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

" β " transformation.

Jordi Gali book, page 42 enter image description here

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.

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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

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

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  • $\begingroup$ I don't understand what you said ? ıs there any problem in my post? $\endgroup$ – Engin YILMAZ Jan 21 '17 at 19:28
  • $\begingroup$ Please ask a self-contained question that lives without external references. $\endgroup$ – FooBar Jan 21 '17 at 19:41
  • $\begingroup$ Ok. I will delete external references.. $\endgroup$ – Engin YILMAZ Jan 21 '17 at 19:44
  • $\begingroup$ Denesp, now I understood. Sorry for my fault. $\endgroup$ – Engin YILMAZ Jan 21 '17 at 20:41
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    $\begingroup$ 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. $\endgroup$ – Chris tie Jan 22 '17 at 12:59
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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.

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  • $\begingroup$ 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" $\endgroup$ – Engin YILMAZ Jan 22 '17 at 11:58
  • $\begingroup$ Dear Alessandro, I added mathematical expressions to my question. $\endgroup$ – Engin YILMAZ Jan 22 '17 at 12:20
  • $\begingroup$ I've edited my answer. Hope it is clear now $\endgroup$ – PhDing Jan 22 '17 at 13:41
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    $\begingroup$ @Alessandro - good answer. However, you should edit to account for time t expectations over next period vars. $\endgroup$ – 123 Jan 22 '17 at 15:18
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    $\begingroup$ @Alessandro You are detailed man ! Thanks for your help ! $\endgroup$ – Engin YILMAZ Jan 23 '17 at 13:49

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