List of risk-averse utility functions

In the context of optimal portfolio allocation, I am looking for a (possibly exhaustive) list of risk-averse utility functions verifying part of the so-called Inada conditions.

Essentially, I am looking for functions $$U : \mathcal{D}=[0,\infty) \to \Bbb{R}$$ verifying $$\frac{\partial U}{\partial x}(x) > 0\quad, \quad \forall x \in \mathcal{D}$$ $$\frac{\partial^2 U}{\partial x^2}(x) < 0\quad,\quad \forall x \in \mathcal{D}$$ $$\quad \lim_{x \to 0^+} \frac{\partial U}{\partial x}(x) \,\,\,= +\infty$$ $$\lim_{x \to +\infty} \frac{\partial U}{\partial x}(x) = 0$$

I could easily come up with the famous family of isoelastic utility functions (CRRA) among which power utility and the logarithmic utility.

However I was wondering whether there existed any other well known instances of such functions. I would be glad if someone could point me towards an answer or a rigorous approach for finding them.

1 Answer

What do you mean by "rigorous approach for finding them"? You have the four conditions and every function which fulfills those conditions is a risk-averse utility function. This is all there is; what else do you need?

If you are looking for a description of this set in terms of elementary functions (+,.,polynomials, exp and such) you will be disappointed. The set of functions fulfilling these four requirements is HUGE and will contain vast amounts of functions which cannot be described in these terms. The easiest way to see this is to write the utility functions as double integrals. As you might know most integrals cannot be explicitly solved in terms of elementary functions.

Furthermore, your desire for explicit representations sounds a bit fishy to me. From the perspective of modelling economic reality, all economic content is contained in those four conditions. If you restrict the utility functions further, e.g. by only looking at CRRA, you add further constraints. These constraints need to be justified by economic reasons, otherwise the conclusions you draw from using restricted utility functions are not based on economics.

From this perspective working with stochastic dominance is much better since you reason about large classes of utility functions and not only about artificially restricted families (such as power or exponential utility).

That said, people often include the full HARA class in their discussions.