Heterogeneity of preferences for local public goods: The case of private expenditure on public...

JOURNAL OF PUBLIC ECONOMICS

ELSEVIER Journal of Public Economics 57 (1995) 103-127

Heterogeneity of preferences for local public goods: The case of private expenditure on public education

John Gross Department of Economics, Marquette University, Milwaukee, WI 53233, USA

Received March 1993, final version received February 1994

Abstract

Revealed preference techniques are used to investigate the degree of heterogeneity of local public good preferences by testing the hypothesis that demographic characteristics are correlated with preferences for expenditure on public education. The test relies on a goodness-of-fit index which is on an estimation of wasted consumer expenditure. We use bootstrap techniques to estimate the distribution of the test statistic. Preference heterogeneity is found to be common, however Democrat-Protestant households are found to have similar tastes, and, to a lesser extent, so are households with school-age children where at least one other trait is held in common. Republicans exhibit the greatest degree of preference heterogeneity.

Key words: Local public goods demand; Revealed preference; Consumer heterogeneity; Public education

JEL classification: H72; D12; C14

1. Introduct ion

O v e r a per iod of m o r e than two decades numerous studies have es t imated d e m a n d s for various local public goods. While these studies differ in a n u m b e r of impor tan t ways, one feature they share is the use of demograph ic character is t ics as proxies for consumer preferences. This practice is dictated by the recogni t ion that consumer preferences are unlikely to be homoge - neous . Observab le soc io-economic characterist ics are appeal ing proxies for these differing, unobservab le preferences .

0047-2727/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0047-2727(94)01440-Y

104 J. Gross / Journal of Public Economics 57 (1995) 103-127

Bergstrom and Goodman (1973) used community characteristics such as percent non-white, percent over aged 65, and percent owner-occupied housing as taste proxies. Wyckoff (1984) included household variables such as children in public schools, blue or white collar occupation, and head-of- household 's education after high school. Several studies estimated public spending demand using the same data employed here, data rich in demographic details. Among these are Bergstrom et al. (1982), Gramlich and Rubinfeld (1982), and Rubinfeld et al. (1987). These studies estimated demand for local public goods using a number of demographic characteristics as explanatory variables. These included religious affiliation, political party affiliation, presence of school-age children, education of head-of- household, status as a public employee, and a number of other traits, though the exact list varied from study to study.

Several factors which complicate the estimation of demand for local public goods have been pointed out in the literature; among these are the correct specification of the tax price, the existence of Tiebout bias, differences in community production functions, and the observability of consumer demands. An equally important but largely unaddressed issue is heterogeneity of consumer preferences. If heterogeneity can be accounted for through the use of demographics as proxies, then this need not be a deep concern, so long as the correct traits are identified. However , if demographics are not suitable proxies for differences in taste, traditional estimates of price and income elasticities may not be reliable. If preference heterogeneity is pronounced, it may overwhelm other estimation difficulties.

Indirectly, studies which estimate demand for local public goods may provide insight into the link between traits and tastes, in that the value of a demographic characteristic as a proxy for taste may be inferred from the statistical significance of its estimated coefficient. To the extent that we can draw such inference, the use of socio-economic traits as taste proxies has had mixed success. It is not uncommon to find, in studies where several characteristics are used, that at least some of their estimated coefficients are not statistically different from zero.

A fundamental goal of local public goods demand studies is the estimation of price and income elasticities. This is typically carried out through estimation of a demand equation in which dummy demographic variables are used to account for differences in tastes. If consumers have the same utility function, and hence the same demand function, there is no need for a ' taste' proxy at all. If consumers have preferences that are similar, in the sense that they generate demand functions with the same mathematical form varying only by the value of a coefficient, then the standard approach is entirely appropriate, so long as the correct mathematical form is estimated. However , consumer preferences may differ in ways that prevent a single form from adequately representing their demands, even if certain parame-

J. Gross / Journal o f Public Economics 57 (1995) 103-127 105

ters are allowed to vary. Given these considerations, the best we can say about the suitability of these coefficients for testing the hypothesis that demographic traits are correlated with consumer preferences is that they are joint tests of several hypotheses. At a minimum these are: the appro- priateness of estimating a single mathematical form; the functional restrictions imposed by the chosen form; and the hypothesis that traits and tastes are correlated.

We propose to test for observable traits which identify consumers with like preferences, and to identify preferences heterogeneity where it exists. From the preceding discussion it is clear that we require methods that place no functional restrictions on the underlying utility functions. To that end, we devise tests based on revealed preference methods. Simple but powerful results of revealed preference theory may be summarized as follows: If cross-sectional data are found to be consistent with revealed preference then a single utility functional exists which would yield the same data as the solution to the maximization problems associated with the observed prices and incomes. Conversely, if the data are not consistent with revealed preferences, no non-satiated utility function could be found which is consistent with the data.

Through what can be thought of as a filtering process, we construct subsets of data consisting of observations on consumers who are homogeneous with respect to demographic traits. The main characteristics for which we filter correspond roughly to the demographic traits used as proxies in prior demand estimation studies. These are: households where school-age children are present, households with school-age children who attend public schools, Democrats , Republicans, Catholics, Protestants, heads-of-households who completed high school, heads-of-households with some college education, and Afr ican-American heads-of-households. A revealed preference goodness-of-fit index is computed for each of these subsets. Larger values of this index indicate greater preference heterogeneity. If single traits are not well correlated with tastes, it may be that multiple traits are. An important innovation of this study is the investigation of multiple traits as correlates with tastes. We apply simultaneous filters to create subsets of data for households with as many as four demographic traits in common.

The problem is modeled as one in which consumers face two choice variables, per-pupil educational expenditure in public schools, and expenditure on all other goods. Thus, we test for the existence of a utility function over these two goods which could have generated the observations for a subset of consumers with demographic characteristics in common.

