Are municipalities Tieboutian clubs?

ELSEVIER Regional Science and Urban Economics 26 (1996) 203-226

Are municipalities Tieboutian clubs?

Eric J. Heikkila

School of Urban and Regional Planning, University of Southern California, Los Angeles, CA 90089-0042, USA

Received 1 January 1993; revised version received 6 August 1995

Abstract

This paper seeks empirical confirmation of the hypothesis that municipalities are Tieboutian clubs. Using data from the 1990 Census for Los Angeles County, the work proceeds in three stages. First, factor analysis identifies the basis vectors to describe census tracts. Secondly, an analysis of variance for each factor shows that municipal boundaries reinforce club distinctions along four dimensions: urban scale, ethnicity, household type, and economic class. Finally, using cluster analysis we conclude that the structure of clubs here is highly fragmented. We conclude that municipalities are indeed Tieboutian clubs, although there may be further spatial clustering at a higher level of aggregation.

Keywords: Tiebout; Club theory; Urban; Urban structure; Los Angeles

JEL classification: R13; R14

1. Introduct ion

This paper conducts an exploration into the structure of one of North America 's most complex and fascinating urban areas, Los Angeles. Of particular interest in this enquiry is the presence of a myriad of communities with their own distinctive characters. With well over 100 local governments, Los Angeles County seems to epitomize a Tieboutian world in which resident consumers of local public goods flock to those communities that offer the vector of attributes that best suit their own preferences. And as Tiebout (1956, fn. 12) himself noted: "not only is the consumer-voter

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204 E.J. Heikkila / Regional Science and Urban Economics 26 (1996) 203-226

concerned with economic patterns, but he desires, for example, to associate with 'nice' people. Again, the greater the number of communities, the closer he will come to satisfying his total preference function, which includes non-economic variables." An important implication of the Tiebout model is the prediction that communities will be relatively homogeneous with respect to the preferences of their residents for these economic and non-economic local public goods. The purpose of this paper is to provide some empirical insight into the outcome of this process in contemporary Los Angeles.

This study differs from earlier works in several important ways. First, rather than focusing on a single variable such as income, our analysis examines a broad range of socio-economic data taken from the 1990 Census. To do this we employ a statistical technique, factor analysis, that is not often used by economists but that has wide currency in other social sciences. Factor analysis is used to reduce a dataset of 64 variables to a more manageable 16 composite variables, or factors, where each factor represents a distinct aspect of the socio-economic fabric of the county. Taken together, these 16 factors provide a rich descriptive basis for the study area. Second, this paper actively relates its findings to their geographic context to gain insight into the spatial clustering phenomenon that is central to Tiebout's hypothesis. Third, this paper addresses the issue of jurisdictional homogeneity, an aspect of the Tiebout model that is not always considered. We apply an analysis of variance test to determine whether heterogeneity between municipalities exceeds that within municipalities. The test is similar in spirit to earlier work done by Eberts and Gronberg (1981) and Munley (1982), although it differs in methodology. A final distinctive aspect of this paper is the use of cluster analysis to assess the degree of fragmentation within the region. We find that, on balance, there is strong evidence to support the claim that municipalities are Tieboutian clubs, but that this evidence is not conclusive. There is clearly strong evidence of a tendency for similar characteristics to cluster together geographically, and intra- munici- pal homogeneity signficantly exceeds inter-municipal homogeneity. How- ever, this evidence may also be consistent with a tendency for like groups to cluster at higher levels of aggregation. 1

The paper proceeds in several related stages. Following a literature review in the Section 2, Section 3 describes the data that are used in the factor analysis in Section 4. The method outlined there is similar to that employed by Heikkila (1992), but with two important differences. First, Heikkila (1992) uses these techniques for simple descriptive purposes and does not seek to establish links to club theory or to employ any statistical tests of the sort developed here. Secondly, the empirical application in the earlier paper

1 I am indebted to an anonymous referee for clarifying this point, which is developed more extensively in Section 6.

E.J. Heikkila / Regional Science and Urban Economics 26 (1996) 203-226 205

uses data that are far too aggregated to be of use here. Accordingly, the data for this paper are at the census tract level. Factor scores are mapped in Section 5 to assess the geographic distribution of selected factors. The analysis of variance in Section 6 tests for relatively homogeneous dis- tributions of each factor within municipalities. Section 7 examines the degree of fragmentation in the clustering of like census tracts. The concluding section assesses the question framed in the title, 'Are municipalities Tieboutian clubs?', in light of our findings.

2. Tieboutian clubs

The seminal work by Tiebout (1956) on local public goods has spawned a profusion of theoretical and empirical articles seeking to expand upon his original insight into the structure of metroplitan areas. Tiebout's paper was a response to Samuelson's (1954) observation that the marginal efficiency conditions for a pure public good violate the conditions that characterize competitive market equilibria. Tiebout showed that a quasi-spatial adjust- ment mechanism (what has since been termed 'voting with one's feet') could, under certain conditions, lead to equilibria that satisfy the efficiency conditions for a local public good. In a Tieboutian world households with similar preferences congregate in municipalities that cater for these preferences at competitive costs. An irresistible metaphor is a giant shopping mall with countless 'municipality boutiques' offering specialized products or services for a well-defined clientele. Indeed, Tieboutian economics has many parallels with the general economic literature on multiproduct firms with fixed plant costs (Brueckner and Lee, 1988), and Tiebout (1956) himself wrote that: "there is no way in which the consumer can avoid revealing his preferences in a spatial economy. Spatial mobility provides the local public- goods counterpart to the private market's shopping trip."

