Journal
of Public Economics
33 (1987) 245-260.
North-Holland
DISENTANGLING
THE DEMAND
FUNCTION
FROM THE
PRODUCTION
FUNCTION
FOR LOCAL PUBLIC SERVICES
The Case of Public Safety
Robert
M. SCHWAB
University of Maryland,
Ernest
The Catholic
Received
College Park, MD 20742, USA
M. ZAMPELLI*
University of America, Washington, DC 20017, USA
March
1985. revised version
received December
1986
In this paper we develop a theoretical framework and an econometric
model which allow us to
separate the effect of income and other socioeconomic
variables on the demand for publicly
provided goods from their effect on the price of those goods. We apply our approach in a study
of expenditures
for police protection in a cross-section
of 73 U.S. cities and counties. Our results
suggest that studies of public expenditures
which fail to incorporate
community
characteristics
in
production
and cost functions can yield very misleading results.
1. Introduction
Income and other socioeconomic
variables have played an important
role
in virtually all empirical studies of expenditures
on publicly provided goods.’
Typically, these variables are incorporated
on the demand side with income
measuring purchasing power and socioeconomic
variables proxying tastes for
the public good.’
This demand
oriented
approach
fails to recognize
that income
and
socioeconomic
characteristics
may also affect the production
of publicly
provided goods. In an important
paper, Bradford, Malt and Oates (1969)
draw a distinction
between the output produced directly by the public sector
(termed ‘D-output’) and the output that is of primary concern to the citizen*We thank
Bruce Hamilton
and Wallace Oates for their many helpful comments
and
suggestions.
We also thank
Ellen Roche and Anne Stephens
for their excellent
research
assistance.
This research
was funded by a Sloan Foundation
Grant
to the University
of
Maryland.
‘See Inman (1979) and Gramlich (1977) for excellent reviews of this literature.
‘Recently, Bradbury et al. (1984) have examined the effect of socioeconomic
variables on the
cost of public goods. Their approach is somewhat different from ours.
0047-2727/87/$3.50
0
1987, Elsevier Science Publishers
B.V. (North-Holland)
246
R.M. Schwab and E.M. Zampelli,
Demand for local public services
consumer (termed ‘C-output’). D-output is a function of purchased inputs; Coutput is a function of D-output and the community
environment.
Consider,
for example, public safety. By combining
police officers and
other police inputs, a local government
produces a vector of D-output
which
might include the number
of city blocks patrolled,
the number
of intersections provided with traffic control, and so on. The citizen, however, is
concerned
with the level of public safety, i.e. C-output,
which depends not
only on D-output
but on the community’s
characteristics
as well. For
example, poor communities
burdened
by high unemployment
may need to
hire more police ofticers to provide a given level of public safety than
wealthier jurisdictions.
The observation
that community
characteristics
enter the production
functions for local public goods has at least three important
implications.
First, it has often been argued that local communities
use zoning and other
land use controls as a tool to exclude lower-income
households
whose tax
payments will be less than the cost of providing the services they consume;
Hamilton
(1975) uses the term ‘fiscal zoning’ to describe these exclusionary
measures. But local zoning regulations
can also serve as a mechanism
to
control the composition
of a community’s
population
in an effort to increase
the production
of local public goods; Oates (1981) calls this ‘public-goods
zoning’. While this view may raise some difficult normative questions, it does
offer a provocative further explanation
of zoning.
Second, a substantial
body of literature beginning
with Tiebout (1956) has
argued that allowing people to choose their preferred level of local public
goods by ‘voting with their feet’ is efficiency enhancing.
However,
if
community
characteristics
enter local public goods production
functions, this
may not be true. It is plausible that for some goods, efficient consumption
while efficient production
may
may require
homogeneous
communities,
require heterogeneity;
overall efficiency would then require a balancing
of
these two conflicting objectives.3
Third, the theoretical
foundation
of most of the empirical work on the
determinants
of local public expenditures
becomes problematic
if community
characteristic
enter public goods production
functions.
Typically,
expenditures have been viewed as the outcome of a political process in which the
utility of the median voter is maximized
subject to a resource constraint.
