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
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
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
(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
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
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
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
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
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
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.
Athanasios, Noulas, S. Ray, and S. Miller, "Returns to Scale and
Input Substitution for Large U.S. Banks," Journal of Money, Credit
and Banking, 22:1, 1990, 94-108.
Chapman, Jeffrey, W. Hirsch, and S. Sonenblum, "Crime
Prevention, the Police Production and Budgeting," Public Finance,
30:2, 1975, 197-215.
Craig, Steven, "The Impact of Congestion on Local Public Good
Production," Journal of Public Economics, 32:3, 1987, 331-353.
Darrough, Masako N., and J. M. Heineke, "Law Enforcement
Agencies as Multiproduct Firms: An Econometric Investigation of
Production Cost," Public Finance, 34:2, 1979, 176-195.
DeBoer, Larry, "Economies of Scale and Input Substitution in
Public Libraries," Journal of Urban Economics, 32:2, 1992, 257-268.
Duncombe, William, and J. Yinger, "An Analysis of Returns to
Scale in Public Production, With an Application to Fire
Protection," Journal of Public Economics, 52:1, 1993, 49-72.
Gardner, John, "City Size and Municipal Service Costs," in
Urban Growth Policy in a Market Economy, G. Tolley, ed., Academic Press,
New York, 1979, 51-61.
Gyimah-Brempong, Kwabena, "Elasticity of Factor Substitution in
Police Agencies: Evidence from Florida," Journal of Business and
Economic Statistics, 5:2, 1986, 257-265.
-----, "Economies of Scale in Municipal Police Departments: The
Case of Florida," Review of Economics and Statistics, 69:2, 1987,
-----, "Production of Public Safety: Are Socioeconomic
Characteristics of Local Communities Important Factors?" Journal of
Applied Econometrics, 4:1, 1989a, 57-71.
-----, "Demand for Factors of Production in Municipal Police
Departments," Journal of Urban Economics, 25:2, 1989b, 247-259.
Kirlin, John, "Impact of Contract Service Arrangements Upon the
Los Angeles Sheriff's Department and Law Enforcement Services in
Los Angeles County," Public Policy, 21:4, 1973, 554584.
Mehay, Stephen, "Evaluating the Performance of a Government
Structure: The Case of Contract Law Enforcement," Institute of
Government And Public Affairs Reports, University of California Los
-----, "Intergovernmental Contracting for Municipal Police
Services: An Empirical Analysis," Land Economics, 55:1, 1979,
-----, "Economic Incentives Under Contract Supply of Local
Government Services," Public Choice, 46:1, 1985, 79-86.
Phillips, Llad, "Factor Demands in the Provision of Public
Safety," in Economic Models of Criminal Behavior, J. M. Heineke,
ed., North Holland, Amsterdam, 1978, 211-258.
Pindyck, Robert, "Interfuel Substitution and the Industrial
Demand for Energy: An International Comparison," Review of
Economics and Statistics, 61:2, 1979, 169-179.
Renner, Tari, "Trends and Issues in the Use of Intergovernmental
Agreements and Privatization in Local Government," International
City Management Association Baseline Data Report, 1989.
Shoup, Donald, and S. Mehay, Program Budgeting For Urban Police
Services, Praeger Publishers, New York, 1972.
UPCLOSE California Databook, UPCLOSE Publishing, El Granada,
Votey, Harold, and L. Phillips, "Police Effectiveness and the
Production Function for Law Enforcement," Journal of Legal Studies,
1, 1972, 423-436.
Walzer, Norman, "Economies of Scale in Municipal Police
Services: The Illinois Experience," Review of Economics and
Statistics, 2:4, 1972, 423-436.
Zellner, Arnold, "An Efficient Method for Estimating Seemingly
Unrelated Regressions and Tests for Aggregation Bias," Journal of
the American Statistical Association, 57:298, 1962, 348-368.
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 firstname.lastname@example.org
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