Scale economies and police department consolidation evidence from Los Angeles.
This paper tests for scale economies for police departments operating in Los Angeles County. Intergovernmental sharing in providing police services prevails within the county, suggesting that scale economies exist in producing this public good. Empirical analysis here rejects the hypothesis that police output is produced under increasing returns, which would be an efficiency explanation for the large degree of consolidation found within the county.

Police (Economic aspects)
Economies of scale (Testing)
Finney, Miles
Pub Date:
Name: Contemporary Economic Policy Publisher: Western Economic Association International Audience: Academic; Trade Format: Magazine/Journal Subject: Business; Economics Copyright: COPYRIGHT 1997 Western Economic Association International ISSN: 1074-3529
Date: Jan, 1997 Source Volume: v15 Source Issue: n1
Accession Number:
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The cost structure of government production is an important policy issue given the possible cost savings that local governments can achieve by consolidating the provision of public goods. Gyimah-Brempong (1987, 1989a) suggests that such consolidation would be an inefficient move by local governments. The research indicates that police output is produced under decreasing returns to scale, inferring that jurisdictions should consolidate to produce police services only if the respective constituencies' preference for consolidation justifies the increased cost.

A preference for consolidation would not likely explain the degree of intergovernmental sharing found within Los Angeles County. While an estimated 6% of U.S. cities consolidate in some manner to produce police services (Renner, 1988), over 45% of Los Angeles jurisdictions provide police service through intergovernmental agreement. More significantly, over 90% of Los Angeles jurisdictions that have incorporated since 1954 engage in such intergovernmental arrangements (39 out of 43). The disproportionate amount of police consolidation in Los Angeles County raises the question of whether such governmental sharing may be, at least within the region, economically efficient. This paper examines this question by estimating the cost structure of Los Angeles area police departments.

In Los Angeles, many municipalities contract with the county sheriff's department for police services. Mehay (1979, 1985), Kirlin (1973), and Shoup and Mehay (1972) explore the impetus for the contractual arrangements. Mehay (1979) finds that production of police output in the contractual cities is less than that of cities with independent departments - sometimes substantially less. The study suggests that whatever cost savings that may arise from the arrangements originate from the relatively low police output produced through the agreements. This could not serve as a motive for consolidation. Mehay (1985) and Kirlin (1973) suggest that market incentives generated by the contractual arrangements may increase the efficiency with which contract providers produce police services. Shoup and Mehay (1972) imply that the municipal governments are motivated to contract with the county because the county subsidizes them. However, no empirical study has tested the hypothesis that suburban Los Angeles jurisdictions disproportionately consolidate because the average cost for an individual department producing police services in the area falls as its output grows.

Testing for scale economies is complicated by the fact that one can define police output in a number of ways. Empirical studies use such intermediate output definitions as arrests (Gyimah-Brempong, 1987, 1989b), clearances (Darrough and Heineke, 1979), and clearance rate (Chapman et al., 1975) while defining final output measures as some transformation of the crime rate (Gyimah-Brempong, 1989a; Phillips and Votey, 1972; Craig, 1987). Varying definitions may be the reason why these studies produce disparate results on the empirical question of police scale economies.

This study tests for scale economies using two separate output definitions: the intermediate good of arrests and the inverse of the crime rate (sometimes referred to in the literature as "safety"). The analysis here follows Gyimah-Brempong (1987, 1989a, 1989b) in employing the translog cost function to test for the underlying scale relationship in police output. This functional form is among the least restrictive to estimate costs.


This study utilizes a single product translog equation to estimate the cost function of the Los Angeles area police departments. It models police departments as maximizing output subject to a cost constraint. The study presents separate models utilizing the inverse of the crime rate and arrests as alternative definitions of output.

