An Empirical Analysis of the Relationships Between Family Income Inequality, City Size, and Population Trends STEPHEN NORD* city size and direction of change in population in such a way as to lead to systematic variations in both the magnitude of family income inequality as well as underlying causal factors. Three arguments suggest such an interaction.
While policy makers and economists have long been interested in income inequality, little research has been conducted to examine the variance in inequality among cities. Most studies have merely included cities in their analyses of income inequality through some proxy measure for "urbanization" in observations of counties, states, and nations. While a few studies have examined the variance in income inequality between SMSA's [Murray, 1969; Frech and Burns, 1971; Farbman, 1975; Danziger, 1976; and Long, Rasmussen, and Haworth, 1977], conspicuously absent from the literature is empirical research aimed at extending to small cities the generalizations regarding city size and income inequality.' Moreover, those generalizations that do exist in the literature are contradictory: Duncan and Reiss [1956], Aigner and Heins [1967], A1-Samarrie and Miller [1967], and Richardson [1973], for example, conclude that income inequality is inversely related to city size, while Farbman [1975] and Long, e t al [1976] have concluded exactly the reverse. This study reaches a compromise in these conclusions by postulating and examining the relationship between family income inequality, city size, and growing and declining cities.
First, Tiebout [1956] postulates that families respond to services provided by the public sector by locating in communities where public services are provided in accordance with their preferences. Families with strong preferences for schools or police protection, for example, have a tendency to locate together and tax themselves to provide their desired levels of those public services. While other factors are recognized as being important, this simple theory is widely accepted as an explanation of the manner by which the public sector affects family location patterns within and among cities. It follows that occupational subgroups highly supportive of public services will likely hold stronger locationat preferences for municipalities more responsive to their desires. Greer and Orleans [1968] found that individuals within the professional and salaried managerial occupations provide the most active political support for municipal public services and that their activity declines as city size increases. In another study, Beaton [1972] found that the public sector in large cities is less sensitive to relative concentrations of these same occupational subgroups and responds in their budget process by providing relatively lower per capita expenditures on public services. These subgroups who are supportive of public services should therefore have stronger tocational household preferences for smaller cities where they are able to influence their elected officials in the provision of public services. Since professional
I. The General Arguments
The hypotheses examined in this study are that family income inequality interacts with *Northern Illinois University. I wish to thank the Board of Editors of this Journal for its helpful comments and suggestions on an earlier draft of this paper. Murray [ 1969] examinesthe variancein inequality in smaller cities within the boundaries of 15 SMSA's and concludes an indeterminant relationship between inequality and the size of municipalities within the SMSA's.
51
52
ATLANTIC ECONOMIC JOURNAL
and salaried managerial occupations exhibit a wide range of incomes with the highest earnings normally realized by relatively few individuals [Mincer, 1958; and Miller, 1963], we postulate that these occupational subgroups will exert more dispersion in the distribution of family income in smaller as opposed to larger cities through their household tocational choices. Secondly, cities declining in population experience different public expenditure requirements than cities growing in population. Declining cities, for example, have been shown to allocate relatively larger proportions of their budgets to the protection and control of abandoned physical plants [Sternlieb and Burchell, 1973]. In cities growing in population, on the other hand, the private sector is found to provide significant contributions to the provision of public services such as fire, police, and sanitation services [Weicher, 1970]. Cities growing in population thus have greater flexibility and" are likely to be more responsive to the preferences of their more politically active professional and salaried managerial populations in the types and quantities of services provided through the public sector. It is likely that these occupational groups will respond to this variance in the provision of public services in declining and growing cities in their household locational choices. Therefore, it is expected that professional and salaried managerial households wilt also prefer to locate in growing cities which, ceteris paribus, will result in a relatively greater dispersion in family income in these cities. Finally, agglomeration economies may exert pressures causing inequality to vary with city size. Small cities, particularly those not adjacent to metropolitan areas, are likely to provide few employment opportunities which, eeteris paribus, will produce greater inequality relative to larger cities. Cities with larger populations should sustain greater employment opportunities from specialization and diversification and exhibit greater equality. However, beyond some city size, it is likely that the dispersion in income
