points are very close to one another in the input space does not at all imply that the maximally productive technique for the first point singled out by the production function will look anything like the maximally productive technique singled out by the production function for the second technique. What is more, the assumption of continuity is arbitrary and quite likely false**.
Let me give you a real and historically important example to
illustrate this claim. The ante-bellum
Southern plantations were essentially large farms worked by slave labor,
producing first indigo, rice, and tobacco, then, of course, cotton. The slaves were forced to work long hours in
the fields, guarded and threatened, if they disobeyed, by white overseers armed
with whips and clubs. After cotton took
over as the premier cash crop, the plantation owners devised an especially
brutal but also especially productive method for organizing the slave
labor. Instead of simply sending the
slaves into the fields to cultivate the land, plant the crops, weed them, and
harvest them, the owners would line the slaves up at one end of a field and
move them forward in tandem, like a living machine. This was sometimes called the gang method. To keep the slaves in rows all moving at the
same pace, the owners hired a white employee who became known as a slave
driver. His job was to stand behind the
line of slaves as they moved down the rows, whipping them to keep them to drive
them in a line. This was a much more
efficient mode of production. Looked at
from the point of view of a production function, a small increase in one input
[labor] made possible a totally different technique of production that mapped
onto an output point quite distant from the output point corresponding to the
same set of inputs without the incremental amount of labor. In other words, the production function for
cotton was

**not**continuous!
After Emancipation, by the way, when the plantation owners
were forced to hire the former slaves as wage laborers, they tried to get the
freedmen and freedwomen to use the gang method, but even when they offered
higher wages, the now free workers refused.

Why on earth does any of this matter? Well, as I said when I began, I find this concept
of a production function rather lovely, and I lie spelling it out clearly so
that you can appreciate it also. But
there is an important ideological reason as well [surely you are not
surprised.] Neo-classical economists who
talk knowingly about marginal productivity are appealing to the application in
Economics of a famous theorem by the great 18

^{th}century Swiss mathematician Leonhard Euler. This theorem concerns a species of continuous real functions known as*homogeneous functions*, and most particularly a sub-species of these known as*linear homogeneous functions*. I shan’t go into that here, inasmuch as I have bored you far too long. Suffice it to say, if the production function of a capitalist economy is not linear homogeneous, then none of the congratulatory self-justifying things the economists want to say about capitalist economies have any reason to be believed.
Well, I feel much better.
Now I can go to Paris with a light heart. Thank you for listening.

## 7 comments:

Enjoy your trip to France.

Sometimes dealing with life in another language, in this case, in French, can free one at least temporarily of obsessions.

Dear Prof. Wolff,

Very interesting post (as always). For reasons relating to my own work on sweatshops, I was intrigued by the following aside: "After Emancipation, by the way, when the plantation owners were forced to hire the former slaves as wage laborers, they tried to get the freedmen and freedwomen to use the gang method, but even when they offered higher wages, the now free workers refused."

I have not heard this before and was wondering if you could direct me to a good discussion of it.

Best,

Michael

Great blog! (Andrew Sullivan eat your heart out! I really dislike that guy.) After reading just this blog, I don't understand (although I could offer nasty explanations) why economist would not be compelled to take Marx seriously.

Safe travels.

A most interesting discussion. Professor Wolff is an academic treasure who truly contributes to the public good. And I think Leiter's blog underestimated him when the political philosophers were ranked. I put him in the top ten, and his knowledge of economics probably surpassed that of the late great G.A. Cohen.

Thank you for the kind words.

What about mixtures of activities? e.g. a 3/4 time, wage-earning overseer who runs the gang-method during his 3/4 of a day, and in the remaining quarter day, the next-best method is used without him. By adjusting the overseer's hours, we can mix the activities in proportion, which would give us a production function that mostly continuous (at upper limits from mixtures, at the bottom there might is some minimum time at an activity to yield productive gain) and approximately linear.

Having worked at a large company the idea of mixtures, competing teams with different methods working at the same goal, seems more than plausible.

I would need to know a great deal more about the actually running of a cotton plantation to know whether this mixture was feasible. I am not sure how one would run the gang system part of the time but not all of the time.

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