# 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.