<p>A formal assumption and twice ignoring Jensen's inequality can lead to the approximation.</p>
<p><strong>FORMAL ASSUMPTION</strong></p>
<p><span class="math-container">$$
{\rm Cov} \left[C_{j,t+1}^{-\sigma}, \,\frac{R_{t+1}}{P_{t+1}}\right] =0.
$$</span>
Technically, this does not mean that we assume that the two components are independent, only that their covariance is zero. They may exhibit non-linear dependence. The fact that we are looking at a non-linear function of consumption (reciprocal and raised to a power) may be seen as a supporting &quot;argument&quot;.</p>
<p><strong>TWICE IGNORING JENSEN'S INEQUALITY</strong></p>
<p>...by approximating
<span class="math-container">$$E_t\left[\frac{1}{C_{j,t+1}^{\sigma}}\right] \approx \frac{1}{\left[E_tC_{j,t+1}\right]^{\sigma}}.$$</span></p>
<p>Depending on the value of <span class="math-container">$\sigma$</span>, the one blind-eye may tend to offset the effect of the other blind-eye, or intensify it.</p>
<p>Accept these steps at your own peril.</p>

CC BY-SA 4.0
2022-09-07T13:11:03.550Z
Thank you so much, very well explained.
Kwame Brown
CC BY-SA 4.0
2022-09-07T04:01:13.580Z
2022-09-07T04:01:13.580Z
Alecos Papadopoulos
<p>I am working through a textbook that derives the following first order
(Euler) equation between periods <span class="math-container">$t$</span> and <span class="math-container">$t+1$</span>. We end up with:
<span class="math-container">$$
-C_{j,t}^{-\sigma}+\beta E_{t}\left\{ \left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)\left[\left(1-\delta\right)P_{t+1}+R_{t+1}\right]\right\} =0
$$</span>
The variables carry the usual meanings, as <span class="math-container">$P,R$</span> and <span class="math-container">$C$</span> represent
prices, interest rates and consumption respectively, and <span class="math-container">$t$</span> indexes
time. Next period prices and interest rates are unknown, leading to
the expectations operator. The author then simplifies this to:
<span class="math-container">$$
\left(\frac{E_{t}C_{j,t+1}}{C_{j,t}}\right)^{\sigma}=\beta\left[\left(1-\delta\right)+E_{t}\left(\frac{R_{t+1}}{P_{t+1}}\right)\right]
$$</span>
This seems to me sloppy use of the expectations operator without any
additional assumptions given. For instance, simplying the first expression,
one gets:
<span class="math-container">\begin{align*}
C_{j,t}^{-\sigma} &amp; =\beta E_{t}\left\{ \left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)\left[\left(1-\delta\right)P_{t+1}+R_{t+1}\right]\right\} \\
 &amp; =\beta E_{t}\left(\left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)\left(1-\delta\right)P_{t+1}\right)+\beta E_{t}\left(\left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)R_{t+1}\right)\\
 &amp; =\beta\left(1-\delta\right)E_{t}\left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)+\beta E_{t}\left(\left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)R_{t+1}\right)
\end{align*}</span>
From here, there is no more room for simplification unless one makes
restrictive (and unrealistic) assumptions on the nature of the random
variables at <span class="math-container">$t+1.$</span> In particular, consumption tomorrow will depend
on relative prices and interest rates, so we cannot use the fact that
<span class="math-container">$E[XY]=E[X]E[Y].$</span> How can we simplify the above expression to obtain
what the author has done? Thank you for your time.</p>

CC BY-SA 4.0
2022-09-07T16:47:22.517Z
This is the first time I have seen this weird simplification, Would be interesting to know what textbook you are referring to. Alecos answer, as good as it is, shows you what assumptions you would need to make to get there. But, as Alecos says, it doesn't tell you whether you should make these assumptions, or even if they are sensible. That's your call.
BrsG
<optimization>
Simplfying Euler equation under expectations
Kwame Brown
# Simplfying Euler equation under expectations

