Euler equation guess and verify So I have the Euler condition written as follows: $$\frac{1}{k_t^a-k_{t+1}}=\frac{a\beta k_{t+1}^{a-1}}{k_{t+1}^a-k_{t+2}}$$
and it says that $k_{t+1}$ takes the form $gk_t^a$, where g is an unknown to be determined. I know the result is $k_{t+1}=a\beta k_t^a$, but don't know how the derivation goes to reach this result.
From the hint, I got that $$\frac{k_{t+2}-k_{t+1}^a}{k_{t+1}-k_t^a}=a\beta k_{t+1}^{a-1}$$
So I guess the LHS is g=$a\beta k_{t+1}^{a-1}$?
Am I doing it right or not?
Hint:
Note that
\begin{equation}
k_{t+1}=gk_t^a \quad\Rightarrow\quad g=\frac{k_{t+1}}{k_t^a}=\frac{k_{t+2}}{k_{t+1}^a}.
\end{equation}
Rearrange your Euler condition into a form such that you can use $g$ to sub-out the above ratios. Then it should be straightforward to solve for $g$.