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3.5

# Elasticity

Whether bigger police budget and more police officers cause crime to fall has been our main research question so far, but just as important is by how much crime falls. But what would be a good way to measure this?

Elasticity is a widely used by economists to quantify how responsive one factor is with respect to the change in another factor.

If we want to know how responsive the rate of crime is with respect to the number of police working hours, we would construct the elasticity in the following way:

elasticity=(percent change in the rate of crime)/(percent change in the number of police working hours)

For example, if we observe that crime rate falls by 5 percent after police working hours increased by 10 percent, we would say that

“the elasticity of crime rate with respect to police working hours is equal to -0.5 (= $\frac{-5}{10}$).”

An important advantage of elasticity is that it is unit-independent. Suppose we simply compare the change in crime rates with the change in police working hours instead. We then would have to specify the units we used for crime rates and police working hours (increasing police hours by 2,000 hours led to a reduction of 100 crimes, for example). On the other hand, elasticity is simply a ratio of two percent changes and does not involve a unit.

Let’s look at the magnitudes of the estimated effects of police on crime using elasticity. Draca, Machin and Witt (2011) find that the elasticity of crime with respect to police presence is approximately -0.3. Evans and Owens (2007) conclude that the elasticity of crime with respect to the police force size is equal to -0.26 for property crimes and -0.99 for violent crimes. Aaron Chalfin and Justin McCrary report that the elasticity with respect to the police force size is -0.17 for property crimes and -0.35 for violent crimes.

Overall, existing research evidence suggests that a 10 percent increase in police would result in a 2-4 percent crime drop.

## References:

• Chalfin, Aaron, and Justin McCrary. “Are US Cities Under-Policed? Theory and Evidence.” Review of Economics and Statistics, forthcoming.
• Draca, Mirko, Stephen Machin, and Robert Witt. “Panic on the Streets of London: Police, Crime, and the July 2005 Terror Attacks.” American Economic Review 101.5 (2011): 2157-2181.
• Evans, William N., and Emily G. Owens. “COPS and Crime.” Journal of Public Economics 91.1 (2007): 181-201.