Maximizing $q_1+q_2$ subject to $x_{11}q_1^b+x_{21}q_2^b\leq k_1$ and $x_{12}q_1^b+x_{22}q_2^b\leq k_2$. How thorough do I need to be in my argument?

Question

This is more about what I have to do than about how to do it.

I have the following maximization problem:

$$ \begin{array}{ll} \text{maximize}&q_1+q_2\\ \text{subject to}& x_{11}q_1^\beta+x_{21}q_2^\beta\leq k_1\\ & x_{12}q_1^\beta+x_{22}q_2^\beta\leq k_2\\ & q_1\geq 0\\ & q_2\geq 0\\ \end{array} $$ where $x_{11},x_{22},k_1,k_2\geq0$, $x_{12},x_{21}>0$ and $\beta>1$.

Under these conditions, the slope of the constraints can be shown to be strictly decreasing from $0$ to $-\infty$. I know that there are different types of solutions to this problem depending on the values of $k_1$ and $k_2$.

1. Do I need to prove that at least one of the two first constraints is binding or is it enough to note that the feasible set is compact?
2. When just one of the constraints is binding, do I need to prove that the slope of that constraint at the optimum is $-1$?
3. When both constraints are binding, two situations can arise (with respect to the slope of the constraints at the optimum): either one of the slopes is $-1$ and the other is different from $-1$ or one is larger and the other smaller than $-1$. Does any this need to shown or is it enough to enunciate it?
4. Can I argue that if the optimum is such that the slopes of the constraints are different from $-1$ then both constraints need to be binding?

Answer 1

Let $\gamma_{1}$ be the curve given by $x_{11}q_{1}^{\beta}+x_{21}q_{2}^{\beta}=k_{1}$ and let $\gamma_{2}$ be the curve given by $x_{12}q_{1}^{\beta}+x_{22}q_{2}^{\beta}=k_{2}$.

The function $f(q_{1},q_{2})=q_{1}+q_{2}$ does not have points for which the partial derivatives with respect to $q_{1}$, $q_{2}$ are both zero, so the the maximum and minimum of $f$ are attained on the boundary of the region defined by your constraints.

At this point you can use Lagrange multipliers to solve your problem, or just find the maximum and minimum of $f$ directly using single variable calculus on the three, or four parts of the boundary.

(If $\gamma_{1}$ and $\gamma_{2}$ intersect, then the boundary consists of the line given by $q_{1}=0$, the line given by $q_{2}=0$ and the curves $\gamma_{1}$ and $\gamma_{2}$. If $\gamma_{1}$ and $\gamma_{2}$ do not intersect, then the boundary consists of the line given by $q_{1}=0$, the line given by $q_{2}=0$ and the curve $\gamma_{1}$, or the curve $\gamma_{2}$).

I hope this helps!