Consider a crime that, if committed, would impose a cost of $$100$ on the victim. Suppose that there are two potential offenders, one who gains $$40$ from the crime and one who gains $$80.$ Assume that the probability of apprehension is fixed at $\mathrm{.}5.|($a$)$ Is it efficient for either of the offenders to commit the crime? $($b$)$ Suppose the fine is set equal to the actual harm $($that is$,$ f $=$ $$100).$ Will either of the offenders commit the crime? $($c$)$ Calculate the optimal harm$-$based fine and show that it achieves the efficient outcome. $($d$)$ Calculate the optimal benefit$-$based fine for each offender and show that they also achieve the efficient outcome. $($Assume that when indifferent, offenders do not commit crimes.$)$