# Is this a case of nonseparable utility (across states of nature)?

There are two states of nature: summer (hot) and winter (cold).
I have a utility function indexed by states of nature: $$u(\cdot;summer)$$ and $$u(\cdot;winter)$$.
There are two good to choose between: ice tea and hot tea.
In the summer it is hot and I prefer ice tea: $$u(ice \ tea;summer)>u(hot \ tea;summer)$$.
In the winter it is cold and I prefer hot tea: $$u(ice \ tea;winter).

Is that a valid example of a utility function that is nonseparable across states of nature?

Edit: summer vs. winter may imply different times, yet the example was meant to imply different states of nature while holding the time fixed. That was a poorly thought through choice on my end; sorry. Replace summer with shine and winter with rain for a less confusing example. (I am not doing that above to keep the already posted answer and comments relevant.)

• I made this up from scratch without knowing much about considerations of states of nature in utility theory, so a rookie mistake or two is likely lurking somewhere. Dec 3, 2021 at 20:41
• I later found this and especially this to be helpful. Dec 6, 2021 at 18:54

I think the core issue with this question (and the other related one the OP posted Nonseparable utility across states of nature: an intuitive example) is we need to clarify what is meant by "separable".

Unfortunately, "separable" is among the most overused adjectives across formal theories, in econ and beyond (including in pure math itself, see https://en.wikipedia.org/wiki/Separability). It's also commonly used in casual speech about formal models as an informal allusion to some sort of invariance (the video linked in the related question is a good example).

So it's impossible to make progress on any question about "separability" without defining more formally what "separable" is supposed to mean, which I believe will show that the answer to the OP's question is unavoidably definition-specific.

Fundamentals (keeping things simple here, for illustration purposes only)

• Two states: $$S = \{s, s'\}$$.
• Set of outcomes $$X$$.
• Preference represented by $$u$$ over the set of all $$[(s,a), (s',b) | s'']$$, with $$a,b \in X$$ and $$s''$$ representing the state you are currently in (either the state that materialized, if only one state can, or the current state of nature if states represent successive events).

Within-state Separability

(The name is made-up. I don't claim this is standard terminology or a good choice thereof. There are also many variants of the property you could think of, e.g., by holding one of the two outcomes fixed in the "other" state.)

$$u[(s,a), (s',c)|s] \geq u[(s,b), (s',d)|s] \text{ for some c,d \in X}$$

implies

$$u[(s,a), (s', e)|s] \geq u[(s,b), (s',f)|s] \text{ for all e,f \in X},$$

and

$$u[(s,c), (s',a)|s'] \geq u[(s,d), (s',b)|s'] \text{ for some c,d \in X}$$

implies

$$u[(s,e), (s', a)|s'] \geq u[(s,f), (s',b)|s'] \text{ for all e,f \in X}$$

In words, Within-state Separability says that conditional on being in state $$s^*$$, your preferences over what happens in that state are independent of what [could have happened/has happened/will happen] in the other state.

However, Within-state Separability allows preferences to be "state-dependent" in the sense of having $$u[(s,a), (s',c)|s] > u[(s,b), (s',d)|s]$$ but $$u[(s,c), (s',a)|s'] < u[(s,d), (s',b)|s']$$.

In this sense (which seems to be close to what @tdm has in mind), the example your provide fails to be non-separable. As @tdm suggests, with this definition of separability, a non-separable preference would require something like $$u[(s,a), (s',c)|s] > u[(s,b), (s',c)|s]$$ but $$u[(s,a), (s',d)|s] < u[(s,b), (s',d)|s]$$, i.e., the outcome you [had/will have/could have had] in state $$s'$$ impacts the way you rank outcomes in state $$s$$ (conditional on being in state $$s$$).

Between-state Separability

(Similar warning applies)

$$u[(s,a), (s',c)|s] \geq u[(s,b), (s',d)|s] \text{ for some c,d \in X}$$

if and only if

$$u[(s,c), (s', a)|s'] \geq u[(s,d), (s',b)|s']$$

In words, Between-state Separability says that, conditional on being in state $$s$$, you prefer getting outcome $$a$$ over $$b$$ in state $$s$$ if and only if you would also prefer $$a$$ over $$b$$ in state $$s'$$ conditional on being in that state (and provided what you [had/will have/could have had] in the other state is held fixed).