Section 2 details the hypothesis-testing procedures. We construct a test statistic based on an estimate of the expenditure wasted by each consumer if all are maximizing a common utility function. The distribution of this statistic is estimated using bootstrap methods. This method is quite general


and can easily be used to test for preference heterogeneity in purely private goods setting.

Section 3 is a discussion of the Michigan data employed and some of the problems they present. Chief among these is that approximately 42 percent of the consumers surveyed reported that they were dissatisfied with current levels of expenditure. Three different methods of employing these qualitative responses are used in computing the reported results.

The empirical results are given in Section 4. In all, 57 subsets of data are created through the application of single and multiple filters. In addition, each subset is treated with all three methods for employing the information contained in the qualitative responses, generating a total of 171 subsets of data.

2. Testing the hypothesis

For each subset of data, selected according to one or more demographic characteristics, we test the following null hypothesis:

Null hypothesis: Data selected according to consumers' demographic traits display no greater consistency with the axioms o f revealed preference than would be expected if consumers facing the same budget were selected at random.1

Afriat's theorem (Afriat, 1967, 1976; Diewert, 1973; Diewert and Parkan, 1985; and Varian, 1982), states that the data generated by maximization of a single utility function will be consistent with the Generalized Axiom of Revealed Preference (GARP). Rose (1958) shows that the Weak Axiom of Revealed Preference (WARP) and GARP are equivalent if there are only two consumption goods. Theoretically, the revealed preference restrictions must be met exactly. In practice, such a requirement is far too restrictive to allow for meaningful tests.

Revealed preference methods are particularly appealing for the investigation of preference heterogeneity for local public goods. Or, put another way, local public goods data are well suited to revealed preference tests of differences in tastes. A violation of revealed preference can only occur whenever budgets intersect. For private goods, cross-sectional data often exhibit little variation in relative prices, creating few budget intersections. For local public goods, the price is the tax price which is potentially different

~The requirement that the null hypothesis is stated in terms of the observed budget constraints will become clear below.


for each consumer, providing considerable price variation and many budget intersections.

We devise a measure of goodness-of-fit for observed data that is an estimate of the fraction of expenditure by each consumer if all are attempting to maximize the same utility function. 2 Thus, for a given set of observed budgets, larger values of this metric are associated with a poorer fit. 3 Using bootstrap methods we estimate the frequency with which a value of the goodness-of-fit measure larger than that observed is expected under the null.

2.1. Construct ion o f the goodness-of - f i t index

For any set of data containing violations, a subset of exactly consistent data may be constructed by removing some of the observations involved in violations. Loosely speaking, we achieve the minimal removals by removing those observations involved in the largest number of violations. Using methods based on this idea, we partition the data into two disjoint subsets. 4 The violator set is formed by the removal of the minimal number of observations necessary to leave its complement, the consis tent set, exactly consistent with revealed preference.

The logic behind the goodness-of-fit index is simple. If violations are due to small maximization errors, observational errors, or errors in recording data, we expect the data to be nearly consistent. That is, the preferences behind the violator set will be very similar to the preferences represented by the consistent set. Conversely, if inconsistency is the result of differing preferences then the tastes giving rise to the violator set will be likely to be quite different from those of the consistent set.

We construct the index as if the violator set and the consis tent set arise f rom the same preferences. This enables us to compute an index of the fraction of expenditure 'wasted' in purchasing a given level of utility for each observation. Each observation in the consis tent set is assigned an index value of 0. For observations in the violator set, we compute upper and lower bounds of the minimal expenditure required to purchase a level of utility equivalent to the one currently enjoyed if the observation arose from preferences which match those of the consistent set. 5 We average these over-

2 See Gross (1995) for a more detailed discussion of these methods . 3This metric is shown in Gross (1991) to be superior to a metric based on a simple

expendi ture index for capturing differences in tastes in revealed preference tests, and is similar to a metric proposed by Varian (1990) as a goodness-of-fit measure for parametric estimation.

4 m detailed description of the partit ioning algorithm and Pascal source code implement ing it are available f rom the author upon request.

5 These indices are computed with algori thms based on Varian's (1982) revealed preference me thods for comput ing the over- and under-approximat ions of the direct compensat ion funct ion for consistent data.


and under-approximations of wasted expenditure, then divide this average by total expenditure to obtain an approximation of the fraction of expenditure wasted in achieving a given level of utility.

Let awi represent our approximation ( 'a ' is a reminder of the approximate nature of the calculation) of the fraction of expenditure wasted by consumer i in maximizing utility if her preferences are those of the consistent set. Using these indices, we compute a summary a summary index, ~, which is the arithmetic mean of the individual indices. Eq. (1) gives the computation of this index for m observations where m is the number of observations in the consis tent set and the violator set combined:

= ~] awi ( 1 )

i=1 Fn

For data that are exactly consistent with revealed preference, ~ =0 . Larger differences in tastes and proportionally larger violator sets will be associated with larger index values.

2.2. Boots t rapping the distribution o f the test statistic

Derivation of the analytical form of the distribution of 6 under the null hypothesis is impractical. Thus, we turn to the bootstrap, due to Efron (1979, 1982). Bootstrap tests resample observed data in such a way as to obtain an empirically derived distribution of the test statistic. Theoretical and Monte Carlo results indicate that the bootstrap approach is a valid and effective method for a~proximating the distribution of a test statistic with an unknown distribution.

Under the null hypothesis 0 ~< 6 < 1 is a random variable with density function FIBc- The subscript BC refers to the fixed set of budget constraints. The distribution of any metric calculated with revealed preference methods depends on the location and frequency of budget intersections. To see this, consider a data set containing only two observations. If the two budgets do not intersect, no violation of revealed preference is possible for any underlying behavior. For such data, an observed value of 6 = 0 carries no information since the same value would be observed under any hypothesized behavior. In general, a particular value of 6 or any revealed preference

6 Several articles on the bootstrap test, in addition to Efron's, have appeared in the statistics literature. See, for example, Singh (1981), Bickel and Freedman (1981), Freedman (1981), and Athreya (1987). In the economics literature, Srivastava and Singh (1989) give a bootstrap method for estimating the constant term in Cobb-Douglas models.