Expenditures
are thus seen as a function of the median voter’s income, the
median voter’s tax price, and a set of environmental
characteristics
representing the jurisdiction’s
tastes. The tax price variable reflects the median voter’s
tax share and input prices. The Bradford, Malt and Oates argument suggests,
however, that this price variable represents the price of D-output
rather than
%ee Oates (1977, 1981) for a discussion
of these points.
R.M. Schwab and E.M. Zampelli, Demand for local public services
241
the price of C-output.4
But clearly it is the latter which is of prime
importance
since it is C-output,
and not D-output,
which enters the voter’s
utility function.
If this is true, virtually
all empirical
estimates
of the
determinants
of demand are based on models which have been misspecitied.5
In particular,
Hamilton
(1983) has convincingly
argued that failure to
recognize the dual role of socioeconomic
variables may lead to serious error
in estimating
the income elasticity of demand for public services. Suppose
income, both in its own right and as a proxy for community
characteristics,
enters the production
function
for some public service. If we mistakenly
ignore income’s role in the technology,
we would incorrectly
attribute
any
change in expenditure
resulting from a change in income to income’s impact
on demand.
Our goal in this paper is to disentangle
the role of income and the
environment
on the demand for publicly provided goods from their role in
the production
of these goods. The paper is organized as follows. Section 2
presents a theoretical framework which develops Hamilton’s
thesis in terms
of the income and price elasticities of demand and what we call the ‘income
elasticity
of the price of publicly
provided
goods’.
We develop
an
econometric
model which allows us to estimate the three elasticities
even
when the publicly produced output cannot be observed.
We apply our approach
to a study of the demand and supply of public
safety in a cross-section
of 73 communities.
The data and variables
are
described
in section 3 and the findings presented
in section 4. A brief
summary and conclusions
are offered in section 5.
2. Theoretical
framework
The significance of income’s dual role becomes apparent when comparing
the traditional
theoretical
model with one that follows from the framework
adopted by Bradford, Malt and Oates and by Hamilton.
Initially, we present
a model which is consistent
with Borcherding
and Deacon’s (1972) wellknown paper.
Consider a local public good which is produced under constant returns to
scale and assume income Y does not enter this good’s production
function.
The price of the good P, i.e. the cost of the resources required to produce
one unit of output, is therefore independent
of community
income and the
level of output. Demand Q is a function of income and price and expenditure
“If population,
to capture the effects of congestion
on the cost of local public goods, is
included as a determinant
of tax price then tax price is not strictly a measure of the price of Doutput. It remains, however, an inappropriate
measure of C-output.
‘See Dynarski, Schwab and Zampelli (1986) for a further discussion on this point.
248
R.M. Schwab and E.M. Zampelli, Demand for local public services
E is equal to price times quantity.
This simple model can be written
Q = Q(p, Y),
P=P(W,,
as
(1)
W,,...),
E=PQ,
(3)
where the w are the input prices. In this model it is clear that the income
elasticity of demand is equivalent to the income elasticity of expenditure.
Now consider a model in which income is not only a determinant
of
demand but also a factor in the production,
and therefore the price, of the
local public good. Retaining
the assumption
of constant
returns to scale in
the purchased inputs, we replace eq. (2) with
P=P(Y,
w,, w, )... ).
(2’)
Total differentiation
of (1) (2’), and (3), holding all exogenous
variables
other than income constant, yields the total effect of changes in income on
expenditure:
dlnE/dln
Y=alnQ/aln
Y+(l+i3lnQ/alnP)alnP/~lnY.
(4)
Eq. (4) can be interpreted
as follows. Suppose income rises by 1 percent.
Demand, and therefore expenditure,
rises by the income elasticity of demand
8ln Q/aln Y. Price changes by 8 In P/din Y the income elasticity of the price
of the publicly
provided
good. Expenditure
changes
by an additional
(1 + a In Q/a In P) (a In P/a In Y) in response to this price change.
Note that eq. (4) is a simple and convenient
way to represent Hamilton’s
explanation
of the flypaper effect and other anomalies.
If the public good is
normal,
demand
is price inelastic,
and the income elasticity
of price is
negative, then the income elasticity of expenditure
is less than the income
elasticity of demand.
Technology and demand
A major goal of this paper is to develop an econometric
model which
decomposes
the total effect of income on expenditures
along the lines of eq.
(4). If suitable measures of Q, i.e. C-output, were available we could develop
a very general model within the traditional
demand and supply framework.
