Saturday, October 22, 2011

There's no consistent realistic utility over wealth

Kind of a technical point, but I'm tired of making it in discussion forums.

Utility over wealth is usually considered concave: an extra 10 dollars make more difference to one's life when one's total wealth is 100 dollars than when it is 100 000 dollars. This is an outcome of optimal use of limited resources: spend scarce dollars on the essentials, abundant ones on frivolities.

On the other hand, response to price is also usually assumed concave: a price increase of 1 dollar leads to a stronger response for a initial price of 10 dollars than for 10 000 dollars.

These two statements are mutually inconsistent, and both are true. In other words, there's no single utility function that captures both, but both phenomena exist in reality. This means that you can't have a utility function over wealth that is true in the world. It's fairly obvious since price is negative wealth, and you can't have a function that is concave both in one of its variables and its symmetric.

Formally, if $u(\cdot)$ is the utility over wealth $w$, the response to price $p$ is given by $u(w-p)$ since paying $p$ decreases the wealth $w$ by that amount. If $u(\cdot)$ is a concave function of its argument, then $u(w-p)$ is concave in $w$ and convex in $p$; if $u(\cdot)$ is a convex function, then $u(w-p)$ is convex in $w$ and concave in $p$. Both concave is an impossible case.

But... but... but... I hear some empirical modelers say, we use a concave utility function over price all the time, typically $\log(p)$.

Yes. I know. That's why I'm writing this post. Of course since price is a negative term in wealth, your "concave" function is actually convex in wealth, as the term enters as $- \log(p)$, a convex function.

Math doesn't lie. All you need to do is pay attention.