J. Gross / Journal of Public Economics 57 (1995) 103-127 109

metr ic mus t be c o m p a r e d with values expected under the main ta ined hypothes is for the same n u m b e r of observat ions with the same budgets .

I m p l e m e n t a t i o n of a boots t rap test involves genera t ing an empirical dis t r ibut ion of the statistic by resampling the data to p roduce repea ted samples consis tent with the null. Since the distr ibution of the test statistic is condi t ional on the observed budgets , our boots t rap approach assigns a consumpt ion bundle , consistent with the null, to each of the original budgets . This enables us to obtain a null consistent value of the wasted expendi tu re index.

In a two-good fixed budge t model , the fract ion of total expendi ture devo t ed to one good fully de termines a consumpt ion bundle. Thus , for each budge t , we de te rmine a null hypothesis consumpt ion by drawing with r ep l acemen t f rom the set of observed fractions of total expendi ture devo ted to public educat ion. By relying on observed values, we assure that the boo t s t r ap distr ibution reflects realistic cho ices ] Repea t ing the process for each budge t , we obta in a data set consistent with the null, which in turn yields an empirical ly der ived value of the test statistic, ~.

Suppose we derive n such data sets d l , . . . , d" with associated observed values o f ~ 1 , . . . , ~ , , an empir~.cally derived distribution. The boots t rap es t imate of F(@)lBc is given by P~, the f requency with which the ~'s are less than ~. Read ing # as ' n u m b e r of ' we can write the boots t rap es t imator as

P£ (2) n

Eq. (2) gives an est imate of the probabil i ty that the index of wasted expendi ture would be no greateAr, for the same budgets , under the null hypothesis . Thus a value of P ~ = 0 . 1 indicates that the probabi l i ty of observ ing a value as low as ~ under the null is es t imated to be 0.1. Such results indicate that observed preference he te rogene i ty is lower than that expec ted if consumers were selected randomly .

7 An alternative to a bootstrap approach is to employ methods similar to those used by Bronars (1987). He constructs a test of the power of GARP in which he assumes that, under the null, the consumption bundle is drawn from a uniform distribution along the entire budget constraint. While reasonable in that context, such an approach here would be equivalent to the assumption that, under the null, consumers are as likely to devote 95-100 percent of total expenditure on public education (choosing perhaps to starve) as to spend 2-7 percent. We carried out a few trials using this method of deriving a distribution and found that rejection of the null using such distributions is a virtual certainty. Indeed, values of the test statistic as well as violation counts derived in this manner were typically an order of magnitude greater than observed values.

110 J. Gross / Journal o f Public Economics 57 (1995) 103-127

3. Data

The data consist of 2001 observations on Michigan residents collected in November 1978. Detailed accounts of the survey used are given by Courant et al. (1979, 1980). Among other information, consumers were asked their income, type of employment, number and age of children, religious preferences, political preferences, estimated property tax payments, status as a ren te r /homeowner , race, and whether they wanted more, less or about the same level of per-pupil expenditure on public education. If respondents indicated that 'more ' was desired, a follow-up question asked if they were willing to pay higher taxes in order to receive the increased spending. Only respondents who answered 'yes' to both questions were coded as wanting 'more ' .

Of the 2001 observations, only 854 are complete. Varian (1988) shows that the axioms of revealed preference place no restriction on unobserved prices or quantities. Thus, only complete observations may be used. We further restrict the analysis to homeowners. Of the 854 complete observations only 8 were non-homeowners, leaving a total of 846 observations.

Quanti ty of public education is taken as expenditure per student in the local school system. For the 846 observations employed, values of this variable range from $1245 to $2606 with a mean of $1721. Multiplying per-pupil expenditure by the tax price gives an individual's total expenditure on publicly provided education. The remaining good is a composite good with a price of one dollar, computed as total income minus expenditure on education. This variable ranges in value from a minimum of $926 to a maximum of $399,030. 8

3.1. Goods aggregation

We employ a simple two-good aggregation scheme, dictated by the available data (as is usually the case) and by technical considerations. Thus, data are tested for the existence of a common utility function which can explain observed choices over a single public good (public educational expenditures) and a composite good.

Data which give prices and consumption levels of a single public good and consumers ' incomes are relatively easy to obtain, certainly far easier than data on prices and consumption levels of a large number of public and private goods. Empirical studies usually involve estimation of demand functions for which the prices of most other goods are not included as

8 The distribution of income suggested by these figures is not as disperse as the maximum private goods expenditure of $399,030 suggests. Only one of the 846 respondents reported an income over $120,000. Approximately 72 percent of the reported incomes in our sample were between $10,000 and $40,000 and roughly 89 percent fell between $5000 and $50,000.


explanatory variables. Thus, we may wish to know if data such as these could arise as the solution to a single maximization problem.

As noted in Section 2, for two-good data we need only test for consistency with WARP. The introduction of a third good necessitates a test for consistency with GARP, vastly increasing the computational demands of the revealed preferences tests.

3.2. Tax pr ice speci f icat ion

Respondents were asked to estimate both their property tax payments and the market value of their homes. Bergstrom et al. (1982) constructed two tax price variables based on these answers. Differences in estimates obtained using the two variables were small, but they report slightly better fits using the variable based on the consumer's notions of their property tax payments, thus we use this version. This variable was calculated by dividing the ith respondent 's estimated property tax bill, T B i, by the tax rate, tr, in her community of residence, giving an approximation of the assessed value of the respondent 's home, i.e. T B i / t r = A V i. AV~ was then divided by T A V or total assessed value in i's community, giving her tax share. Multiplying the number of students in the community by the tax share gives the tax price for public education, or the change in i's tax bill due to a one dollar per student increase in educational expenditure. Values of the tax price variable range from a minimum of $0.02 to a maximum $6.82 with a mean of $0.55.

