# Expected Discounted Utility

Expected discounted utility is one of the most common ways to represent preferences over risky consumption plans. Consider an agent, sitting at time $t$, who will receive a consumption stream $c$ until $T$: \begin{equation} U_t(c)= E_t \left[ \sum \limits_{s=t}^T \beta^{s-t}u_s(c_s)\right] \end{equation} Where $\beta$ is the discount factor and $u$ is a within-period utility function. A problem with expected discounted utility is that it cannot separate preferences for smoothing over time, and smoothing across states.
Consider the following example: You are stranded on an island at $t=0$. A man comes in a boat and offers you a choice of two deals (1) Every morning he comes and flips a coin, if it comes up heads, you get a bushel of bananas that day (2) He flips a coin today, if it comes up heads you get a bushel of bananas every day until time $T$, and if it comes up tails you get no bananas until time $T$. It’s initiative that plan 2 is riskier than plan 1, but under expected discounted utility, for any $\beta$ and $u$ the agent is indifferent between the two plans: \begin{equation} U(Plan 1) = \sum\limits_{t=0}^T \beta^t \frac{u_t(1)+u_t(0)}{2}= U(Plan 2) \end{equation}

# Recursive Utility

The only way to even partially separate preferences for smoothing over time, and preferences for smoothing across states is to use recursive utility (see Skiadas 2009 for a complete proof - this is an if and only if relationship). Recursive utility has two ingredients, the aggregator, which determines preferences over deterministic plans (time smoothing) $f(t,c_t,\upsilon_t \left(U_{t+1}(c)\right))$ and the conditional certainty equivalent $\upsilon_t(c)$ (state smoothing). The steps below formulate expected discounted utility as recursive utility. For simplicity, drop the dependence of all functions on time, so we can remove all the subscript $t$’s. Now, propose a desirable property for the utility function - normalization. Consider any deterministic plan $\alpha$, then a utility is normalized if $\bar{U}(\alpha)=\alpha$. Normalize utility $U$, the expected discounted utility defined above, as $\bar{U}(c)=\psi^{-1}(U(c))$ where $\psi_t(\alpha)=\sum\limits_{s=t}^T \beta^{s-t} u(\alpha)$. Basically, $\psi$ gives the discounted utility of deterministic plan $\alpha$, so $\psi^{-1}$ gives the deterministic $\alpha$ required to make the agent indifferent between potentially risky plan $c$ and deterministic plan $\alpha$.
For expected discounted utility, the aggregator is: $f(t,x,y)=\psi^{-1}_t (u(x)+\beta \psi_{t+1}(y))$. The intuition is that with expected discounted utility, the agent’s utility from plan $c$ is a weighted average of their consumption today, and the utility of the equivalent deterministic plan until $T$. For utility to be normalized, the aggreator must satisfy $f(t,\alpha,\alpha)=\alpha$ for any deterministic plan $\alpha$. Put this into the equation above to solve for $\psi$: $f(t,x,x)=\psi_t^{-1}( u(x) + \beta \psi_{t+1}(x)) = x$. Then, apply $\psi_t$ to both sides: \begin{equation} u(x) + \beta \psi_{t+1} (x) = \psi_t(x) \end{equation} Fix $\psi_T=u$, and interpret terminal consumption value $c_T$ as consuming $c_T$ for the rest of time (equivalently, imagine letting $T$ go to infinity). This implies we can drop the subscripts on the $\psi$: \begin{equation} u(x)=\psi(x)-\beta\psi(x) \end{equation}

Rearranging yields $\psi(x)=(1-\beta)^{-1}u(x)$ and $\psi^{-1}(x)=u^{-1}((1-\beta)x)$. Putting this back into our expression above for $f(t,x,y)$ implies: \begin{equation} f(t,x,y)=u^{-1}((1-\beta)u(x)+\beta u(y)) \end{equation} Given the way the aggregator is defined, we can see that $f$ depends on the curvature of $u$ - in other words, the within period utility function $u$ will influence preferences for smoothing over time. This also gives intuition for how to make an agent not indifferent between deal (1) and deal (2) described above - $f$ needs to be defined independently of $u$ (or $\upsilon$).

# Conclusion

Recursive utility is a general framework, with expected discounted utility as a special case. For a deeper look at recursive utility, see Asset Pricing Theory by Costis Skiadas.