The Tieboutian model has subsequently been subsumed by a broader class of models, called club theory. As described by Hochman et al. (1995), club theory is concerned with the formation of optimal consumption (sharing) and production groups, their characteristics, and their sustainability under a competitive market regime. Alternatively, according to Cornes and Sandier (1986), a club is defined as a voluntary group deriving mutual and excludable benefit from sharing production costs, members' characteristics, and/or excludable goods. Viewed in this light the Tiebout model is akin to situations in which individuals are partitioned into homogeneous clubs.

The Tiebout literature has moved in several interesting directions. One strand seeks to identify the conditions under which optimal clubs are mixed rather than homogeneous in composition (Berglas, 1976, 1984; Brueckner and Lee, 1989). Most articles look to characterize the efficiency conditions


of club systems. Cornes and Sandler (1986) present a taxonomy of eight distinct club categories on the basis of whether a club membership is homogeneous or not, partitioned (non-overlapping) or not, and whether the utilization of the club good is fixed or variable. What is conspicuously missing from almost all of this, including the original Tiebout article, is the spatial element. To paraphrase Hochman et al. (1995), the literature is replete with 'flying clubs'. That is to say, they are not rooted in geography but are instead represented in an abstract quasi-space. Hochman et al. (1995) follow Fujita (1989) and Hochman (1982) in placing clubs along a one-dimensional strip of land of finite length. They represent urban structure as a system of independent spatial-club agglomerations at specific locations along the strip of land. By doing so they neatly merge the local public goods tradition of Tiebout with an economic theory of central places as expanded upon by Eaton and Lipsey (1982) and others.

Another strand of the Tiebout literature is concerned with formulating and testing hypotheses stemming from the Tiebout model. Initially, following the famous article by Oates (1969), authors focused on detecting the Tiebout mechanism through capitalization effects. The reasoning behind this approach is that, in keeping with Tiebout's second assumption, individuals are assumed to have full knowledge of differences in revenue and expenditure patterns offered by competing local governments. Consequently, property values in localities offering attractive revenue-expenditure packages would capitalize the net benefits of these packages. Epple et al. (1978) commented on a host of papers preceding theirs by demonstrating, using a theoretical model, that it is exceedingly difficult to derive econometrically testable results through capitalization effects. They concluded that no one had yet successfully implemented such a test, i.e. one that would necessarily entail joint estimation of the set of structural equations that determine housing purchases and locational choice.

Following this, Eberts and Gronberg (1981) argued that there are really two Tiebout hypotheses 2 namely the capitalization effect discussed above, and homogeneity within jurisdictions. They test for homogeneity by using an entropy measure of income developed by Theil (1967). This particular measure can be decomposed into two additive components, inequality within (w) jurisdictions and inequality between (b) jurisdictions. They then take the ratio w/b as a measure of the relative extent of heterogeneity within jurisdictions for 33 standard metropolitan statistical areas (SMSAs) using data from the 1970 Census. This measure is regressed on SMSA size, number of municipalities, and other explanatory variables. They find empirical confirmation of the Tiebout homogeneity hypothesis by showing

2 In their recent survey of empirical literature in the Tiebout tradition, Dowding et al. (1994) identify 11 testable implications stemming from the Tiebout model.


that the percentage of within-district inequality decreases with the number of local jurisdictions (school districts) within SMSAs. Munley's (1982) approach is similar in spirit to that of Eberts and Gronberg (1981), in that he also regresses a measure of heterogeneity on, among other variables, the number of local jurisdictions within an area, and he too finds corroborat ion for the Tiebout 'homogeneity hypothesis' . Dowding et al. (1994) claim that these results are consistent with the null hypothesis because a greater number of subsets for a given population will automatically lead to increased homogenei ty within each subset, even if the sorting mechanism is purely random, i.e. even in the absence of any Tiebout effect. A third paper in a somewhat similar vein is by Grubb (1982). He uses data from the 1960 and 1970 Censuses for the Boston SMSA to test whether communities are moving towards increasing or decreasing homogeneity. The results obtained were mixed. As noted in the introduction, this paper is best placed within the context of this latter group of papers that address the Tiebout homogeneity hypothesis.

3. The data

The basic objective of the next three sections is (i) to characterize each census tract in terms of a broad set of descriptive variables, (ii) to identify, without reference to location, sets of census tracts that would appear to be similar and treat these sets as candidate 'clubs', and (iii) to compare the spatial manifestation of these clubs with municipal boundaries. Census tracts are convenient spatial units to employ here. Our data set details N = 1890 census tracts within Los Angeles County, compared with 129 municipalities. Census tracts carry the obvious advantage of having a wealth of data at tached to them and are sufficiently small and numerous that they could easily coalesce into municipal-like format ions-or not. Table 1 reports the means and standard deviations for the 64 variables used in this analysis and that fall under the following general headings:

• age • education • ethnicity • gender • occupation • previous residence • industry • housing tenure • mode of commuting • commuting time • language • urbanization • household type • type of dwelling • income group • household finance

208

Table 1 Variable

E.J. Heikkila / Regional Science and Urban Economics 26 (1996) 203-226

names and definitions

Variable Definition Mean Standard name deviation

% Persons by age

A G E T 0 1 4 Up to 14 years old 21.4 8.7 AGE1524 15 to 24 years old 15.3 7.5 AGE2539 25 to 39 years old 27.6 8.3 AGE4064 40 to 64 years old 25.1 9.6