Several authors have considered problems associated with this simple specification of the tax price. (For examples, see Rubinfeld, 1987; Wildasin, 1989; a n d Crane, 1990.) The ' true' marginal tax price is, from one perspective, the price perceived by the consumer. From another point of view, the ' t rue ' marginal tax price fully reflects grant programs, deductibility of local taxes for Federal tax payments, tax 'relief' programs, and all capitalization effects. Neither consumers' perceptions nor these other effects are observable in this sample.

The literature dealing with tax price specification problems is largely theoretical in nature. Reliable empirical estimates of the various effects mentioned above are not available, nor are estimates that capture true consumer perceptions. Indeed, both sorts of estimation will most likely require new data sets. It is almost a certainty that the price perceived by consumers and the one that reflects all appropriate programs and capitalization effects differ. 9 Until more is learned about the exact nature of these

9 At the time of the survey two state programs that affected marginal tax prices had been in effect for approximately four years. Each year the formulae involved changed. Bergstrom et al. raised the possibility that most consumers did not really understand the impact of these programs on the marginal tax price of public education that they faced. In that study, when the appropriate calculations to compute the correct marginal tax price were applied to each observation, they found that estimated results were changed very little.


effects and consumers ' pe rcep t ions , a s imple vers ion of the tax pr ice var iab le is appropr i a t e .

3.3. I ncome stratification

Unles s p re fe rences are homothe t i c , the des i red f ract ion of total expendi -

tu re a l loca ted to publ ic educa t ion will vary with c o n s u m e r income, l° A

s imple e x p e n d i e n t which enables us to avoid bui lding in a p r e sumpt ion of h o m o t h e t i c i t y is to par t i t ion the obse rva t ions into two groups , ~1 those with

annua l i ncomes be low $20,000 at the t ime of the survey and those with

i n c o m e s o f $20,000 and above . The boo t s t r ap es t ima to r is de r ived by

d rawing a f rac t ion of expend i tu r e on public educa t ion for each obse rva t ion f r o m the co r r e spond ing dis t r ibut ion of fract ions. 12

T h r e e h u n d r e d and e igh ty- th ree consumers r epo r t ed annual incomes

be low $20,000, 447 r e p o r t e d h igher incomes . Fo r the lower group , the

m i n i m u m fract ion of total expend i tu re was 0.0037 while the m a x i m u m was

0.3596, with a m e a n of 0.0752. The h igher i ncome group t e n d e d to d e v o t e smal le r f ract ions of total expend i tu re to public educa t ion . Fo r this g roup ,

the smal les t f rac t ion was 0.0041 with a m a x i m u m of 0.2264 and a m e a n of

only 0.0378, about half that of the lower i ncome g r o u p ) 3

3.4. E m p l o y i n g qualitative responses

R e v e a l e d p r e f e r ence theory rests on the assumpt ion that obse rved

b e h a v i o r is uti l i ty maximiz ing , yet roughly 42 pe rcen t of those su rveyed

10 Our initial approach to this estimation implicity assumed homotheticity. We are grateful to Ted Bergstrom and Steven Bronars for pointing this out.

11 We could, of course, have partitioned the data into three, four, or 100 such groups. The larger the number of such groupings the smaller the variance of the bootstrap distribution. In the limit, if each consumer were in a unique group, each would be given its own consumption bundle in the bootstrap procedure and thus every estimate of ~ would be the same as the observed value ~. By choosing two groups were able to capture differences due to income variation while keeping the likelihood that any budget would receive its own consumption bundle in the bootstrap procedure acceptably low.

12 For example, suppose the ith budget is for a household with annual income of $25,000 with a tax price of $1.10. As this corresponds to the higher income group, we draw, with replacement, from the higher income distribution. Let our draw be 0.08. We compute a null hypothesis quantity of education (expenditure per pupil) as $25,000(0.08)/(1.10) = $1818. The quantity of the composite good will be $25,000(1 - 0.08) = $23,000.

13A few fractions of income spent on public education were so small or large that we considered them very suspect. We dropped the lowest and highest 1 percent of these values in each group, leaving a total of 830 fractions. Removing these values tends to reduce the variance of the bootstrap distribution of the ~:'s and shift it toward zero, making it slightly less likely that the null will be rejected.


repor ted dissatisfaction with the level of public education expenditure in their communities. We used three methods of incorporating this information.

3.4.1. Removal. Since revealed preference tests presume observed behavior is maximizing, the simplest expedient is to remove from consideration observations from consumers who report dissatisfaction with their consumption bundle. 14 As this is a large fraction of our observations, even a majority in some cases, removal frequently created stratifications too small to be of value.

3.4.2. Adjustment. Our second approach was to assign each dissatisfied respondent a new consumption bundle more in line with preferred amounts. The problem with this approach is that, for these data, we only know the direction of more pleasing bundles. Recognizing this limitation, we applied a simple reasonableness argument that allowed us to bound the adjustment.

A level of dissatisfaction that is quite small would be unlikely to induce a dissatisfied response, whereas a very large difference between desired and actual quantities would provide a compelling motivation for a Tiebout-like move to a new community, even when significant mobility constraints exist.

We adjusted each dissatisfied respondent 's consumption bundle in the desired direction by the amount afforded, at her own tax price, if her expenditure on local public education were changed by 1 percent of her income. We chose this as an amount large enough to be likely to lead to a dissatisfied response, yet small enough that the incentive to undertake a costly relocation could reasonably be presumed to be absent. We stress that we do not suppose that these adjusted bundles are in any sense correct. These results are provided for comparative purposes.

3.4.3. Fitting. Suppose we are concerned about a dissatisfied consumer's response only if it is involved in a violation. We could adjust such a consumption bundle in the desired direction and, if this removes the violation, choose not to count it. While this is a compellingly simple idea, it is problematic. As stated, it does not bound the change in the consumer's bundle. It provides no guidance if the adjustment removes one violation but induces others. Finally, if an observation is in violation with other dissatisfied consumers, it provides no insight into which is adjusted first and by how much.

