Penalty and minimization of a social cost

Question

It is part of broader question in economics however it is about minimizing the expression (which depicts social cost of a crime): the expression is $$\min \left\{x+[c+p(x)wy]\left[1-\frac{p(x)y}{y^{\max}} \right]\right\}$$ I need to find a minimum value of $x$ knowing that $w=0$. I do not know whether it is possible if it is not i will obviously delete this question considering it irrelevant.

Answer 1

When $w=0$, the function is $$x+c-c\frac{y}{y^{max}}p(x)$$

Assuming everything else is constant with respect to $x$ except $p(x)$ then the whole function is convex as long as $p(x)$ is concave. In this case, the first order condition is sufficient for a minimum. This is:

$$1=c\frac{y}{y^{max}}p'(x)$$

$$p'(x)=\frac{1}{c}\frac{y^{max}}{y}$$

It's hard to interpret this without knowing more about the problem.