@Giskard i just realized what it's wrong. I thought that the implicit theorem function will show a different relationship implied by the third first order condition but it does'nt

@Giskard could you briefly explain what is incorrect? we have an optimization problem with 3 endogenous variables and 1 restriction, so we have 4 first order condition. I only substituted one in the other, until I reached a system of 2 equations with 2 endogenous variables

And aswering your question @asvecon, you are right, $\beta$ appear only in the first equation, But $h$ appear in both, remember that $c_{1}=R_{c_{1}}= y-h + \frac{ w(h)}{1+r} - \frac{ c_{2}}{1+r}$, so $R_{c_{1}}$ is a function of h. So $\beta$ affect $h$.

Now, if $ h \neq r $ this means that there is no perfect mobility between the two types of investment or the two uses for your savings. If there were perfect mobility, nobody would invest in the market that returns $ r $ or $ h $, depending on which one was less. So I assume that this agent has access to a possibly international market that is not integrated into international markets. In addition, the rate that is higher is what all your savings will go to.

Yes, you are right @Giskard, you can save at r competitive. But that does not change that to invest you have to save; h is "invested" or "saved" (you have to reduce $ c_ {1} $ to invest in $ h $ as it is a two-period model only) in competitive markets otherwise the payment function would be $ w (h, w) $, since the size of your investment would affect the return of $ h $. Almost nothing changes from what I said in the answer. But i forgot that you can invest in two assets.

Ha! i just realized that you solved the problem, but i hope that you may find my answer valuable anyway.

$h$ appears in two first order conditions, you need to solve them to see of what depends $h$

If it's not clear that the dependence exists. You can just substitute the restriction on the utility function for some $c_{i}$ and derivate for $h$ and $c_{2}$, and it's more clear that a dependence exists. Or just choose some utility function and solve it.

Yes, because you save to invest in h or to consume in period two.