% Persons by ethnicity

W H T N H White non-Hispanic 44.7 32.0 HISP Hispanic 33.8 27.6 B L K N H Black non-Hispanic 10.7 19.7 CHN Ethnic Chinese 2.8 6.8 J A P A N Ethnic Japanese 1.7 4.0 K O R E A Ethnic Korean 1.5 3.4

% Persons aged 16+ by occupation (by place of residence)

CLERIC C R A F T D E L I V F A R M HOUS M A C H I N M A N A G PROF P R O T E C S A L E S SERV TECHN

Administrative support including clerical 17.5 7.8 Precision production, crafts and repair 11.0 6.4 Transportation and material moving 3.4 3.0 Farming, forestry, and fishing 1.4 3.3 Private household services 1.0 1.6 Machine operators, assemblers and inspectors 8.3 8.6 Executive, administrative and managerial 13.1 8.4 Professional speciality 14.1 10.0 Protective services 1.7 3.1 Sales 11.5 6.0 Other service occupations 37.3 12.1 Technicians and related support 3.4 3.1

% Persons aged 16+ by industry (by place of residence)

FIRE Finance, insurance and real estate 7.6 5.1 PRIM Agriculture, forestry, fisheries and mining 1.7 4.3 SECON Manufacturing and construction 26.4 11.7 T R A D E Wholesale and retail trade 20.0 7.1 TRCOM Transportation and Communication 7.0 4.8

% Persons by transportation mode to work

D R A L O N Drive alone 70.0 15.5 PUB Public transportation 6.4 8.9

% Person by language spoken at home

E N G L I S H English 58.0 25.0

% Persons by household type

M A R C H Married with children 27.6 14.2 M A R N C H Married no children 25.2 12.8 FHOUSCH Female household with children 6.9 6.7 FHOUSNCH Female household no children 6.3 5.7 MHOUSCH Male household with children 2.3 3.4 MHOUSNCH Male household no children 3.2 2.8


Table I (continued)

Variable Definition Mean Standard name deviation

% Persons by income group

INC-LTIO Less than $10,000 12.4 10.2 INCIO-19 $10,000 to $19,000 14.5 8.8 INC20-29 $20,000 to $29,000 14.7 8.0 INC30-39 $30,000 to $39,000 13.6 8.3 INC40-74 $40,000 to $74,000 28.5 12.5 INC75-99 $75,000 to $99,000 7.8 7.9 HO0-ISO $100,000 to $150,000 5.1 7.2

% Persons aged 18+ by education attained

N OC OL L Low education 51.0 22.2 POST-SEC High education 21.8 16.8

% Persons by sex

PERM Male 49.7 6.2

% Persons by residence in 1985

FORCNTR Foreign country 7.1 6.6 SCOUNT Same county 35.1 11.4 S S T A T Same state 3.9 5.1 SHOUSE Same house 49.0 15.2

% Households by housing tenure

RENTER Rental accommodation 44.0 26.4

% Persons by travel time to work

TIMEOol4 0 to 14 min 21.8 10.1 TM15-29 15 to 29 min 34.7 9.2 TM30-59 30 to 59 min 32.4 9.9

% Households by type of location

UINURB Urban inside urban area 98.0 13.3 UOUTURB Urban outside urban area 0.1 3.2 R F A R M Rural farm 0.0 0.3 R N F A R M Rural non-farm 1.9 12.8

% Households by units in structure

ONED One unit detached 55.4 30.6

Other household indicators

MEDINC Median household income 39412.8 19817.4 PCINC Per capita income 17475.0 14029.6 MEDVAL Median home value 232352.0 121520.1 M G R E N T Median gross rent 658.2 220.7 PERSPH Persons per household 3.4 6.0


While this data set is quite complete from a census standpoint, it lacks information on local public goods. There are several reasons for this, the most compelling being that such data are not available by census tract, but would instead tend to be kept by jurisdiction. A second reason has to do with the nature of the hypothesis that we are testing. To include data by municipalities would tend to skew our results in favor of finding clubs that coincide with municipal boundaries. It would be more compelling to arrive at that conclusion using census-level data alone. Finally, while the original Tiebout hypothesis was formulated in terms of local public goods, club theory has since progressed beyond that. In particular, a number of authors have developed club models where members consume both the shared good and the characteristics or attributes of the other members. These are often referred to as discriminatory clubs (see, for example, Cornes and Sandier, 1986; De Serpa, 1977; Ng and Tollison, 1974; Tollison, 1972). The analogy to non-economic clubs is most appropriate, where the membership of a club is often the primary attractive force for new members. Moreover, peer effects have been shown to dominate the production function of important local public goods like education (Arnott and Rowse, 1987; Dynarski et al., 1989) or safety from crime (Heikkila and Craig, 1990). The importance of peer effects has also been recognized in recent theoretical developments in club theory (Brueckner and Lee, 1989).

4. Factor analysis

A central premise of this paper is that a proper test of the Tiebout hypothesis must encompass a broad range of socio-economic characteristics. Studies that focus solely on income cannot hope to capture the richness or complexity of large diverse metropolitan areas. Certainly income is only one of many dimensions that residents in Los Angeles, for example, must consider in making their residential location decisions. While the 64 variables described in the preceding section provide a fairly extensive descriptive basis of our study area, the resulting data set is also a bit unwieldy.

