<p>No, I don't think so. Let <span class="math-container">$a_i$</span> be your choice (e.g. ice tea) in situation <span class="math-container">$i$</span> (e.g. summer).</p>
<p>Separability over states says that if for some <span class="math-container">$b_j$</span>, <span class="math-container">$a_i$</span> and <span class="math-container">$c_i$</span>:
<span class="math-container">$$
u(a_i, b_j) \ge u(c_i, b_j),
$$</span>
then for all <span class="math-container">$d_j$</span>
<span class="math-container">$$
u(a_i, d_j) \ge u(c_i, d_j).
$$</span>
Notice that both comparisons keep the choice in some state fixed but this choice changes. So your preference of <span class="math-container">$a$</span> over <span class="math-container">$c$</span> in state <span class="math-container">$i$</span> does not depend on what you have in state <span class="math-container">$j$</span>.</p>
<p>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.</p>
<p>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.</p>

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2021-12-04T17:31:15.807Z
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?
Richard Hardy
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2021-12-04T17:34:55.443Z
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.
Richard Hardy
CC BY-SA 4.0
2021-12-07T11:54:16.153Z
@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.
afreelunch
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2021-12-07T12:17:43.077Z
@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.
Richard Hardy
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2021-12-07T14:52:13.800Z
@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.]
afreelunch
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2021-12-07T14:54:47.483Z
@RichardHardy As a minor comment, you should be careful to distinguish separability across times and separability across states of the world. Summer vs winter are different times, not different states of the word (unless you are imagining, e.g., that I don't know what the season will be when I wake up after a sleep of many months...)
afreelunch
CC BY-SA 4.0
2021-12-07T14:59:44.797Z
@afreelunch, good catch! I am well aware of the difference between separability across time vs. across states, yet my example does not show this well at all. I should have used rain vs. shine instead of summer vs. winter. I have added a corresponding note at the end of the OP.
Richard Hardy
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2021-12-07T15:07:23.427Z
@afreelunch, regarding the need to separate states from outcomes: I think without it we get some sort of tautology or something. Given a utility function that has a single argument, I do not think you can construct a case where the utility in a particular state of the world depends on the utility in a different state of the world if the states are the same as outcomes; there are not enough degrees of freedom in the model for that. (Sorry for imprecision of my formulation; I am still working things out.)
Richard Hardy
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2021-12-07T20:26:44.540Z
Could you please indicate what definition of nonseparable utility you are using and refer to its source? Thank you.
Richard Hardy
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2021-12-08T14:23:56.050Z
@RichardHardy see here for the standard definition of separable preferences in economics [excellent notes by Ted Bergstrom](https://econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/separabilitynotes.pdf) (see definition 1).
tdm
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2021-12-04T13:16:35.570Z
2021-12-04T13:16:35.570Z
tdm
<p>I think the core issue with this question (and the other related one the OP posted <a href="https://economics.stackexchange.com/questions/48581/nonseparable-utility-across-states-of-nature-an-intuitive-example">Nonseparable utility across states of nature: an intuitive example</a>) is we need to clarify what is meant by &quot;separable&quot;.</p>
