How Much Crime Reduction Does the Marginal Prisoner Buy? Rucker Johnson Goldman School of Public Policy UC Berkeley Steven Raphael Goldman School of Public Policy UC Berkeley

Deriving long-run equilibrium in incarceration rates as a function of observed transition probabilities

Alternative simulation of the evolution of U.S. incarceration rates based on 1980 starting values and observed transition probabilities

A simple non-behavioral model of the incapacitation effects of prison on crime

Basic identification problem highlighted in the existing literature Based on the derivation above, it’s easy to show that dS * 2 /dc, dCrime * /dc >0 Based on the derivation above, it’s easy to show that dS * 2 /dc, dCrime * /dc >0 Shocks to underlying criminality will induce positive covariance between crime rates and incarceration rates operating through the criminality parameter c. Shocks to underlying criminality will induce positive covariance between crime rates and incarceration rates operating through the criminality parameter c. Criminality is unobservable Criminality is unobservable

Basic identification strategy: isolate variation in incarceration along the dynamics adjustment path between equilibrium in response to shocks to the transition probability parameters

t=0 t=1 S*, t=0 S*, t>0 Incarceration rate Time since shock

We can derive a similar equilibrium adjustment path for crime Note, the first term in crime adjustment path is positive yet diminishing in time, t. The second term is equal to the equilibrium crime rate for t>0. Together, the two components indicate that an increase in c causes a discrete increase in crime above the new long-term equilibrium and then adjusts to the new equilibrium from above.

t=0 t=1 Crime*, t=0 Crime*, t>0 Crime rate Time since shock

t=0 t=1 S*, t=0 S*, t>0 Incarceration rate Time since shock Crime rate C*, t=0 C*, t>0

Change from t=0 to t=1 for both crime and incarceration are positive. Change from t=0 to t=1 for both crime and incarceration are positive. Crime rate reflects positive effects of change in criminality as well as the negative effect of increased incarceration. Crime rate reflects positive effects of change in criminality as well as the negative effect of increased incarceration. Change from t>0 to t+1 will be negative for crime and positive for incarceration Change from t>0 to t+1 will be negative for crime and positive for incarceration Decline in crime rate is driven by an increasing incapacitation effect alone. Increase in incarceration is driven by the system catching up to the new equilibrium value with a lag (the key to our identification strategy Decline in crime rate is driven by an increasing incapacitation effect alone. Increase in incarceration is driven by the system catching up to the new equilibrium value with a lag (the key to our identification strategy

Deriving explicit expressions for the periodic changes in incarceration and crime for t=0 and t=1 where ΔS t =S t+1 -S t Changes in the incarceration rate

Expression for change in crime from t=0 to t=1 Partial incapacitation effect associated with contemporaneous increase in incarceration in response to criminality shock Increases in crime caused by increased criminality holding incarceration to the previous equilibrium level We observe the change in crime and the contemporaneous change in incarceration and wish to estimate the incapacitation effect, c1. We do not observe the second term however, and thus in a regression of the change in crime on the change in incarceration, it will be swept into the error. Change in incarceration will be positively correlated with the error term

Expression for change in crime from t=1 to t=2 Change in crime for this period driven only by the increase in incarceration rate associated with the incarceration rate adjusting upwards to it’s new equilibrium in response to last period’s shock. This suggests the following identification strategy: use last period’s shock to predict how the incarceration rate will change between now and next period. Instrument the actual change in incarceration rate with the predicted change, thus isolating variation in incarceration associated with the dynamic lagged adjustment

Deriving explicit expressions for the periodic changes in incarceration and crime for t=0 and t=1 where ΔS t =S t+1 -S t Changes in the incarceration rate

Data State level panel covering the period 1978 to State level panel covering the period 1978 to Data on crime (7 part 1 felony offenses) from from the Uniform Crime Reports Data on crime (7 part 1 felony offenses) from from the Uniform Crime Reports Prison totals, total admissions, and total releases by state and year come from the Bureau of Justice National Prisoner Statistics program. Prison totals, total admissions, and total releases by state and year come from the Bureau of Justice National Prisoner Statistics program. Population totals come from the Census bureau as do a number of state-level demographic measures. Population totals come from the Census bureau as do a number of state-level demographic measures. Regional economic indicators come from either the Bureau of Labor Statistics or the Bureau of Economic Analysis. Regional economic indicators come from either the Bureau of Labor Statistics or the Bureau of Economic Analysis.

