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Sunday, April 27, 2014


In the wake of the Piketty tsunami, there have been a good many references to marginal productivity, particularly to the claim that the stratospheric salaries of the "supermanagers" are justified on the grounds that they are merely earning their marginal product, which is therefore [this is always a very large leap] fair.  The low wages of common workers are explained by their low marginal productivity.

I am now going to do a complicated dance on a high wire without a net.  If I go splat, I hope some of you more knowledgeable than I will clean up the mess.  I should explain that in the Spring of 1978, during one of my very rare sabbatical semester leaves, I sat in on Donald Katzner's graduate microeconomics course at the University of Massachusetts, plowing my way through Henderson and Quandt and doing all the exercises [though not handing them in to be corrected by the long-suffering Katzner, who put up graciously with my presence.]  My knowledge of the subject of marginal productivity derives entirely from that thirty-six year old experience, so you can see this is a real reach for me.  I mean, I was out of my depth when I was fifty-four and in my prime [as Miss Jean Brodie would say.]  At this point, I am desperately twiddling my toes to tread water and stay afloat.  Ok, enough excuses.

The concept of marginal productivity is somewhat misleadingly simple, at least as it is explained to undergraduate students taking an elementary Micro course.  The idea is supposed to be this:  There are a number of inputs into any production process -- land, raw materials, tools, and of course labor.  Clearly [this is so far from being clear that it is actually almost always false, but never mind that for a moment], if we hold all but one of the inputs constant, and add one more unit of the remaining input, there will be some increase in output.  That increase is called the output at the margin, or the marginal output, and this marginal increase in output attributable to the addition of one unit of a factor is called that factor's marginal product.  We could as easily ask how much output would be lost if we held all but one of the inputs constant and decreased the remaining input by one unit.  Intuitively, this will not in general be the same quantity as the quantity of output gained by the addition of an extra unit, but [and here we go again with the little assumptions on which total ideological rationalizations are erected] if we assume continuity of inputs and outputs and continuity of the production function of the process [which is not the same thing at all as the assumption of continuity of the inputs and outputs, but never mind for the moment], then as we make the units smaller and smaller, the difference between the two vanishes.

Everybody still with me?

Now comes the thing that gives economists secret erotic thrills and convinces them that they are not merely scientists but Philosopher Kings fit to rule a modern society.  An eighteenth century Swiss chap named Leonhard Euler [a very big deal in Mathematics] proved a theorem about what are called homogeneous functions.  Briefly, to make this as simple as possible, a homogenous function in a number of variables is a function in which the sums of the powers to which the variables are raised in each term are equal.  Thus, if we have a function in three variables, f(x,y,z) = (x2yz3 + xy4z + x1/2y1/2z5) then the function is homogenous of order 6, because if you add up all the little superscripts or powers in each term (including the 1's, which I did not bother to put in), in each term they total 6.

Brief pause to permit the math averse to get a breath of fresh air or read a Dickinson poem. 

Euler proved that if we take the partial derivatives of the function, f, and multiply each one by the value of that variable at a given point x = (x1 , y1, z1), then the value of the function at that point times the order of the homogeneity is equal to the sum of all those partial derivatives multiplied by the value of the variable at that point.  Or, in symbols, where m = the order of homogeneity:


mf(x1, y1, z1) = (df/dx1)x1 + (df/dy1)y1 + df/dz1)z1


Now, when the order m = 1, the function f is called a linear homogeneous function, and in this case:


f(x1, y1, z1) = (df/dx1)x1 + (df/dy1)y1 + df/dz1)z1


Well, the term on the left is simply the output of the product for some level of inputs of x, y, and z, and the each term on the right can be interpreted as the marginal product of that factor, because it is the amount of output yielded by holding everything else constant and increasing or decreasing that input marginally.  So the total output of the product for the inputs x1, y1, z1 is just exactly equal to the sum of their marginal products.  Now, the income or receipts from selling the product is of course the quantity sold times the price.  If we pay each input into the production of that output an amount proportional to its marginal product, then the entire monetary return from selling the product will just exactly be used up paying each factor of production its marginal product.  In the simplest case, there are two factors of production, capital and labor, and assuming that the society-wide production function relating inputs of capital and labor to total social output is linear homogeneous, then clearly the fair, efficient, rational, and one might go so far as to say divinely ordained thing to do is to pay capital its marginal product in the form of profit, and pay labor its marginal product in the form of wages.  Ta Da!

