Crime and Punishment

Model

We can represent the utility from a crime with $$ U = \sum_i{ \left[ -C_i \cdot A_i \cdot {\left( \frac{P}{P_0} \right)}^{-\epsilon_i} - {\left( \frac{P}{P_0} \right)}^{-\delta_i} \cdot p_i \cdot A_i {\left( \frac{P}{P_0} \right)}^{-\epsilon_i} \cdot \left( k_0 + L_i \cdot k_1 \right) \right] } - k_2 \cdot P $$ where $i$ denotes the crime and

Computing the number of optimal number of police turns to to be pretty mathematically complicated; however, it is much simpler to compute whether we should have more or fewer police by looking at the sign of the derivative. This implies we should have more police if and only if $$ \sum_i{ \frac{A_i}{P_0} \left[ \epsilon_i \cdot C_i \cdot {\left( \frac{P}{P_0} \right)}^{-\epsilon_i - 1} + \left( \epsilon_i -\delta_i \right) {\left( \frac{P}{P_0} \right)}^{-\delta - \epsilon_i - 1} \cdot p_i \cdot \left( k_0 + L_i \cdot k_1 \right) \right] } \gt k_2 $$

If we can estimate these coefficients, this makes it easy to find the optimal number of police using a computer.

# of Crimes ($A_i$)

Here are some estimates for the percent of crimes reported to police Criminal Victimization:

To determine these numbers, people were surveyed and asked whether they were victims of these crimes and, if so, whether they reported them to police. Presumably, the percent of homicides reported to police is near 100%, but the above method obviously won’t work for them.

If we use the FBI data for homicide and the survey data for the other crimes we get estimates for each crime Crime in the United States:

Sentence Length ($L_i$)

Sentence length is tricky. The years dished out rarely match the years actually served. Moreover, because years served can be quite long, it’s hard to get recent data. Still, it’s depressing that the most recent data I could find comes from 1995 Prison Sentences and Time Served for Violence:

I was really surprised to find how little variance there was in these averages. Killing someone apparently warrants only 1.6 times as much prison time as robbing them - who knew?

Cost of Crimes ($C_i$)

Estimating the social cost of crimes is tricky [citation needed], but various brave researchers have tried The Cost of Crime to Society: New Crime-Specific Estimates for Policy and Program Evaluation:

Unfortunately, these costs (1) include the cost of prison and (2) exclude the quality-of-life lost by prisoners. Fortunately, the study included estimates for costs to the justice system, so we can just subtract those out (TODO: this excludes prosecution costs):

Probability of Being Caught ($p$)

The FBI reports the proportion of reported crimes ending in arrest Percent of Offenses Cleared by Arrest or Exceptional Means. If we multiply these by the probability of a crime being reported, we get the probability of being caught. I should note, that I’ve taken the clearance rate for rape as a proxy for the clearance rate for sexual assault.

Shared Values $\left( P_0, k_2, k_1 \right)$

Federal, state, and local governments employ approximately 885,000 law enforcement officers Law enforcement in the United States. Their wages tend around the median wage, but it’s important to note that for every two law enforcement officer, there is about one additional employee working behind the scenes. Actual cost per officer averages \$116,500 at the local level Local Police Departments, 2007, which we’ll use for our estimate for $k_2$.

Finally, a prisoner costs about \$32,000 to incarcerate per year, so that’s our estimate for $k_1$.

Elasticities

Analysis

Sentence Length

We can represent the utility from a crime with $$ U = -C \cdot A \cdot {\left( \frac{L}{L_0} \right)}^{-\epsilon} - p \cdot A \cdot {\left( \frac{L}{L_0} \right)}^{-\epsilon} \cdot \left( k_0 + L \cdot k_1 \right) $$ where $\epsilon$ is now the elasticity of crime to sentence length and $L_0$ is the current average sentence length.

In this model, the ideal sentence length is given by

This equation qualitatively makes sense:

• $L_0$ disappears, because what is doesn’t affect what should be.
• As $\epsilon$ increases, prison becomes more effective at reducing crime, and the optimal sentence length increases.
• As the probability of being caught increases, the probability of a socially costly prison sentence increases, so sentences become less net-effective. This may not happen in real life, because as $p$ increases, we’d expect sentences to become more effective as a deterrent - that is to say as $p$ increases, we might expect $\epsilon$ to increase.
• As the cost of the crime goes up, the optimal sentence length increases.
• As the cost of a year of prison goes up, the optimal sentence length decreases.

One thing that might be surprising is that $A$ - the absolute number of crimes - doesn’t affect the result. However, if you look at the original equation again, this makes sense. If crime doubles, then the cost to the victims double, but the (personal and financial) cost of housing prisoners also doubles - resulting in no change in optimal sentencing.

In reality, of course, a temporary crime-surge has the additional cost that more prisons must be built and, given that the surge ends eventually, will eventually be under-used. This suggests that the optimal sentence may shrink a bit during crime surges and grow during more peaceful times - the exact opposite of what most people intuitively want.

Finally, it’s worth noting that this elasticity parameter is doing a lot of work for us. It incorporates a variety of factors such as the effect of

• taking criminals off the street
• deterring future crime
• hardening criminals in prison

Okay, without further ado, let’s get to estimates between crime and sentence length. The elasticity between crime and sentence length is about 0.2 and is probably due primarily due to incapacitation effects rather than deterrence effects Criminal Deterrence: A Review of the Literature. Using this value, we compute the following optimal sentence lengths:

This suggests that we should probably have more severe punishments for murder, that we’re relatively close to optimal for rape/sexual assault, and that we should punish other crimes significantly less.

While these qualitative results for rape/sexual assault can change quite a bit depending on the assumptions, the huge discrepancies between typical and optimal sentences for the non-sexual crimes leaves huge room for error, which makes me pretty confident in the conclusions.

That being said, I see these estimates as being very rough - especially the estimate for murder. Murder is likely to be a crime of passion rather than crimes like robbery and burglary, indicating to me that the elasticity of murder to sentence length might be lower than most other crimes.

Conclusions

Ultimately, then, it looks like we have good evidence that sentences should be longer for homicide and shorter for other non-sexual crimes. There is also some decent evidence that we should hire more police.