When we take our Lagrangian and we include non-negativity constraints. If a variable $x = 0$ do we take FOC first or set $x=0$ first?
E.g.
$Max \; L(x, y, λ) = f(x,y) - λ_1(g(x,y) - k) - λ_x(-x) - λ_y(-y)$
Or alternately written as :
$Max \;(x, y, λ) = f(x,y) + λ_1(k - g(x,y)) +λ_x(x) + λ_y(y)$
In the case where one of our variables is actually = 0, E.g. $x=0$ hence $λ_x > 0$
- Do we differentiate first getting the FOC.
$\frac{\partial d}{\partial x} = f_x - λ_1g_x + λ_x \le 0$
- Or do we take $x = 0$ first and then derive our FOC.
$\frac{\partial d}{\partial x} = f_x - λ_1g_x\le 0$
The reason i ask is that i assumed it would be the first way. Because otherwise what is the point of the Lagrange multiplier for the non-negativity constraint. Either it's 0 because $x > 0$ or it disappears because we get $λ(0) = 0)$ in the Lagrangian.
The mark scheme for this question I have asked here would suggest it's actually the second approach? This tiny thing ruined the rest of my analysis, as the following questions relied on there being no $λ_κ$.
This problem is really bugging me as exam commentaries are incredibly vague as to when they subtly do and don't use non-negativity constraints. Please let me know your thoughts, thanks!