# Policy 1: Change Sentence Lengths

### Description

An increase of sentence lengths will deter some people from committing crimes in the first place through general deterrence. We can calculate this crime reduction benefit in dollars as: $\mathit{Benefit}_{GD} = \overline{c}x\eta_{GD}*crime$ Here $$\overline{c}$$ is the cost of an average crime and the other variables are as defined above. An $$x$$% increase in sentence lengths will reduce crime through general deterrence by $$x\eta_{GD}$$. This will yield a total reduction of crime of $$x\eta_{GD}*crime$$, which has a total value (in dollars) of $$\overline{c}x\eta_{GD}*crime$$.

One may use a somewhat similar calculation for the incapacitation benefit of an $$x$$% increase in all sentence lengths: $\mathit{Benefit}_{Inc} = \overline{c}x\lambda*prison$ Here, instead of an elasticity, I introduce $$\lambda$$, the number of additional crimes a prisoner would be responsible for annually if released (above the replacement crime rate). $$prison$$ is simply the prisoner count. Thus, increasing all sentences by $$x$$% is the equivalent of increasing the prison population by $$x*prison$$. This will result in $$x\lambda*prison$$ averted crimes, with an attendant cost savings of $$\overline{c}x\lambda*prison$$.

The calculation for specific deterrence is a bit more complicated, since specific deterrence does not appear to have an impact for all sentence lengths, but only for relatively short sentence lengths. $Benefit_{SD}=\overline{c}*\underset{\text{Change in recidivism rate}}{\underbrace{x*\overline{s}_{effective}*\frac{drecid}{ds}}}*\underset{\substack{\text{Number of crimes for which}\\\text{specific deterrence is effective}}}{\underbrace{\lambda*prison_{effective}}}$ Here the subscript $$effective$$ indicates the sentencing interval over which specific deterrence is effective; $$\frac{drecid}{ds}$$ is the amount the recidivism rate is reduced as the sentence length $$s$$ increases within the effective interval; $$\overline{s}_{effective}$$ is the mean sentence length in the effective interval and thus $$x*\overline{s}_{effective}$$ is the change in sentence length due to an $$x$$% overall increase in sentences. For small $$x$$’s $$x*\overline{s}_{effective}*\frac{drecid}{ds}$$ yields a good approximation for the decrease in recidivism rate due to specific deterrence. $$prison_{effective}$$ is the number of prisoners in the effective interval, so $$\lambda*prison_x$$ denotes the number of crimes they would commit annually if not incarcerated. Multiplying this by change in recidivism rate yields the expected total reduction in crime. Multiplying this by $$\overline{c}$$ gives the benefit of this reduction in dollars.

The net benefit will simply be the sum of the benefits of crime reduction by each of the three mechanisms.

The costs of increasing all sentences by $$x$$% come primarily from increased expenditures for imprisonment, and are substantially easier to compute. $Cost=(CP+VoF)*x$ Where $$CP$$ is the total expenditure on imprisonment and $$VoF$$ is the value of freedom to a prisoner.

### Parameters

Select a set of values from the dropdowns below to pre-fill the parameters or use custom values. For more about the papers this calculator refers to, please consult the references section.

 $$\overline{c}$$, mean crime cost $$x$$, % increase in sentence length [0,1] $$\eta_{GD}$$, general deterrence benefit of 1% increase in sentence length $$\eta_{Inc}$$, incapacitation benefit of 1% increase in sentence length $$\overline{s}_{effective}$$, mean sentence length in the effective interval $$\frac{drecid}{ds}$$, marginal reduction in recidivism rate as sentence length $$s$$ increases effective share, proportion of sentence where specific deterrence is effective # of crimes committed/year # of prisoners/year # of jail inmates/year $$CP_{day}$$, cost/prisoner/day $$VoF_{day}$$, value of freedom/prisoner/day $$\lambda$$, # of potential crimes committed/year if released

### Results

 Total Benefit: Not Calculated General Deterrence: Not Calculated Specific Deterrence: Not Calculated Incapacitation: Not Calculated Cost: Not Calculated Net Benefit: Not Calculated Net Benefit/Prisoner: Not Calculated