Factor analysis is a well-known technique to reduce the dimensionality of a data set, in this case from 64 to some smaller number J by producing composite variables called factors that represent patterns of covariance appearing within the original data set (Harmon, 1976). More formally, it is posited that each variable V~ can be represented as a linear combination of the J common factors plus a unique component Ui (Kim and Mueller, 1978):

V~=~,bqFj+U~, i=1 ,2 , . . . , 64 and j = l .... , J , J

(1)


or, in matrix algebra notation:

V = B F + U . (2)

The set of estimated coefficients b~j comprises the factor pattern matrix B, which in turn is one of the primary outputs from a factor analysis. The J factors (in our case, as reported below, J = 16) define a J-dimensional vector space that is a subset of the original N-dimensional vector space.

Table 2 summarizes the results of the factor analysis. The first column lists the factors in the order that they were extracted- in declining order of the eigenvalues shown in the second column. An eigenvalue in this context is interpreted as the amount of variance explained by the factor in question, where each of the original variables is normalized with unit variance. Thus, for example, factor one explains variance equivalent to that of 12.9 variables f rom the original data set. The 16 factors retained jointly, explain over 70% of the variance attributable to the original 64 variables. J

The third and the fourth columns of Table 2 ( 'factor label' and 'associated variables' respectively) work in conjunction with one another. Specifically, the third column contains factor labels, or names, that reflect the patterns of association detailed in the fourth column. These names are, of course, provided by the user and not by the computer program that generates the factor pattern matrix. A factor label summarizes what group of variables are most closely associated with the factor in question. In this application, the factor labels in the third column are based on the variables listed in the fourth column.

The fourth column lists those variables from the original data set that are most closely correlated, positively or negatively, with each of the retained factors. Specifically, for any row corresponding to factor j in Table 2, the number in parentheses next to any variable i in the fourth column indicates the coefficient b~j from the factor pattern matrix. Table 2 reports all variables for which b~j > 0.5 in absolute value. By examining which variables appear in the fourth column we can find interpretations for each of the factors, which in turn can be thought of as composite variables. 4 The factor label in the third column is a name, or label, that summarizes our interpretat ion. Thus, the factor analysis not only provides a way to reduce the data set, but also provides insight into the patterns of correlation between the original set of descriptive variables. This in turn can help us to

3 Following s tandard procedure, only factors with an eigenvalue of 1.0 or greater were re ta ined, where an eigenvalue is an est imate of the variance explained by the corresponding factor, and where 1.0 is the amoun t of variance explained on average by any one of the original variables. The total of retained eigenvalues in this case yields a final communal i ty es t imate of 45.33, thus explaining over 70% (=45 .33 /64 .0) of the variance within the original data set.

4 As we show below, each factor is actually a linear combinat ion of the communa l componen t s of the original variables.


Table 2 Factor analysis and analysis of variance results

Factor Eigen- Factor label Associated variables ANOVA value F-stat

One 12.9 White-collar elite (+) POST-SEC (+0.90); MEDVAL (+0.80); 8.14"* PROF (+0.80); PCINC (+0.78); MANAG (+0.76); WHTNH (+0.73); MEDINC (+0.72); 1100-150 (+0.65); SERV (+0.64); ENGLISH (0,63); AGE4064 (+0.59); FIRE (+0.57)

Hispanic labor ( - ) NOCOLL (-0.87); HISP (-0.78); SECON (-0.72); MACHINE (-.072); AGET014 (-0.64); CRAFT (0.62); DELIV (-0.55)

Two 5.6 Maids (+) PUB (+0.73); HOUS (+0.65); SFORCNTR 6.26** (+0.65); RENTER (+0.60); 1NCI0-19 (+0.56); INC-LTIO (+0.56)

Suburbia ( - ) DRALON (-0.68); 1NC40-74 (-0.68); ONED (-0.53)

Three 3.1 Early middle age ( - ) SFHOUSE (-0.84); AGE2539 (-0.72); 2.59* SCOUNT (-0.76)

Four 2.4 Rural-urban (+) RNFARM (+0.93); RFARM (+0.64); 6.44** UINURB (-0.92)

Five 2.4 Marriedwithchildren(+) MARCH (+0.79) 4.28**

Six 2.3 Single black mothers (+) BLKNH (+0.73); FHOUSCH (+0.59) 3.81"*

Seven 2.1 Clerical networks (+) CLERIC (+0.81); TRCOM (+0.72) 2.62*

Eight 2.1 Countryside (+) PRIM (+0.92); FARM (0.89) 1.57"

Nine 1.9 Youngadults(+) SSTAT (+0.67); TIMEO-14 (+0.57); 2.30* AGEI524 (+0.56)

Ten 1.8 Retail(+) TRADE (+0.83); SALES (+0.68) 2.01"

Eleven 1.6 Serious commuters (+) TM30-59 (+0.87); TM15-29 (-0.61) 2.38*

Twelve 1.6 Barracks (+) PERSPH (+0.81); PERM (+0.68) 1.09

Thirteen 1.5 East Asians (+) JAPAN (+0.70); CHN (+0.56) 7.55**

Fourteen 1.5 Working middle class (+) INC30-39 (+0.80) 1.56"

Fifteen 1.3 Protective services (+) PROTEC (+0.80) 1.75"

Sixteen 1.2 Small town (+) UOUTURB (+0.71) 19.02"*

Note: Final communality estimate (total variance explained, or sum of eigenvalues, for sixteen retained factors): 45.33.

* Significant at the 1% level ( F > F c = 1.53). ** F > 3.0 (as discussed in text).

b e t t e r u n d e r s t a n d t h e s o c i o - e c o n o m i c c o m p o s i t i o n o f t h e s t u d y a r e a . T h e

f i na l c o l u m n o f T a b l e 2 s u m m a r i z e s t h e r e s u l t s o f t h e a n a l y s i s o f v a r i a n c e

t h a t is r e p o r t e d u p o n in S e c t i o n 6.