Our fitting method maintains the spirit of this idea while avoiding these difficulties. We set reasonable (though admittedly arbitrary) bounds on the

14 Thanks to Steven Bronars and Melissa Faumulari for convincing us to try this approach.


size of adjustment. No one is permitted to receive an adjustment greater than one which would cost (or save) 10 percent of income) 5

Roughly, each dissatisfied observation involved in a violation is adjusted, in the desired direction, the minimal amount between 0 and 10 percent of its income to minimize the violations in which it is involved. The adjustment is computed for all dissatisfied observations before being applied to any) 6

In practice, this method leaves most consumption bundles for dissatisfied consumers unchanged. The largest proportion of dissatisfied respondents whose consumption bundles were adjusted was 18.91 percent, for the stratification involving all 846 observations. This represents only 7.8 percent of the total observations for that stratification. The adjustments themselves were small as well. The largest adjustment in absolute terms was $660, 3 percent of total expenditure for that household. No adjustment was greater than 4 percent of total expenditure, and only 5 of the 57 subsets contained an observation to which a 4 percent adjustment was applied. The average adjustment for dissatisfied respondents is again largest for the complete data, 0.29 percent of total expenditure, or 0.12 percent of total expenditure for all respondents. 17

4. Empirical results

We test the hypothesis that data selected according to demographic traits display no greater consistency with revealed preference than would be expected if consumers facing the same budgets were selected at random.

We filtered the data creating subsets for which the observations have one characteristic, or more, in common. Data were filtered for religious affiliation, political affiliation, level of education of reported head-of-household, the presence of school-age children in the household, and the presence of school-age children who only attend public schools. Some subsets formed by the application of as many as four filters were quite small. For example, we have only nine observations for Catholic-Republicans with children in public school where the head-of-household's highest level of education is high school.

In all, we created 57 subsets, each of which was then treated by each of the three procedures for handling the qualitative information on consumer satisfaction, yielding 171 subsets in all. For each, we report N, the number

15 AS we see below, our results would be entirely unchanged if we had limited this proportion to 4 percent and only slightly changed had we limited it to 3 percent.

~6 Upon request, the author will supply implementing Pascal source code and a description of the fitting procedure.

17 Details of the outcomes of the fitting procedure for each of the stratifications used are available from the author.


A

of observations, P~:, the bootstrap estimate of the frequency with which a value no greater than ~ would be expected under the null, 18 and V, the number of violations of WARP. 19 In order to reduce table clutter we do not report the values of ~.20

Results from all 171 stratifications are given in Tables 1-3. Table 1 gives results from all 57 data sets after applying the 1 percent adjustment to the consumption of each dissatisfied consumer. Table 2 gives the results obtained from analysis of the data derived by applying the fitting procedure discussed in Section 3. Finally, Table 3 gives the results for the same 57 stratifications after removal of the dissatisfied responses.

The volume of information in these tables threatens to overwhelm any at tempt to draw general conclusions. However , upon inspection we see that in a number of cases values of P~: are less than 0.1, a value often taken as 'significant'. We do not claim that a particular value of P~: is necessarily 'critical' with regard to the hypothesis test. The manner in which the qualitative information was used, the values removed from the empirical distribution of expenditure fractions, the number of simulations used to compute the bootstrap estimator, and the number of income classes used all affect the bootstrap distribution and hence values of P£. Nevertheless, some intefresting patterns emerge when the results are sorted according to values of P£. In Tables 4, 5, and 6 we give the stratification for which fi~ 4 0 . 1 0 . 21

Table 4 gives results for the 1 percent adjusted data, Table 5 gives results for the fitted data, while Table 6 gives the results for only satisfied consumers.

Fewer stratifications appear in Table 4 than in Table 5, but the same pat tern appears in both. Few single characteristics show up, and none appear in both tables. Thus, our results indicate that single demographic characteristics do a poor job of identifying consumers with similar tastes. This indicates that preference heterogeneity may be a serious problem not easily corrected by the use of simple traits as taste proxies.

Multiple stratifications fare better in selecting consumers with similar

~SThe Pascal source codes for the procedures used in this study are available f rom the author . We thank Hal Varian and Lee Redding for subrout ines borrowed with little change f rom NonPar which compute the upper and lower bounds on the expenditure function.

~9 If all budgets intersect, an extremely unlikely occurrence, the max imum possible number of violations of W A R P is found by comput ing the number of combinat ions of n objects (budget constraints) taken two at a t ime. The algorithm we use for counting violations of W A R P actually double counts so that if observat ions i and j are in violation with one another the a lgori thm counts two violations. Thus , in each case, the max imum number of violations of W A R P is given by (~) × 2. For 846 observations, if all budgets intersect, the max imum possible violations of W A R P is 714,870. However , even if all budgets do intersect, it is typically not possible for each budget to allow a violation with all other budgets.

~0 Tables including values of ~ and sizes of consistent sets for each stratification are available f rom the author .

2~ We have no quarrel with the reader who feels that a larger or smaller value is more appropria te , but we 'net ' some interesting groups at this particular level.


