October 25: Here are some key features of a quasiconcave function f(x).
- It has convex upper level sets. The set of points x such that f(x) >= a is convex for any number a.
- It has convex indifference curves if it is a utility function. If f(x) is strictly monotonically increasing, the function g(x) such that f(x)=a is a convex function.
Conjecture: Iff function f(.) is quasiconcave, there exists an increasing transformation g(.) such that g(f(.)) is concave.
I'd start to prove the conjecture this way. Let x and y be points in the upper level set of f(.), which means f(x)>=a and f(y)>=a. Since f(.) is quasiconcave, the upper level set is convex, which means that f(mx+ (1-m)y) >=a too. What we need to show first is that there exists some increasing function g() such that
g(f(mx+ (1-m)y)) >= mg(f(x)) + (1-m)g(f(y)). I think we need to start by assuming that f(x) \neq f(y), and that they are both on the boundary of that convex upper level set. Then we can see how g has to affect those two levels of f differently.
If the conjecture is true, then maybe we can think of quasiconcavity as being the equivalent of concavity for functions that are just defined on ordinal, not cardinal spaces.
October 26. Why, though, do we worry about quasi-concavity at all in economics? Why not just assume that utility functions are concave? The conventional answer would be that utility is ordinal, not cardinal. That is a bad answer for three reasons. First, even if it is ordinal, we could say, "It's only the ordinal properties of a utility function that affect decisions. Therefore, for convenience, let's say that whatever function you start with, you have to use a monotonic transformation to make it concave before we start working with it." Second, we might say, "Since only ordinal properties matter, let's assume utility is concave for convenience." Third, we might accept cardinality. Everybody uses von-Neumann Morgenstern cardinal utility in their models anyway, making only a brief nod, if any, to ordinality. But a risk-averse agent has concave utility. For these reasons, I wonder why it's worth making our graduate students learn about quasi-concavity. The opportunity cost is that they're not learning about something more useful such as the CAPM or the Coase Theorem.
Maybe quasi-concavity comes up in enough other contexts to be important. I know Rick Harbaugh has a paper on comparative cheap talk where it comes up. In Varian, it comes up first in production functions, where it allows you to have convex input sets for a given output without requiring diminishing returns to scale, as true concavity would.
October 27. Yet another thought. Margherita Cigola has done work on defining quasiconcavity in ordinal spaces, on lattices. Convexity has to be defined specially there. She uses a different (equivalent in R space) definition of quasiconcavity:
f(mx + (1-m)y) >= mf(x) + (1-m)f(y)
I like that because it is closer to the definition of concavity.
Or another, suitable when the function is differentiable: f is quasiconcave if whenever there is a maximum (i.e., the first derivatives are zero), the matrix of second derivatives is negative definite. MR suggested that, for the single-dimensional x case. I'm not sure it does generalize that way.
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