However, Between-state Separability allows preferences in one state to depend on what you get in the other state in the sense of having $$u[(s,a), (s',c)|s] > u[(s,b), (s',d)|s]$$ but $$u[(s,a), (s',e)|s] < u[(s,b), (s',f)|s]$$.

If separability is understood as Between-state Separability (which seems to be closer to the definition of separability you have in mind), then the example you suggested is non-separable.

• Thank you very much for your thorough answer! I am trying to wrap my head around a utility function of the form $u[(s,a),(s′,c)|s]$ and it is not easy (why condition on $s$? why derive utility from something that happens in a different state than the actual one?). But in the meantime, what about the following: Within-state Separability allows preferences to be "state-dependent" <...> In this sense <...>, the example your provide fails to be separable. Did you mean nonseparable? Also, is conditioning not missing from the left side of the expression right below (Similar warning applies)? Dec 8, 2021 at 15:55
• Thanks for catching the typos. You were right in both cases. I made the corrections. Dec 8, 2021 at 16:22
• "why condition on 𝑠? why derive utility from something that happens in a different state than the actual one?". I am not claiming it's a good model or that you should ever assume people derive utility in that way. It just seemed useful for clarification purposes to set up a general framework where preferences could in principle depend on a) what you would get in state $s$, b) what you would get in state $s'$ and c) what state you are in. Then it's the job of properties like the ones discussed in my answer to clarify invariances (i.e., what matters to the decision-maker, and what doesn't). Dec 8, 2021 at 16:28
• That makes sense and is very helpful. Thank you so much! Dec 8, 2021 at 16:55

No, I don't think so. Let $$a_i$$ be your choice (e.g. ice tea) in situation $$i$$ (e.g. summer).

Separability over states says that if for some $$b_j$$, $$a_i$$ and $$c_i$$: $$u(a_i, b_j) \ge u(c_i, b_j),$$ then for all $$d_j$$ $$u(a_i, d_j) \ge u(c_i, d_j).$$ Notice that both comparisons keep the choice in some state fixed but this choice changes. So your preference of $$a$$ over $$c$$ in state $$i$$ does not depend on what you have in state $$j$$.

So for example, if you prefer ice tea in summer over hot tea in summer when you have ice tea in the winter, then you should also prefer ice tea in summer over hot tea in the summer when you have hot tea in the winter.

A violation of separability would be that your preferences of ice tea versus hot tea in summer depends on what you have in the winter. For example, you prefer ice tea over hot tea in summer when you have hot tea in winter, but you prefer hot tea over ice tea in summer when you have ice tea in the winter.

• Thank you! I presume this is correct. First, there is something in the terminology that puzzles me. States are summer and winter, not ice tea and hot tea. However, your last paragraph suggests my preference in the summer should not depend on what I have in the winter. That is not a dependence on a state (winter) but on a choice/action/outcome in winter. Is that so? Is such wording not confusing? Second, I wonder about a utility function with a single argument. Is it possible to have nonseparable utility with such a function, or do we need at least two arguments of the function? Dec 4, 2021 at 17:31
• The latter question is related to the fact that in macro-finance, utility often has a single argument, that is, amount of consumption. Yet nonseparable utility is introduced in that context. Perhaps they introduce a new argument of the function there (e.g. amount of leasure); I guess it is not unreasonable to speculate that that is the case there. Dec 4, 2021 at 17:34
• @RichardHardy I think you are right that the wording is confusing. Even under expected utility, my utility will depend on the state. For example, if I win the lottery, then my utility will be higher than if I don't win the lottery! The question, rather, is whether my utility in a particular state of the world depends on the utility I get in some different state of the world -- even though at most one of these states will be realised. Dec 7, 2021 at 11:54
• @afreelunch, I think we may need to be more careful in distinguishing between states and outcomes. Winning a lottery vs. losing a lottery are outcomes, and they do not have to coincide with states of nature such as summer and winter. If I preferred winning to losing in summer but losing to winning in winter, we would be closer to nonseparability. But according to tdm, this is not enough. The idea is expanded in a couple of threads I found to be quite helpful: see links in my comment under the OP. Dec 7, 2021 at 12:17
• @RichardHardy I was considering a situation where are there are two states of the world (i.e. outcomes): winning the lottery vs not winning the lottery. Why would we need to distinguish states and outcomes in this example? [To be clear, there are different cases where we can distinguish states and outcomes -- I just don't see how the distinction is relevant here.] Dec 7, 2021 at 14:52