Why utility rather than expected utility in Cochrane's "Asset Pricing"?

Question

Cochrane "Asset Pricing" Chapter 1 p. 6 says

We model investors by a utility function defined over current and future values of consumption, $$ U(c_t,c_{t+1}) = u(c_t) + \beta \mathbb{E_t}[u(c_{t+1})] $$ where $c_t$ denotes consumption at date $t$.

Later, this utility is maximized subject to a sort of a budget constraint

$$ \max_{\xi}\ u(c_t) + \mathbb{E}_t[\beta u(c_{t+1})]$$ where \begin{aligned} c_t &= e_t - p_t \xi, \\ c_{t+1} &= e_{t+1} + x_{t+1}\xi. \end{aligned}

I am used to maximization of expected utility rather than raw utility. Moreover, the expression on the right hand side of $U(c_t,c_{t+1})$ looks just like expected utility where the expectation is conditional on the information available at time $t$: $$ u(c_t) + \beta \mathbb{E_t}[u(c_{t+1})] = \mathbb{E_t}[( u(c_t) + \beta u(c_{t+1})]. $$ Question: Why does Cochrane not call $U(c_t,c_{t+1})$ expected utility then?

Answer 1

As mentioned in the comments this comes down to stylistic choices, since as you correctly pointed out:

$$u(c_t)+\beta E_t[u(c_{t+1})]=E_t[u(c_t)+\beta u(c_{t+1})]$$

However, in principle both expressions are correct. The first expression states that the $U_t(c_t, c_{t+1})$ is a composite function of present utility of consumption $u_t(c_t)$ and expected utility of future consumption $E_t(u_{t+1}(c_{t+1}))$, whereas the second expression is directly expected utility of present and future consumption. When we condition on presently known information those two are equivalent here.

The notation that Cochrane uses is not used everywhere but there does not seem anything wrong with it either from mathematical perspective or economically.