The major purpose here is to analyze the relationship between police inputs and outputs across jurisdictions. Estimating the cost function allows one to infer the technical relationship between police inputs and the production of police services. However, Brempong (1989a) notes that social inputs also may contribute to the production of police services. Socioeconomic characteristics may influence returns to scale relationships. Excluding the variables may bias the remaining parameter estimates. The analysis therefore models two social variables as part of the inputs used to produce police services. With five inputs and alternative measures of a single output, equation 1 represents the cost function:

(1) [Mathematical Expression Omitted]

where TC is total cost, Q is output, Wi is the price of police input i, [S.sub.m] is the un-priced socioeconomic input m, and the [Beta]'s represent parameter estimates. The police inputs are uniformed ([W.sub.1]) and civilian ([W.sub.2]) labor as well as a measure of capital ([W.sub.3]).

Empirically testing a number of social variables in the cost function identifies two socioeconomic variables: the inverse of the municipality poverty rate ([S.sub.1]) and percentage of homes that are owner occupied ([S.sub.2]). (Combinations of other variables tested in equation 1 include the percentage of the adult population with a high school education and population density.) The activities of the citizenry may facilitate the output produced by police departments in the form of either safety or arrests. Although empirical testing selected the socioeconomic variables, one easily can reason why these social characteristics would impact the production function of police departments. Citizens who occupy self-owned housing should be more willing to take private measures to keep their property and neighborhood secure. Seemingly, a population less likely to live in poverty also produces less crime and more readily participates with police to avert crime.

The derivative of equation 1 with respect to the police input prices generates cost share equations for each of the inputs. To estimate the cost function above, the analysis uses Zellner's Seemingly Unrelated Regression (SUR) procedure, which utilizes information from the correlation between equation 1 and the independent input cost share in order to increase estimation efficiency.

Satisfying the conditions of a cost function requires restricting the translog equation. An equation serving as a cost function must be homogeneous of degree one in input prices and have symmetric second order cross effects. These conditions restrict the sum of the [B.sub.2] coefficients to equal one and the parameter estimates for the interaction terms to sum to zero.


This study uses data from 14 of the 47 independent municipal police departments in Los Angeles County operating over the period 1989-1992. Although data are available for all 47 jurisdictions, missing wage information for individual jurisdictions in particular years precludes using a larger sample. Additionally, this paper focuses on the cost structure of suburban departments and therefore excludes the city of Los Angeles. The sample is a pooled cross section of 14 departments over a four-year period, which generates 56 data points.

The multiple observations by department allow the analysis to control for unobserved characteristics by individual police jurisdiction. Fixed effects by individual departments are accounted for by a within estimator in which the (log) data are differenced from their jurisdiction-specific mean. This procedure is equivalent to estimating costs by including a dummy variable for each police department, though without the corresponding loss in degrees of freedom. A department's fixed effect may arise from city specific institutional practices and laws that may impact the police production function. Additionally, this study does not control for variables such as fringe benefits and departmental use of computers although the fixed effect estimator may partially account for these variables.

Data used to estimate the cost functions are year specific except for the socioeconomic variables. The inverse of the poverty rate ([S.sub.1]) and percentage of housing owner occupied ([S.sub.2]) are from the 1990 census. Differencing the data causes the non-interacted terms of these two variables to drop out (equation 1). The last two terms of equation 1 involve the social input variables since they interact with the wage and output variables that are year specific. The socioeconomic variables help determine cross sectional differences in police department costs but do not determine year-to-year differences for individual jurisdictions.

Two police department outputs modelled are the final output of safety and the intermediate good of arrests. The safety variable is a qualitative index measured by the inverse of the crime rate in each city. The crime rate is the ratio of the FBI major crime index to the jurisdiction's population. (The FBI composite index encompasses willful homicide, forcible rape, robbery, aggravated assault, burglary, motor vehicle theft, larceny theft and arson.) Both the crime and arrest data are obtained from the California Department of Justice. Yearly population by jurisdiction is from the California Databook, which compiles estimates by the California Department of Finance.