It would not be an exaggeration, I think, to say, that when economists looking for some way to justify low wages hit on this interpretation of Euler's Theorem, it was as though the heavens had opened up and the heavenly choir had started to sing to them.  Pythagoras could not have asked for more.  Pure Mathematics justified, indeed required, that capital get its profits and labor be satisfied with its wages.

Sigh.  It really is a thing of beauty, this little bit of formal by-play.  And thanks to the fact that most Americans are essentially illiterate when it comes to math, it hasn't been difficult for economists, who do really know better, to spend three quarters of a century or so teaching their idiot students in Intro Econ courses that minimum wage laws and taxation of profits and such like will just disturb the divine harmony of marginal productivity and produce unimaginable inefficiencies and counter-productive consequences.  The grad students learn better, but they also learn not to talk too much about what they have learned, except to other initiates with doctorates in Economics.

Here's the thing.  It does not take much additional math to prove some rather unsettling conclusions about economies governed by linear homogeneous production functions in which, to be sure, Euler's Theorem applies.  One thing that falls out of the math is that such economies exhibit what is called constant returns to scale, which means there is nothing to be gained by expanding the scope of production in a company.  That is equivalent to saying that the economy is in long run equilibrium, which means no capitalist has any incentive to enter or exit any particular line of production.  It is also true that in such an economy there is a zero rate of profit [not a zero rate of interest -- that is something different].

Say what?  Does this sound at all like the American economy, or indeed any capitalist economy that has ever existed?  Well, er, no.  But the math, the math, look at the math, isn't the math beautiful?  And it is absolutely one hundred percent valid, that math.

It is all a fraud.  A beautiful fraud, an elegant fraud, but a fraud nonetheless. 

Nobody in America is paid his or her marginal product, save by accident, and we wouldn't know it if it happened, because there is no plausible way of calculating the society's production function.  What is more -- this is a topic for another post -- the very notion of a continuous production function is about as close to nonsense as you can get.  [The great Joan Robinson and her associates in Cambridge, England pointed this out long ago.]

Well, if all of this doesn't drive you away from this blog permanently, I will be very grateful.  I promise to return to more graceful and enjoyable topics tomorrow.


Magpie said...

Excellent exposition, Prof. Wolff!

Your last equation f(x, y, z) = sum(diff(f(x,y,z), x..z)*(x..z)) is also the notorious trinity formula, if I am not mistaken. So, with Euler and the production function economics went back in time to Adam Smith.

Your own exposition is reminiscent of John Bates Clark (after whom the medal was named). In "The distribution of wealth: a theory of wages, interest, and profits" Clark conceived what for decades has been the standard Micro101 marginal productivity explanation: given a farm, if the farmhands work empty-handed (i.e. they don't use capital, or whatever capital used is negligible), we increase the number of farmhands and measure the output. The change in output, is the contention, is due to a change in labour only.

I have my own objection to that argument, but Joan Robinson and others came up with a simpler and humorous objection: if my production function uses shovels and muscular men to produce holes, the addition of a single shovel, without its corresponding strong man, will produce no additional output (i.e. hole). Similarly, with the addition of another man, without his own shovel.

California Prof said...

Beautifully done, professor. This little gem is a Wolff classic.

Anonymous Coward said...

I'm not sure that "management" counts as a production input.

Robert Paul Wolff said...

Sure it does. Ask yourself whether an automobile plant could run without some sort of oversight and coordination. That is the function of management. That doiesn't mean it should be paid a thousand times as much as the work on the production line.

Anonymous Coward said...

I stand corrected.