This is the problem I've been given.
Assume the consumers’preference over current and future consumption is given by $U(c,c')=c^\frac{1}{2}*c'^\frac{1}{2}$.
Further assume there is no government.
1) solve for the optimal decisions: $c,c′$ and $s$ ?
This is as far as I have come
$\max U(c,c')=c^\frac{1}{2}c'^\frac{1}{2}$
s.t. $c+\frac{c'}{1+r}=y+\frac{y'}{1+r}$
$c, c',s \ge 0 $
$L=c^\frac{1}{2}*c'^\frac{1}{2}+λ(c+\frac{c'}{1+r}-y-\frac{y'}{1+r})$
(1)derv of $c$: $\frac{1}{2}c^{-\frac{1}{2}}*c'^\frac{1}{2}-λ=0 $
(2)derv of $c'$:$\frac{1}{2}c^\frac{1}{2} c'^{-\frac{1}{2}}-λ\frac{1}{1+r}=0$
(3)($λ$):$c+\frac{c'}{1+r}=y+\frac{y'}{1+r}$
Divide equation 1 by 2
$\frac{ \frac{1}{2}c^{-\frac{1}{2}}c'^\frac{1}{2}} {\frac{1}{2}c^\frac{1}{2}c'^{-\frac{1}{2}}}= \frac{λ}{\frac{λ}{1+r}}$ = $\frac{c'}{c}=1+r$
I would appreciate if someone could give some help on how to continue so I can find the optimal decisions for $c, c', s$