Such a model would consist of a set of simultaneous
equations
specifying
cost, factor demands, and the demand for output. In such a model, very few
a priori restrictions
would have to be imposed on the structure of produc-
R.M. Schwab and E.M. Zampelli, Demand for local public services
249
This would allow us to address a number of interesting
issues such as
returns to scale in the provision of local public goods.
Unfortunately,
appropriate
measures of C-output
rarely (if ever) exist. Coutput is the citizen’s perception
of the quantity
and quality of the public
output. Therefore, by its very nature, it is subjective, multi-dimensional,
and
unobservable.
In some cases, measures of some elements of C-output
are available. For
example, test scores capture certain dimensions
of education
and various
crime rates are related to public safety. But for other aspects, no measures
exist. Even if one could quantify each component
it would be very difficult to
combine them in an overall index of C-output.
Moreover,
the available measures of the elements of C-output
are often
suspect. For example, measures
of output in virtually
all studies of the
production
of public safety are based on reported crimes. But as Carr-Hill
and Stern (1973) explain, this can lead to the perverse result that the
addition of police officers (increased inputs) raises the crime rate (decreased
output). Ceteris paribus, additional
police personnel
presumably
reduce the
number of crimes committed. However, they may also increase the fraction of
crimes that are reported and detected so that the reported crime rate rises.
These problems
strongly suggest the use of an approach
that does not
require an explicit measure of C-output.
We offer such an approach
below,
building on Hulten’s (1984) insightful and innovative
work on productivity
growth in the state and local sector.
Suppose all of the socioeconomic
variables, including
income, enter the
production
function
for Q as Hicks neutral
technological
change.
The
technology can then be represented as
tion6
Q=G(Z,,Z,
,.,. )-‘F(L,,L,
,... ),
(5)
where the Zi are the socioeconomic
variables and the Li are the purchased
inputs. Under constant returns to scale in the purchased inputs, unit cost can
be written as
P=G(Z,,Z,
,... )C(W,,W,
,... ).
(6)
In light of Sheppard’s lemma, which shows that the derivatives of the cost
function
equal the factor demands,
total differentiation
of C(W,, W,, . . )
yields:
dlnC=CsidlnV$,
I
(7)
where the si are each factor’s share of unit cost.
%I a very interesting paper, Baum (1986) uses this approach
public education.
to study the production
of local
250
R.M. Schwab and E.M. Zampelli, Demandfor local public services
Eq. (7) can be used as the basis for a Divisia index of input prices. We can
approximate
the difference in cost between city j and city k by replacing
logarithmic
differentials
with differences in logarithms
and using average
shares:
In Cj-lnCk=~+(~~+s$)(ln
W{--ln
Wf).
Diewert (1976) has shown that (8) is exact if C(W,, W,,. . .) is translog.
Divisia index can be computed by setting C equal to 1 for a base city.
We specify the environmental
variables’ role in production
as’
G(Z,,Z,,...
A
)=exp
The demand side of the model is characterized
by a linear expenditure
system, a specification
which appears frequently in the state and local public
finance literature.’
In this model, expenditure
E is a linear function
of an
intercept, ‘taste’ variables Xi, and the price of the local public good:9
E=a,+CyiXi+Ct,Y+a,’
I
=a,+
CyiXi+Z,Y+a,eXp
I
(. >
C/?iZi
C.
‘Two considerations
argue in favor of the functional
forms we have chosen. First, if price is
independent
of socioeconomic
characteristics,
then P will be equivalent
to the Divisia index.
Second, (9) and (10) allow us to identify income’s separate effects on cost and demand.
‘See lnman (1971), McGuire (1978), and Follain (1979) for example.
‘This notion of tax price can be extended two ways. First, communities
can reduce the cost of
local public goods by claiming local taxes as an itemized deduction
on their federal taxes.