T h e f i r s t f a c t o r l i s t e d in T a b l e 2 p o i n t s t o a l a r g e g r o u p o f h i g h l y


intercorrelated variables and, as indicated by the corresponding eigenvalue, accounts for almost 13 variables' worth of variance. 5 The variables that are positively correlated with factor one include post-secondary education (POST-SEC), median value of homes (MEDVAL), professional occupations (PROF), per capita income (PCINC), managerial or executive occupations (MANAG), white non-hispanics (WHTNH), median income (MEDINC), annual household income above $100,000 (1100-150), service occupations (SERV), English speaking (ENGLISH), upper middle age (AGE4064), and finance, insurance and real estate occupations (FIRE). Factor one points to and is representative of this constellation of intercorrelated variables. In the next section, factor scores are defined as the values associated with factors, where the factors are themselves composite variables derived from the original data set. With this definition in mind, Table 2 tells us that those census tracts that have higher (lower) than average scores for factor one also tend to have higher (lower) than average scores for POST-SEC and each of the other variables listed above. Thus, factor one points to census tracts whose residents are well-to-do white professionals, hence the term 'white- collar elite'. In the following section the geographic distribution of factor scores for this factor is represented in mapped format.

Factor one points also to a second group of variables that are positively correlated with each other but negatively correlated with factor one and with the group of variables listed in the previous paragraph. This second constellation of variables, which we have labelled 'hispanic labor ' , points to census tracts that are characterized by low education (NOCOLL), hispanics (HISP), manufacturing and construction-related employment (SECON), machine operators (MACHIN), young children (AGETO14), crafts (CRAFT) and delivery services (DELIV). Census tracts that have lower (higher) than average scores for factor one will tend to have higher (lower) than average scores for these variables. Note that there is no special significance attached to positive vs. negative correlation in this context. Factor one might just as easily have been defined in terms of its negative. If this were indeed the case, then the 16 factors would still span the same vector space and none of the results reported here would differ in any significant way. In either event, we see that 'hispanic labor' and 'white-collar elite' appear to fall at opposite ends of the socio-economic spectrum.

Factor two is called 'maids' in its positive aspect and 'Suburbia' in its negative aspect, and is a discernible echo of the socio-economic dichotomy alluded to by factor one. 'Maids' has positive loadings for use of public t ransportat ion (PUB), provision of household services (HOUS), recent

5 The impor tance of the large eigenvalue (12.9) should not be overstated. It tells us that the original data set contains many variables that are highly intercorrelated. That in itself does not signify that this factor is in some sense 'more impor tant ' than others.


immigrant (FORCNTR), renters (RENTER), and lower-income groups (INC10-19; INC-LTIO). Meanwhile, 'suburbia' has positive loadings on single-family dwelling units (ONED), middle-income ranges (INC40-74) and drive-alone commuters (DRALON). Unlike factor one, no ethnic correlates appear here.

In a similar manner, each of the 16 retained factors is examined in terms of its correlates and is named accordingly. Each of the names is necessarily arbitrary, and the reader is of course free to examine the data in Table 2 to determine whether some other names are more to his or her fancy. In each case the name chosen should convey some sense of what the correlates are. As indicated in Table 2, the factors are quite wide-ranging in their interpretations, including 'early middle age' (factor three), ' rura l -urban ' (factor four), 'married with children' (factor five), 'single black mothers ' (factor six), 'clerical networks' (factor seven), 'countryside' (factor eight), 'young adults' (factor nine), 'retail ' (factor 10), 'serious commuters ' (factor 11), 'barracks' (factor 12), 'East Asians' (factor 13), 'working middle class' (factor 14), 'protective services' (factor 15) and 'small town' (factor 16). Taken as a whole, these 16 factors provide a rich basis for describing what is undeniably a very complex urban region. The next step is to see how these factors manifest themselves spatially for evidence of Tiebout-type clustering.

5. Geographical distribution of factor scores

Factors are, in effect, composite variables constructed from the original data set. From Eq. (2) it follows immediately that

F= B- I (v - U), (3)

which states that factors are constructed by applying the inverse of the factor pat tern matrix to the communal components of the original variables. Just as the original variables have values for each census tract, so too do the composite factors. These factor scores are routinely calculated by many commonly used statistical packages and are presented here in mapped format (Figs. 1 and 2) for the first two factors. There are three shades in each thematic map. The darkest shade indicates a factor score greater than one, while the lightest shade indicates a factor score less than minus one. Because the original set of variables was converted to standardized form with unit variance and zero means, the maps indicate those census tracts that are estimated to have factor scores at least one standard deviation above or below the mean. For the purpose of this analysis, we treat the set of outliers (i.e. those census tracts with factor scores exceeding one in absolute value) for each factor as candidate clubs.


White Collar Elite / Hispanic Labor

• -2.63 to -1.00 [ ] -1.00 to 1.00 • 1.00 to 4.15

Fig. 1. Distr ibution of factor values (see text).


Maids / Suburbia

• -3.57 to -1.00 • -1.00 to 1.00 • 1.00 to 4.93

Fig. 2. Distribution of factor values (see text).


