How Much Crime Reduction Does the Marginal Prisoner Buy?

1. 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

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

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

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

11. 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 • Shocks to underlying criminality will induce positive covariance between crime rates and incarceration rates operating through the criminality parameter c. • Criminality is unobservable

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

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

14. 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.

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

16. Incarceration rate Crimerate S*, t>0 C*, t>0 S*, t=0 C*, t=0 t=0 t=1 Time since shock

17. 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. • 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

18. Deriving explicit expressions for the periodic changes in incarceration and crime for t=0 and t=1 where ΔSt=St+1-St Changes in the incarceration rate

20. 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

21. 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

22. Deriving explicit expressions for the periodic changes in incarceration and crime for t=0 and t=1 where ΔSt=St+1-St Changes in the incarceration rate

23. Data • State level panel covering the period 1978 to 1998. • 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. • 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.

24. Constructing the instrument

32. Comparison of these results to those from previous research • Our violent crime-prison elasticity estimates range from -0.09 to -0.15 and property crime estimates range from -0.28 to -0.39. • Levitt (1996) estimates range from -0.38 to -0.42 for violent crime and -0.26 to -0.32 for property crime.