Second, a fraction of local taxes are exported to non-residents,
e.g. businesses owned by nonresidents pay a part of the property tax. The deductibility
issue is potentially
troubling since it
implies that the price of local public goods is a decreasing function of income if either (i) the tax
system is progressive, or (ii) the probability
of itemizing is an increasing function of income (as it
would be if, for example, the probability
of homeownership
were an increasing
function
of
income). We chose to ignore this issue for two reasons. First, as Feldstein and Metcalf (1986)
have argued, it is not clear how deductibility
should be treated. It could be argued that since
only 30 percent of all taxpayers itemize and itemizers are likely to demand higher levels of local
public goods (because of their higher incomes and lower tax prices), the median voter mode1
suggests that deductibility
has no impact on local spending. Alternatively,
some studies have
assumed that the relevant price reflects the average tax rate in a community
(adjusted for the
fraction of voters who itemize). Second, the best available empirical evidence lends little support
to the hypothesis that deductibility
affects spending. In particular,
Feldstein and Metcalf (1986)
and lnman (1985) lind either no relationship
or a perverse relationship
between the level of local
taxes or the composition
of local taxes and the price of local public goods net of federal income
tax considerations.
R.M. Schwab and E.M. Zampelli, Demand for local public services
251
Eq. (10) decomposes
the total effect of income on expenditure
into its two
components.
Suppose income is Z,. Then (i) a In Q/aln Y is equal to LY~(
Y/E);
(ii) (1 + a In Q/a In P) is equal to c(,(P/E); and (iii) 8 In P/a In Y is equal to /I1 Y.
3. The data
Our study analyzes per capita expenditures
for police protection
in 73
cities and counties in fiscal 1978. We chose this sample in order to take
advantage of certain data described below.
Conceptually,
expenditures
in this model are the cost of purchasing
the
services of the factors of production
used in producing
public safety for one
year. They include total labor compensation
and the implicit rentals on the
police capital stock but specifically exclude capital expenditures.
As Inman
(1979) notes, capital expenditures
have almost no role to play in this type of
model.
Our data on labor compensation
are based on the U.S. Department
of
Justice survey of employment
and expenditures
in the criminal
justice
system.”
We impute the rentals on police vehicles as follows. The Police
Executive Research Forum (1978) and the Police Foundation
(1978) provide
data on the number of motorcycles,
motor scooters, patrol wagons, marked
cars, and unmarked
cars for each of the police departments
in our sample.
We developed estimates of the purchase price of each type of vehicle from
data supplied by the Montgomery
County, Maryland
police department.
We
assumed a cost of capital of 20 percent.”
Our wage rates for sworn and non-sworn
police personnel are also based
on the U.S. Justice Department
survey. The Divisia index thus includes three
factors of production
(two types of police personnel and vehicles), but only
the two wage rates vary across cities.”
We used a number of socioeconomic
variables in our study. Per capita
income, the unemployment
rate, and population
are calendar 1977 data. All
other socioeconomic
variables, including percent non-white, the homeownership rate, and the percent of adults that are high school graduates, are taken
from the 1980 U.S. Census.
We include per capita general revenue sharing and per capita state and
federal criminal justice grants to examine the impact of intergovernmental
transfers on police protection expenditures.
Given the limitations
of our data,
we are forced to assume that the criminal justice grants are either nonmatching or fully utilized matching grants.
“‘See U.S. Department
of Justice (1979) for a discussion of this survey.
“This figure assumes ‘one hoss shay’ depreciation,
a zero percent real interest rate, and a
useful life of 5 years.
“We assume that vehicles are sold in national markets and therefore have the same price in
all cities. Even though their price does not vary, vehicles must be included in the index to
maintain the correct factor shares.
252
4. Estimation
R.M. Schwab and E.M. Zampelli, Demand for local public services
and findings
In order to provide a benchmark,
we first estimate our model under the
traditional
assumption
that the price of the publicly
provided
good is
independent
of the community’s
income.’ 3 Following
Borcherding
and
Deacon (1972) we include population,
as a measure of the ‘publicness’ of the
good, as one of the determinants
of price. We argue below that there are
other interpretations
of the population
parameter.
The percent
of the
population
that is non-white
and the homeownership
rate are included as
taste variables.
Estimates of this model are presented in the first column of table 1. As
shown, the income slope is insignificant
and the income elasticity of demand
evaluated
at sample means is only 0.065. Therefore,
if we rely on the
traditional
model, we would conclude that the demand for public safety is
virtually unrelated to income.
There are several other implications
of the estimates presented in the first
column of table 1. First, the coefficient on general revenue sharing is very
troubling;
it is hard to believe that if a community
receives an additional
dollar of revenue sharing that it will spend an additional
$1.17 on police
protection.
