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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?

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    $\begingroup$ This seems to be more a question about Mr. Cochrane and his stylistic choices than about economics. But the right-hand side is clearly expected utility, and the left-hand side does not make much sense- but is not used anyways. $\endgroup$ Feb 22 at 13:37
  • $\begingroup$ I will probably discover the answer somewhere later in the book, but I found the discrepancy between the language/text and the formulas quite striking. $\endgroup$ Feb 22 at 13:38
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    $\begingroup$ The theory here uses expected utility. Not all authors are equally careful when it comes to formal details. $\endgroup$ Feb 22 at 13:42
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    $\begingroup$ @1muflon1, it is weird and I think incorrect to say someone's utility is a function of consumption today and expected consumption tomorrow. People do not derive utiity from expected vaules of consumption, only from actual consumption. In that sense I think it is justifiable to have a harsh comment. $\endgroup$ Feb 22 at 16:38
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    $\begingroup$ @RichardHardy but note that is not what Cochrane is stating there. He states that the overall utility $U$ is sum of present utility of consumption $u_t(c_t)$ and expected utility of future consumption $E_t(u_{t+1}(c_{t+1}))$. Is it non-standard? Yes! Does it offend my sense of aesthetics? A bit. Is it Incorrect? I don't think that it is incorrect to have composite utility function which is sum of present utility that is known and expected utility from future consumption - I consider it less elegant as just having $E[U]$ but I would not go as far as saying its nonsense. $\endgroup$
    – 1muflon1
    Feb 22 at 16:43
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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.

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    $\begingroup$ This kind of notation shows up quite a bit. Small $u$ is a flow utility and $U$ is an aggregation of utility over time, like a stock of utility. See this question here economics.stackexchange.com/q/22475/59 and my answer here economics.stackexchange.com/a/27378/59 . $\endgroup$
    – jmbejara
    Feb 22 at 19:51
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    $\begingroup$ @jmbejara thanks for the comment, I don't know why but I remember usually seeing people using $E(U)$ but I guess it might differ in subfields or perhaps I just thought so because it was notation used in something I read recently, I will scratch the non-standard part from my answer $\endgroup$
    – 1muflon1
    Feb 22 at 20:09
  • $\begingroup$ Yeah, conventions probably differs by field. Thanks! $\endgroup$
    – jmbejara
    Feb 22 at 22:24

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