On law enforcement
A Model in two parts
1 Modeling the incidence of criminality
We consider a model with a large number of households with preferences over consumption [$] C [$] and jail [$] J [$] according to [$] U=E_B\left[u\left(C\right)-J\right] [$] where [$] u [$] is strictly concave increasing in [$] C. [$] The household is endowed with lawful income of [$] y [$] and has the option of committing a burglary to steal [$] B [$] units of consumption. If the household is convicted, he serves jail time that yields [$] J [$] units of disutility (we can think of disutility from jail as being a function [$] j\left(s\left(B\right)\right) [$] where [$] s\left(B\right) [$] is a policy function prescribing sentences based on the magnitude [$] B [$] of the crime, and [$] j\left(\cdot\right) [$] as the utility function of jail time. However, we consider the extensive margin where [$] B [$] is fixed.)The judicial authority investigates a share [$] q [$] of the population and the investigation leads to a conviction rate of [$] r_1\lt 1 [$] among those who are investigated and committed the crime (that's a sensitivity of [$]r_1[$]), and [$] r_2\lt r_1 [$] among individuals who are investigated but innocent (that's a specificity of [$]1-r_2[$]). Therefore, the probability of being convicted given that the household commits the crime is [$] p_1=q_1r_1 [$] and the probability of conviction given hat the household does not commit the crime is [$] p_2=q_2r_2, [$] where [$]q_1,q_2[$] are the probabilities of investigating an innocent and guilty person respectively, which are functions of the total share of the population [$]q[$] that is investigated, as will be defined in section 2 below. [update: this paragraph has been edited to correct some of the notation]
The household's budget constraint is [$] C\leq y+B [$] if he commits the crime, and [$] C\leq y [$] otherwise. The household choice has two regimes, one where he commits crimes with probability 1, and one where he commits crime with probability 0, where utility from the former is [$] u\left(y+B\right)-p_1 J [$] and the utility from the latter is [$] u\left(y\right)-p_2 J. [$]
Borrowing from the indivisible labor literature1 and assuming a functional form for the utility function, we can rewrite the above in terms of a representative agent that chooses a probability of committing crime [$] \alpha [$] according to
\begin{align*}
\max_{\alpha}~ ln\left(C\right)-\alpha p_1J-\left(1-\alpha\right)p_2J&\\
subject~to~C\leq y+\alpha B&
\end{align*}
Solving yields [$$] \alpha^*=\frac{1}{J\left(p_1-p_2\right)}-\frac{y}{B} [$$] This model captures our intuitions about criminal justice. For example, it is plainly apparent from the solution that increasing penalties [$] J, [$] all else equal, reduces crime rates. It is often assumed that stepping up investigations--that is, targeting a larger share of the population for investigations--will result in a reduction in crime rates. To examine this we take the derivative of [$] \alpha [$] with respect to the investigation rate [$] q: [$] [$$] \frac{\partial \alpha}{\partial q}=\frac{J}{\left(J\left(p_1-p_2\right)\right)^2}\left(\frac{\partial p_2}{\partial q}-\frac{\partial p_1}{\partial q}\right) [$$] which is less than zero if and only if [$$] \frac{\partial p_1}{\partial q}>\frac{\partial p_2}{\partial q}. [$$]
2 Modeling the profiling decision
To understand that last derivative, we need a model of police profiling. Mixing models of heterogeneity with a representative agent framework can be problematic, but lets assume that utilities are such that this is valid. Assume that individuals are heterogeneous in such a way that their probability [$] \tilde{\alpha} [$] of committing a crime--as measured by a hypothetical social planner--is distributed according to a continuously differentiable distribution function [$] F\left(\tilde{\alpha}\right) [$] with support [$] \left[0,1\right]. [$] The judicial authority prioritizes investigations of individuals so that individuals with the highest [$] \tilde{\alpha} [$] probabilities are investigated first, followed by progressively lower probability types until they've exhausted their investigative resources--that is, until the share of the population being investigated equals the policy parameter [$] q. [$] Thus we can write [$$] q\equiv 1-F\left(\bar{\alpha}\right) [$$] where [$] \bar{\alpha} [$] is the lowest probability type to be investigated. Therefore, we have that\begin{align*}
p_1&=\underbrace{\left(1-F\left(\bar{\alpha}\right)\right)}_{q}\underbrace{\int^{1}_{\bar{\alpha}}\tilde{\alpha}f\left(\tilde{\alpha}\right)d\tilde{\alpha}}_{\alpha}r_1\\
p_1&=\underbrace{\left(1-F\left(\bar{\alpha}\right)\right)}_{q}\underbrace{\left(1-\int^{1}_{\bar{\alpha}}\tilde{\alpha}f\left(\tilde{\alpha}\right)d\tilde{\alpha}\right)}_{1-\alpha}r_2
\end{align*}
where [$] f\left(\tilde{\alpha}\right) [$] denotes the density function of [$] \tilde{\alpha} [$] and therefore the first derivative of [$] F\left(\tilde{\alpha}\right). [$] Hereafter we will write [$] E_\bar{\alpha} [$] to denote [$] \int^{1}_{\bar{\alpha}}\tilde{\alpha}f\left(\tilde{\alpha}\right)d\tilde{\alpha}, [$] which is the expected criminality of the population being investigated.
