ChapterZero

In my economics class, we’re discussing price elasticity, a measure of how the quantity of a good demanded varies with the price of that good. Specifically, if is the demand curve, the quantity demanded of a good as a function of the good’s price, the price elasticity between a point and a point on is defined as

This is an interesting definition because it is not symmetric; the price elasticity going from to is not the same as the price elasticity going from to . Economists view this as a drawback for obvious reasons, so they replace the divisors and with the respective midpoint values, and call this is the midpoint price elasticity:

Not surprisingly, considering the fact that we’re dealing with the ratio of the percentages of two quantities, and not the quantities themselves, even for a linear , the elasticity between two points a fixed distance apart varies depending on the points.

The first thought that came into my mind when I heard about price elasticity was differential geometry: why not take the limit of or even more interestingly, as , and then investigate it as a property of curves?

Just shooting from the hip, here are some problems I thought of (here, elasticity refers to either or ): what does a curve of constant elasticity look like? Can it be found as a function of curvature, torsion, etc.?

Update: I took the limit, and found — I think I’ve seen this somewhere before. More later.