Crime and Punishment


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

$A_i$The number of occurrences of that crime per year
$C_i$The average number of QALYs lost per occurence
$P$# of police
$P_0$\$ of police today
$\epsilon_i$Elasticity of that crime to the # of police
$\delta_i$Elasticity of that crime's clearance rate to the # of police
$p_i$The probability of being caught if you commit that crime
$L_i$# of years spent in prison if convicted
$k_0$The QALYs lost by going to prison
$k_1$The QALYs lost per year in prison
$k_2$The annual cost per officer

Computing the number of optimal number of police turns ot 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 ($A_i$)

Right off the bat, we are faced with a dilemma: should A be the number of crimes committed or the number of crimes reported to police? In English, do we believe that more police reduces the number of unreported crimes? For instance, suppose we find that when we double police, the number of reported burglaries drops by 20%. What do we expected happened to the number of unreported burglaries? Did they also drop by 20%? Maybe some burglaries that went unreported before are reported now - messing up our data even more.

I’m going to assume that all crimes (reported and unreported) drop by equal amounts - an assumption I make because it seems reasonable that before committing a crime, a person doesn’t know whether it will be reported. Less justifiably, I’m going to assume that the proportion of crimes reported to police remain unchanged.

Here are some estimates for the percent of crimes reported to police "title": "Criminal Victimization, 2016":

Rape & Sexual Assaults22.9%
Aggravated Assault58.5%
Motor Vehicle Theft29.7%

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 "title": "Crime in the United States":

Rape & Sexual Assaults323,450
Aggravated Assault1,084,340
Motor Vehicle Theft585,500

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 "title": "Prison Sentences and Time Served for Violence":

CrimeAvg Years Served
Sexual Assault2.92

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 "title": "The Cost of Crime to Society: New Crime-Specific Estimates for Policy and Program Evaluation":

Rape & Sexual Assault\$240,776
Aggravated Assault\$107,020
Motor Vehicle Theft\$10,772

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:

Rape & Sexual Assault\$214,297
Aggravated Assault\$98,379
Motor Vehicle Theft\$6,905

Probability of Being Caught ($p$)

The FBI reports the proportion of reported crimes ending in arrest "title": "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.

CrimeCaught Change
Rape & Sexual Assault8.7%
Aggravated Assault31.6%
Motor vehicle theft10.5%

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

Federal, state, and local governments employ approximately 885,000 law enforcement officers "title": "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 "title": "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 $k_1$.



Estimates of crime~police elasticities are pretty difficult to nail down, but the most authoritative probably come from Aaron Chalfin and Justin McCrary "title": "The Effect of Police on Crime: New Evidence from U.S. Cities, 1960-2010" [show]:

Murder0.666 (0.238)
Rape0.255 (0.219)
Robbery0.559 (0.117)
Assault0.099 (0.127)
Burglary0.225 (0.089)
Larceny0.083 (0.067)
Motor Vehicle Theft0.343 (0.101)


For the elasticity of clearance~police, we turn to Pare, Felson, Ouimet (2007) "title": "Community Variation in Crime Clearance: A Multilevel Analysis with Comments on Assessing Police Performance":

"When types of crime are adequately controlled in the total sample, workload does not have an effect. Workload did have a small, negative effect on the clearance of misdemeanor offenses, however."

We should also look at Cordner (1989) "title": "Police agency size and investigative effectiveness" and Jackson, Boyd (2005) "title": "Minority‐threat hypothesis and the workload hypothesis: A community‐level examination of lenient policing in high crime communities" later.


Principal Analysis

To perform the analysis, I only looked at homicide, rape/sexual assault, robbery, and assault. I left out burglary, theft, and vehicle theft because of incomplete data. However, these three crimes are each a tenth as important as the four I did include, so they probably don’t change the results significantly. In any case, ignoring these will bias the results towards fewer police officers.

Using these estimates, we can estimate with the models given above that the number of police should be increased - probably by about 35%.

Sensitivity Analysis

The obvious criticism is that this analysis is only as accurate as our estimates of coefficients. For this reason, I considered how far off each coefficient estimate would have to be in order for this analysis to recommend fewer police. Any of these would flip the results:

  • if the # of crimes is less than 60% the estimates
  • if the cost of crimes were less than half the estimates
  • if the elasticities of crimes to police were all 0.13+ lower
  • if the elasticities of clearance to police were all 0.47+ points higher

Interestingly, the qualitative results don’t change based on average sentence length, current clearance rates, or the cost of prison. The reason is that the costs of crime on society dwarf the costs of the prison system.

I’m skeptical that our estimates for the # of crimes are significantly off or that Pare, et al. missed an effect size of 0.47, so this leaves two roots to argue against our analysis: elasticity of crime to police and the costs of the crimes.

If you examine the above table, the standard errors for our elasticity estimates range from 0.12 to 0.24. So, in order to achieve the 0.13 threshold to support fewer police, we need our z-score to be roughly -0.75. The odds for this are about 3-to-1, but we'd need to achieve that for each crime, making it unlikely that our estimates are that far off due to sampling error.

Ultimately, then, this leaves the only compelling arguments against increasing police methodological ones: either you have to think the study estimating the elasticity of crimes to police or the one estimate the cost of crime are overestimates significantly due to analytical failings.

I have no comment on this at the moment.

Sentence Length

We can represent the utility from a crime with $$ U = -C \cdot A \cdot {\left( \frac{L}{L_0} \right)}^{-\epsilon} - p \cdot A \cdot {\left( \frac{L}{L_0} \right)}^{-\epsilon} \cdot \left( k_9 + L \cdot k_1 \right) $$ where $\epsilon$ is now the elasticity of crime to sentence length and $L_0$ is the current average sentence length.

$$ L^* = \frac{\epsilon}{1 - \epsilon} \cdot \frac{C + p \cdot k_0}{k_1 \cdot p} $$

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. According to a literature review by Chalfin and McCrary, the elasticity between crime and sentence length is about 0.2, which are probably primarily due to incapacitation effects rather than deterrence effects "title": "Criminal Deterrence: A Review of the Literature". Using this value, we compute the following optimal sentence lengths:

Murder5.9 years33 years
Sexual Assault & Rape2.9/5.4 years5.8 years
Robbery3.7 years0.4 years
Assault2.4 years0.7 years
Burglary?0.1 years
Theft?0.0 years
Vehicle Theft?0.2 years

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 other crimes leaves huge room for error, which means I’m 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.


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.

Works Cited [show]