will rise from the attractions of poor and displaced workers on the one hand, and the housing of major corporations with their supportive highly trained and compensated personnel at the opposite end of the earnings spectrum. II. The Model To examine our arguments of the interaction of income inequality with city size and the direction of population change, we have partitioned 1,079 cities ~ in the Mideast region ~ of the United States with populations exceeding 2,500 into observations of four strata of growing and declining cities for separate regression analysis. Each regression model reported is found to satisfy the homoscedasticity requirement of a constant variance in the distribution of residuals and thus defines appropriate sets of cities susceptible to valid applications of least squares regression analysis. 4 Gini coefficients (Gini) have been calculated from the Bureau of Census 15-class size distributions of income for each of the cities and serve
The SMSA is not used as a unit of observation in this study. We recognize the logic of maintaining the SMSA as a unit of observation as it represents a local labor market defined by commuting distances; however, communities in which incomes are earned are of lesser interest to this study than assessing the impact of professional and salaried managerial related household locational preferences on city variances in inequality. 3Including New York, New Jersey, Pennsylvania, Delaware, Maryland, and the District of Columbia. Cities that underwent boundary changes between 1960 and 1970 were deleted from the sample. 4While each of our models satisfies the homoscedasticity requirement, an aggregated model of all 1,079 cities was found to produce heteroscedastic residuals and thus misleading results from inefficient regression estimates and standard errors. In order to remove this obstacle, we subdivided our observations on the basis of the variable suspected of causing heteroscedasticity [Klein, 1961, pp. 196-7]. In this way we partition the 1,079 cities in the Mideast region of the United States into four strata of growing and declining cities for separate regression analysis. The procedure and results of the tests for homoscedasticity are available from the author upon request.
NORD: FAMILY INCOME, CITY SIZE, AND POPULATION as our dependent variable measuring family income inequality. ~ The measure of city size employed as an explanatory variable is the quadratic form of total city population (Pop). In line with our arguments above, we expect population to capture the effects of specialization and diversification on inequality and postulate that it will exert an equalizing effect in strata of smaller cities (OGini/OPop < 0) and raise inequality in our largest strata of cities (OGini/ ~Pop > 0). The professional and salaried managerial occupational subgroups found by Greer and Orleans to be the most supportive of municipal public services are specified as an explanatory variable (Prof-Man percent) and is constructed as their percentage representation of the labor force within each respective municipality. From our arguments that city size and population trends affect family income inequality through household locational preferences, we expect the professional-salaried managerial class index to have a higher mean value (Mean Prof-Man percent) in smaller cities and for every strata of growing cities. With the highest concentration ratios of any other occupational subgroups, it is reasonable to further postulate, ceteris paribus, that the professional-salaried managerial class index will exacerbate family income inequality in these same cities more than in larger and declining cities. Consistent with the literature, we have selected five additional explanatory variables to examine which previous studies have generally found to be significant determinants Of income inequality. Briefly, the specification of the remaining explanatory variables in our models is as follows: the percent of blacks in the adult population (Percent Black) and the percent of
s Midpoints are used in calculating the Gini coefficients for each of the closed intervals while a Pareto curve is fitted to the open-ended interval to estimate the mean measure of income for the class distribution of income exceeding $50,000.
53
families with female heads of households (Percent Female Head) will likely increase the dispersion in a family income caused by a concentration of lower marginal incomes, and are thereby expected to augment family income inequality. Mean family income (Income) is included as an explanatory variable on the basis that the upward movement of incomes in an expanding economy has resulted in a greater concentration of the labor force at middle income occupations [Fitzwilliams, 1964] which, ceteris paribus, should lessen the dispersion in incomes and exert a stronger central tendency in the distribution of family incomes. Education is often cited as a means by which individuals may enhance their earnings through the development of marketable skills and training. 6 We expect an inverse relationship between the median number of school years completed by the adult population (Education) and family income inequality. Finally, manufacturing employment is generally the most visible source of high wage employment for poorer groups, and we therefore postulate that the percent of the labor force employed in manufacturing (Percent Mft) will narrow the dispersion in income. In the next section, the following model is estimated separately for each of the four strata of growing and declining cities: Y=a+blxl
+b2x2 + . . . b n x n
+U
where Y is our measure of family income inequality; xl ... xn are the independent variables, and u represents random variation.
III. Empirical Analysis Table 1 displays the regression coefficients, t-ratios, beta values, and other statistical information for strata of cities growing in population from 1960 to 1970. Table 2 displays the
6This argument finds substantial support in human capital studies [Becker, Chiswick, Mincer, et al] which view earnings as a flow of discounted returns to investment in human capital.
