On Karush of the Karush-Kuhn-Tucker theorem

People like Paul Krugman would call this type of post “wonky”. You have been warned.

In preparation for teaching Mathematics for Economists I for graduate students tomorrow, I decided to see what the Web has on Karush, of the Karush-Kuhn-Tucker theorem. I should have done it earlier! Even though I had been writing Karush-Kuhn-Tucker in my lecture notes almost from the first time I taught mathematics for economists, I did not have much to offer my students on the mystery of why Karush’s work was never published. Why I never searched online about it before I do not understand. Nevertheless, I hit a goldmine in my search today: a paper by R.W. Cottle that offers a lot of fascinating details on Karush and Kunh-Tucker, along with photos of the protagonists and excerpts of letters between Kuhn and Karush, when Kuhn belatedly discovered Karush’s master’s thesis, which pretty much anticipated the Kuhn-Tucker theorem, and hastened to give credit to Karush to fix the historical record. And how did Kuhn discover that Karush had done this? By reading Akira Takayama’s massive 1974 book on Mathematical Economics.

Call me a hopeless nerd, but I find this fascinating. A bit of human interest to liven up all the constrained maximization we do for our work, in one way or another, every day.

The URL where I discovered the paper is
http://www.math.uiuc.edu/documenta/vol-ismp/41_cottle-richard.pdf

My encapsulation of mathematical modeling in economics

I wrote this in a comment on Google+ today, in a discussion of the death of Elinor Ostrom. I kind of like it, so I am putting it here, too:

Economic theory building proceeds by using abstractions to encapsulate what are thought to be important basic principles of interactive decision-making and then uses mathematics as a language for analysis to reach conclusions. There is also an empirical side that tries to find appropriate values for some of the parameters that are used in theorizing, with disputed success. The social side of human behavior is quite important in some parts of mathematical economic modeling; economic network analysis and the analysis of information diffusion and information cascades being some prominent examples that spring to mind.

More praise for Peter Diamond

Oh, and a very sensible take on the use of mathematics in economics, from Mark Thoma:

It is really hard to convince new graduate students that mathematics without the underlying economic intuition, i.e. technique for the sake of technique, is pretty useless. It’s the economics that are important — mathematics is simply a tool that allows us to better understand the economic content of the models we work with — the math itself is not the point of the exercise. In fact, the best models are the ones that are boiled down to the essentials so that they isolate important phenomena in a way that makes them transparent. Mathematical complexity is not always the best way to reach this goal. Models should be as complex as needed to highlight the essential issues, to use Krugman’s term they should be “elegant,” and additional complexity beyond that point detracts from their elegance and obscures rather than clarifies the central features of the model. Sometimes a high degree of complexity is required, but not always.

This quotation comes from this post, which is definitely worth reading in its entirety as it also quotes at length some very a propos praise of Peter Diamond that puts the above quotation in good context.

Brad DeLong Says Economic Theory Does Not Exist

In a column that ran today in the Project Syndicate, Brad DeLong said this:

One of the dirty secrets of economics is that there is no such thing as “economic theory.” There is simply no set of bedrock principles on which one can base calculations that illuminate real-world economic outcomes. We should bear in mind this constraint on economic knowledge as the global drive for fiscal austerity shifts into top gear.

Unlike economists, biologists, for example, know that every cell functions according to instructions for protein synthesis encoded in its DNA. Chemists begin with what the Heisenberg and Pauli principles, plus the three-dimensionality of space, tell us about stable electron configurations. Physicists start with the four fundamental forces of nature.

Economists have none of that. The “economic principles” underpinning their theories are a fraud – not fundamental truths but mere knobs that are twiddled and tuned so that the “right” conclusions come out of the analysis.

I am of two minds about this. I certainly feel that the beautiful economic theories that have been created with the help of some serious mathematics in the last few decades have yielded valuable insights. Yet on the other hand, these insights are far from telling us unambiguously important things about economic reality and from giving us good recipes for economic policy. I don’t even feel we understand, as economists, how such a basic thing such as economic trade can emerge, based on trust among people. So we have ended up with “theory” as a plaything of political interests. For such reasons, I share DeLong’s frustration. Yes, Paul Seabright has written the wonderful book The Company of Strangers, but still we don’t have a good grasp of the fundamentals of economic trade at the level of really basic theory! Naturally, I am trying to do something about this in ongoing research with my long-time collaborator, Rob Gilles, or I would not be justified in airing my complaints on this theoretical lacuna.

But is it really beauty and some insights of doubtful empirical relevance versus abandoning all hope of having an economic theory? I sincerely hope not. We have, in game theory, the mathematical theory of networks, and in the technology of simulation, some tools that should allow us to build a better theory. One that, although it will always be subject to criticism and will always create the longing for something better, will not be so easily dismissed as nonexistent by a leading economist.

I understand that DeLong is concerned about macroeconomics and one can read his nonexistence claim in terms of macroeconomic theory. But that is a cop-out. If we had a good theory of economic fundamentals, we would be able to build an, at least existent, theory of macroeconomics, by DeLong’s standards. So I addressed my remark here to basic economic theory, not its macroeconomic special case.

Clairaut’s Theorem

For some reason that I do not fully understand, the equality of cross partial derivatives gets all munged up, in naming at least, when taught in the typical math econ class, including the one when I was an undergraduate student. In reading this great expository blog on mathematics, I finally found a reference to a named theorem about it. And the name is not Young’s Theorem, as I have been taught. I learn something every day. Wikipedia has something to say about this, too.