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

Abstract

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.

Appendix

In the appendix, I consider an extended model of incorporating the private benefit of the offender into social welfare and demonstrate the non-maximality of the optimal fine.

The social welfare function can be defined as

$$ W=\int\limits_{p(m, r^{e})f}^{\infty}(b-h) g(b)db -m. $$

(9)

Note that the equilibrium offense rate still satisfies Eq. (4).

Now, the partial derivative of (9) with respect to f yields

$$ \frac{\partial W}{\partial f}=(h-pf)g(pf)\left(p+p_2 \frac{\partial r^{*}}{\partial f}f\right)=\frac{(h-pf)gp}{1+gfp_{2}} $$

(10)

by using (5). Then, since h > pf, ∂W/∂f < 0 if 1 + gfp ₂ < 0. This implies that lowering the fine increases the social welfare.