<p>As mentioned in the comments this comes down to stylistic choices, since as you correctly pointed out:</p>
<p><span class="math-container">$$u(c_t)+\beta E_t[u(c_{t+1})]=E_t[u(c_t)+\beta u(c_{t+1})]$$</span></p>
<p>However, in principle both expressions are correct. The first expression states that the <span class="math-container">$U_t(c_t, c_{t+1})$</span> is a composite function of present utility of consumption <span class="math-container">$u_t(c_t)$</span> and expected utility of future consumption <span class="math-container">$E_t(u_{t+1}(c_{t+1}))$</span>, 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.</p>
<p>The notation that Cochrane uses is not used everywhere  but there does not seem anything wrong with it either from mathematical perspective or economically.</p>

CC BY-SA 4.0
2021-02-22T19:51:32.517Z
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 https://economics.stackexchange.com/q/22475/59 and my answer here https://economics.stackexchange.com/a/27378/59 .
jmbejara
CC BY-SA 4.0
2021-02-22T20:09:03.263Z
@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
CC BY-SA 4.0
2021-02-22T22:24:05.853Z
Yeah, conventions probably differs by field. Thanks!
jmbejara
CC BY-SA 4.0
2021-02-22T17:06:19.827Z
2021-02-22T20:09:30.670Z
2021-02-22T20:09:30.670Z
1muflon1
<p>Cochrane &quot;Asset Pricing&quot; <a href="http://assets.press.princeton.edu/chapters/s7836.pdf" rel="nofollow noreferrer">Chapter 1</a> p. 6 says</p>
<blockquote>
<p>We model investors by a <em>utility function</em> defined over current and future values of consumption,
<span class="math-container">$$ 
U(c_t,c_{t+1}) = u(c_t) + \beta \mathbb{E_t}[u(c_{t+1})]
$$</span>
where <span class="math-container">$c_t$</span> denotes consumption at date <span class="math-container">$t$</span>.</p>
</blockquote>
<p>Later, this utility is maximized subject to a sort of a budget constraint</p>
<blockquote>
<p><span class="math-container">$$
\max_{\xi}\ u(c_t) + \mathbb{E}_t[\beta u(c_{t+1})]$$</span>
where
<span class="math-container">\begin{aligned}
c_t &amp;= e_t - p_t \xi, \\
c_{t+1} &amp;= e_{t+1} + x_{t+1}\xi.
\end{aligned}</span></p>
</blockquote>
<p>I am used to maximization of <em>expected</em> utility rather than raw utility. Moreover, the expression on the right hand side of <span class="math-container">$U(c_t,c_{t+1})$</span> looks just like <em>expected</em> utility where the expectation is conditional on the information available at time <span class="math-container">$t$</span>:
<span class="math-container">$$
u(c_t) + \beta \mathbb{E_t}[u(c_{t+1})] = \mathbb{E_t}[( u(c_t) + \beta u(c_{t+1})].
$$</span>
<strong>Question:</strong> Why does Cochrane not call <span class="math-container">$U(c_t,c_{t+1})$</span> <em>expected</em> utility then?</p>