<p>Unfortunately, &quot;separable&quot; is among the most overused adjectives across formal theories, in econ and beyond (including in pure math itself, see <a href="https://en.wikipedia.org/wiki/Separability" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Separability</a>). 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).</p>
<p>So it's impossible to make progress on any question about &quot;separability&quot; without defining more formally what &quot;separable&quot; is supposed to mean, which I believe will show that the answer to the OP's question is unavoidably definition-specific.</p>
<p><em><strong>Fundamentals (keeping things simple here, for illustration purposes only)</strong></em></p>
<ul>
<li>Two states: <span class="math-container">$S = \{s, s'\}$</span>.</li>
<li>Set of outcomes <span class="math-container">$X$</span>.</li>
<li>Preference represented by <span class="math-container">$u$</span> over the set of all <span class="math-container">$[(s,a), (s',b) | s'']$</span>, with <span class="math-container">$a,b \in X$</span> and <span class="math-container">$s''$</span> 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).</li>
</ul>
<p><strong>Within-state Separability</strong></p>
<p><em>(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 &quot;other&quot; state.)</em></p>
<p><span class="math-container">$$ u[(s,a), (s',c)|s] \geq u[(s,b), (s',d)|s] \text{ for some $c,d \in X$}$$</span></p>
<p>implies</p>
<p><span class="math-container">$$ u[(s,a), (s', e)|s] \geq u[(s,b), (s',f)|s] \text{ for all $e,f \in X$},$$</span></p>
<p>and</p>
<p><span class="math-container">$$ u[(s,c), (s',a)|s'] \geq u[(s,d), (s',b)|s'] \text{ for some $c,d \in X$}$$</span></p>
<p>implies</p>
<p><span class="math-container">$$ u[(s,e), (s', a)|s'] \geq u[(s,f), (s',b)|s'] \text{ for all $e,f \in X$}$$</span></p>
<p>In words, Within-state Separability says that conditional on being in state <span class="math-container">$s^*$</span>, your preferences over what happens in that state are independent of what [could have happened/has happened/will happen] in the other state.</p>
<p>However, Within-state Separability allows preferences to be &quot;state-dependent&quot; in the sense of having <span class="math-container">$u[(s,a), (s',c)|s] &gt; u[(s,b), (s',d)|s]$</span> but <span class="math-container">$u[(s,c), (s',a)|s'] &lt; u[(s,d), (s',b)|s']$</span>.</p>
<p>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 <span class="math-container">$u[(s,a), (s',c)|s] &gt; u[(s,b), (s',c)|s]$</span> but <span class="math-container">$u[(s,a), (s',d)|s] &lt; u[(s,b), (s',d)|s]$</span>, i.e., the outcome you [had/will have/could have had] in state <span class="math-container">$s'$</span> impacts the way you rank outcomes in state <span class="math-container">$s$</span> (conditional on being in state <span class="math-container">$s$</span>).</p>
<p><strong>Between-state Separability</strong></p>
<p><em>(Similar warning applies)</em></p>
<p><span class="math-container">$$ u[(s,a), (s',c)|s] \geq u[(s,b), (s',d)|s] \text{ for some $c,d \in X$}$$</span></p>
<p>if and only if</p>
<p><span class="math-container">$$ u[(s,c), (s', a)|s'] \geq u[(s,d), (s',b)|s']$$</span></p>
<p>In words, Between-state Separability says that, conditional on being in state <span class="math-container">$s$</span>, you prefer getting outcome <span class="math-container">$a$</span> over <span class="math-container">$b$</span> in state <span class="math-container">$s$</span> if and only if you would also prefer <span class="math-container">$a$</span> over <span class="math-container">$b$</span> in state <span class="math-container">$s'$</span> conditional on being in that state (and provided what you [had/will have/could have had] in the other state is held fixed).</p>
<p>However, Between-state Separability allows preferences in one state to depend on what you get in the other state in the sense of having <span class="math-container">$u[(s,a), (s',c)|s] &gt; u[(s,b), (s',d)|s]$</span> but <span class="math-container">$u[(s,a), (s',e)|s] &lt; u[(s,b), (s',f)|s]$</span>.</p>
<p>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.</p>