54
ATLANTIC ECONOMIC JOURNAL
results of a set of two-tailed student's t tests indicate that the professional-salaried managerial used to determine if the net regression coeffi- class index is the most important factor associcients of an independent variable is significant- ated with greater income inequality in cities ly different from one another across each city within our models with populations to 50,000 size stratum. ~ Tables 3 and 4 repeat this pattern and has its largest exacerbating effect in strata for the four strata of cities declining in popula- of cities with smaller populations. '° tion. Table 5 completes the statistical analysis; Comparison of the mean measurement of it exhibits the results of a set of tests used to family income inequality for each strata of determine if the net regression coefficients growing and declining cities additionally reveals for the growing cities can be considered to be greater income inequality for every strata of significantly different from the corresponding growing cities with populations to 50,000 and coefficients of declining cities. The regression greater income inequality for cities declining coefficients for each of the models conform than growing in population in our largest strata to our hypothesized partial effects of the ex- of cities. Complementing the larger relative conplanatory variables and the R2's are quite centrations of professional and salaried manahigh. 9 In general, smaller residual variances ap- gerial occupational subgroups found in each pear for strata of smaller and growing cities. strata of growing cities are larger beta values Examination of the mean measures of fam- indicating the greater role played by the proily income inequality for 1969 (Mean Gini 69) fessional-salaried managerial class index in reveals some interesting relationships between augmenting family income inequality in cities income inequality, city size, and secular trends growing in population. in population, In both growing and declining The regularity with which the professionalcities, income inequality appears to be greatest salaried managerial class index is 'simultaneously for cities with populations 2,500 to 10,000, de- significant and in direct correlation with the clines for strata of cities with populations to mean measures of inequality, together with 50,000, and rises in cities exceeding 50,000 in the results that the regression coefficients genpopulation. As expected, the mean value of the erally test significantly different from one professional-managerial class index (Mean Prof- another among the models, are persuasive to Man Percent) varies inversely with city size; this variable's importance to the relationship that is, there is a larger relative concentration of between inequality, city size, and population professional and salaried managerial occupa- trends uncovered in this study. tional subgroups in smaller cities. With respect While these results appear generally supportto this occupational variable, the beta values ive of our arguments that the household locational choices of professional and salaried Beta values enable comparisons of the relative im- managerial related families for smatler and portance among independent variables by essentially growing cities induce income inequality to syssimplifying the net regression coefficients to a common denominator. The beta values are calculated by tematically vary with city size and population multiplying the ratio of the standard deviation of the trends, the regression results suggest significant independent variable to the standard deviation of the dependent variable by the net regression coefficient of the respective independent variable. ~°In light of the variance in the explanatory values 8For a discussion of the procedure of the compari- of a number of variables between the strata of cities, sons of net regression coefficients from separate re- nonlinear regressions were performed and yielded very gression equations, see Steel and Torrie [1960,p. 173]. similar results. Therefore, the results of this study can9Inspection of the regression results omitting high- not be resolved by nonlinear specifications of the ly correlated variables indicates that multicoUinearity models. The results of these nonlinear regressions are available from the author upon request. is not a serious problem in these regressions.
