Mainstream production theory without Cobb-Douglas assumptions?

Economics Economics Questions
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I guess this somewhat relates to my previous threads on the law of returns. In mainstream production theory firm output is often treated in terms of a production function relating output to factors, e.g. Q(for output), being some function, e.g. of Q=AK^0.3L^0.7. These functions are picked for 2 reasons. They allow for constant returns to scale, an assumption that I think quite rightly is usually employed, lest economists wish to take on the laws of thermodynamics as well as those pesky a priori “Austrians” (Note I am strictly talking about physical output. I think if one quantifies output in monetary terms, as is often done, then this clouds the issue altogether, and we could have all kinds of returns to scale). Secondly, such production functions do obey the law of returns, though, since they trivially have a uniformly diminishing marginal productivity(given the fact that the 2nd order derivative of the above function w.r.t K or L will yield negative values for all K or L>0.

Now while I do agree that these production functions, often labelled “Cobb-Douglas” production functions do successfully replicate the above mentioned requisite properties, it seems strange to me that these are the only types of production functions ever considered. At least these are the only ones I have encountered in my studies so far. I was wondering if anyone knows of any neoclassical treatments that deal with production theory, but more broadly, dealing with functions that obey the above properties but are not limited to simply the type of Cobb-Douglas function I’ve labelled above?

One reason I think that I think there might be an exclusive focus with Cobb-Douglas functions is mistaking the law of returns with its corrollary. The actual law of returns is that when other factors are held constant then variation in one in isolation will eventually lead to the point that average returns with respect to that factor are maximised (as stated by Mises, Rothbard and I think by Stigler too). A consequence of this is that the same must be the case for the marginal productivity with respect to that factor holding other factors constant, causing marginal productivity to be eventually diminishing.

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I agree with you on law of returns; neo-classicals do think diminishing returns is derived in this way (which goes back to Wickstead’s and then Pareto’s treatment’s of production).

There are no neoclassical treatments that significantly deviate from such functions, because of situation similar to what is called the Integration Problem, which goes back all the way to Pareto in 1906 (book and article) and 1909 (book).

Pareto declared order of consumption (path of integration of marginal utility components) is fixed, before any choices are made in regard to quantities, when talking about ordinal utility (which, of course, makes his utility quasi-ordinal).

Similarly, production functions need to have partial derivatives that can be integrated, and at same time, partial derivatives are chosen according to how easy it is to fit statistical data into them on aggregate scale. It was question of ordinal utiltiy for Pareto, and it is question of getting functions useful in statistics for modern-neoclassical economics. Wickstead’s discussion 1880’s treatment of production function, which is basically what Cobb-Douglas is related to, fits these criteria.

That’s why “impatience” coefficient instead of income dependent time-preference mainfold is used in Kyndland’s modification of Solow’s model, for instance.

In all cases, simple functions are used because if they simply began with partial derivatives, they might find mathematical tools they don’t know are required.

Hayek said it, “most economists are bad mathematicians, who think most mathematics is statistics.”

If one deals with difficult linear production functions, most economists switch over to operations analysis, restricting themselves to one variable for these functions, e.g., Paul Samuelson and his functions of one variable: labour, which is somehow transmuted into cars, houses, and bread (in what way? directly, apparently… )

Edit:

What do you think is a useful production function? Try working with that–ignore Cobb-Douglas in publications. That’s what I am doing.

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Thank you for an excellent and edifying response.

I did not know of Pareto’s connection with this in regard to treating utility, and moreover Wicksteed’s. I didn’t comment on this but since you brought it up, it has been my suspiscion that part of the reason these functions are popular is that they make the mathematics nicer. The old adage of the economist stuck on an island with a can of beans and a chemist and physicist who both propose practical solutions to opening the can, while the economist proposes imagining a can opener definitely comes to my mind with this kind of thing.

I guess it makes sense in a way. In the natural sciences and engineering mathematics ultimately has some sort of practical utility. When engineers encounter problems that cannot be solved straightforwardly in an analytic fashion they either need to think outside the box or start employing Newton-Raphson, Runge Kutta or some other computational method to gain an approximate solution. When economists think they might encounter such a problem they pretend its linear so they can keep writing equations.

Makes sense in a way. An engineer/physicist caught simplifying practical problems like this would probably get fired. For an economist, they are practically getting paid to mentally masturbate. Pretty sweet deal I guess.

The connection with statistics makes sense. the wikipedia entry on these functions does suggest there have been problems with the lack of microfoundations for these functions, despite their widespread use. I know Mr Cobb and Douglas did some statistical work allegedly verifying this, which is why they have their namesake and not simply labelled “homogenous”, as they are in mathematics. Seems fishy and dubious to me.

I guess I’m rather agnostic on what would make a good production function, that would and should depend entirely on the situation, as long as they both obey CRS and the law of returns in the wider sense as I’m sure we both acknowledge.

Also, that’s rather interesting with Samuelson treating Labour as a single factor. Was that that the same time Harrod and Domar did pretty much the same thing with their growth theory if I’m not mistaken? Speaking of the childish use of ODEs by engineer/mathematician wannabes, that nonsense I think is a great example.

Edit: Can you tell me where you got that Hayek quote?

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For thelion, or anyone who happens to know - Cobb & Douglas, in their original article, state that “The theory referred to (due to J.B. Clark, Wicksteed et al.) states that Production, Labor and Capital are so related that if we multiply both Labor and Capital by a factor m then Production will be increased m times, that is Production is a first-degree homogeneous function of Labor and Capital…” http://www.aeaweb.org/aer/top20/18.1.139-165.pdf

Exactly which book or article does “JB Clark, Wicksteed et al.” refer to in this case?

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Any other thoughts?

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http://socserv.mcmaster.ca/econ/ugcm/3ll3/wicksteed/wickess.pdf

Wicksteed 1894.

He also discusses this in his large work in 1910, Common Sense of Political Economy, if I remember correctly, but not in his 1888 Alphabet, yet, where he translated into english the Austrian “marginal utility.” He was OK with all this because he believed in cardinal utility.