So..I understand up to step 4..but then there are these things I dont get, to start with , it says on (5) that the utility function depended only on the ratios p1/w p2/w ?? why does it say that? Uitlity function doesnt depend on this ratios but on the result of question (4).
Then, if I consider that the indirect utility function depends on these ratios p1/w p2/w, and they are q1 and q2, then why the langrangian is set up as the square root of the sum of 1/q1 and 1/q2? why not q1+q2 , or p1/w+p2/w?
Cheers!
Given , in (4), because the result$$v(p_1, p_2, w) = \sqrt{\frac w {p_1}+\frac w {p_2}}$$ holds, we can say that the result of $v$ depends on 2 ratios $\frac {p_1} w$ and $\frac {p_2} w$ because the value of $v$ changes when these 2 ratios change, in (5). Of course, we can say that $v$ changes with the change of any one of the 3 variables $p_1,p_2 $ and $w$ but each of these changes affect the ratios. Also, we know that $v$ changes based on the reciprocal of the 2 variables. (5) then lets the ratios be 2 (I think you can say parametric) values $q_1 $ and $q_2$. This is how the utility function depend on the ratios.Perform the substitution and you get the constrained minimization problem with constraints.
$\text{min } v(q_1, q_2) = \sqrt{\frac 1 {\frac{w}{p_1}} + \frac 1 {\frac{w}{p_2}}} = \sqrt{\frac 1 {q_1} + \frac 1 {q_2}} \\\text{subject to } q_1x_1 + q_2x_2 = 1$
Take the above equations and then you can create the Langrangian for this optimization problem
$\mathcal{L}(q_1, q_2, \lambda, x_1, x_2) \\= v(q_1,q_2) - \lambda(\text{set constraint to 0 on one side of the constraint}) \\\sqrt{\frac 1 {q_1} + \frac 1 {q_2}}-\lambda(1-q_1x_1 - q_2x_2)$
I believe the FOC are intuitive since they are just simple differentiation.