If we examine the map in Fig. 1 we see that 'white-collar elite' are clubs distributed most prominently along the panhandle to the west, representing Malibu and other prestigious locations along the coastal mountain range. This band of exclusivity extends eastward along the mountain crest, taking in the Pacific Palisades, Belair, Brentwood and Beverly Hills. A near continuation of this same band runs from La Canada Flintridge through parts of Pasadena and into San Marino. Another noticeable concentration of 'white-collar elite' is found on the southwest tip, taking in the ocean bluffs of Palos Verdes and the beach communities to the north of that. The remaining areas of the 'elite' are scattered throughout the county. The overall distribution of these candidate club members certainly does corre- spond to what one normally thinks of as the economically elite areas within Los Angeles County. The extreme negative values for factor one show that 'hispanic labor' is concentrated in East Los Angeles, extending south along the eastern perimeter of South Central Los Angeles and extending east into Pico Rivera and other San Gabriel Valley locations. Other pockets of 'hispanic labor' are found in the north San Fernando Valley and in the Signal Hi l l /Long Beach area to the south.

Fig. 2 shows that the extreme positive values for factor two, 'maids', are concentrated in the downtown and surrounding areas. This part of Los Angeles is in fact well known as a landing point for legal and illegal immigrants. Also, recall tha tpubl ic transit use is highly correlated with this factor, and the downtown area is a focal point for bus routes. What is perhaps more surprising, at least initially, is the concentrations of 'maids' found to the west in many of the same census tracts that housed the 'white-collar elite'. From this we may infer either that many people who provide household services are live-in domestics or that there are very sharp distinctions between residential districts and classes within census tracts. As one might expect, the negative values of factor two, corresponding to 'suburbia' , are located in the periphery of the county, particularly to the north of the mountains where land is relatively abundant and so one finds higher proport ions of single-family dwelling units and solitary middle- income commuters.

Similar maps can be produced for each of our 16 factors, but two is sufficient for our purposes to demonstrate the basic technique. To recapitu- late, our exploration of the socio- economic structure of Los Angeles County began with a large data set containing 64 census variables for almost 2000 census tracts. Factor analysis was used to reduce the original set of 64 variables to 16 factors that account for 70% of the variance in the original data set. Each factor can in turn be explained or interpreted in terms of the original variables that are most closely correlated with it. If we think of each factor as a composite variable, then the corresponding factor scores are like weighted averages of the scores (values) for the origina~ data set. By


examining how these factor scores are distributed geographically, we gain insight into the spatial and socio-economic composition of the urban area in question.

The maps in Figs. 1 and 2 are quite instructive. Although the factor analysis methodology as employed here is spatially neutral, each of the maps shows strong visual evidence of geographic clustering. Similar clustering was evident in the maps of the other factors that were not published here. Were this not the case, it would be difficult to argue that municipalities are the natural gathering places of these census tracts cum 'club members'. Thus, evidence of spatial clustering is a kind of necessary but not sufficient condition to establish the importance of municipalities. It is possible, however, that the clustering that takes place is independent of municipal boundaries. To test for this possibility an analysis of variance is performed in the next section.

6. Analysis of variance

As noted earlier, Eberts and Gronberg (1981) conduct their test of Tiebout's homogeneity hypothesis by using an entropy-based income inequality measure developed by Theil (1967) that allows total inequality to be decomposed into two additive components, w and b, where w measures inequality within jurisdictions and b measures inequality between jurisdictions. They regress (w/b) on a number of explanatory variables, including number of jurisdictions within the metropolitan area, for 34 SMSAs in seven states. They find, ceteris paribus, that intra-jurisdictional inequality is significantly lower for those SMSAs that have more jurisdictions. The approach taken in this paper is similar in spirit but relies instead on an analysis of variance technique that is better suited to data for one metropolitan area and which avoids the problem of statistically generated partitioning noted by Dowding et al. (1994).

Denoting Ykj as the score of factor j for municipality k, analysis of variance (ANOVA) allows us to delineate variation between municipalities from variation within municipalities. More specifically, the deviation of Yki from its observed county-wide mean y~ can be expressed in terms of two components, (Ykj ---Ykj) + (-fikj ---Yj), where Ykj is the mean value of Ykj for municipality k. This kind of decomposition is familiar to economists in the context of regression analysis where Yk~ is analogous to the estimate generated by a regression model. In our case the 'predicted' value for the factor score of a census tract is simply the mean value for the municipality in which it resides. The interpretation of the F-statistic in this analysis of variance is perfectly analogous to its role in regression analysis. A large F-ratio indicates that there is relatively little variance in factor scores within


municipalities compared with between municipalities. More specifically, consider a regression model that expresses the y value, or factor score, for census tract ] in municipality k as

(Ykj --Yj) = /30 +/3~kj + eke, (4)

where ekj is a normally distributed random error term with zero mean and uniform constant variance. In this context the analysis of variance tests the null hypothesis that /3j = 0 and /30 = 0. A sufficiently large value of the F-statistic allows us to reject the null.

The final column of Table 2 reports the F-statistics for the analysis of variance. In our case all but one of the 16 factors are significant at the 1% level. This is strong confirmation that similar census tracts tend to congregate together with reference to municipal boundaries. To further analyze these results, it is instructive to consider the seven factors for which F > 3.0. These are the factors that show the strongest evidence of clustering within municipal boundaries, and suggest four major classifications that are reinforced by municipal boundaries, as shown in Table 3 (where some factors appear twice).

