Nonseparable utility across states of nature: an intuitive example

I am new to nonseparable utility across states of nature as found in some macro-financial models (discussed in this YouTube video lecture by John Cochrane). I do not find the notion intuitive. Could anyone suggest an intuitive example or point to a reference that includes one? (Cochrane does not seem to provide any examples in the video.)

• Hi @RichardHardy. My understanding of the best intuitive example is when behaviour is characterised by habit formation. home.cerge-ei.cz/petrz/fm/f400n26.pdf. There are many implications of habit formation, but one of I am aware of is in terms of the transmission of monetary policy. Dec 1 '21 at 17:37
• @EB3112, your link is about time separability with which I do not have a problem at the moment. My question is about separability across states of nature. Dec 1 '21 at 17:58
• I think people disagree on whether intuitive examples exist or not. But there is a huge literature on non-expected-utility preferences, mostly outside macro. Dec 1 '21 at 18:25
• This is quite a famous example: everything2.com/title/Machina%2527s+Paradox Dec 3 '21 at 18:42
• @afreelunch: You should write an answer out of it. Dec 4 '21 at 3:36

I'm not aware of any intuitive justification for the state-non-separability in Epstein-Zin preference. However, as both @MichaelGreinecker and @afreelunch alluded to, there are micro/behavioral explanations for why utility is non-separable across states.

Habit formation is the prime example of time non-separability, and it is modeled as current utility being a function of (a set of) past consumption. Likewise, any model in which utility in one state is a function of (a set of) other states would feature state non-separability. Theories that explicitly incorporate disappointment / regret / elation / rejoice are such examples.

If a decision maker (DM) ends up in a state that is worse than what she had anticipated, she may feel disappointed; similarly, if the DM ends up in a state that is better than her initial anticipation, she would feel elated. Naturally, disappointment and elation are a function of not only the realized state, but also the unrealized ones.

For example, Loomes and Sugden (1986) posit the following criterion for evaluating lotteries: $$\begin{equation} U(L)=\sum_{s\in S}p_s\biggl[u(s)+D\bigl(u(s)-\bar u\bigr)\biggr],\quad\text{where}\;\bar u=\sum_{s\in S}p_su(s). \end{equation}$$ Hence, in state $$s$$, the DM gets not only the state utility $$u(s)$$, but also extra (dis)utility depending on how $$u(s)$$ compares to the overall expected utility level $$\bar u$$. The function $$D(\cdot)$$ respects the sign of $$u(s)-\bar u$$. So DM experiences disappointment (and disutility) when she is in a state $$s'$$ such that $$u(s')<\bar u$$. Since $$\bar u$$ is a function of all states, this formulation makes state non-separable.

In a similar spirit, Gul (1991) derive a criterion with the certainty equivalent as the reference level for disappointment and elation. However, the state non-separability in Gul's criterion is in terms of the probabilities of the states, which show up as decision weights that are nonlinear in the probabilities, rather than wealth levels in those states.

More generally, reference-dependent preferences typically exhibit state non-separability.

• Cannot we simplify the expressions and represent the utilities of each state without using disappointment and elation? E.g. consider a case of three possible outcomes with utilities $(8,10,12)$ with the middle one being the "default" (expected) state. Why interpret them as $(8,10,12)=(9,10,11)+(-1,0,1)$ where the latter part is due to disappointment ($-1$) and elation ($+1$)? I.e. how can we distinguish between the two possible representations if we only observe the final numbers $(8,10,12)$? (OK, we do not observe utility – these numbers – directly but somehow get to them.) Dec 6 '21 at 18:19
• I think my comment reflects that perhaps your answer (or just my understanding of it) is missing a distinction between outcomes and states of nature. Without it, $(9,10,11)+(−1,0,1)$ collapses to $(8,10,12)$ without any way of distinguishing them observationally. Then $(9,10,11)+(−1,0,1)$ is hard to justify since we could start adding even more elements such as $(9,10,10)+(−1,0,1)+(0,0,1)$ and making up additional effects and interpretations of what all these elements are. Dec 6 '21 at 18:23
• Perhaps the answer in this thread is relevant. Dec 6 '21 at 18:37
• @RichardHardy: Sorry but I'm a bit confused. I thought you wanted an intuitive example that shows utility functions may be non-separable in states. I believe disappoint/elation is such an example: utility in one state is a function of other states. Whether or not one can redefine states and outcomes in such a way that utility is now separable in this alternative definition of states/outcomes is to me a different question. Let me know if I misunderstood anything. Dec 7 '21 at 1:44
• You are right that I am looking for an intuitive example. My problem with yours is that I am not sure it is a valid one. Given what I have read here, here and here, I suspect it is not. Now I do not yet understand the topic well enough to be sure about my suspicion. Thus I turn to you with additional questions/comments. I have tried to show that one need not redefine states to turn this example from nonseparable to nonseparable. Dec 7 '21 at 7:39