I am working on labor demand models where firms have to choose the optimal level of employment by maximizing profits. In particular, I have faced the following problem:
Maximize with respect to $l$ the following function with $w$ and $A_h>A_l$ constants:
$$\Pi=A_h\log(l)-wl-max\{0,c\left(l-\dfrac{A_l}{w}\right)\}$$
Is there any analytical approach (direct argument) to solve this problem or should it be solved by inspection (by cases)?
I HAVE TRIED:
If $\dfrac{A_h}{w+c}<\dfrac{A_l}{w}<\dfrac{A_h}{w}$ then $l^*=\dfrac{A_l}{w}$
If $\dfrac{A_l}{w}<\dfrac{A_h}{w+c}$ then $l^*=\dfrac{A_h}{w+c}$
If $\dfrac{A_l}{w}>\dfrac{A_h}{w}$ then it violates the assumption that $A^h>A^l$.
So a robust method to go about this is by using constraints. as in maximize the two problems, $$A_{h}log(l)−wl-\lambda (l-\frac{A_l}{w})$$ $$A_{h}log(l)−wl-(1-\lambda)(l-\frac{A_l}{w})$$
But this is just like cases, and you get the same answer you got. I guess I would like a more specific description of where help is required because yo have a complete characterization.