Friday, November 12, 2010

Corner Solutions and Johnny Cash

In economics, a corner solution is (roughly speaking) the solution to an optimization problem that involves being up against a binding constraint such that the quantity of one of the arguments is zero. Suppose I have a budget of $50 to spend on apples and pears. Apples are more expensive, but I like them more than pears. Then again, the more apples I eat, the less I enjoy each one. An interior solution (the opposite of a corner solution) is when I trade off my enjoyment for each, and buy some amount of apples and some amount of pears. A corner solution would be if I buy only apples. It implies that, if had some amount more money, I'd buy even more apples. Given my budget, all I want is apples.

What, you may be asking, has this got to do with Johnny Cash? Well, I recently downloaded the song above. When choosing which songs to play, my attention is like a budget constraint that operates sequentially - I can only play one song at a time. I get enjoyment from each song, but the enjoyment diminishes with each successive play.

When my music collection is in equilibrium, the interior solution is that songs will be played with certain probability according to how much I enjoy them. The songs I enjoy will be played on average more, the songs I enjoy less will be played less, and these probabilities reflect the relative enjoyment of each song. At the margin, my enjoyment of each song that gets played is the same. The songs I like more in general I hear more so that I'm more sick of them, and I enjoy them as much as the songs I like less in general but are fresh each time due to being played less.

But suppose I come across a wicked new song? The equilibrium is temporarily disrupted.

The interior solution to this problem is the following:
Prob(Play Johnny Cash's "Devil's Right Hand") = 1
Prob(Play Anything Else) = 0

So far we're up to 17 plays in a row!

No comments:

Post a Comment