It is difficult to explain this result, but we can offer the following
hypothesis. The formula for distributing
general revenue sharing favors the
older, larger, declining industrial central cities.i4 We might expect the economic
environment
in these cities (everything else equal) to lead to high crime rates,
i.e. the price of public safety in cities that receive disproportionately
large
amounts
of revenue sharing may be high. We consistently
found the price
elasticity
of the demand
for public safety to be low. Therefore,
general
revenue sharing may be positively
associated
with police expenditures
in
part because revenue sharing acts as a proxy for the environment.
We present
evidence below to support this hypothesis.
Second, the coefficient on population
in the price term is positive and
significant;
we obtain similar estimates
of this coefficient in virtually
all
versions of the model we examined. All of the variables in the price term
have been scaled by their means so that their coefficients equal the elasticities
of price when evaluated at the sample means; therefore a 1 percent increase
in population
is associated
with a 0.119 percent increase in the price of
public safety.
Borcherding
and Deacon would argue that this is implausible.
In their
framework, if the elasticity of price with respect to population
were - 1, then
we should conclude that the good is a pure public good; if the elasticity were
0, then we should conclude
that it is a pure private good. But in the
13All versions of the model are estimated using non-linear
specified in per capita terms, we assume that heteroscedasticity
‘%ee Nathan et al. (1975) for a discussion of this point.
least squares. Because the model is
is not a serious issue.
253
R.M. Schwab and E.M. Zampelli, Demand for local public services
Table 1
Parameter
(1)
estimates.”
(2)
(4)
(3)
(5)
Demand terms
Constant
Per capita
- 8.8793
(-0.41109)
income
0.000589
(0.35681)
- 64.741
(- 1.8058)
0.0091207
(2.4137)
-24.851
(- 1.0323)
- 62.522
(- 1.9298)
0.005344
(2.9382)
0.011002
(3.1701)
95.389
(1.9130)
- 0.0028261
( - 0.73565)
Price
40.079
(3.8105)
134.33
(1.8257)
39.664
(3.8786)
88.834
(2.0591)
6.9357
(0.91163)
Per capita trim.
just. grants
0.82871
(1.4488)
0.84161
(1.6726)
0.75006
(1.5520)
0.76218
(1.5625)
0.90735
(1.9121)
Per capita
sharing
1.1694
(4.3979)
1.1673
(4.5248)
1.0519
(4.4247)
1.0586
(4.1957)
0.67660
(2.9602)
0.29143
(2.0526)
0.28679
(2.1077)
0.11303
(0.85735)
0.12042
(0.98118)
revenue
oA non-white
-0.43519
y0 home ownership
( - 2.2600)
- 0.39201
( - 2.0062)
% high school
graduates
Unemployment
rate
- 0.93488
( - 3.2248)
-0.19989
-0.21703
(- 1.2154)
(- 1.0714)
- 0.75280
( - 2.094)
-0.55243
(-2.3415)
-0.53659
(-2.1370)
( - 0.58000)
-0.23015
2.9408
(3.1531)
2.8581
(3.1845)
-0.28013
(-0.14795)
0.11975
(2.5097)
_
0.11996
(2.5424)
0.12865
(3.6067)
- 0.79084
1.0848
(2.3683)
Cost terms
Population
Per capita
0.11891
(2.2028)
_
income
0.12599
(2.4316)
- 1.2089
( - 2.2469)
(- 1.7528)
y0 non-white
_
0.56195
(3.9282)
y0 home ownership
_
0.45009
(1.2035)
_
- 0.90374
(- 2.2085)
% high school
graduates
_
Unemployment
_
_
0.425 11
(1.5399)
Elasticities at sample means
Price elasticity
demand
of
-0.13953
-0.11675
Income elasticity
demand
of
0.065037
Income elasticity
price
of
0
- 1.2089
0
Income elasticity
expenditures
of
0.065037
- 0.0607
0.59013
Log-likelihood
value
-292.5815
“The dependent
variable
t-statistics are in parentheses.
1.0071
- 290.7786
in all equations
-0.13426
-0.14733
0.59013
1.2148
-0.31205
- 0.79084
-283.2158
is per capita
-0.29110
1.0848
0.53014
-281.8558
expenditures
0.45696
- 272.4459
on police
protection;
254
R.M. Schwab and E.M. Zampelli, Demand for local public services
Bradford, Malt and Oates framework this result is sensible. Population
is a
proxy for the propensity
to commit crimes as well as a measure of the
publicness
of public safety and therefore there is no basis to predict its
sign.’ 5
Third, percent non-white and the homeownership
rate are both significant;
these results are consistent with other studies.16 In this model we are forced
to interpret these variables as taste variables.
