Crime, prosecutors, and the certainty of conviction

It has been laid down as a maxim on this subject, that the certainty of punishment is more effectual than the severity of punishment, in deterring offenders.

Leigh Hunt, in: ‘The Examiner’, No 365, 25 Dec. 1814, p. 829.

Abstract

This paper tests predictions of a structural, augmented supply-of-offenders model regarding the relative effects of police, public prosecution and courts, respectively, on crime. Using detailed data on the different stages of the criminal prosecution process in Germany, empirical evidence suggests that public prosecutors and their influence on the probability of conviction play a major role in explaining the variation of crime rates, while the impact of the severity of punishment is small and insignificant.

Appendix

See Tables 8, 9, 10, 11, 12 and 13.

Table 8 Descriptive statistics

Table 9 Estimation in levels

Table 10 Property crimes (robbery, serious theft, petty theft, fraud); residuals adjusted for serial correlation

Table 11 Violent crimes (murder/manslaughter, rape and indecent assault, serious assault); residuals adjusted for serial correlation

Table 12 Property crimes, estimation of individual crime categories

Table 13 Violent crimes, estimation of individual categories

1.1 Proof of theoretical results presented in Sect. 2 (Eq. 2.5)

• a. Maximize expected utility:

$$ \begin{aligned} E\left( U \right) &= p(1 - p_{s|c} )U\,\left[ {A + L^{b} \left( {t_{\ell } } \right) + G\,\left( {t_{i} } \right)} \right] +\, pp_{s|c} U\left[ {A + G\left( {t_{i} } \right) - F\,\left( {t_{i} } \right)} \right] \\ & \quad + \left( {1 - p} \right)U\left[ {A + L\left( {t_{\ell } } \right) + G\left( {t_{i} } \right)} \right] \\ \end{aligned} $$

(A1)

Thus, three different payoffs need to be distinguished. Define

$$ \,Y_{1} \, = \,A\,\, + \,\,L\,\left( {t_{\ell } } \right)\,\, + \,G\,\,\left( {t_{i} } \right),\,Y_{2} = \,A\,\, + \,L^{b} \,\left( {t_{\ell } } \right)\, + \,G\,\left( {t_{i} } \right),Y_{3} \, = \,A\,\, + \,\,G\,\left( {t_{i} } \right)\,\, - F\,\left( {t_{i} } \right) $$

Using the implicit function theorem, we first define

$$ \begin{aligned} E_{i} &= \frac{\partial E\left( U \right)}{{\partial t_{i} }} = p(1 - p_{s|c} )U^{\prime } \left( {Y_{1} } \right)G^{\prime } \left( {t_{i} } \right) + pp_{s|c} U^{\prime } \left( {Y_{2} } \right)\left[ {G^{\prime } \left( {t_{i} } \right) - F^{\prime } \left( {t_{i} } \right)} \right] \\ & \quad + \left( {1 - p} \right)U^{\prime } \left( {Y_{3} } \right)G^{\prime } \left( {t_{i} } \right) = 0 \\ \end{aligned} $$

The second-order condition is

$$ \begin{aligned} E_{ii} &= p(1 - p_{s|c} )U^{\prime \prime } \left( {Y_{1} } \right)G^{\prime } \left( {t_{i} } \right)^{2} + p(1 - p_{s|c} )U^{\prime } \left( {Y_{1} } \right)G^{\prime \prime } \left( {t_{i} } \right) \\ \quad + pp_{s|c} U^{\prime \prime } \left( {Y_{2} } \right)\left[ {G^{\prime } \left( {t_{i} } \right) - F^{\prime } \left( {t_{i} } \right)} \right]^{2} + pp_{s|c} U^{\prime } \left( {Y_{2} } \right)\left[ {G^{\prime \prime } \left( {t_{i} } \right) - F^{\prime \prime } \left( {t_{i} } \right)} \right] \\ \quad + \left( {1 - p} \right)U^{\prime \prime } \left( {Y_{3} } \right)G^{\prime } \left( {t_{i} } \right)^{2} + \,\left( {1 - p} \right)U^{\prime } \left( {Y_{3} } \right)G^{\prime \prime } \left( {t_{i} } \right) \\ \end{aligned} $$

(A2)

$E_{ii} \, < \,0 \Leftrightarrow$ assumptions 1 to 3 hold, i.e.

