Concavity, the Mystery Property (1998)

by Alexander "Sasha" Volokh
in the style of "Macavity, the Mystery Cat" by T.S. Eliot (1885-1965); can also be sung to the tune of "Macavity" from the musical Cats

Concavity's a property of power and ubiquity.
Its presence gives you maxima, its absence is iniquity.
It's the bafflement of econ students, problem sets' despair:
For when you differentiate, concavity's not there!

Concavity, concavity, there's nothing like concavity,
In microeconomics it's more requisite than gravity.
We wave our hands pretending that exceptions are quite rare,
Lest, when we build the Hessian, concavity's not there!
You may make a function giving goods a constant budget share --
But little have you proven if concavity's not there!

Concavity, concavity, there's nothing like concavity.
Why, folks with convex functions are exemplars of depravity.
They like x' and x, but they get worse off when they share --
How do they do it? No one knows! Concavity's not there!

We also say convexity is how things should behave,
But all that means is, for that set, the complement's concave.
When marginal production shows no sign of diminution,
Or x of p and w has more than one solution,
Or Kuhn and Tucker giggle while you moan and rend your hair,
It's no use optimizing, for concavity's not there!

So say x squared + y squared is the function that you get.
Now maximize the function on a compact budget set.
You take in hand your maximand, you solve its first conditions,
But find all other bundles have superior positions. . . .
Alas, it's but a minimum! You sigh and gravely say:
"It's all that non-concavity that's leading me astray!"

Concavity, concavity, there's nothing like concavity.
Not having it's more painful than a root canal or cavity.
I'd rather be attacked by an echidna or a bear
Than have to face the problem that concavity's not there.
And they say that of all properties they like to teach to us,
Like monotony or smoothness or defined upon R+,
They are but as an amoeba, insignificant as gauze,
When compared with this condition -- the Napoleon of Laws!

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