Risk-Averse Utility Functions, Inada Conditions, and Portfolio Choice
Summary
The document considers utility functions for optimal portfolio allocation that are increasing and strictly concave, with marginal utility tending to infinity near zero wealth and to zero as wealth grows. It notes that isoelastic utility, including power and logarithmic forms, satisfies the requested pattern, then discusses why there is no short exhaustive list of qualifying functions.
The answer explains that the conditions admit a very large set of functions, many without elementary closed forms, and argues that choosing a narrower family such as CRRA imposes additional modeling assumptions that need economic justification. It points to the broader HARA class and recommends stochastic dominance as a way to reason across classes of utility functions. The response is conceptual; it does not derive examples or prescribe a particular portfolio model.
Key ideas
- The stated conditions require positive marginal utility, strict concavity, and specified limits at low and high wealth.
- Isoelastic utility, including power and logarithmic forms, is one familiar qualifying family.
- The set of functions satisfying the conditions is too broad to enumerate as a practical list.
- Restricting utility to a named family adds assumptions that should be economically justified.
- Stochastic dominance can support comparisons across broad classes of utility functions.
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# List of risk-averse utility functions
# 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.
## Answer by g g (score 2)
https://quant.stackexchange.com/a/30260
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.