These results are consistent with the Tiebout hypothesis but they do not necessarily prove that clustering is taking place at the municipal level only. Consider Fig. 3. On the left is a stylized representation of Los Angeles County in which clustering of census tracts takes place at the municipal level only. There are many constituent municipalities of two types. Within each municipality there is perfect homogeneity (either grey or black), but there is no evidence of clustering of similar types at any higher level of aggregation. The fight-hand side of Fig. 3 is similar to the left-hand side, but with one notable difference-clustering takes place within two larger areas. There is

Table 3

Type of classification Corresponding factors

Urban scale 'Small town' (factor 16) 'Rural-urban' (factor 4) 'Maids/suburbia' (factor 2)

Ethnicity

Household type

Economic scale

'White-collar elite/hispanic labor' (factor 1) 'East Asian' (factor 13) 'Single black mothers' (factor 6)

'Married with children' (factor 5) 'Single black mothers' (factor 6)

'White-collar elite/hispanic labor' (factor 1) 'Maids/Suburbia' (factor 2)

220 E.J. Heikkila Regional Science and Urban Economics 26 (1996) 203-226

l m u n ~

(b)

O tyl~ one O t ~ t~m

Fig. 3. Alternative patterns of clustering for Los Angeles county: (a) clustering within municipalities; (b) clustering within larger areas.

still perfect homogeneity within municipalities but these jurisdictions are absorbed within the larger areas, where the latter may not coincide with any jurisdictional boundaries. The important point here is that the analysis of variance test just reported cannot distinguish between the case on the left and that on the right. In both cases the relationship between intra-municipal and intra-county variance, as measured by the F-statistic, is precisely the same. This is a problem that also undermines the results of Ebert and Gronberg (1981) and Munley (1982), although that point is not brought out in their work. 6 Thus, the statistical evidence brought to bear by these earlier authors and in this paper is best viewed as weak support for the Tiebout hypothesis. It satisfies a necessary but not a sufficient condition for establishing municipalities as Tieboutian clubs.

Two considerations lead us to believe that the results reported above probably understate the importance of municipal boundaries in explaining the variance of factor scores within the county. First, the City of Los Angeles has many distinctive enclaves or districts that are themselves comparable in scale and diversity to many of the other municipalities within the county. Nonetheless, the City of Los Angeles has been treated here as just another municipality with no allowance made for its great size and diversity. Had we, for example, treated the 35 community development areas as 'pseudo-municipalities', there would be more scope for finding a relatively high degree of homogeneity within municipalities, thus leading to even higher F-ratios and stronger results. Secondly, as noted earlier, data on

6 Evidently, they did not have the good fortune to obtain the services of the same astute anonymous referee as I did.

E.J. Heikkila / Regional Science and Urban Economics 26 (1996)203-226 221

municipal services have not been included here. Because municipalities are typically the entities responsible for delivering these services, we can only presume that the results reported here tend to understate the importance of municipalities.

7. Cluster analysis

In Section 5 we implicitly defined and mapped groups of census tracts on the basis of their commonality with respect to factor scores. One drawback of that approach is that it places disproportionate emphasis on outliers. Census tracts gain membership into a candidate club by virtue of their deviations from the norm with respect to the factor in question. An implication is that the method produces sets that are neither exhaustive nor non-overlapping. Census tracts that do not deviate from county-wide averages will not be included in any groups, while those that are outliers with respect to many variables or factors will likely be included in several groups. This is not necessarily a bad thing; we often characterize objects with reference to their unique, outstanding or defining features.

The analysis of variance in the previous section clearly indicates that intra-municipal homogeneity exceeds intra-county homogeneity for a wide range of factors. However, as pointed out in the discussion centered around Fig. 3, the problem is that those results alone do not allow us to delineate the level of aggregation at which clustering of similar types occurs. Cluster analysis can help here. Simply stated, it is a procedure to group similar objects together into sets of objects known as clusters. While there are innumerable variants to the procedure (see Aldenderfer and Blashfield, 1984) the fundamental steps are the same: (i) measure the objects along some suitable scale(s), (ii) calculate the distance between all object pairs, and (iii) place similar (i.e. close) objects together into clusters. Cluster analytic techniques differ with respect to the type of scales that might be used, the method for calculating distance, and the procedures for forming clusters. The method described here is a common one and makes use of the factor scores developed earlier. Using factor scores rather than the original set of variables helps to avoid 'double counting' highly correlated variables.

The procedure is iterative. After all N = 1890 census tracts have been measured in terms of the retained factors, the distance between each tract in this 16-dimensional vector space is calculated using simple Euclidean norms. The two objects with the smallest distance between them in characteristics space are deemed to be the most similar and are then joined together to form a group or cluster, thus yielding 1889 objects. The procedure begins anew, where the centroid of the newly formed cluster is used as the reference point when recalculating distances between all object pairs. Again


the two most similar objects are joined together to form a single group. As the procedure continues iteration by iteration, sometimes single census tracts will be merged with other single tracts. Sometimes two groups of objects will merge to form a larger group. And sometimes single census tracts will be pulled into a group that has already formed. The procedure continues for N = 1890 iterations, after which there will be one giant cluster with 1890 members. At each step the clusters are mutually exhaustive and non-overlapping.

The pattern of mergers can be instructive. For example, it might happen that a few distinctive clusters emerge at an early stage and then gain in prominence with each iteration only to be merged in the final rounds. Such a pattern would strongly suggest the existence of clustering within larger areas, as characterized on the right-hand side of Fig. 3. Alternatively, it might happen that one great cluster emerges early on and that individual census tracts join in one by one, resulting in a large agglomeration with no distinct groupings. This may be termed the 'great blob hypothesis'. Such a finding would suggest relative uniformity across the county, and would not appear to be consistent with our analysis of variance results. A third possibility is that a great deal of pairing may take place at lower levels with larger groups formed only reluctantly, as it were, in the later stages. Such an outcome would suggest a great deal of fragmentation among groupings and would lend support to the hypothesis that municipalities are Tieboutian clubs.

