<p>There is no trick in that Euler equation. In the New Keynesian model, the Euler equation for consumption is derived from the first order condition for $B_t$, the bond holding. You have to pay attention to the fact that bonds appear in the budget constraint at two moments in time $t$ and $t+1$. Thus you have to compute the foc as
\begin{gather}
B_t:\beta^t\lambda_tQ_t-\beta^{t+1}\lambda_{t+1}=0
\end{gather}
Then substituting $\lambda$ using the foc for $C_t$ and simplifying the betas, you obtain
\begin{gather}
Q_t=\beta E_t\{\frac{U_{c_{t+1}}}{U_{c_t}}\frac{P_t}{P_{t+1}}\}=0
\end{gather}</p>

<p><strong>EDIT</strong>: In the NK model, household maximizes its utility function choosing $C_t$, $N_t$ and $B_t$. Thus, you have to compute the FOC for these three variables, i.e.
\begin{align}
B_t:&amp;\beta^t\lambda_tQ_t-\beta^{t+1}E_t\{\lambda_{t+1}\}=0 \\
C_t:&amp;\beta^tU_{c,t}-\lambda_tP_t=0 \\
N_t:&amp;\beta^tU_{n,t}-\lambda_tW_t=0
\end{align}</p>

<p>In order to compute the consumption Euler equation, you need to substitute $\lambda_t$ from the $C_t$ foc into the $B_t$ foc. Then rearranging and simplifying $\beta$ as $\beta^{t+1}/\beta^t=\beta$ you get the equation you wrote above. </p>

<p>You get only one $\beta$ because it is constant over time, it is not index by $t$ as the other variables, it is a constant. Thus whatever time is $C$, you will have the same $\beta$. I think it is well explained in Gali too.</p>

CC BY-SA 3.0
2017-01-22T11:58:14.523Z
Dear Alessandro, Can you write solution step by step ? Especially your explanation "Then substituting λ using the foc for CtCt and simplifying the betas, you obtain"
Engin YILMAZ
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2017-01-22T12:20:05.583Z
Dear Alessandro, I added mathematical expressions to my question.
Engin YILMAZ
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2017-01-22T13:41:14.353Z
I've edited my answer. Hope it is clear now
PostDocing
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2017-01-22T15:18:46.807Z
@Alessandro - good answer. However, you should edit to account for time t expectations over next period vars.
123
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2017-01-22T15:23:02.367Z
@123 Thanks for making me notice that ;)
PostDocing
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2017-01-22T18:27:47.140Z
@Alessandro, forgot Pt en second FOC. Please can you write "simplifying the betas", is that mean beta=beta (t+1) / beta(t)
Engin YILMAZ
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2017-01-23T08:30:28.483Z
@EnginYILMAZ Thank you, edited. Yes, I mean that since $\beta$ is the same in every moment in time, you can simplify it as you written
PostDocing
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2017-01-23T10:59:22.717Z
Ciao @Alessandro My question is assumption of "In the baseline calibrations of the model’s preference parameters it is assumed β = 0.99" and for this (0.99) need to βt+1 is less than βt. Is it possible ?
Engin YILMAZ
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2017-01-23T12:23:00.243Z
@EnginYILMAZ $\beta$ is a constant parameter, thus it is always equal to 0.99. The computation would be $0.99^{t+1}/0.99^t$ that is equal to 0.99, i.e. your $\beta$
PostDocing
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2017-01-23T12:48:02.963Z
@Alessandro t is "power of variable β" or "time index of variable β" ?
Engin YILMAZ
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2017-01-23T13:04:37.763Z
@EnginYILMAZ Power! It just means that if you are discounting utility $t$ period on, you have to rise your discount factor to the power of $t$
PostDocing
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2017-01-23T13:49:25.823Z
@Alessandro You are detailed man ! Thanks for your help !
Engin YILMAZ
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2017-01-22T11:05:39.327Z
2017-01-23T08:29:29.043Z
2017-01-23T08:29:29.043Z
PostDocing
<p>Derivation euler equation in gali book, I don't understand </p>