Average monthly salaries for police officers ([W.sub.1]) and civilians ([W.sub.2]) are from the U.S. Department of Commerce annual Survey of Government Employment. Estimating yearly salaries involves multiplying the wage data by 12. This study follows Phillips (1978) and Gyimah-Brempong (1987, 1989a, 1989b) in proxying the cost of capital ([W.sub.3]) by the price of the average police automobile. The automobile used is a Chevrolet Caprice. Ideally, the cost of fixed structures also would be estimated in the cost function. However, controlling for this fixed cost is difficult given variation in timing and financing of long-term investments.

Total expenditures by municipal police department for the 1989-1992 period are from the California Office of State Controller. The number of police and civilian personnel by department - needed to calculate input shares - is from the California Department of Justice.

The analysis deflates the total cost and input price variables by the yearly composite price index for the Los Angeles metropolitan area published by the U.S. Department of Commerce. This adjustment converts yearly changes in those variables into real terms. Table 1 summarizes the expenditure and salary data in real terms (with 1979 as the base year).


Table 2 presents the results of the SUR regression for model 1 in which output is the inverse of the crime rate (safety) and model 2 in which output is arrests. Although this analysis does not report an [R.sup.2] for the SUR model, the F-statistics indicate that both equations are highly significant. The F-value of approximately 14.7 for each equation indicates that the null hypothesis that the parameters are jointly zero would be rejected at any conventional level of significance.

This study tests for scale economies in police departments. Measurement of returns to scale is one minus the output elasticity: 1 - [Delta]ln(TC)/[Delta]ln(Q). The output elasticity is calculated through equation 2. Given the within transformation of the data in equation 1, the variable ln[W.sub.i] in equation 2 consists of yearly deviations in the input prices from their department specific means. The two non-interacted socioeconomic variables, [S.sub.m], which do not vary by year, drop out in deriving the equation. Because the variable, ln[W.sub.i], sums to zero, the expected output elasticity for the full sample is the value of [B.sub.1] for the respective models.

(2) [Delta]lnTC/[Delta]lnQ = [[Beta].sub.1] + [summation of] [[Beta].sub.4i] ln[W.sub.i] where i=1 to 3

The results indicate that the Los Angeles police departments produce safety and arrests under decreasing returns to scale. The Los Angeles suburban jurisdictions produce safety with estimated average returns to scale of -0.125. They generate arrests with returns equal to -1.65. (Regressions were performed with squared output added as an independent variable. This term would allow for the possible change in returns to scale over different output ranges. The squared term always is highly insignificant and, to preserve degrees of freedom, the analysis excludes it.) Both returns estimates differ from zero (which would indicate constant returns) at the 5% level of significance. The output elasticities used to calculate the returns to scale values indicate that the average department's costs would rise by 26.6% if it increased arrests by 10%. Costs would rise by 11.3% if the jurisdiction increased safety by 10%.

The decreasing returns to scale found for Los Angeles jurisdictions correspond to the recent results for Florida police departments (see Gyimah-Brempong, 1987, 1989a). Gyimah-Brempong - whose methodology the present study follows - finds that production of arrests and safety are on average under decreasing returns. However, the returns to scale estimates found in the analysis here contradict the findings of Chapman's et al. (1975) that the Los Angeles central city produces police services under strongly increasing returns to scale. The difference between the returns estimates for suburban Los Angeles jurisdictions obtained here and those in Chapman et al. (1975) for the central city may have a number of causes. The difference in time periods as well as the possible distinction between producing police services in urban and suburban environments may account for the disparate findings. The studies also utilize dissimilar methodologies. The Chapman et al. (1975) methodology for directly estimating the production function limits the possibility for input substitution.

This study's returns to scale estimates strongly suggest that consolidation would be inefficient. Of the two models, the estimated returns with respect to arrests is more germane to the consolidation question since arrests are a more quantitative measure than is safety. Returns calculated from an equation such as model 1 actually represent returns to quality as opposed to a quantitative measure of production scale (Duncombe and Yinger, 1993). Per capita expenditure (expenditure/population) in 1992 for the 14 sampled police departments averaged $177.36. This study's output elasticity for arrests indicates that consolidating the 14 independent jurisdictions would have cost a police department $472.78 per capita to produce the same total number of arrests as the 14 departments produced in 1992. In that year, the Los Angeles jurisdictions that contracted with the county sheriff's department spent an average $77.32 per capita for police services.