The estimates in the first column explicitly ignore the potential
effect of
income in the production
of public safety. In the second column we take
income’s dual role into account by including it in the price term.
The results are strikingly different. Income as a determinant
of demand is
now positive and significant;
income as a determinant
of price is negative
and significant.
The income elasticity of expenditure
at the sample means is -0.061.
We
can decompose income’s total effect in light of eq. (4). A 1 percent increase in
income increases demand, and therefore expenditure,
by 1.01 percent. It also
lowers the price of public safety by 1.21 percent; this lower price lowers
expenditure
by 1.07 percent given the price elasticity of -0.117.
Thus, the
total effect of income on expenditure
is roughly 0, as we concluded
from
column (l), but for an entirely different reason. The income elasticity
of
demand is not 0; instead, income’s effect on expenditure
through demand is
almost exactly offset by its effect on price. The estimated
coefficients
on
percent non-white
and the homeownership
rate are very similar in both
models.”
4.1. Extensions
of the basic model
The first two models we presented
included
only three socioeconomic
variables. We now introduce
two additional
variables, the percent of adults
that are high school graduates (a measure of education)
and the unemployment rate (a measure of the local economy).
We present a new estimate of the traditional
model where only population
enters the price term in column (3). Income is now significant; the income
elasticity of demand at the sample means is 0.590. Our education
variable is
’ 5Baumol (1967) argues that the price of many local public goods will rise as population
rises.
“See Bergstrom and Goodman
(1973) for example.
“It has been pointed out to us that the service responsibilities
of local governments
vary
widely, even for a service provided everywhere
such as police protection.
For example, state
police provide services in some areas which are provided
by city governments
in others.
Everything
else equal, we would expect to find that those communities
with greater responsibilities spend more on police protection.
A simple indicator of local responsibility
is the state’s
share of state and local police expenditures
in each community’s
state. If we add this variable to
the model summarized
in the second column of table 1, its coefficient is negative (as expected)
and significant. The sign, magnitude, and significance of the other coefficients are unchanged.
R.M. Schwab and E.M. Zampelli, Demand for local public services
255
negative
and significant
and the unemployment
rate is positive
and
significant.
We can explain the differences between the estimates in column (1) and
those in column (3) in terms of omitted variables bias. In column (1) we
exclude education
and unemployment
from the model and therefore force
income to serve as a proxy for these other socioeconomic
variables. Income
is positively
associated
with the education
variable
in our sample and
negatively
associated
with unemployment.
Omitting
the two new variables
therefore masks the effect of income acting in its own right inasmuch
as
decreases in education
and increases in unemployment
are associated with
to note that the
increases
in expenditure.
Here again, it is important
traditional
model would force us to conclude that well educated people have
a weaker taste, and the unemployed
a stronger taste, for public safety.
In column (4) we allow income to enter the price term and include the two
new variables as taste terms. Thus, the relationship
between the estimates in
columns (3) and (4) parallels the relationship
between the models in (1) and
(2). The results are not unexpected.
The income elasticity of price is still
negative and significant (at the 10 percent level), though somewhat smaller
than in column (2). The income elasticity of demand evaluated at the sample
means is 1.21 which is greater than the estimated elasticity in column (2)
(because of omitted variables bias) and column (3) (because of income’s effect
on price).
In column (5) we include population
and income as well as the four
socioeconomic
variables in the price term. The estimates in column (5) differ
markedly from our earlier findings. The coefficient on income as a demand
variable is negative but not significantly
different from 0. Income in the price
term is now positive and significant;
at the sample means, a 1 percent
increase in income raises the price of public safety by 1.08 percent. The
education
variable is negative and significant in the cost term; percent nonwhite is positive and significant. The ownership rate and unemployment
are
both positive in the price term but neither is significantly
different from 0.
Here again we can explain these results in terms of omitted variables bias.
