Suppose that a certain fixed subset of a society consists of identical individuals who are willing to commit crimes $($crime elements$)$ under some circumstances. Each individual i in this subset must choose a share of his time si to devote to crime. Doing so leads to benefit of B$($si$),$ but the expected cost is a penalty F times the probability of being caught p$($si$,$ s$),$ where s is the average of time devoted to crime by members of this criminal elements $($s$=$E$($s$)).$ Assume that

B$\text{'}>0,$ B$"0,$

$>0.>0,$ and $<0.$

a$)$ Why would one expect that $>0,$ and a$2$p $<0?$ Discuss these two assumptions in terms of optimal responses by police.

$(6$ marks$)$

b$)$ Prove that a given criminal commits more crime when the aggregate crime rate $($S$)$ is higher.

$(7$ marks$)$

c$)$ From the point of view of criminals $($ignoring the costs to crime victims$),$ is the aggregate crime rate Pareto efficient? If so$,$ provide a proof. If not, discuss whether the criminals could create rules and$/$or institutional arrangements that could improve their expected outcomes $($again$,$ without regard to their victims$).$

$(6$ marks$)$