Utility Function Domains and Preference Ordering
Summary
The document asks whether CRRA utility functions such as logarithmic and power utility must be defined only for positive wealth, while CARA exponential utility can be defined over all real wealth. It explains that the relevant domain depends on the problem and on which wealth or consumption outcomes are possible. Utility represents preferences: an agent weakly prefers one outcome when its utility is at least as high as another’s.
The answer distinguishes cardinal utility, where utility values are treated as measurable and comparable, from ordinal utility, where only rankings matter. In the ordinal setting common in quantitative finance, any monotonic transformation that preserves the preference ordering can serve as a utility representation. The discussion therefore emphasizes consistency between the function’s domain, the modeled outcomes, and the preference ordering rather than prescribing one domain for every utility function. It offers conceptual guidance but does not derive domain restrictions for particular economic models or address how to handle outcomes outside a chosen function’s mathematical domain.
Key ideas
- Utility functions represent preference orderings by assigning higher values to preferred outcomes.
- Cardinal utility treats utility levels as measurable or comparable, while ordinal utility uses rankings.
- In ordinal utility, monotonic transformations preserve the represented preference ordering.
- A utility function’s domain depends on the problem and the outcomes being modeled.
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Full text
# Domains of Utility Functions
# Domains of Utility Functions
I am learning a mathematical finance course and the lecturer didn't provide us with a rigorous definition of utility functions.
He just shows us (by simple Calculus) that the utility functions of every CRRA type agent are $u(x)=\ln x$ or $\frac{x^{\gamma -1}}{\gamma -1}$, where $x$ is the wealth of the agent involved and $\gamma$ is the relative risk aversion constant.
He also shows that the utility functions of every CARA type agent is $u(x)=- \exp(-\alpha x)$, where $x$ is the wealth of the agent involved and $\alpha$ is the absolute risk aversion constant.
He didn't talk about the domains for both cases.
It is clear that the domain of $- \exp(-\alpha x)$ can be $\mathbb R$. However, if restricted in real analysis, the domains of $u(x)=\ln x$ and $\frac{x^{\gamma -1}}{\gamma -1}$ should be $\mathbb R^+$. This leads to my question that if different utility functions have different domains?
Remark: We can also define $u(x)=\ln x$ or $\frac{x^{\gamma -1}}{\gamma -1}$ on $\mathbb R^-$ by definition from complex analysis, but I guess this is not desired.
## Answer by Kermittfrog (score 1, accepted)
https://quant.stackexchange.com/a/70503
As always, the answer depends a bit on the problem domain.
Commonly, utility functions only 'make sense' in terms of preference orderings, i.e.
$$B \succsim A \quad \Leftrightarrow \quad u(B)\geq u(A) \tag{1}\label{1}$$
If the agent weakly prefers $B$ over $A$, then their utility of $B$ must be weakly larger than that of $A$.
There are two types of utility functions, cardinal and ordinal. Both map preferences to $\mathbb R$ (or some subdomain of it). The cardinal theory speaks of "utils" as a measurable / comparable quantity, whereas in the ordinal world (implicitly found in most of the quant finance world, IMO), only the ordering (i.e. rank) of consumption bundles / lotteries are relevant. Both worlds are compatible with \eqref{1}, of course.
All that really matters (in this discussion) is the monotonicity of the utility function or of any transformation of the utility function. More is better, independent of the domain of the utility function - as long as \eqref{1} is satisfied.
HTH?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.