NORD: FAMILY INCOME, CITY SIZE, AND POPULATION variations in the relative importance of a number of independent variables between the strata of cities. That is, relative concentrations of professional and salaried managerial occupational subgroups are not the only factor associated with the variance in income inequality with city size and trends in population. Our measure of population size is significant in five of the eight regression models. Population is significant and negatively related to inequality in each strata of growing cities with populations to 50,000. Moreover, the beta values indicate that larger negative relationships exist between population and inequality for smaller cities. These results are supportive of the argument that the greater levels of specialization and diversification that generally accompany population growth are conducive to an environment favorable to greater equality for cities with populations up to 50,000. Complementing this effect, manufacturing employment appears consistently significant in reducing inequality in these same strata of cities. The greatest reduction in inequality probable to occur for any given percentage increase in population or manufacturing employment is in the smallest strata of growing cities. The elasticity calculated at the mean suggests that a I0 percent increase in population in the smallest strata of growing cities will result in a 4 percent decline in measured family income inequality. Similarly, a I0 percent increase in manufacturing employment is predicted to reduce inequality by nearly 6 percent in these same cities. While these results are in agreement with the Kuznets' hypothesis that greater equality accompanies industrialization within the process of economic growth, we find that increases in population in cities beyond 50,000 in population result in a rise in inequality ~' and manu1~To the author's knowledge, city population size, in either linear or quadratic form. appears as an explanatory variable in only three determinant analyses of inequality for SMSA's.The studies by Farbman [ 1975 ] and Long, et al [1977] find significant positive relationships between inequality and population size,
55
facturing employment declines in importance in lessening inequality. Several additional inferences can be made from our regression results as to the reasons family income inequality appears to interact with city size and population trends. The percent of blacks in the adult population and the percent of female heads of households explain more of the variance in family income inequality in larger cities and citDs declining in population. Comparison of the beta values for cities growing and declining in population reveal both these variables to possess greater explanatory power as the city population size range increases and for every strata of cities declining as opposed to growing in population. It is of interest to point out that both blacks and female heads of households explain more of the variance in income inequality than the professional-salaried managerial class index in cities exceeding 50,000 in population. The negative effect which education is consistently found to have upon income inequality also appears to vary in significance with city size and population trends. Larger negative relationships between inequality and education are observed in both growing and larger strata of cities. Elasticities range from -0.12 for the smallest strata of declining cities to -0.60 for the largest strata of growing cities. Presumably from the greater opportunities for skilled types of employment, education thus possesses larger potentials for lessening inequality in larger cities and cities sustaining growth in population. Our results support the familiar equalizing effect of rising incomes, '2 but here too the efwhile Danziger [1976] concludes that size has no significant effect on inequality. ~2The origin of this relationship is with Kuznets [1955] in his reporting anegative relationship between income and inequality in his study of developing countries. This relationship has been supported by Aigner and Heins [1967], A1-Samarrie and Miller [1967], Conlisk [ 1967 ], et al using state data, Farbman [ 1973 ] using country data, and Frech and Burns [1971], Farbman [1975], and Long, Rasmussen, and Haworth [1976] using SMSA data.
56
ATLANTIC ECONOMIC JOURNAL
fect appears to be influenced by city size and trends in population. Examination of the regression results reveals that income explains more of the variance in the distribution of family income in both the strata of smaller and growing cities. The range in elasticities indicates that a 10 percent increase in income in the two smallest strata of growing cities will lessen inequality by nearly 6 percent whereas an equivalent percentage increase in income in the largest strata of declining cities will result in a 1 percent decline in inequality. A postulation supported by these results is that for any given increase in income, the equalizing effect, ceteris paribus, will be less in cities with larger populations and in those cities, particularly larger cities, which are experiencing a loss in population. Moreover, we then observe the adverse effects on income inequality generated by blacks and female headed households increasing with city size and declining populations, while the equalizing effect of rising incomes, hitherto ti~e "harbinger" of greater equality, diminished in these same cities. IV. Conclusion On the basis of the above results several conclusions may be drawn regarding the relationships between family income inequality, city size, and population trends. First, income inequality is found to be greatest for small cities with populations 2,500 to 10,000, declines for middle-sized cities with populations t0,000 to 50,000, and rises for cities with populations exceeding 50,000. Second, cities growing in population exhibit greater inequality than cities de-
clining in population through the 50,000 range, while declining cities exhibit greater inequality than growing cities with populations exceeding 50,000. Several factors were found to be associated with these relationships. The household locational preferences of professional and salaried managerial occupational subgroups for smaller and growing cities exert relatively greater positive effects on income inequality within these same strata of cities. While this factor appears significant in explaining the greater inequality for growing as opposed to declining cities with populations to 50,000 and the falling levels of inequality as city size increases to the 50,000 range, other factors gain importance and contribute to the rise in inequality in our largest strata of cities. Blacks and female heads of households contribute relatively more to income inequality in both larger cities and cities declining in population. Moreover, education is tess significant in reducing inequality in cities declining in population, while the equalizing effects of income and manufacturing employment are less significant in both declining and larger cities. These results may be used, with caution, to speculate about the likely effects which changes in a number of explanatory variables may have upon income inequality within cities of different sizes and stages of growth and decline. We believe that our results establish the argument for further analysis of income inequality as it relates to broader ranges of population and other attributes of cities.