Income plays two roles in the production
of public safety. In part, it acts as a
measure of citizens’ propensity
to commit crimes; as such, we would expect
increases in income to lower the cost of public safety. But as Ehrlich (1973)
and Carr-Hill
and Stern (1973) argue, increases in a community’s
income
also act as an inducement
to criminal
activity by increasing
its potential
return; all else equal, it is more profitable to commit a burglary in a wealthy
community
than in a poor one. In Carr-Hill
and Stern’s terminology,
it
raises the ‘swag’.’ *
‘*See McPheters
point.
and Stronge
(1974) and Popp
and Sebold (1972) for further
evidence
on this
256
R.M. Schwab and E.M. Zampelli, Demand for local public services
When including only income and population
as determinants
of the price
of public safety, as in columns
(2) and (4), the effect of income on the
propensity
to commit crimes dominates
its effect on the return to criminal
activity, and we find that the income elasticity of price is negative. However,
if we include other variables such as education which serve as better proxies
for the propensity
to commit crime, as in column (5), we are left primarily
with income’s effect on the potential
gains from breaking
the law. The
income elasticity of price is therefore positive.
It is interesting to note that education and unemployment
are insignificant
as taste variables in column (5) and therefore the traditional
model apparently
leads to a mistaken interpretation
of their role. Education is negatively related
to expenditures
on police protection,
not because well educated people care
less about public safety, but because they reduce the cost of providing public
safety.
Finally,
when the additional
socioeconomic
variables
are included
the
coefficient on revenue sharing falls to 0.677 compared to values greater than
1 in all other versions of the model. We interpret this pattern as support for
our hypothesis that revenue sharing funds are positively related to the price
of public safety; as we introduce
other determinants
of price such as
education
we come closer to purging the revenue sharing variable of these
effects, though the coefficient remains implausibly
large.
4.2. An assessment
of alternative
interpretations
In this paper we have attempted to estimate the production
technology for
local public goods, even though
the output
of these goods cannot
be
measured. We have dealt with this problem by specifying a cost function in
eqs. (6) and (9) and a demand function in the first line of (10) which together
lead to an estimating
equation
in the second line of (10) that allows us to
separate the effects of the environmental
variables on demand from their
effects on cost.
One reasonable
criticism of our approach
is to note that this estimating
equation is consistent
with other demand and cost equations
and therefore
the interpretation
of our results turns on our assumptions
of a linear
expenditure
equation
and a Hicks neutral
technology;
if either of these
assumptions
is incorrect,
then our conclusions
about the nature
of the
demand and supply of local public goods are also incorrect. In an extreme
case, one might argue that income and the other environmental
variables do
not enter the production
function and that we have simply stumbled onto a
better functional form for the demand equation.
At some very fundamental
level, it will never be possible to fully address
this particular
criticism. Since we cannot measure output, we will never be
able to offer any direct evidence on the technology, i.e. we can never directly
R.M. Schwab and E.M. Zampelli, Demand for local
public services
251
estimate the production
function
or the cost function. We will always be
forced to rely on indirect evidence.
This problem, however, is not unique to our study. Consider the evidence
on the congestability
of local public goods, a topic that has received a good
deal of attention in the local public finance literature. Following Borcherding
and Deacon (1972) and Bergstrom
and Goodman
(1973), studies typically
specify the per capita consumption
of a local public good as aggregate
production
divided by population
raised to some parameter CLIf c1is 0, then
the good is a pure public good; if c( is 1, then it is a pure private good. Since
consumption
cannot be observed, u is usually estimated
by specifying an
expenditure
equation
in which the coefficient on population
can be interpreted as a function
of this crowding
parameter
and the price elasticity.
Estimates of LXtypically center on 1, and the conclusion
that local public
goods are in fact private goods in widely cited.
This conclusion,
however, is entirely dependent
on the specification
of the
technology
and the form of the expenditure
equation.
Edwards (1985) has
recently explored a variety of functional
forms for the congestion
relationship; he finds that the extent of crowding is quite sensitive to the choice of
functional
form. Moreover, other interpretations
of the relationship
between
per capita spending and population
can be offered. Oates (forthcoming),
for
example, has argued that past estimates are evidence that large cities provide
a wider range of public services. Alternatively,
Borcherding,
Bush and Spann
(1977) suggest that bureaucratic
power is likely to be positively correlated
with population
so that the observed
relationship
can be explained
by
Niskanen-type
behavior. Since no one can measure local public goods, it will
be very difficult to find an approach that avoids these problems. We would
argue that, in general, the same is true for all other environmental
factors,
including income.
