# On the uniqueness of utility functions for both risk and time

I have a question regarding the uniqueness of preference functionals under risky and dynamic settings. Two well known models to represent preferences for both settings are the Expected Utility Model and the Discounted Time-Separable Utility respectively. Formally

Risk:

$$V(f)=\sum_{\forall s \in S} p(s) u(f(s))$$

Where $S$ is the set of states and $p(s)$ is the probability of state $s$, and $u(\cdot)$ a cardinal utility function.

Dynamics:

$$W(c)=\sum_{t=0}^{T} \mathcal{D}(t) v(c_t)$$ Where $\mathcal{D}(t)$ is a discount factor and $v(\cdot)$ is a cardinal utility function.

We know that both $u(\cdot)$ and $v(\cdot)$ are cardinal utility functions, and I understand why. But while every representation theorem of this sort of preferences mention the uniqueness of the functions $u(\cdot)$ and $v(\cdot)$(i.e unique up to positive affine transformations), they are silent about the uniqueness of $V(\cdot)$ and $W(\cdot)$. My question is: are the functions $V(\cdot)$ and $W(\cdot)$ ordinal or cardinal? .

My intuition tells me that they should be ordinal, as they only rank objects of choice and the attitudes towards risk and time are captured respectively by the function $u(\cdot)$ in the case of risk, and the preference for consumption smoothing is given by the function $v(\cdot)$.

For example if $V(f)>V(g)$ then, it should also be that for every increasing function $H(\cdot)$, $H(V(f))>H(V(g))$.

Nevertheless by the linearity of the Expected Utility Functional, i.e,

$$h\Bigg(\sum_{\forall s \in S} p(s) u(f(s)\Bigg)=\sum_{\forall s \in S} p(s) h\big( u(f(s)\big)$$

[So if this is correct, $h(\cdot)$ should be affine, making $V(f)$ cardinal. ]

I suspect that my intuition might be wrong (same applies to the dynamic setting), so I was wondering if you guys could help me clarify this question.

Thank you very much!

• @TheoreticalEconomist This observation is not correct. For instance, the functional $HV(f)=(V(f))^3$ preserves the ordering between risky acts in spite of the fact that $x \rightarrow x^3$ is not affine. The functional form has no cardinal properties, only ordinal ones.
– Oliv
Dec 28 '16 at 10:59
• @Oliv Of course; I stand corrected. Dec 28 '16 at 11:02
• @Oliv TheoreticalEconomist thank you both. Im still struggling to see how the ordinality of $V(f)$ coexists with the linearity property of the Expected Utility Functional that made me think in the first place that $h$ should be affine. If anyone could enlighten me on this would be great! On the same line I suspect Oliv that you also think that $W(\cdot)$ is ordinal aswell? Dec 28 '16 at 13:39
• You are right, $W$ is not cardinal. In the expected utility theory, the relative valuation of $A$ vs $B$, equal to $u(A)-u(B)$, is identified (in a cardinal sense) by trade-offs of the form "how does the decision-maker value a 1% increase in the probability of receiving good A vs good B". For the complete functional, no such thought experiment is possible since we cannot increase by 1% the probability of receiving the act $f$ vs the act $g$.
– Oliv
Dec 28 '16 at 17:17