NORD: FAMILY INCOME, CITY SIZE, AND POPULATION
57
TABLE 1 REGRESSION RESULTS FOR 4 STRATA OF CITIES GROWING IN POPULATION FROM 1960 TO 1970 Independent Variable
Popa,b
% Black
% Female Head
Income
Education
% Mft
Prof-Man %
Mean Pro f-Man % Mean Gini 69 Std,Error of Estimate Degrees of Freedom R2 Adjusted R 2 F Score
2,500 to 10,000
1970 City Population Size Ranges 10,000 25,000 to 25,000 to 50,000
Over 50,000
- 1,509 a (2.93) [-0.426]
- 1.062 c (2.14) [-0.392]
- 1.591 c (1.89) [-0.274]
0.327 c (1.83) [0.197]
0.061 e (I. 80) [0.541]
0.105 c ( 1.92) [0.318]
0.167 c (2.21 ) [0.326]
0.215 a (3.27) [0.619]
0.279 c (1.69) [0.4I 8] - 1.017 d (2.43) [-0.579] -0.019 e (1.81) [-0.216] - 1.749 a (3.96) [-0.639]
0.001 (0.86) - 1.002 c (2.25) [-0.512] -0.117 c (1.98) [-0.294] - 1.021 c (1.92) [-0.521]
0.213 c (2.14) [0.354] -0.812 c (2.06) [-0.415] -0.307 a (2.63) [-0.332] -0.672 c (1.85) [-0.318]
0.291 d (2.74) [0.522] -0.505 c (1.89) [-0.393] -0.4138 (2.71) [-0.625] -0.481 c (1.77) [-0.219]
0.401 d (3.06) [0.813]
0.318 a (2.97) [0.792]
0.219 a (2.81 ) [0.733]
0.022 c (2.14) [0.504]
.191 .325 7.1 50.0 .65 .59 30.7
. I64 .371 4.5 15.0 .71 .63 10.4
.217 .385 5.2 431.0 .77 .74 91.5
.201 ,338 18.6 138.0 ,73 .68 48.2
Sources: U.S. Bureau of the Census, Characteristics of the Population, 1970: GeneralPopulation Characteristics, 1970: GeneralSocial and Economic Characteristics, 1970.
aAbsolute values of t-ratios in parentheses. bBeta values of significant regression coefficients in square brackets. cStatistically significant at the 5 percent level. dStatistically significant at the 1 percent level.
ATLANTIC ECONOMIC JOURNAL
58
TABLE 2 NET REGRESSION COEFFICIENTS AMONG THE 4 STRATA OF CITIES GROWING IN POPULATION FROM 1960 to 1970" ii
Independent Variable
i
None Test Significantly Different
Pop % Black % Female Head Income Education % Mft Prof-Man % *The level of significance is 0.05 for a two-tailed test.
At Least One Tests Significantly Different
All Test Significantly Different
X X X X X X X
NORD: FAMILY INCOME, CITY SIZE, AND POPULATION
59
TABLE 3 REGRESSION RESULTS FOR 4 STRATA OF CITIES DECLINING IN POPULATION FROM 1960 TO 1970 Independent Variable Popa, b
% Black
% Female Head
Income
Education
% Mft
Prof-Man %
Mean Prof-Man % Mean Gini 69 Std. Error of Estimate Degrees of Freedom R2 Adjusted R 2 F Score
a'b'e'dSee
Table 1.
2,500 to 10,000
1970 City Population Size Ranges 10,000 25,000 to 25,000 to 50,000
Over 50,000
-0.0003 (0.91) -0.109 c (2.11 ) [0.592] 0.308 c (1.91) [0.4671
-0.018 (1.26) -0.516 e (1.99) [0.479] 0.102 c (2.05) [0.316]
0.031 (1.15) -0.608 e (2.43) [0.4121 0.716 c (2.21) [0.297]
0.473 c (1.89) [0.261] 1.372 a (4.96) [0.712] 0.983 a (3.41) [0.489]
-0.904 e (2.16) [-0.506] -0.008 e (1.67) [-0.1891 -0.982 c (1.94) [-0.612] 0.018 a (2.91) [0.679]
-0.826 c (1.93) [-0.481] -0.012 e (1.81) [-0.1521 -0.506 (1.58) 0.026 a (2.67) [0.613]
-0.617 c (2.00) [-0.3241 -0.036 e (1.97) [-0.2101 -0.109 c (1.79) [-0.554] 0.011 c (2.45) [0.586]
-0.274 e (1.72) [-0.318] -0~ t 03 e (2.09) [-0.3721 0.062 (1.08) 0.004 e (1.86) [0.425]
.193 .329 9.1 82.0 .69 .65 87.1
.168 .316 3.7 23.0 .63 .59 19.6
.131 .384 6.t 29.0 .68 .62 12.8
.199 .379 17.2 255.0 .72 .68 16.4
ATLANTIC ECONOMIC JOURNAL
60
TABLE 4 NET REGRESSION COEFFICIENTS AMONG THE 4 STRATA OF CITIES DECLINING IN POPULATION FROM 1960 TO 1970"
Independent Variable Pop % Black % Female Head Income Education % Mft Prof-Man %
None Test Significantly Different
At Least One Tests Significantly Different
All Test
Significantly Different
X X X X X X X
*The level of significance is 0.05 for a two-tailed test.
