Jordi Gali Euler Equation Beta Derivation euler equation in gali book, I don't understand 

" β " transformation.

Jordi Gali book, page 42


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.







and I expect to different " β " for "Ct"  and for "Ct+1" in (2.18)
But there is only one " β " in (2.18)


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
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:&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}

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.