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2021-12-08T15:55:54.453Z
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)*?
Richard Hardy
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2021-12-08T16:22:53.497Z
Thanks for catching the typos. You were right in both cases. I made the corrections.
Martin Van der Linden
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2021-12-08T16:28:48.073Z
"why condition on ? 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).
Martin Van der Linden
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2021-12-08T16:55:38.370Z
That makes sense and is very helpful. Thank you so much!
Richard Hardy
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2021-12-08T13:58:39.667Z
2021-12-08T16:21:23.020Z
2021-12-08T16:21:23.020Z
Martin Van der Linden
<p>There are two states of nature: summer (hot) and winter (cold).<br />
I have a utility function indexed by states of nature: <span class="math-container">$u(\cdot;summer)$</span> and <span class="math-container">$u(\cdot;winter)$</span>.<br />
There are two good to choose between: ice tea and hot tea.<br />
In the summer it is hot and I prefer ice tea: <span class="math-container">$u(ice \ tea;summer)&gt;u(hot \ tea;summer)$</span>.<br />
In the winter it is cold and I prefer hot tea: <span class="math-container">$u(ice \ tea;winter)&lt;u(hot \ tea;winter)$</span>.</p>
<p>Is that a valid example of a utility function that is nonseparable across states of nature?</p>
<p><strong>Edit:</strong> 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 <em>summer</em> with <em>shine</em> and <em>winter</em> with <em>rain</em> for a less confusing example. (I am not doing that above to keep the already posted answer and comments relevant.)</p>

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2021-12-03T20:41:29.437Z
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.
Richard Hardy
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2021-12-06T18:54:08.750Z
I later found [this](https://economics.stackexchange.com/questions/134/can-the-machina-paradox-be-solved-by-expanding-the-choice-set) and especially [this](https://math.stackexchange.com/questions/779974/von-neumann-morgenstern-independence-axiom-vs-savage-independence-theorm) to be helpful.
Richard Hardy
<utility>
Is this a case of nonseparable utility (across states of nature)?
Richard Hardy
# Is this a case of nonseparable utility (across states of nature)?

(Asked by: username_0 on 2021-12-03)

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.)

**Tags:** utility
> **Comments:**
> **username_0**: 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.  
> **username_0**: I later found [this](https://economics.stackexchange.com/questions/134/can-the-machina-paradox-be-solved-by-expanding-the-choice-set) and especially [this](https://math.stackexchange.com/questions/779974/von-neumann-morgenstern-independence-axiom-vs-savage-independence-theorm) to be helpful.  

---

## 2 answers

### Answer by username_3 on 2021-12-08. Score: 4 | Accepted answer

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](https://economics.stackexchange.com/questions/48581/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.

> **Comments:**
> **username_0**: 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)*?  
> **username_3**: Thanks for catching the typos. You were right in both cases. I made the corrections.  
> **username_3**: "why condition on ? 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).  
> **username_0**: That makes sense and is very helpful. Thank you so much!  

---

### Answer by username_1 on 2021-12-04. Score: 4

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.

> **Comments:**
> **username_0**: 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?  
> **username_0**: 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.  
> **username_2**: @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.  
> **username_0**: @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.  
> **username_2**: @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.]  
> **username_2**: @RichardHardy As a minor comment, you should be careful to distinguish separability across times and separability across states of the world. Summer vs winter are different times, not different states of the word (unless you are imagining, e.g., that I don't know what the season will be when I wake up after a sleep of many months...)  
> **username_0**: @afreelunch, good catch! I am well aware of the difference between separability across time vs. across states, yet my example does not show this well at all. I should have used rain vs. shine instead of summer vs. winter. I have added a corresponding note at the end of the OP.  
> **username_0**: @afreelunch, regarding the need to separate states from outcomes: I think without it we get some sort of tautology or something. Given a utility function that has a single argument, I do not think you can construct a case where the utility in a particular state of the world depends on the utility in a different state of the world if the states are the same as outcomes; there are not enough degrees of freedom in the model for that. (Sorry for imprecision of my formulation; I am still working things out.)  
> **username_0**: Could you please indicate what definition of nonseparable utility you are using and refer to its source? Thank you.  
> **username_1**: @RichardHardy see here for the standard definition of separable preferences in economics [excellent notes by Ted Bergstrom](https://econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/separabilitynotes.pdf) (see definition 1).  

---

