A note on the non-maximality of the optimal fines when the apprehension probability depends on the offense rate

This paper reconsiders the problem of optimal law enforcement when the apprehension probability depends on the offense rate as well as policing expenditures. A natural consequence of such an apprehension probability is the possible multiplicity of the equilibrium due to strategic complementarity, and the actual offense rate is realized by the self-fulfilling nature of the offense rate. If people believe that lowering the fine will lead to a lower offense rate, each individual will be less inclined to commit an illegal activity due to their expectation of a higher apprehension probability. Thus, the maximal fine is not socially optimal in this case.