# Why utility rather than expected utility in Cochrane's “Asset Pricing”?

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?

• 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. – Michael Greinecker Feb 22 at 13:37
• 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. – Richard Hardy Feb 22 at 13:38
• The theory here uses expected utility. Not all authors are equally careful when it comes to formal details. – Michael Greinecker Feb 22 at 13:42
• @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. – Richard Hardy Feb 22 at 16:38
• @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. – 1muflon1 Feb 22 at 16:43

$$u(c_t)+\beta E_t[u(c_{t+1})]=E_t[u(c_t)+\beta u(c_{t+1})]$$
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.
• 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 . – jmbejara Feb 22 at 19:51
• @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 – 1muflon1 Feb 22 at 20:09