Continuation value versus utility in asset pricing

Question

Is there a difference between continuation value ($V_t$) and utility ($U_t$) except for a possible scaling / difference in units? My question refers to the consumption-based asset pricing literature.

In standard time additive power utility settings, people seem to only talk about utility (e.g. $U_t=u(C_t)+\beta E_t[u(C_{t+1}]$). In recursive utility / Epstein-Zin-Weil settings, people often refer to a continuation value (e.g. $V_t=((1-\beta)C_t^{1-\rho}+\beta (\mathcal{R}_t(V_{t+1}))^{1-\rho})^{1/(1-\rho)}$).

It seems to me that both are fairly similar. The only reference I could find on the topic is the asset pricing book from Back (2010), in which (intuitive definitions) utility seems to be a utility measure in "utility units" while continuation value seems to be a utility measure in "consumption good units", and both are related via $U_t=u(V_t)=\frac{V_t^{1-\gamma}}{1-\gamma}$.

(Please note: Back confusingly talks about a continuation utility and uses a different notation, the one posted here is inherited from standard references. Also, you can find his book with this link.)

Answer 1

The term "continuation value" is usually used to refer to an aggregation of utility in future periods. In the case of $U_t = u(C_t) + \beta E_t[u(C_{t+1})]$, $u_t$ is the flow of utility per period and $U_t$ is an aggregate of $u_t$ over all time periods $t$ and $t+1$. The quantity $\beta E_t[u(C_{t+1})]$ is the expected discounted continuation value and is in the same units as $u(C_t)$.

Recursive utility works similarly. The difference is that utility over multiple time periods can't be easily aggregated. In the expected utility case, it's aggregated using a weighted sum of expectations. To see this, I'll show how you can write an expected utility function in recursive form. This should make the continuation value used in discussions of recursive utility easier to understand.

Again, let $u_t$ be the flow of utility per period. Suppose that $U_t$ is an aggregate of $u_t$ over all time periods from time $t$ into the future. Sometimes $u_t$ is referred to as the "per-period utility function". An economic actor making decisions in time $t$ maximizes $U_t$, where $u_\tau$ for $\tau = t, t+1, t+2,...$ are simply the components that make up $U_t$. Consider the case of CRRA utility: $$ U_0 \equiv E_0 \sum_{t=0}^\infty \beta^t \frac{C_t^{1-\gamma}}{1-\gamma}. $$ Let $u_t = \frac{C_t^{1-\gamma}}{1-\gamma}$ and let $$ U_t = E_t \sum_{\tau=t}^\infty \beta^{\tau-t} \frac{C_\tau^{1-\gamma}}{1-\gamma}. $$ Then, $$ U_t = E_t \sum_{\tau=t}^\infty \beta^{\tau-t} u_\tau $$ and we can represent utility here recursively, $$ U_t = u_t + \beta U_{t+1}. $$ In the expected utility case of CRRA, $U_t$ is simply a weighted sum of future per-period utility. Epstein-Zin utility is a case in which $U_t$ is an aggregate of future per-period utility, but it cannot be expressed as a simple weighted sum. Rather, $U_t=((1-\beta)C_t^{1-\rho}+\beta (\mathcal{R}_t(U_{t+1}))^{1-\rho})^{1/(1-\rho)}$, where $\mathcal R$ is a "certainty equivalent" operator.

Answer 2

I can't speak to Back's usage (I can't access the book through the link), though in general, I believe the "utility function" refers to the standard meaning of the term (the ultimate function that agent's are trying to maximize), while "continuation value" refers to the Bellman Equation solved for the next period.

A more abstract example of both the "utility function" and "value function" appearing together can be found in this set of notes from Peter Ireland in section 3 (example 2) after the optimality conditions are solved for.