In our particular
study, we can respond to this line of argument
in two
ways. First, the interpretation
of many
of our results is much more
straightforward
if income and the other socioeconomic
variables are allowed
to enter the production
function. For example, it is more plausible to argue
that the cost of providing
public safety is lower in communities
where the
population
is well educated and the unemployment
rate is low than to argue
that poorly educated people and the unemployed
have a greater preference
for public safety.
Second, reported
crimes are often used as a measure of public safety
though, as we argued above, they are a highly imperfect measure. If we are
willing to put aside these shortcomings
for the moment, then we can offer the
following test of our results. We can construct
alternative
measures of the
‘productivity’
of the environment
for each of the communities
in our sample
by evaluating
eq. (9) using estimates of the p’s from table 1. Note that larger
values of this index imply a less productive environment,
while smaller values
258
R.M. Schwab and E.M. Zampelli, Demand for local public services
imply a more productive
environment.
We can construct
a measure
of
purchased inputs by deflating expenditures
in each community
by the index
of factor prices, and we can construct
a measure of the required input per
unit of output by dividing this input index by the ‘safety’ rate, i.e. the inverse
of the (reported) crime rate. If the approach taken in this paper is sensible,
then we should find that an index of the environment
which includes income
and other socioeconomic
variables would better explain variations
in this
input-output
ratio than would an index based solely on population.
The results are shown in table 2. The dependent
variable is the log of our
input-output
ratio. The independent
variable in column (3’) is an environmental index based on estimates in column (3) of table 1 and which therefore
only includes
population.
This index is unable
to explain why different
communities
are required to purchase different amounts of inputs to produce
a unit of output. The slope coefficient has the correct sign but is insignificant;
R2 is only 0.036.
An index that includes income as well as population
performs much better.
The independent
variable in (4’) is based on the coefficients in column (4) of
table 1. The slope coefficient in this equation
has the correct sign and is
significantly
different from zero; R2 rises to 0.201. The index in (5’) is based
on the estimates in column (5) of table 1 and therefore reflects a wide range
of socioeconomic
variables as well as populations
and income. Here again
the slope coefficient is significant and RZ rises further.lg
We recognize that this test is less than perfect. For example, it suffers from
an errors-in-variables
problem and our measure of output leaves much to be
desired. We would argue, however, that it represents
evidence
that our
approach is a promising way to begin to understand
the environment’s
role
in the production
of local public goods.”
Table 2
Community
characteristics
and
0utput.a
(3’)
Slope
Intercept
1.1038)
(1.6214)
1.2062
(11.868)
R2
“t-statistics
0.0360
input
(4’)
1.3686
(4.2193)
per
unit
of
(5’)
(0.6746)
(5.4493)
2.2663
(10.002)
0.1602
(0.7204)
0.2005
0.2949
in parentheses.
t9We have performed a non-nested
test of the hypothesis
that the index in (5’) is the correct
measure of productivity;
we are unable to reject this hypothesis. In contrast, a non-nested test of
the hypothesis that the index in (3’) is easily rejected. Details of these tests are available upon
request.
“We have done a similar analysis using reported crime rates for specific crimes (e.g. murder,
robbery) rather than the total crime rate as a measure of output. The results are very similar to
those reported in table 2 and are available upon request.
R.M. Schwab and E.M. Zampelli, Demand for local public services
259
5. Summary and conclusions
In this paper we develop a theoretical
framework
and an econometric
model to separate the effect of income on the demand for publicly provided
goods from its effect on the price of those goods. We apply our approach to
a study of expenditures
for police protection
in a cross-section
of 73 cities
and counties.
We believe the results provide strong support for the Bradford, Malt and
Oates, and Hamilton
hypotheses. Studies of public expenditures
which fail to
incorporate
income
and other socioeconomic
variables
correctly
in the
production
function can yield very misleading results.
This paper is certainly not the final word on the demand and production
of local public goods. Some of the results of our study are troubling,
and
other studies that examine other services using other specifications
may well
come to different conclusions.
However, we believe we have established
a
strong case that the issues and questions
raised in this paper should no
longer be ignored in the public expenditure
literature.
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