• $U^{\prime\prime}\,\, < \,0\,\,\,\,\,$

• $G^{\prime\prime}\left( {t_{i} } \right)\, < \,0$

• $F^{\prime\prime}\left( {t_{i} } \right) > 0$

• b. The effect of detection and conviction:

$$ \frac{{\partial t_{i} }}{\partial p}\,\,\, = \,\, - \frac{{\frac{{\partial E_{i} }}{\partial p}}}{{\frac{{\partial E_{i} }}{{\partial t_{i} }}}}\,\, = \,\, - \,\frac{{(1 - p_{s|c} )\,U^{\prime}\left( {Y_{1} } \right)G^{\prime}\left( {t_{i} } \right)\,\, + p_{s|c} U^{\prime}\left( {Y_{2} } \right)\left[ {G^{\prime}\left( {t_{i} } \right) - F^{\prime}\left( {t_{i} } \right)} \right]\,\,\, - \,U^{\prime}\,\left( {Y_{3} } \right)\,G^{\prime}\,\left( {t_{i} } \right)}}{{E_{ii} }} $$

$$ = \, - \,\frac{{G^{\prime}\,\left( {t_{i} } \right)\,\left( {(1 - p_{s|c} )U^{\prime}\,\left( {Y_{1} } \right)\, - U^{\prime}\,\left( {Y_{3} } \right)} \right)\, + \,p_{s|c} U^{\prime}\,\left( {Y_{2} } \right)\,\left( {G^{\prime}\left( {t_{i} } \right)\, - F^{\prime}\left( {t_{i} } \right)} \right)}}{{E_{ii} }} $$

(A3)

$$ \frac{{\partial t_{i} }}{\partial p}\, < 0 \Leftrightarrow $$

• $G^{\prime } \left( {t_{i} } \right)\, - F^{\prime } \left( {t_{i} } \right) < 0$ (see assumption 4)

• $Y_{3} - Y_{1} = L\,\left( {t_{\ell } } \right) - L^{b} \left( {t_{\ell } } \right) > 0 \Rightarrow (1 - p_{s|c} )U^{\prime } \left( {Y_{1} } \right) - U^{\prime } \left( {Y_{3} } \right) < 0$ (assumption of ‘stigma’)

• c. The effect of non-custodial sentencing:

$$ \frac{{\partial \,t_{i} }}{{\partial (1 - p_{s|c} )}}\,\, = \,\, - \,\frac{{\frac{{\partial \,E_{i} }}{{\partial (1 - p_{s|c} )}}}}{{E_{ii} }} = \, - \,\frac{{p\,U^{\prime}\,\left( {Y_{1} } \right)\,G^{\prime}\,\left( {t_{i} } \right)\, - pU^{\prime}\,\left( {Y_{2} } \right)\,\left[ {G^{\prime}\,\left( {t_{i} } \right)\, - \,F^{\prime}\left( {t_{i} } \right)} \right]}}{{E_{ii} }} = \, - \,\frac{\left( + \right)\,\,\, - \,\,\,\left( - \right)}{\left( - \right)}\,\, > \,0 $$

Again, the unambiguous sign depends on the validity of Assumption 4.

• d. The effect of punishment:

Stricter punishment means increasing costs of crime at given crime intensity t _l . A sufficient condition for growing sentencing costs is an increase in $F^{\prime } \left( {t_{i} } \right)$. Studying the ceteris-paribus effect of elevated (marginal) cost of imprisonment, $\bar{F}^{\prime },$ on crime activities, we obtain:

$$ \frac{{\partial \,t_{i} }}{{\partial \bar{F}^{\prime } }} = - \frac{{\frac{{\partial \,E_{i} }}{{\partial \bar{F}^{\prime } }}}}{{E_{ii} }} = - \frac{{\left( { - \,p\,p_{s|c} \,\,U^{\prime } \left( {Y_{2} } \right)} \right)}}{{E_{ii} }} < 0 $$