Stories

Concavifying the Quasiconcave

A paper of mine as an example of the Lakatos process that bridges econ and math:

I'd long had a conjecture that a function being quasiconcave was the same as saying you could take it and blow it up with a monotonic transformtion to be concave. (The ultimate real idea is that if it's quasi concave, it has a single peak, but we take unique maximand as being the conclusion of the theorem and make quasi-concavity the assumption, tho we could run it the opposite way too.)

  I told a geometry professor, Chris Connell, my idea at coffee after church one day, and he suggested we work on it together. Immediately, we realized that we had to exclude the case of  a plateau at the single peak. The other conditions we needed were basically that unless the function is monotonic, so the peak is at the min or max x-value, it can't be "too flat" or "too steep". Chris says that the most interesting thing mathematically is a more subtle condition: the derivative can't change too quickly-- there's a bounded variation condition. (See the "magnifying class" style Figure 8  at http://rasmusen.org/papers/quasi-connell-rasmusen.pdf, my favorite diagram in any of my papers-- economists never get to use sine waves). We would never have thought of that except for having to do the proof. 

  The main points come up  in one dimension, but we generalize to  geodesic metric spaces-- infinite dimensions, fractals, graphs, etc. And we do the special case of differentiable manifolds, though we then discovered it has been essentially done already by somebody else. We also found that it was too mathy for econ journals, so we cut out almost all the examples, cut out the less elegant extensions, shifted from Lakatos style to laconic math style, and published in a math journal instead.  The final, more correct but less readable version, http://rasmusen.org/papers/quasi-short-connell-rasmusen.pdf, is in the Journal of Convex Analysis.  

  So it's a great illustration of Lakatos's points. 

Sun and Moon, a Parable (from Pastor Bayly)

imagine Moon says to Sun, "I have glory too, you know! From now on you take the night & I take the day!" Sun responds: "You right. I've been such a glory hog. From now on, I take the night & you take the day."

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