Operating on the theory that when you dodge a bullet, the thing to do is to put another target on your chest, I am going to write something now about the concept of a production function, which, when I first encountered it, puzzled me for some long while, until I thought it through. I think in an odd way I benefited from never actually taking an economics course [although I did teach Introductory Microeconomics at UMass -- see my Autobiography!] I am encouraged in this foolhardy undertaking by Magpie, who, in his very kind response to my rant about marginal productivity, referenced John Bates Clark and Joan Robinson.
The first thing I had to get clear in my head when I started studying Economics was that economists -- whether classical or neo-classical -- never seem to talk about how stuff is actually made in factories or on farms or in mines or in offices. Indeed, I think it is fair to say they don't know much at all about how stuff is made. The closest they come is to specify the collection of physical inputs and the amount of labor required to produce one unit of this or that -- the "unit input coefficients" as they are sometimes called. But how all those inputs are combined exactly is left to someone else -- the engineers, presumably, who, even though they are located in the Faculty of Science, are at the absolutely bottom of the status hierarchy there and can therefore be ignored.
Now, here is the really important thing to understand. All economists simplify, just as do all physicists and all geologists and all microbiologists. If they did not, if they went on endlessly about every single detail of every market or factory or consumer or producer or sector of the economy, they could never carry out any interesting reasoning at all. But it matters a great deal just how one decides to simplify. To be specific, the Classical Political Economists [Smith, Ricardo, Marx] assume that at any given time, there is one dominant technique for making each commodity that is produced in a factory or on a piece of agricultural land. They know this is not quite true, of course. There might be two or three ways to make straight pins or to set up the power looms in a fabric factory or to use oxen and seed corn and plows on a field, but they just abstract from that, figuring that one of the available techniques will be more profitable, and that fact will become known to all the entrepreneurs in that line of business, who will then shift to the best technique. They also know that from time to time, innovations are introduced, so that for a while there will be two techniques going rather than one, but they expect that an innovation will either make a bigger profit for its innovator, in which case it will be adopted throughout the industry, or it will prove less profitable and be dropped.
Neo-classical economists make exactly the opposite "simplification." They assume that there are actually an infinite number of ways in which inputs can be combined to produce the output of an industry. These different ways will not only use different amounts of the same inputs in varying and differing ways, they may also use all manner of alternative inputs.
Now, if you think about it at a very high level of abstraction [where economists, like Sherpas, feel most comfortable], there are a finite number of possible inputs into any line of production, and each of them, including labor, can be added in any number of different ways. For example, suppose you are in the business of growing wheat. On an acre of arable land, you can throw a cupful of seed at the acre of land from a corner of the acre, cover the acre with tarpaulins so that no water gets to the ground, and hire one worker to look at the field every month and cross himself piously. That is indeed one way of growing wheat. The inputs here are a cupful of seed, one man-day of labor per month, and a stack of tarpaulins big enough to cover an acre. The output using the inputs in this manner will be zero bushels of wheat. But there are many other ways of combining those exact same inputs for the purpose of growing wheat. One of them, for example, is to have the man walk across the field for a day the first month poking holes with his fingers and planting the wheat seeds, and stacking the tarpaulins in a corner of the field so they do not get in the way of whatever rain should come, and then spending a day a month weeding. The output will, we may assume, be several bushels of wheat. There are infinitely many possible combinations of all the available inputs [which, of course, includes cyanide, firecrackers, Hope Rain Dance outfits, and much more besides], and many many different ways of technically combining each set of possible inputs, and each combination with each way of combining them has some amount of output of wheat associated with it.
A Wheat Production Function is a mapping of each of the infinite number of possible combinations of inputs onto the maximum amount of wheat producible by that combination of inputs, using one or another of the finite but very numerous possible techniques that can be run with that combination of inputs.
Now, here is the crucial thing to grasp, which it took me a very long time to figure out, since none of the books I was reading talked this way at all. To say that the production function is continuous, and therefore differentiable, is to say that if you hold all but one of a set of possible inputs constant, and increase the remaining input a little bit, the maximum amount of wheat producible with this slightly altered set of inputs will also be a little bit bigger. BUT THERE IS NO ASSUMPTION BEING MADE AT ALL THAT THE MAXIMALLY PRODUCTIVE TECHNIQUE WITH THAT SLIGHTLY ALTERED SET OF INPUTS WILL LOOK ANYTHING LIKE THE MAXIMALLY EFFECTIVE TECHNIQUE WITH THE ORIGINAL SET OF INPUTS.
Go back and look at the John Bates Clark example cited by Magpie. Clark imagines that a group of men are working on a field of grain, and supposes that if you add one man [a small increase of one of the inputs] you can grow a little bit more grain [presumably because when one man gets tired the extra hand can step in and keep the work going, or something like that.] But that example encourages us, quite incorrectly, to assume that the new technique, with the extra man, will in some natural way look quite like the old technique without him. And the concept of a production function totally abstracts from any such natural way of thinking about the matter.
Joan Robinson pointed out that if you add one shovel instead of one man, you will not increase the output because there will be no one to use the shovel. [This was not quite fair, of course. The shovel might come in handy if one of the workmen broke the handle of his shovel, but never mind.] But even Robinson took it for granted that the techniques would look quite similar.
In order for economists to use Euler's Theorem and all the other really nifty stuff they hijacked from elementary calculus, they have to assume that the production functions they are working with are continuous, and hence partially differentiable. But when you set aside Clark's just-so stories of men working a field, there is no reason at all to suppose that the mapping of vectors of inputs onto the scalar of maximum output is in any way, shape, or form continuous, even as an approximation. No reason at all.
Let me give you a real historical example that played a rather important role in the American South in the latter years of slavery and the early years of freedom. Large Southern plantations used an organization of agricultural labor called the gang system. Slaves were lined up in a row at one end of a field and moved forward in a coordinated fashion planting or weeding or harvesting. This was much more efficient, for obvious reasons [they could come to the end of a row and wheel in a coordinated fashion], but to maintain the order of the line required an overseer, a slave driver with a whip, who walked behind the line of men and women and whipped them if they lagged. [Yes, Virginia, there really were slave drivers -- not office bosses who demand that secretaries finish their typing before going home, but men with whips who drove the slaves like oxen.] After liberation, when the slaves were freed [and for a brief time actually had something resembling the rights of free men and women], they refused to work in this fashion, regardless of the pay offered by desperate plantation owners. This was a marginal change in input -- one worker fewer, namely the slave driver -- but it resulted in a large change in output, not a marginal change.
The concept of a production function, which is central to modern Economics, is loaded with [or, as people like to say these days, fraught with] difficulties and implausibilities, none of which, so far as I can tell, is ever mentioned in Economics textbooks or classes.
Well, that concludes my rant about production functions. But I thought I would add one little note about the claim economists make that their field is scientific. Martin Feldstein is one of the most influential economists in America. Currently a Professor at Harvard, he served for four years as Reagan's Chairman of the Council of Economic Advisors. Feldstein in those days taught Economics 10, the enormous intro course with lectures by himself and a gazillion sections taught by grad students and young Instructors. While he was away in Washington, some leftie grad students succeeded in persuading the person who stood in for Feldstein to allow them to teach a certain number of "alternative," or as Krugman would say heterodox, sections in which something other than orthodox economic theory was presented to the lucky students. When Feldstein returned, he discontinued that practice, saying, "The purpose of Economics 10 is to teach students that the market works."
NOT HOW THE MARKET WORKS, BUT THAT THE MARKET WORKS. Savonarola could not have put it better!