TABLE 5 NET REGRESSION COEFFICIENTS FOR GROWING CITIES TO DECLINING CITIES*
Independent Variable
Pop % Black % Female Head Income Education % Mft Prof-Man %
2,500 to 10,000
1970 City Population Size Ranges 10,000 25,000 to 25,000 to 50,000
X X
X X
X X
*The level of significance is 0.05 for a two-tailed test.
X X X
X X
X
Over 50,000
X
X X X
NORD: FAMILY INCOME, CITY SIZE, AND POPULATION
61
REFERENCES D. Aigner and A. Heins, "On the Determinants of Income Inequality," American Economic Review, 57, March 1967, pp. 175-81. A. A1-Samarrie and H. Miller, "State Differentials in Income Concentration," American Economic Review, 57, March 1967, pp. 59-72. W. Beaton, "The Determinants of Police Protection Expenditures," National Tax Journal, 28, June 1974, pp. 335-49. J. Conlisk, "Some Cross-State Evidence of Income Inequality," Review of Economics and Statistics, 49, February 1967, pp. 115-8. S. Danziger, "Determinants of the Level and Distribution of Family Income in Metropolitan Areas, 1969," Land Economics, 52, November 1976, pp. 467-78. O. Duncan and A. Reiss, Jr., Social Characteristics o f Urban and Rural Communities, 1950, New York, 1956. M. Farbman, "Income Concentration in the Southern United States," Review of Economics and Statistics, 55, August 1973, pp. 333-40. , "The Size Distribution of Family Income in U.S. SMSA's," Review of Income and Wealth, June 1975, pp. 217-37. J. Fitzwilliams, "Size Distribution of Income in 1963," Survey of Current Business, 44, April 1964, pp. 3-11. H. Frech and L. Burns, "Metropolitan Interpersonal Income Inequality," Land Economics, February 1971, pp. 104-6. S. Greet and P. Orleans, "The Mass Society and
the Parapolitical Structure," The New Urbanization, S. Greer etal, New York, 1968, pp. 201-21. L R. Klein, An Introduction to Econometrics, Englewood Cliffs, New Jersey: Prentice-Hall, Inc., t962. S. Kuznets, "Economic Growth and Income Inequality," American Economic Review, 45, March 1955, pp. 1-28. J. Long, D. Rasmussen, and C. Haworth, "Income Inequality and City Size," Review of Economics ~nd Statistics, 59, May 1977, pp. 244-6. H. Miller, Trends in the Income of Families and Persons in the United States: 1947 to 1960, Washington, 1963. J. Mincer, "Investment in Human Capital and the Personal Distribution of Income," Journal of Political Economy, 66, August 1958, pp. 281-302. B. Murray, "Metropolitan Interpersonal Income Inequality," Land Economics, 45, 1969, pp. 121-5. H. Richardson, The Economics of Urban Size, Westmead, 1973. R. Steel and J. Torrie, Principles and Procedures of Statistics, New York: McGraw-Hill Company, 1960. G. Sternlieb and R. Burchell, ResidentialAbandonment: The Tenement Landlord Revisited, New Brunswick, 1973. C. Tiebout, "A Pure Theory of Local Expenditures," Journal of Political Economy, 64, October 1956, pp. 416-24. J. Weicher, "Determinants of Central City Expenditures: Some Overlooked Factors and Problems," National Tax Journal, 23, December 1970, pp. 379-96.