Table 1 All stratifications: Adjusted by 1 percent

Characteristic N V A

e~

All African-American Catholic College College, Catholic College, Protestant Democrat Democrat, Catholic Democrat, Catholic, high school, kids Democrat, Catholic, high school, kids public school Democrat, Catholics, kids Democrat, Catholics, kids public school Democrat, college Democrat, high school, kids Democrat, high school, kids public school Democrat, high school Democrat, kids Democrat, Protestant Democrat, Protestant, high school, kids Democrat, Protestant, high school, kids public school Democrat, Protestant, kids Democrat, Protestant, kids public school Democrat, kids pub school High school High school, Catholic High school, Protestant Kids Kids, Catholic Kids, Catholic, high school Kids, college Kids, high school Kids, Protestant Kids, Protestant, high school Kids public school Kids public school, Catholic Kids public school, Catholic, high school Kids public school, College Kids pub school, high school Kids public school, Protestant Kids public school, Protestant, high school Protestant Republican Republican, Catholic Republican, Catholic, high school, kids Republican, Catholic, high school, kids public school Republican, Catholic, kids

846 67

278 161 44

101 295 116 31 24 40 34 33 73 63

196 102 161 37 36 59 53 87

524 189 298 327 116

83 59

220 194 124 276

89 61 46

186 178 117 507 184 44 11 9

19

940 6

106 26

4 6

110 16 0 0 0 0 0

10 8

64 14 32

2 2 4 2

10 328

60 118 182 26 22

0 104 62 22

136 18 14 0

76 54 22

332 84 12 4 4 4

0.83 0.49 0.94 0.69 0.99 0.55 0.03 0.53 0.48 0.63 0.29 0.38 0.62 0.84 0.72 0.48 0.65 0.00 0.10 0.06 0.02 0.05 0.62 0.97 0.57 0.16 0.79 0.96 0.94 0.23 0.98 0.25 0.32 0.89 0.92 0.98 0.34 0.93 0.06 0.72 0.28 0.84 1.00 1.00 1.00 1.00

J. Gross / Journal o f Public Economics 57 (1995) 103-127

Table 1 (Contd.)

Characteristic N V

117

A

P~

Republican, Catholic, kids public school 14 4 Republican, college 58 4 Republican, high school 103 32 Republican, high school, kids 39 12 Republican, high school, kids pub school 33 12 Republican, kids 65 16 Republican, kids pub school 53 14 Republican, Protestant 136 36 Republican, Protestant, high school, kids 28 2 Republican, Protestant, high school, kids public school 25 2 Republican, Protestant, kids 47 6 Republican, Protestant, kids public school 41 4

1.00 0.88 0.91 0.99 0.99 1.00 0.99 0.36 0.91 0.93 0.90 0.97

Table 2 All stratifications: Fitted

Characteristic N V P~

All 846 372 0.03 African-American 67 2 0.19 Catholic 278 24 0.00 College 161 4 0.50 College, Catholic 44 0 0.41 College, Protestant 101 2 0.49 Democrat 295 52 0.15 Democrat, Catholic 116 6 0.29 Democrat, Catholic, high school, kids 31 0 0.56 Democrat, Catholic, high school, kids public school 24 0 0.62 Democrat, Catholic, kids 40 0 0.31 Democrat, Catholics, kids public school 34 0 0.44 Democrat, college 33 0 0.54 Democrat, high school 196 30 0.18 Democrat, high school, kids 73 6 0.42 Democrat, high school, kids public school 63 4 0.35 Democrat, kids 102 8 0.62 Democrat, Protestant 161 16 0.01 Democrat, Protestant, high school, kids 37 0 0.08 Democrat, Protestant, high school, kids pub school 36 0 0.04 Democrat, Protestant, kids 59 0 0.03 Democrat, Protestant, kids pub school 53 0 0.06 Democrat, kids public school 87 6 0.22 High school 524 148 0.02 High school, Catholic 189 14 0.09 High school, Protestant 298 54 0.07 Kids 327 60 0.19 Kids, Catholic 116 4 0.05


Table 2 (Contd.)

Characteristic N V A

P~

Kids, Catholic, high school Kids, college Kids, high school Kids, Protestant Kids, Protestant, high school Kids, public school Kids public school, Catholic Kids public Kids public Kids public Kids public Kids public Protestant Re ~ublican Re ~ublican, Re ~ublican, Re ~ublican, Re ~ublican, Re ~ublican, Re ~ublican, Re ~ublican, Re vublican, Re ~ublican, Re ~ublican, Re ~ublican, Re ~ublican, Re ~ublican, Re ~ublican, Re ~ublican, Re ~ublican,

school, Catholic, high school school, college school, high school school, Protestant school, Protestant, high school

Catholic Catholic, high school, kids Catholic, high school, kids public school Catholic, kids Catholic, kids public school college high school high school, kids high school, kids pub school kids kids, public school Protestant Protestant, high school, kids Protestant, high school, kids public school Protestant, kids Protestant, kids public school

83 4 0.18 59 0 0.27

218 32 0.57 194 16 0.08 124 8 0.07 276 38 0.02

88 0 0.00 61 0 0.04 46 0 0.40

186 18 0.44 174 14 0.47 117 8 0.54 507 160 0.02 184 24 0.15 44 0 0.43 11 0 0.97 9 0 0.95

19 4 1.00 14 0 0.92 58 2 0.91

103 6 0.55 39 2 0.91 33 2 0.86 65 4 0.95 51 4 0.82

136 22 0.08 28 2 0.95 25 2 0.93 47 6 0.98 41 4 0.96

Table 3 All stratifications: Satisfied-only

Characteristic N A

v

All African-American Catholic College College, Catholic College, Protestant Democrat Democrat, Catholic Democrat, Catholic, high school, kids Democrat, Catholic, high school, kids public school Democrat, Catholic, kids

481 136 0.03 27 0 0.33

161 10 0.28 86 2 0.70 22 0 0.90 61 2 0.65

163 8 0.21 69 2 O.53 16 0 0.75 13 0 0.82 22 0 0.59


Table 3 (Contd.)