<p>" β " transformation.</p>

<p>Jordi Gali book, page 42
<a href="https://i.sstatic.net/PYceI.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/PYceI.png" alt="enter image description here"></a></p>

<p>There is no explanation gali book</p>

<p>the notes which are prepared by Drago Bergholt (Page 6)</p>

<p>explain FOC for "Ct"  (2.13)
and
(2.18) explain Euler equation</p>

<p>Writer uses FOC for "Ct" and FOC for "Ct+1" to form euler.</p>

<p><a href="https://i.sstatic.net/v70QA.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/v70QA.png" alt="enter image description here"></a></p>

<p><a href="https://i.sstatic.net/XzvjP.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/XzvjP.png" alt="enter image description here"></a></p>

<p><a href="https://i.sstatic.net/2MGhp.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/2MGhp.png" alt="enter image description here"></a></p>

<p>and I expect to different " β " for "Ct"  and for "Ct+1" in (2.18)
But there is only one " β " in (2.18)
<a href="https://i.sstatic.net/8K5KG.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/8K5KG.png" alt="enter image description here"></a></p>

<p>My question is assumption of "In the baseline calibrations of the model’s preference parameters it is assumed β = 0.99" and
for this (0.99) need to βt+1 is less than βt. Is it possible ?</p>

<p>Sincerely</p>

CC BY-SA 3.0
2017-01-21T19:28:34.113Z
I don't understand what you said ?  ıs there any problem in my post?
Engin YILMAZ
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2017-01-21T19:41:28.533Z
Please ask a self-contained question that lives without external references.
FooBar
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2017-01-21T19:44:41.173Z
Ok. I will delete external references..
Engin YILMAZ
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2017-01-21T20:41:48.480Z
Denesp, now  I understood. Sorry for my fault.
Engin YILMAZ
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2017-01-22T12:59:38.010Z
Sorry I don't understand what you are asking. Please elaborate a bit more. Or are you asking why we can simplify $\beta=\beta^{t+1}/\beta_t$? In this case, the answer is that $\beta$ is a constant and can therefore be simplified.
Chris tie
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2017-01-22T18:35:13.307Z
@Christie Dear Chris; β=βt+1/βt we are agree ? You said that this is simplification. My question is assumption of "In the baseline calibrations of the model’s preference parameters it is assumed β = 0.99" and and need to βt+1 is less than βt. Is it possible ?
Engin YILMAZ
<macroeconomics><new-keynesian-economics>
Jordi Gali Euler Equation Beta
Engin YILMAZ
# Jordi Gali Euler Equation Beta

(Asked by: username_0 on 2017-01-21)

Derivation euler equation in gali book, I don't understand

" β " transformation.

Jordi Gali book, page 42
[![enter image description here](https://i.sstatic.net/PYceI.png)](https://i.sstatic.net/PYceI.png)

There is no explanation gali book

the notes which are prepared by Drago Bergholt (Page 6)

explain FOC for "Ct" (2.13)
and
(2.18) explain Euler equation

Writer uses FOC for "Ct" and FOC for "Ct+1" to form euler.

[![enter image description here](https://i.sstatic.net/v70QA.png)](https://i.sstatic.net/v70QA.png)

[![enter image description here](https://i.sstatic.net/XzvjP.png)](https://i.sstatic.net/XzvjP.png)

[![enter image description here](https://i.sstatic.net/2MGhp.png)](https://i.sstatic.net/2MGhp.png)

and I expect to different " β " for "Ct" and for "Ct+1" in (2.18)
But there is only one " β " in (2.18)
[![enter image description here](https://i.sstatic.net/8K5KG.png)](https://i.sstatic.net/8K5KG.png)

My question is assumption of "In the baseline calibrations of the model’s preference parameters it is assumed β = 0.99" and
for this (0.99) need to βt+1 is less than βt. Is it possible ?

