I have an Euler equation $ (\frac{ c_ {t+1} }{c_t})^\sigma = \beta (1+r)$,where $c_t$ is the t period consumption,r is interest rate and $\beta$ is discount rate, and a budget constraint $\sum _{t=0}^\infty \frac{c_t}{(1+r)^t}=\sum _{t=0}^\infty \frac{y_t}{(1+r)^t}$, where $y_t$ is income.
My question is how to substitute Euler equation into budget constraint to get the following equation?
$c_t\sum_{j=0}^\infty\frac{[\beta(1+r)]^\frac{j}{\sigma}}{(1+r)^j}=\sum _{t=0}^\infty \frac{y_t}{(1+r)^t}$
The hint is to use the recursive Euler equation to express all future consumptions $c_{t+j}$ as a function of current consumption $c_t$, and to play with the indexes in the sum of the left.
Hint: Separate the consumption time periods $$\begin{align} \left(\frac{ c_ {t+1} }{c_t}\right)^\sigma & = \beta (1+r) \\ \frac{ c_ {t+1} }{c_t} & = \left[\beta (1+r) \right]^\frac{1}{\sigma} \\ c_ {t+1} & = c_t \left[\beta (1+r) \right]^\frac{1}{\sigma} \\ \end{align}$$
Notice that $(\frac{ c_ {t+2} }{c_{t+1}})^\sigma = \beta (1+r)$. So,
$$c_ {t+2} = c_{t+1} \left[\beta (1+r) \right]^\frac{1}{\sigma} = c_{t} \left[\beta (1+r) \right]^{\frac{1}{\sigma} \cdot 2}$$
So
$$\sum _{t=0}^\infty \frac{c_t}{(1+r)^t}$$
can be expressed as a geometric series and simplified. Working that out for yourself should give you the desired answer.