Constructing the instrument

Table 4 OLS and IV Estimates of the Effect of Changes in Incarceration Rates on Changes in Violent Crime Rates Using the Entire State-Level Panel (Dependent Variable=ΔViolent Crime Rate) Specification (1)Specification (2)Specification (3)Specification (4) OLSIVOLSIVOLSIVOLSIV ΔIncarceratio n rate (0.063) (0.166) (0.060) (0.173) (0.050) (0.140) (0.053) (0.189) ControlsNo Yes Year EffectsNo Yes State EffectsNo Yes R2R N1,071 Implied elasticity at the mean

Table 5 OLS and IV Estimates of the Effect of Changes in Incarceration Rates on Changes in Property Crime Rates Using the Entire State-Level Panel (Dependent Variable=ΔProperty Crime Rate) Specification (1)Specification (2)Specification (3)Specification (4) OLSIVOLSIVOLSIVOLSIV ΔIncarceratio n rate (0.338) (0.941) (0.335) (1.015) (0.259) (0.794) (0.271) (1.165) ControlsNo Yes Year EffectsNo Yes State EffectsNo Yes R2R N1,071 Implied elasticity at the mean

Table 6 OLS and IV Estimates of the Effect of Changes in Incarceration Rates on Changes in Individual Part 1 Felony Offenses Specification (1)Specification (2)Specification (3)Specification (4) Dependent Variable OLSIVOLSIVOLSIVOLSIV Δ Murder (0.001) (0.004) (0.001) (0.003) (0.001) (0.004) (0.001) (0.005) Δ Rape (0.004) (0.011) (0.004) (0.012) (0.004) (0.010) (0.004) (0.014) Δ Robbery (0.033) (0.090) (0.033) (0.095) (0.029) (0.082) (0.030) (0.110) Δ Assault (0.037) (0.098) (0.037) (0.104) (0.033) (0.093) (0.035) (0.124) Δ Burglary (0.125) (0.358) (0.127) (0.405) (0.095) (0.288) (0.099) (0.409) Δ Larceny (0.200) (0.532) (0.197) (0.570) (0.161) (0.472) (0.168) (0.669) Δ Motor Vehicle Theft (0.067) (0.182) (0.065) (0.192) (0.059) (0.178) (0.062) (0.262) Control Variables No Yes Year EffectsNo Yes State EffectsNo Yes

Comparison of these results to those from previous research Our violent crime-prison elasticity estimates range from to and property crime estimates range from to Our violent crime-prison elasticity estimates range from to and property crime estimates range from to Levitt (1996) estimates range from to for violent crime and to for property crime. Levitt (1996) estimates range from to for violent crime and to for property crime.

Table 7 First Stage Effect of the Predicted Change in Incarceration Rates Based on Last Period Shock on the Current Change in Incarceration Rates for Three Sub-Periods of the Panel Dependent Variable=ΔIncarceration Rate (1)(2)(3)(4) Time Period: 1978 – 1984 Predicted Δ Incarceration (0.081) (0.082) (0.087) (0.098) F-statistic* (P-value) (<0.0001) (0.0003) (0.0009) 0.26 (0.611) Time Period: 1985 – 1991 Predicted Δ Incarceration (0.074) (0.075) (0.075) (0.099) F-statistic* (P-value) (<0.0001) (<0.0001) (<0.0001) 3.41 (0.065) Time Period: 1992 – 1998 Predicted Δ Incarceration (0.096) (0.113) (0.115) (0.153) F-statistic* (P-value) (<0.0001) (<0.0001) (<0.0001) (0.0002) Controls Variables NoYes Year EffectsNo Yes State EffectsNo Yes