Over-incarceration and disenfranchisement

Abstract

This article presents a model wherein law enforcers propose sentences to maximize their likelihood of reelection, and shows that elections typically generate over-incarceration, i.e., longer than optimal sentences. It then studies the effects of disenfranchisement laws, which prohibit convicted felons from voting. The removal of ex-convicts from the pool of eligible voters reduces the pressure politicians may otherwise face to protect the interests of this group, and thereby causes the political process to push the sentences for criminal offenses upwards. Therefore, disenfranchisement further widens the gap between the optimal sentence and the equilibrium sentence, and thereby exacerbates the problem of over-incarceration. Moreover, this result is valid even when voter turnout is negatively correlated with people’s criminal tendencies, i.e., when criminals vote less frequently than non-criminals.

Appendices

Appendix 1: Proofs of Lemma 1 and Proposition 3

Proof of Proposition 3

(i) $t_{j}^{d}(s_{j-1})$ describes the criminal tendency of the median eligible voter as a function of the sanction and $\overline{s}_{j}(t_{j}^{d})$ describes the sanction that is elected in period j. Given this notation, when ex-convicts are disenfranchised, a steady state is characterized by the condition:

$$\begin{aligned} s^{d}=\overline{s}_{j}\left( t_{j}^{d}(s^{d})\right) \end{aligned}$$

(28)

That an $s^{d}>0$ that satisfies this condition exists follows from the facts that $\frac{d\overline{s}_{j}}{dt}<0$ [(as implied by (14)] and: (a) $\frac{dt_{j}^{d}}{ds}>0$, (b) $\overline{s}_{j}(t_{j}^{d}(0))>0$, and (c) $\underset{s\rightarrow \infty }{\lim }\overline{s}_{j}(t_{j}^{d}(s))-s<0$. The validity of (b) and (c) follow easily from the observations that the sanctions preferred by all $t\ge t^{m}$ are positive and finite. To demonstrate that $\frac{dt_{j}^{d}}{ds}>0$, note that this condition is equivalent to $\frac{\partial V_{j}(t,s)}{\partial s}<0$, owing to the definition of $t^{d}$ expressed in (19). Differentiating $V_{j}$ wrt s reveals that:

$$\begin{aligned} \frac{\partial V_{j}}{\partial s}=\frac{\frac{p^{2}}{2}f[1-p\frac{1-F^{1}}{2}\mu ]\underset{\underline{t}}{\overset{t}{\int }}xgdx-\frac{p^{2}}{2}f\left[ G-p\frac{1-F^{1}}{2}\underset{\underline{t}}{\overset{t}{\int }}xgdx\right] \mu }{\left[ 1-p\frac{1-F^{1}}{2}\mu \right] ^{2}} \end{aligned}$$

(29)

Thus, $\frac{\partial V_{j}}{\partial s}<0$ iff

$$\begin{aligned} \frac{\underset{\underline{t}}{\overset{t}{\int }}xgdx}{\mu }<V \end{aligned}$$

(30)

Proposition 2 demonstrates that $V(t,s)>G(t)$; thus, a sufficient condition for (30) is that $G(t)>\frac{\underset{\underline{t}}{\overset{t}{\int } }xg(x)dx}{\mu }$, which, as demonstrated in (21), holds. Hence, $\frac{\partial V_{j}(t,s)}{\partial s}<0$, and, therefore, a unique steady state sentence $s^{d}$ exists.

(ii) This claim follows directly from Proposition 2 (i): $V_{j}(t,s)>G(t)$ for all s, and, in particular, $V_{j}(t,s^{d})>G(t)$. Thus $s^{d}>s^{n}$. $\square$

Proof of Lemma 1

The inequality in the lemma is

$$\begin{aligned} \frac{\underset{\underline{t}}{\overset{t}{\int }}k(x)l(x)dx}{\underset{\underline{t}}{\overset{\overline{t}}{\int }}k(x)l(x)dx}>\frac{\underset{\underline{t}}{\overset{t}{\int }}k(x)dx}{\underset{\underline{t}}{\overset{\overline{t}}{\int }}k(x)dx} \end{aligned}$$

(31)

which holds if

$$\int\limits_{\underline{t}}^{t} k(x)l(x)dx\left(\int\limits_{\underline{t}}^{t} k(x) dx + \int\limits_{t}^{\overline{t}}k(x)dx\right) >\int\limits_{\underline{t}}^{t} k(x)dx\left(\int\limits_{\underline{t}}^{t} k(x)l(x)dx+\int\limits_{t}^{\overline{t}}k(x)l(x)dx\right) $$