(Asked by: username_0 on 2022-09-06)

I am working through a textbook that derives the following first order
(Euler) equation between periods $t$ and $t+1$. We end up with:
$$
-C\_{j,t}^{-\sigma}+\beta E\_{t}\left\{ \left(\frac{C\_{j,t+1}^{-\sigma}}{P\_{t+1}}\right)\left[\left(1-\delta\right)P\_{t+1}+R\_{t+1}\right]\right\} =0
$$
The variables carry the usual meanings, as $P,R$ and $C$ represent
prices, interest rates and consumption respectively, and $t$ indexes
time. Next period prices and interest rates are unknown, leading to
the expectations operator. The author then simplifies this to:
$$
\left(\frac{E\_{t}C\_{j,t+1}}{C\_{j,t}}\right)^{\sigma}=\beta\left[\left(1-\delta\right)+E\_{t}\left(\frac{R\_{t+1}}{P\_{t+1}}\right)\right]
$$
This seems to me sloppy use of the expectations operator without any
additional assumptions given. For instance, simplying the first expression,
one gets:
\begin{align\*}
C\_{j,t}^{-\sigma} & =\beta E\_{t}\left\{ \left(\frac{C\_{j,t+1}^{-\sigma}}{P\_{t+1}}\right)\left[\left(1-\delta\right)P\_{t+1}+R\_{t+1}\right]\right\} \\
& =\beta E\_{t}\left(\left(\frac{C\_{j,t+1}^{-\sigma}}{P\_{t+1}}\right)\left(1-\delta\right)P\_{t+1}\right)+\beta E\_{t}\left(\left(\frac{C\_{j,t+1}^{-\sigma}}{P\_{t+1}}\right)R\_{t+1}\right)\\
& =\beta\left(1-\delta\right)E\_{t}\left(\frac{C\_{j,t+1}^{-\sigma}}{P\_{t+1}}\right)+\beta E\_{t}\left(\left(\frac{C\_{j,t+1}^{-\sigma}}{P\_{t+1}}\right)R\_{t+1}\right)
\end{align\*}
From here, there is no more room for simplification unless one makes
restrictive (and unrealistic) assumptions on the nature of the random
variables at $t+1.$ In particular, consumption tomorrow will depend
on relative prices and interest rates, so we cannot use the fact that
$E[XY]=E[X]E[Y].$ How can we simplify the above expression to obtain
what the author has done? Thank you for your time.

**Tags:** optimization
> **Comments:**
> **username_1**: This is the first time I have seen this weird simplification, Would be interesting to know what textbook you are referring to. Alecos answer, as good as it is, shows you what assumptions you would need to make to get there. But, as Alecos says, it doesn't tell you whether you should make these assumptions, or even if they are sensible. That's your call.  

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## 1 answers

### Answer by username_2 on 2022-09-07. Score: 3 | Accepted answer

A formal assumption and twice ignoring Jensen's inequality can lead to the approximation.

**FORMAL ASSUMPTION**

$$
{\rm Cov} \left[C\_{j,t+1}^{-\sigma}, \,\frac{R\_{t+1}}{P\_{t+1}}\right] =0.
$$
Technically, this does not mean that we assume that the two components are independent, only that their covariance is zero. They may exhibit non-linear dependence. The fact that we are looking at a non-linear function of consumption (reciprocal and raised to a power) may be seen as a supporting "argument".

**TWICE IGNORING JENSEN'S INEQUALITY**

...by approximating
$$E\_t\left[\frac{1}{C\_{j,t+1}^{\sigma}}\right] \approx \frac{1}{\left[E\_tC\_{j,t+1}\right]^{\sigma}}.$$

Depending on the value of $\sigma$, the one blind-eye may tend to offset the effect of the other blind-eye, or intensify it.

Accept these steps at your own peril.

> **Comments:**
> **username_0**: Thank you so much, very well explained.  

---

