Aggregation Biases and Economies of Scale in the Metropolitan Police Unit Production Function

I. Introduction There may be aggregation biases related to scale when police unit size is measured by the jurisdictional population of the metropolitan police unit. Aggregation biases occur when there are different police production functions for different sizes of police units.(1) The real police production function could be nonhomogeneous. Consequently, changes in the measured cost efficiency of police production may be due to changes in technology,(2) cost may not be continuous or, equivalently, production functions, may shift.(3) Different size related community characteristics may lead to the use of different police technologies. To address the aggregation bias problem, this analysis provides an empirical test for the existence of different production functions related to changes in the scale of metropolitan police units. The literature indicates that police production is commonly related to some measure of the risk of criminal sanction, such as the probability of apprehension, conviction, and/or punishment. To this extent, crime clearance rates -- the ratio of crimes cleared by arrest or other exceptional means to crimes detected and reported -- are readily available data. Moreover, crime solutions -- as measured by-clearance rates -- appear to represent a significant portion of police objectives, as well as to serve as a proxy for risk production. The approach taken in this paper is similar to that of other studies and considers aggregate crime clearance rates as an intermediate police output that is determined by a risk-production function. Final output is the aggregate reported crime rate -- the ratio of detected and reported crimes to population in a metropolitan police jurisdiction -- and is determined by a supply of offense function. Since all crime and clearance rates are aggregate measures, this avoids the problem of assuming functional separability with respect to each crime category. In fact, the results of Darrough and Heineke (1978) indicate that the assumption of functional separability in the police production function is invalid. Review of Prior Literature Recent studies have found economies of scale among police units by using more sophisticated methods and by controlling for differences in output quality (See, for example, Gebelein, 1973; Scicluna, Foot, and Bird, 1981; Darrough and Heineke, 1978; and Chapman, Hirsch, and Sonenblum, 1975). Findings such as these suggest that police units should be consolidated based upon gains in economic efficiency due to scale. If, however, there are aggregation biases in the data, the apparent economies of scale may be incorrect. Moreover, consolidation, based upon the assumption of a homogeneous police production function, may result in the use of technologies that do not fit a community's needs. External factors such as differences in the crime mix, laws and regulations concerning police behavior, funding and training levels, the availability of different types of training, the degree of police-citizen cooperation, urban versus rural location, and the degree of geographic dispersion of the police unit's jurisdiction may lead to the use of different police production technologies. Moreover, particular external technology affecting factors may be related to the jurisdictional size of the police unit's population, such as: a large population as opposed to a small one may be characterized by an urban location, less geographical dispersion, less police-citizen cooperation, more restrictions on police behavior, and a crime mix involving a greater percentage of violent crimes. It may be that a community's size determines, or at least influences, the police unit production function. In addition to the possible influence of external factors on police technology, different size related police production functions may be suggested by the literature. For example, Parks and Ostrom (1982) estimate a police production function for the outputs of arrests for serious crimes and response capacity. …