(32)

or,

$$\begin{aligned} \frac{\underset{\underline{t}}{\overset{t}{\int }}k(x)l(x)dx}{\underset{\underline{t}}{\overset{t}{\int }}k(x)dx}>\frac{\underset{t}{\overset{\overline{t}}{\int }}k(x)l(x)dx}{\underset{t}{\overset{\overline{t}}{\int } }k(x)dx} \end{aligned}$$

(33)

which holds if $l^{\prime }<0$, since

$$\frac{\int\limits_{\underline{t}}^{t} k(x)l(x)dx}{\int\limits_{\underline{t}}^{t} k(x)dx} > \frac{\int\limits_{\underline{t}}^{t} k(x)l(t)dx}{\int\limits_{\underline{t}}^{t}k(x)dx} = l(t) = \frac{\int\limits_{t}^{\overline{t}} k(x)l(t)dx}{\int\limits_{t}^{\overline{t}} k(x)dx} > \frac{\int\limits_{t}^{\overline{t}} k(x)l(x)dx}{\int\limits_{t}^{\overline{t}} k(x)dx} $$

(34)

$\square$

Appendix 2: Excluding criminal benefits

The previous analysis proceeds by comparing sanctions listed in (25). It is worth noting that the only sanction in this list that depends on the definition of social welfare is $s^{o}$, i.e., the optimal sanction. Moreover, as Proposition 4 demonstrates, $s^{d}<\min \{s^{n},s^{\nu },s^{\delta }\}$. Thus, if we let $s^{c}$ denote the sanction that maximizes the sum of all utilities net of criminal benefits (or, equivalently, the cost minimizing sanction), it follows that all results presented in the previous sections are preserved (when criminal benefits are excluded from the social welfare function) if $s^{d}>s^{c}$.

Note that when criminal benefits are excluded, social welfare (previously expressed in (10)) becomes:

$$\begin{aligned} \psi (s)=-\theta (h+p(\sigma +1)s) \end{aligned}$$

(35)

which I will assume is single-peaked to simplify the analysis. Then $\psi$ can be plugged into the median type’s utility as follows:

$$\begin{aligned} U(s,t^{m})&=t^{m}\underset{ps}{\overset{\infty }{\int }}(b-ps)f(b)db-\theta (h+p\sigma s)\nonumber \\&=t^{m}\underset{ps}{\overset{\infty }{\int }}(b-ps)f(b)db+ps\theta +\psi (s) \end{aligned}$$

(36)

Therefore, $U_{s}(s^{c},t^{m})>0$ if

$$\begin{aligned} -\frac{t^{m}}{\mu }p\theta +p\theta +ps\theta _{s}>0 \end{aligned}$$

(37)

This condition is equivalent to:

$$\begin{aligned} \frac{\mu -t^{m}}{\mu }>-\frac{s\theta _{s}}{\theta } \end{aligned}$$

(38)

When this condition holds, it follows that $U_{s}(s,t^{m})>0$ for all $s<s^{c}$ whenever $\psi$ is single peaked, and, thus, $s^{d}>s^{c}$. Note that the right-hand side of (38) denotes the sanction elasticity of the crime rate. Thus, if this value is sufficiently smaller than the gap between the median and the mean criminal tendencies in the population, then the analysis extends to cases where criminal benefits are excluded from the welfare function.

The intuition for this result is somewhat convoluted, but relates to the misalignment between the median voter’s objective and the social objective, which is illustrated in (36). The punishment suffered by criminals is not internalized by the median type [this is denoted by $ps\theta$ in (36)], but his own net criminal benefit (i.e., $t^{m}\underset{ps}{\overset{\infty }{\int }}(b-ps)f(b)db$) is. An increase in the sanction results in more punishment suffered by others [represented by $p\theta$ in (37)] as well as the punishment that the median type expects to suffer [i.e., $-\frac{t^{m}}{\mu }p\theta$ in (37)], and the magnitude of these changes are both proportional to the crime rate, because the crime rate is proportional to the probability with which each individual offends (i.e., $\frac{t}{\mu }\theta$ for a type t individual). But, since the median type has a lesser criminal tendency, the personal costs associated with an increase in s are less than the costs he does not internalize. Thus, increasing the sanction beyond $s^{c}$ leads to net benefits for the median type owing to increases in the severity of punishment. However, these benefits need to be weighed against the benefits generated by a reduction in the crime rate [represented by $ps\theta _{s}$ in (37)], which the median voter does not internalize. These reductions are small compared to the net-benefits that are proportional to the crime rate, if the sanction elasticity of the crime rate is small.