CC BY-SA 4.0
2021-02-22T13:37:14.793Z
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
CC BY-SA 4.0
2021-02-22T13:38:39.457Z
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
CC BY-SA 4.0
2021-02-22T13:42:19.430Z
The theory here uses expected utility. Not all authors are equally careful when it comes to formal details.
Michael Greinecker
CC BY-SA 4.0
2021-02-22T16:02:18.327Z
I agree that this comes to the stylistic choice, but I also think previous comment is bit too harsh saying it does not make sense. It is like saying using sing $2 \cdot  5$ and calling it a *product of integer* is more/less correct than using 10 and calling it *integer*. To be clear most of the literature would use the RHS and write it as $E_t[U]$ instead of LHS $U(u_t,E_t(u_{t+1}))$, but due to expectations being applied only to the argument it could be called just utility with $u_{t+1}$ being expected utility even when it’s completely equivalent to having expected utility of whole function
1muflon1
CC BY-SA 4.0
2021-02-22T16:38:41.140Z
@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
CC BY-SA 4.0
2021-02-22T16:43:47.227Z
@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
CC BY-SA 4.0
2021-02-22T16:49:59.140Z
@1muflon1, I must agree. Thank you for a thoughtful comment! I think you could post it as an answer.
Richard Hardy
CC BY-SA 4.0
2021-02-22T23:52:43.447Z
@1muflon1 The normal way to read $U(c_t,c_{t+1})$ is that $U$ is a function of two numbers, $c_t$ and $c_{t+1}$. Indeed, Cochrane write that utility  is "defined over current and future values of consumption" But $U$ does not just depend on the current and future values of consumption, it depends on the current value and the conditional expaction over all possible future values. You are not going to read that in a math paper.
Michael Greinecker
CC BY-SA 4.0
2021-02-23T00:02:35.497Z
@MichaelGreinecker but in this case $U$ is clearly composite function, so here I think Cochrane is just bit sloppy as clearly the function is actually a composite one $U(u_t(c_t),E[u_{t+1}(c_{t+1})] )$
1muflon1
CC BY-SA 4.0
2021-02-23T00:24:03.033Z
@1muflon1 It is more complicated than that. The expectation is the conditional expectation given $t$, which cannot be chosen independently of $c_t$. And Cochrane's clarification "where $c_t$ denotes consumption at date $t$." certainly suggests that $c_{t+1}$ denotes consumption at $t+1.$ At the next page, he even writes "Consumption $c_{t+1}$ is also random." which clarifies that he talks about the value. I really don't see what is controversial about my statement that it does not make sense.
Michael Greinecker
<asset-pricing><expected-utility>
Why utility rather than expected utility in Cochrane's "Asset Pricing"?
Richard Hardy
# Why utility rather than expected utility in Cochrane's "Asset Pricing"?

(Asked by: username_0 on 2021-02-22)

Cochrane "Asset Pricing" [Chapter 1](http://assets.press.princeton.edu/chapters/s7836.pdf) 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?

**Tags:** asset-pricing, expected-utility

> **Comments:**
> **username_1**: 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.  
> **username_0**: 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.  
> **username_1**: The theory here uses expected utility. Not all authors are equally careful when it comes to formal details.  
> **username_2**: I agree that this comes to the stylistic choice, but I also think previous comment is bit too harsh saying it does not make sense. It is like saying using sing $2 \cdot  5$ and calling it a *product of integer* is more/less correct than using 10 and calling it *integer*. To be clear most of the literature would use the RHS and write it as $E_t[U]$ instead of LHS $U(u_t,E_t(u_{t+1}))$, but due to expectations being applied only to the argument it could be called just utility with $u_{t+1}$ being expected utility even when it’s completely equivalent to having expected utility of whole function  
> **username_0**: @username_2, 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.  
> **username_2**: @username_0 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.  
> **username_0**: @username_2, I must agree. Thank you for a thoughtful comment! I think you could post it as an answer.  
> **username_1**: @username_2 The normal way to read $U(c_t,c_{t+1})$ is that $U$ is a function of two numbers, $c_t$ and $c_{t+1}$. Indeed, Cochrane write that utility  is "defined over current and future values of consumption" But $U$ does not just depend on the current and future values of consumption, it depends on the current value and the conditional expaction over all possible future values. You are not going to read that in a math paper.  
> **username_2**: @username_1 but in this case $U$ is clearly composite function, so here I think Cochrane is just bit sloppy as clearly the function is actually a composite one $U(u_t(c_t),E[u_{t+1}(c_{t+1})] )$  
> **username_1**: @username_2 It is more complicated than that. The expectation is the conditional expectation given $t$, which cannot be chosen independently of $c_t$. And Cochrane's clarification "where $c_t$ denotes consumption at date $t$." certainly suggests that $c_{t+1}$ denotes consumption at $t+1.$ At the next page, he even writes "Consumption $c_{t+1}$ is also random." which clarifies that he talks about the value. I really don't see what is controversial about my statement that it does not make sense.  

---

## 1 answers

### Answer by username_2 on 2021-02-22. Score: 4 | Accepted answer

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.

> **Comments:**
> **username_3**: 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 https://economics.stackexchange.com/q/22475/59 and my answer here https://economics.stackexchange.com/a/27378/59 .  
> **username_2**: @username_3 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  
> **username_3**: Yeah, conventions probably differs by field. Thanks!  

---

