<p>If you set the partial derivative of (OA5) wrt <span class="math-container">$y_i$</span> to zero, the division by 2 can be simplified, and disappears:
<span class="math-container">$$ \frac{ \partial v}{\partial y_i}(y_i,y, a_i,\omega)=0 \iff v_1 + v_{11}y_i+v_{12}y+v_{13}a_i+v_{14}\omega = 0. $$</span>
When you solve this equation in <span class="math-container">$y_i$</span>, you will find the optimal level  <span class="math-container">$y_i^*$</span>, almost as given in (OA6), but the constant term <span class="math-container">$v_1$</span> is actually missing there.</p>

CC BY-SA 4.0
2023-02-23T20:40:46.047Z
Thanks @Bertrand! By the way could you give me more details? So how does the division by2 disappear?
user14261785
CC BY-SA 4.0
2023-02-23T21:24:00.873Z
@user14261785: I have added the first order condition for an inner maximum.
Bertrand
CC BY-SA 4.0
2023-02-24T01:55:44.913Z
I just saw your edited answer. Thanks!
user14261785
CC BY-SA 4.0
2023-02-23T20:15:30.620Z
2023-02-23T21:23:03.917Z
2023-02-23T21:23:03.917Z
Bertrand
<p>I have a very simple question. <a href="https://i.sstatic.net/TMXCM.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/TMXCM.jpg" alt="enter image description here" /></a></p>
<p>I don't understand OA6 equation. Where is the 2, the denominator of <span class="math-container">$\frac{v_{11}}{2}$</span>, gone?</p>

<log-linearization>
a question about second-order log-linear approximation
user14261785
# a question about second-order log-linear approximation

(Asked by: username_0 on 2023-02-23)

I have a very simple question. [![enter image description here](https://i.sstatic.net/TMXCM.jpg)](https://i.sstatic.net/TMXCM.jpg)

I don't understand OA6 equation. Where is the 2, the denominator of $\frac{v\_{11}}{2}$, gone?

**Tags:** log-linearization

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## 1 answers

### Answer by username_1 on 2023-02-23. Score: 2

If you set the partial derivative of (OA5) wrt $y\_i$ to zero, the division by 2 can be simplified, and disappears:
$$ \frac{ \partial v}{\partial y\_i}(y\_i,y, a\_i,\omega)=0 \iff v\_1 + v\_{11}y\_i+v\_{12}y+v\_{13}a\_i+v\_{14}\omega = 0. $$
When you solve this equation in $y\_i$, you will find the optimal level $y\_i^\*$, almost as given in (OA6), but the constant term $v\_1$ is actually missing there.

> **Comments:**
> **username_0**: Thanks @Bertrand! By the way could you give me more details? So how does the division by2 disappear?  
> **username_1**: @user14261785: I have added the first order condition for an inner maximum.  
> **username_0**: I just saw your edited answer. Thanks!  

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