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# RACE ON POLICING: A Stylized Model 2

It is frequently alleged, for instance, that white officers disproportionately single out non-white youths for harassment. Also, collusive agreements between criminal organizations and police may be easier to enforce within a racial group, resulting in fewer successful arrests due to corruption.

Denoting the likelihood of a true arrest as T and a false arrest as F, the relationship between race and arrest probabilities is expressed formally as follows:

where the subscript i refers to the race of the criminal, R is an indicator variable equal to one if the crime is reported to the police and zero otherwise, and Pw and Pn are respectively the number of white and non-white police officers per capita. When R=0, both T and F are zero, i.e. no arrest will be made. When R~ 1, T and F are positive. Increasing the number of police (Pw and PJ is likely to have a positive impact on true arrests, but may have an ambiguous effect on the number of false arrests. White and non-white police may have different effects on T and F, and the effects read more.

The decision of a victim about whether or not to report a crime to the police is assumed to be a function of the likelihood that reporting the crime leads to either a true or false arrest which, from equations 1 and 2, depends upon the size and racial composition of the police force:

where i subscripts the race of the victim, and it is assumed that 6R/dT>0 and dR/dF

Criminals in the model are risk-neutral expected utility maximizers whose decisions about whether to commit crimes depend upon the likelihood of a true arrest, which is a function of victim reporting rates and true arrests conditional on victim reporting:

where reflects the total number of arrests of suspects of race i and the other variables are as described above, but with functional dependences omitted. The total number of arrests is equal to the number of reported crimes (C*R) multiplied by the combined arrest rate for both true and false arrests per reported crime (Г+ F).