Characteristic N V P~:

Democrat, Catholics, kids public school Democrat, college Democrat, high school Democrat, high school, kids Democrat, high school, kids public school Democrat, kids Democrat, kids public school Democrat, Protestant Democrat, Protestant, high school, kids Democrat, Protestant, high school, kids public school Democrat, Protestant, kids Democrat, Protestant, kids public school High school High school, Catholic High school, Protestant Kids Kids, Catholic Kids, Catholic, high school Kids, college Kids, high school Kids, Protestant Kids, Protestant, high school Kids, public school Kids public school, Catholic Kids public school, Catholic, high school Kids public school, college Kids public school, high school Kids public school, Protestant Kids public school, Protestant, high school Protestant Republican Republican, Republican, Republican, Republican, Republican, Republican, Republican, Republican, Republican, Republican, Republican, Republican, Republican, Republican, Republican, Republican,

Catholic Catholic, high school, kids Catholic, high school, kids public school Catholic, kids Catholic, kids public school college high school high school, kids high school, kids pub school kids kids, public school Protestant Protestant, high school, kids Protestant, high school, kids public school Protestant, kids Protestant, kids public school

13 0 0.75 13 0 0.95

116 4 0.00 38 0 0.36 31 0 0.32 47 0 0.28 41 0 0.09 86 0 0.00 15 0 0.58 14 0 0.57 23 0 0.52 20 0 0.53

317 46 0.02 111 4 0.35 175 16 0.30 184 10 0.67 67 0 0.22 50 0 0.12 34 0 0.52

130 6 0.12 108 6 0.05 71 4 0.76

156 8 0.51 55 0 0.13 39 0 0.34 28 0 0.54

111 6 0.13 100 4 0.04 65 0 0.64

299 52 0.02 112 12 0.96 23 0 0.82

7 0 0.94 5 0 0.98

10 0 0.97 7 0 0.96

34 2 0.87 66 2 0.18 26 0 0.67 22 0 0.78 43 0 0.77 35 0 0.35 88 12 0.78 19 0 0.76 17 0 0.75 34 0 0.43 31 0 0.31


Table 4 Adjusted by 1 percent: P~ ~<0.10

A


Democrat 295 110 0.03 Democrat, Protestant 161 32 0.00 Democrat, Protestant, kids 59 4 0.02 Democrat, Protestant, kids public school 53 2 0.05 Democrat, Protestant, high school, kids 37 2 0.10 Democrat, Protestant, high school, kids public school 36 2 0.06 Kids public school, Protestant 178 54 0.06

Table 5 Fitted: P~ ~< 0.10

A

Characteristic N V P~:

All 846 372 0.03 Catholic 278 24 0.00 Democrat, Protestant 161 16 0.01 Democrat, Protestant, high school, kids 37 0 0.08 Democrat, Protestant, high school, kids public school 36 0 0.04 Democrat, Protestant, kids 59 0 0.03 Democrat, Protestant, kids public school 53 0 0.06 High school 524 148 0.02 High school, Protestant 298 54 0.07 Kids, Catholic 116 4 0.05 Kids, Protestant 194 16 0.08 Kids, Protestant, high school 124 8 0.07 Kids, Public school 276 38 0.02 Kids Public school, Catholic 88 0 0.00 Kids, Public school, Catholic, high school 61 0 0.04 Protestant 507 160 0.02 Republican, Protestant 136 22 0.08

Table 6 Satisfied-only: P~: ~< 0.10


All 481 136 0.03 Democrat, high school 116 4 0.00 Democrat, kids public school 41 0 0.09 Democrat, Protestant 86 0 0.00 High school 317 46 0.02 Kids, Protestant 108 6 0.05 Kids public school, Protestant 100 4 0.04 Protestant 299 52 0.02


tastes. Every stratification involving Democrats who are also Protestants shows up in both tables. 22 It is not surprising that multiple filters do a better job of selecting consumers with similar tastes. If demographics tell us anything at all about preferences, holding more traits in common should be associated with greater preferences similarity. We note however that Democrat -Cathol ics , Republican-Protestants , and Republican-Catholics are absent from these two tables. Thus, we cannot conclude, as a general rule, that having both politics and religion in common suggests similar preferences.

Further examination of Tables 4 and 5 reveals a link between the presence of school-age children in the home and preferences, though only if other traits are also shared. Five of the seven stratifications in Table 4 involve school-age children, while 10 of 17 in Table 5 involve school-age children. Apar t from the Democra t -Protes tants , only the trait Protestant in combination with either kids or kids in public school is consistently associated with similar tastes.

The results given in Table 6, for satisfied-only respondents 23 do not show the same pattern as those in Tables 4 and 5. However, a careful examination of the full set of satisfied-only results given in Table 3 reveals no contradic- tions with the patterns found in Tables 4 and 5. In Table 3 we see that roughly 79 percent of the subsets of satisfied consumers are either exactly consistent with revealed preference (36 of 57) or exhibit only two or four violations of revealed preference (9 of 57). All Democra t -Pro tes tan t stratifications are exactly consistent. The full subset of Democra t -Pro tes -

A

tants is both exactly consistent and P~ =0.00. One could not ask for stronger results. Indeed, these results are quite robust with respect to methods of handling the qualitative responses. For all three methods, Democra t -Pro tes tan t s exhibit consistency that is significant at the 0.99 level or higher. Unfortunately, test results indicate that we cannot attach much significance to the more restricted subsets of satisfied-only D em o cra t - Protestants. Large values of fi~ coupled with exactly consistent data indicate

z2 Lest the reader suspect that the fitting procedure is driving these results, we note that it only altered one of the Democra t -P ro t e s t an t subsets. For that subset only 5.33 percent of the dissatisfied were adjusted, 2.48 percent of the total observations, and each by only 1 percent of income.

23 The stratification 'All ' appears in Tables 5 and 6, indicating the full data set exhibits more consistency than if the same consumers had randomly selected fractions of total expenditure. Quick inspection reveals that 'high school ' and 'Protestant ' in these same tables are also much more consistent with revealed preference that random selection of consumers would suggest. Each of these stratifications alone, represents more than half of the total observat ions, indeed 733 out of 846 of the consumers in the full sample are members of one of these two categories, thus the consistency of the full sample almost certainly s tems from the consistency of these two large subgroups.

122 J. Gross / Journal o f Public Economics 57 (199_5) 103-127

that these small subsets offer few opportunities for violations, thus consistency is frequently expected under the null.

