I have a utility function given by,
$$U(c_1,c_2)=c^{\alpha}_1+\beta c^{\alpha}_2$$
I found that the budget constraint is,
$$c_2=m_2+\left(\frac{1+i}{1+\pi}\right)(m_1-c_1)$$
Where $m_1,m_2$ is the income in period $c_1,c_2$, and i is the interest (the person can borrow and save at the same interest, $i$).
I'm now struggling to find the 'relative consumption' in the two periods $c_2/c_1$, where $a$ and $i$ occur (I have to substitute them for $\sigma =\frac{1}{1-a}$ and $1+r≡(1+i)(1+π)$ in the following question).
I've tried using the Langragian for maximizing the utility under the budget constraint, but I am still very puzzled by this question.
Any help?