This study seeks an efficiency explanation for Los Angeles County's extensive consolidation of police services and finds no such efficiency grounds. The analysis estimates that the cost of producing either safety or arrests rises at a faster rate than do the respective outputs. However, the Los Angeles jurisdictions involved in intergovernmental agreements appear to base their decision to consolidate on cost considerations. Police expenditures by the contracting municipalities typically are far below those found in comparably sized cities with independent police departments. This study's finding of decreasing returns implies that whatever quantity of police services the contracting municipalities are presently purchasing could have been produced by the individual jurisdictions at a lower cost.

A possible explanation for the prevalence of police consolidation may lie in the fact that the intergovernmental arrangements are principally with the Los Angeles County Sheriff's Department. The analysis here assumes that the technology utilized by the sampled police departments represent the knowledge embodied in the police forces across the Los Angeles region. This may not be the case. It may be inefficient for Los Angeles jurisdictions to consolidate individually with one another - the efficiency of incorporation through the county police department remaining an open question. Estimating the cost function of the county police force may provide a rationalization for the observed consolidating behavior that this study suggests is irrational.

The author thanks Sanae Tashiro for invaluable assistance in preparing this study as well as Janet Kohlhase, Steve Craig, Sunil Sapra, and two anonymous referees for helpful comments.


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Finney: Assistant Professor, Department of Economics and Statistics, California State University, Los Angeles, 5151 State University Drive, 90032 1-213-343-2937, Fax 1-213-343-5462 E-mail

Summary Statistics for Dependent and Independent Variables

Variable                 Mean          Standard Deviation

Police Expenditure       15350233           20972377
Arrests                  6356               9357
Crime Rate               0.0658             0.0195
Police Salary            50310              5608
Civilian Salary          29655              5044
Capital Price            15330              215
Police Share             0.50               0.0972
Civilian Share           0.12               0.0342
Capital Share            0.38               0.0883
Population               96474              98718
Poverty Rate             0.1411             0.0634
% Homeownership          0.4736             0.1148


Translog Cost Function Parameter Estimates

Variable                   Model 1                 Model 2

Q                 2.6599(*)     (0.759)      1.125(*)      (.4983)
W1               -0.0203        (0.287)       .159         (.2820)
W2                0.0262        (0.108)      -.033         (.1175)
W3                0.994(*)      (0.338)       .874(*)      (.3403)
W1W1              0.5102(**)    (0.290)       .350         (.2796)
W1W2             -0.0011        (0.021)      -.008         (.0220)
W1W3             -0.5091(**)    (0.291)      -.342         (.2784)
W2W2              0.0743        (0.109)       .126         (.1168)
W2W3             -0.0731        (0.114)      -.118         (.1186)
W3W3              0.5823        (0.346)       .460         (.3373)
W1Q              -0.1823(*)     (0.048)      -.061(*)      (.0201)
W2Q               0.0078        (0.022)       .004         (.0103)
W3Q               0.1744(*)     (0.058)       .056(*)      (.0256)
S1Q              -0.6903(*)     (0.264)      -.298(**)     (.0173)
S2Q               0.6903(*)     (0.264)       .298(*)      (.0173)
W1S1             -0.00005       (0.001)      -.00004       (.0011)
W1S2              0.00005       (0.001)       .00004       (.0011)
W2S1              0.000008      (0.0005)      .000002      (.0005)
W2S2             -0.000008      (0.0005)     -.000002      (.0005)
W3S1             -0.0308        (0.173)      -.300         (.1917)
W3S2              0.0308        (0.173)       .300         (.1917)

F - value        14.69                      14.70
F Prob-value       .0001                      .0001
N = 56

Standard Errors are in parenthesis.

* significant at the 5% level
** significant at the 10% level
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