A new study out today through the National Bureau of Economic Research looks at changes in the size of police forces — including by treating the timing of federal grants as a natural experiment — and produces some interesting results (citations removed):

We find that each additional police officer hired abates between 0.06 and 0.1 homicides. . . . The estimates suggest that investments in police manpower can save a life at a cost of between $1.3 million and $2.2 million while it is common for estimates of the value of a statistical life to exceed $7 million. Although the total reduction in homicide is roughly equal across Black and white victims, the decline in homicide is twice as large for Black victims in per capita terms. . . .
We find that investments in police manpower lead to larger numbers of low-level “quality of life” arrests, with effects that imply a disproportionate burden for Black civilians who are arrested. At the same time, we find that arrests for the most serious offenses fall with investments [in] police manpower. On a per capita basis, the decline in index crime arrests that we observe is between 4-6 times greater for arrests involving Black suspects.