Nonseparable utility across states of nature: an intuitive example

Question

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.)

Answer 1

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.