Fig. 4 compares the actual pattern of mergers from our cluster analysis on Los Angeles County census tracts with two extreme possibilities: perfect agglomeration and perfect fragmentation. Fig. 4 plots the size of the largest cluster at each iteration against the iteration number. Thus, for example, in the case of perfect agglomeration, one large cluster emerges in the early

2000

-;- 1 o0 Perfect .

ooo

500

0 v - CO ~ 04 ,O) ¢.D ¢'~ O I"..- ~ ' , - CO

O") " ¢ CO 03 00 03 I ' . - ¢~ I'..- ' , - ¢.O

w-- ~.- ~ T-. T - T-.

Iteration Number

Fig. 4. Maximum cluster size by iteration.

Perfect Fraqmentation


stages and at each iteration absorbs an additional member so that at each iteration the size of the largest cluster is equal to the iteration number. In contrast, the graph corresponding to perfect fragmentation dings to the horizontal axis until the final iterations. With N = 1890 census tracts there are 945 possible individual pairings before the maximum cluster size increases from two to four at iteration i = 946. Four remains the size of the largest cluster for another 472 iterations until iteration i = 1418, when the largest cluster size increases to six. In the case of perfect fragmentation the maximum cluster size does not exceed 100 until i = 1863.

Fig. 4 also shows that the pattern of mergers for our cluster analysis more closely approximates the perfect fragmentation case. This is verified numeri- cally, where the area beneath the Los Angeles County (actual) curve but above the perfect fragmentation curve is 465,781, which is less than 36% of the 1,298,309 area lying between the Los Angeles County and perfect agglomeration curves. This tells us that the clustering pattern in Los Angeles County tends towards fragmentation. With 1890 census tracts and 129 municipalities a perfect Tiebout result would call for an average of 14.65 = (1890/129) census tracts per cluster. We would not expect perfect fragmentation because that would be inconsistent with the results of the analysis of variance that clearly indicate clustering within municipalities or some larger units. Thus, the apparent tendency towards fragmentation in Fig. 4 provides additional, albeit informal, evidence in favor of municipal-level clustering.

8. Concluding observations

Are municipalities Tieboutian clubs? While no single test is conclusive, the weight of the empirical evidence provided here suggests that the answer is a qualified yes. The analysis of variance in Section 6 confirmed that membership in 'municipal clubs' helps to explain variance along four dimensions: urban scale, ethnicity, household type, and economic class. Not only is there significant clustering of census tracts, but municipal boundaries consistently act as spatial templates that reinforce this clustering. Moreover, as argued above, the results from the analysis of variance likely understate the importance of municipalities in this context. The qualification is that similar municipalities themselves may cluster together. The maps in Section 4 that detail the geographic distribution of extreme valued factor scores, together with Fig. 4 from Section 7, which show the pattern of cluster fragmentation, provide visual stimuli that give some idea of the scale and spatial extent of candidate clubs.

These empirical results should help to inform parallel theoretical developments about the role and spatial orientation of Tieboutian clubs. In evaluating these findings one should ask: What is it about municipalities that attracts these clusters of households bound by urban scale, ethnicity,


household type and economic class? What causes such clusters to form in apparent reference to municipal boundaries? Common sense tells us that municipalities offer convenient policy levers such as zoning and other land-use controls that can reinforce the distinctive attributes that delineate one club from another. Urban scale, ethnicity, household type and economic class are all affected by policies regarding development permits, building by-laws, sign by-laws, local taxes and expenditures on local public goods. Local governments offer the machinery that allows Tieboutian clubs to thrive and prosper. Municipal boundaries change much more slowly than the socio-demographic variables employed here. The map of Los Angeles County and its constituent municipalities is largely unchanged from what it was two or three decades ago. This reinforces the mall analogy, whereby the outside walls and internal dividers are relatively durable but where the lease turnover from year to year reflects changes in demographics and tastes.

A priority for future research is further analysis of the spatial configura- tion of municipal clubs. Do they themselves show a tendency to cluster together, and if so does this imply that clustering occurs at a higher level of aggregation? This likely calls for spatial analysis of the kind that is being used increasingly by geographers. Having established now that municipalities are reasonably homogeneous internally, one can apply statistical techniques to detect spatial autocorrelation as described in Griffith (1988) or Odland (1991) to municipal-level data to test for evidence of clustering. Predicting the location of emerging Tieboutian clubs is much more problematic. Metropolitan areas do not burst on the scene in fits of spontaneous and enduring equilibrium. They evolve from one period to the next, with old clubs dying off and new ones forming in their place. Any number of configurations is feasible, so making it difficult to say anything meaningful in a predictive sense. There is much that we can learn from abstract models of clubs, primarily of an intuitive nature. However, if we want to predict what Los Angeles County will look like after the 2000 or 2010 Censuses, it is essential that we begin with an understanding of where it actually is today and then try to take a few tentative steps forward.

Acknowledgements

An earlier version of this work was presented to the North American Regional Science Association meetings in Chicago, November 1992. While assuming full responsibility for any errors that might yet be contained herein, the author is grateful to Ben Chinitz, R.D. Norton, Richard Arnott, John Quigley and two anonymous referees for helpful comments, and to Chrisoula Kantiotou and Steve Vivanco for their timely and skilful research assistance.


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Are municipalities Tieboutian clubs?