Sincerely

**Tags:** macroeconomics, new-keynesian-economics
> **Comments:**
> **username_0**: I don't understand what you said ?  ıs there any problem in my post?  
> **username_1**: Please ask a self-contained question that lives without external references.  
> **username_0**: Ok. I will delete external references..  
> **username_0**: Denesp, now  I understood. Sorry for my fault.  
> **username_2**: Sorry I don't understand what you are asking. Please elaborate a bit more. Or are you asking why we can simplify $\beta=\beta^{t+1}/\beta_t$? In this case, the answer is that $\beta$ is a constant and can therefore be simplified.  
> **username_0**: @Christie Dear Chris; β=βt+1/βt we are agree ? You said that this is simplification. My question is assumption of "In the baseline calibrations of the model’s preference parameters it is assumed β = 0.99" and and need to βt+1 is less than βt. Is it possible ?  

---

## 1 answers

### Answer by username_3 on 2017-01-22. Score: 1 | Accepted answer

There is no trick in that Euler equation. In the New Keynesian model, the Euler equation for consumption is derived from the first order condition for $B\_t$, the bond holding. You have to pay attention to the fact that bonds appear in the budget constraint at two moments in time $t$ and $t+1$. Thus you have to compute the foc as
\begin{gather}
B\_t:\beta^t\lambda\_tQ\_t-\beta^{t+1}\lambda\_{t+1}=0
\end{gather}
Then substituting $\lambda$ using the foc for $C\_t$ and simplifying the betas, you obtain
\begin{gather}
Q\_t=\beta E\_t\{\frac{U\_{c\_{t+1}}}{U\_{c\_t}}\frac{P\_t}{P\_{t+1}}\}=0
\end{gather}

**EDIT**: In the NK model, household maximizes its utility function choosing $C\_t$, $N\_t$ and $B\_t$. Thus, you have to compute the FOC for these three variables, i.e.
\begin{align}
B\_t:&\beta^t\lambda\_tQ\_t-\beta^{t+1}E\_t\{\lambda\_{t+1}\}=0 \\
C\_t:&\beta^tU\_{c,t}-\lambda\_tP\_t=0 \\
N\_t:&\beta^tU\_{n,t}-\lambda\_tW\_t=0
\end{align}

In order to compute the consumption Euler equation, you need to substitute $\lambda\_t$ from the $C\_t$ foc into the $B\_t$ foc. Then rearranging and simplifying $\beta$ as $\beta^{t+1}/\beta^t=\beta$ you get the equation you wrote above.

You get only one $\beta$ because it is constant over time, it is not index by $t$ as the other variables, it is a constant. Thus whatever time is $C$, you will have the same $\beta$. I think it is well explained in Gali too.

> **Comments:**
> **username_0**: Dear Alessandro, Can you write solution step by step ? Especially your explanation "Then substituting λ using the foc for CtCt and simplifying the betas, you obtain"  
> **username_0**: Dear Alessandro, I added mathematical expressions to my question.  
> **username_3**: I've edited my answer. Hope it is clear now  
> **username_4**: @Alessandro - good answer. However, you should edit to account for time t expectations over next period vars.  
> **username_3**: @123 Thanks for making me notice that ;)  
> **username_0**: @Alessandro, forgot Pt en second FOC. Please can you write "simplifying the betas", is that mean beta=beta (t+1) / beta(t)  
> **username_3**: @EnginYILMAZ Thank you, edited. Yes, I mean that since $\beta$ is the same in every moment in time, you can simplify it as you written  
> **username_0**: Ciao @Alessandro My question is assumption of "In the baseline calibrations of the model’s preference parameters it is assumed β = 0.99" and for this (0.99) need to βt+1 is less than βt. Is it possible ?  
> **username_3**: @EnginYILMAZ $\beta$ is a constant parameter, thus it is always equal to 0.99. The computation would be $0.99^{t+1}/0.99^t$ that is equal to 0.99, i.e. your $\beta$  
> **username_0**: @Alessandro t is "power of variable β" or "time index of variable β" ?  
> **username_3**: @EnginYILMAZ Power! It just means that if you are discounting utility $t$ period on, you have to rise your discount factor to the power of $t$  
> **username_0**: @Alessandro You are detailed man ! Thanks for your help !  

---