Thanks to the Leibniz rule, we can differentiate this to get
\begin{align*}
\frac{\partial p_1}{\partial q}&=E_\bar{\alpha}r_1+\bar{\alpha}\left(1-F\left(\bar{\alpha}\right)\right)r_1\\
\frac{\partial p_2}{\partial q}&=\left(1-E_\bar{\alpha}\right)r_2-\bar{\alpha}\left(1-F\left(\bar{\alpha}\right)\right)r_2
\end{align*}
Therefore, using the result derived in the first section, increasing enforcement decreases crime only if
\begin{equation}
E_\bar{\alpha}+\bar{\alpha}\left(1-F\left(\bar{\alpha}\right)\right)\gt \frac{r_2}{r_1+r_2} \label{conditions}
\end{equation}
We can now state two propositions.
Proposition 1.
It is optimal to investigate everyone if and only if [$] E_0\geq \frac{r_2}{r_1+r_2}. [$]
Proof: Sufficiency follows immediately from \eqref{conditions} with [$] \bar{\alpha}=0. [$] Necessity follows from Proposition 2.
Proposition 2.
If [$] E_0\lt\frac{r_2}{r_1+r_2}, [$] then there exists [$] q^*\gt 0 [$] such that [$] \frac{\partial \alpha}{\partial q}\geq 0 [$] for all [$] q\gt q^*, [$] with strict inequality whenever [$] f\left(\tilde{\alpha}\right)\gt 0. [$] That is, there exists a point beyond which further increasing enforcement actually increases crime.
Proof: Denote [$] G\left(\bar{\alpha}\right)\equiv E_\bar{\alpha}+\bar{\alpha}\left(1-F\left(\bar{\alpha}\right)\right). [$] Then [$] G\left(1\right)=1\gt\frac{r_2}{r_1+r_2} [$] because [$] E_1=1 [$] and [$] F\left(1\right)=1. [$] Moreover, we postulated that [$] G\left(0\right)=E_0\lt \frac{r_2}{r_1+r_2}. [$] Since [$] F\left(\bar{\alpha}\right) [$] is continuously differentiable, [$] G\left(\bar{\alpha}\right) [$] is continuous and by the intermediate value theorem there exists a nonempty set [$] A [$] such that for [$] \hat{\alpha}\in A [$] we have [$] G\left(\hat{\alpha}\right)=\frac{r_2}{r_1+r_2}. [$] Let [$] \bar{\alpha}=\min \hat{\alpha}\in A, [$] and [$] q^*=1-F\left(\bar{\alpha}\right), [$] then for all [$] \tilde{\alpha}\in \left[0,\bar{\alpha}\right) [$] we have [$] G\left(\tilde{\alpha}\right)\leq \frac{r_2}{r_1+r_2} [$] with strict inequality if [$] f\left(\bar{\alpha}\right)\gt 0. [$] Furthermore, note that [$] q\equiv 1-F\left(\tilde{\alpha}\right) [$] is monotonically decreasing in [$] \tilde{\alpha}, [$] and is one-to-one when [$] f\left(\tilde{\alpha}\right)\gt 0, [$] which implies that for [$] q\in \left[0,q^*\right) [$] we must have [$] G\left(\tilde{\alpha}\right)\leq \frac{r_2}{r_1+r_2} [$] with strict inequality when [$] f\left(\bar{\alpha}\right). [$] This concludes the proof.
So what do these propositions say about stop-and-frisk? We are compelled to draw conclusions contrary to the beliefs of the New York City police commissioner: economics tells us that we can actually reduce crime by not investigating those individuals who are least likely to commit crimes, because this will reduce the wrongful conviction rate and increase the incentive to avoid committing crime. Stop-and-frisk policies do precisely the opposite: they target investigations indiscriminately at the public, innocent and guilty alike, which will increase the wrongful convictions and obviate the disincentive our justice system aims to place on criminal acts.
So there you have it. Economics tells us that stop-and-frisk causes crime.
1. I'm probably not the first one to have applied the indivisible labor literature to criminality in this way, though I did not do a search. If you know of any papers that I have incidentally duplicated, let me know so I can give credit here.