In addition to investigating those traits associated with preference homo- geneity, we also investigated those associated with the greatest degree of heteArogeneity. In Tables 7, 8, and 9 we give results for data sets with values of PC ~> 0.90.

The most striking finding is the predominance of Republican stratifications in all three tables. Unlike Democrats who are Protestants, Republicans who share religious affiliation do not seem to share utility functions. The adjusted data yield yield many more stratifications with values of PC >I 0.90 than the fitted data, but of the 10 stratifications with P'~ values at least as large as 0.99, nine involve Republicans. For the fitted data, all the stratifications for which PC/> 0.90 are Republican. The same pattern shows up for those consumers who were satisfied, but, as before, we cannot attach much statistical significance to those results.

Table 7 Adjusted by 1 percent: Ps ~ ~> 0.90

A

Characteristic N V Ps~

Catholic 278 106 0.94 College, Catholic 44 4 0.99 High School 524 329 0.97 Kids, Catholic 116 26 0.96 Kids, Catholic, high school 83 22 0.94 Kids, public school, Catholic 89 18 0.92 Kids pub school, high school 186 76 0.93 Kids pub school, Catholic, high school 61 14 0.98 Kids, high school 220 104 0.98 Republican, Catholic 44 12 1.00 Republican, Catholic, kids 19 4 1.00 Republican, Catholic, kids public school 14 4 1.00 Republican, Catholic, high school, kids 11 4 1.00 Republican, Catholic, high school, kids public school 9 4 1.00 Republican, high school 103 32 0.91 Republican, high school, kids 39 12 0.99 Republican, high school, kids public school 33 12 0.99 Republican, kids 65 16 1.00 Republican, kids pub school 53 14 0.99 Republican, Protestant, kids 47 6 0.90 Republican, Protestant, kids public school 41 4 0.97 Republican, Protestant, high school, kids 28 2 0.91 Republican, Protestant, high school, kids public school 25 2 0.93

J. Gross / Journal o f Public Economics 57 (1995) 103-127

Table 8 Fitted: P~:/> 0.90

123


Republican, Catholic, kids 19 4 1.00 Republican, Catholic, kids public school 14 0 0.92 Republican, Catholic, high school, kids 11 0 0.97 Republican, Catholic, high school, kids public school 9 0 0.95 Republican, college 58 2 0.91 Republican, high school, kids 39 2 0.91 Republican, kids 65 4 0.95 Republican, Protestant, high school, kids public school 25 2 0.93 Republican, Protestant, high school, kids 28 2 0.95 Republican, Protestant, kids public school 41 4 0.96 Republican, Protestant, kids 47 6 0.98

Table 9 Satisfied-only: P~/> 0.90

J ~


College, Catholic 22 0 0.90 Democrat, college 13 0 0.95 Republican 112 12 0.96 Republican, Catholic, high school, kids 7 0 0.94 Republican, Catholic, high school, kids public school 5 0 0.98 Republican, Catholic, kids 10 0 0.97 Republican, Catholic, kids public school 7 0 0.96

4.1. Comparison with previous studies

Comparison with prior studies, even those using these same data, is problematic. We stratified our data differently, and were restricted to fewer observations. Of greater importance, other studies using these data were not designed to test for the relationship between tastes and demographic characteristics, just as this one is not designed to estimate demand functions and elasticities.

Despite these difficulties, some useful comparisons are possible. Both Bergstrom et al. (1982) and Rubinfeld et al. (1987) found race to be a statistically significant explanatory variable. African-Americans were found to demand significantly greater levels of public expenditure than other respondents. Rubinfeld and Shapiro (1989) did not find this variable to be statistically significantly for Massachusetts data. Bergstrom and Goodman (1973) used the community variable percent non-white. They estimated demands for ten states for levels of public expenditure in three categories.


Percent non-white was statistically significant in a modest fraction of cases, though it fared better when data for all states were pooled. In our sample, the number of Afr ican-American respondents was 67, barely an adequate number for a meaningful test and far too few to permit further filtering according to additional characteristics, thus our results on this variable are mixed. Once we restrict our observations to Afr ican-Americans who are satisfied with current levels of per-pupil expenditure we find the remaining observations exactly consistent with revealed preference, but the power of the test is a low and these results are not statistically significant.

These studies, as well as Rubinfeld (1977) and Wyckoff (1984) found most, though not all, coefficients estimated on variables which capture the presence of school-age children in the home to be significant. For the Michigan data, coefficients on variables 'Number of kids age 1-5' and 'Number of kids age 6-11' were significant, but those for 'Number of kids age 12-16' and 'Number of kids age 6-16' were not. Rubinfeld and Shapiro (1989) found the coefficients on both variables used, 'Number of kids age 1-5 ' and 'Number of kids age 6-16, ' to be significant. To avoid small samples we did not filter on specific ages of school-age children, opting instead to filter on the presence of any school-age children. Our results too can be considered somewhat mixed. Though not all subsets involving school-age children seem to have arisen from consumers with similar tastes, quite a few d o - s o long as at least one other trait is shared.

The conclusion that Democra t -Protes tants have very similar preferences is not comparable to results from these studies. The use of multiple characteristics to identify consumers with similar tastes is, to our knowledge, new. Our results for Republicans is consistent with the Bergstrom et al. results. They found the effect on demand for per-pupil expenditure of Republican affiliation to be negative, but small, and statistically insig- nificant. This is not surprising in that we found Republicans to have the most diverse preference of any identified group.

Our overall results indicate that demographic traits are not effective in controlling for preference heterogeneity, at least not for the Michigan data. This result calls into question prior results based on these data and points to a possible source of difficulty for all public goods demand estimation.

5. Conclusion

Our results indicate that preference heterogeneity is more likely to be the rule than the exception. The implications of this finding for the estimation of local public goods demands is profound. If preferences are heterogeneous, these differences must be controlled for if estimation is to be reliable. In

J. Gross / Journal o f Public Economics 57 (199_5) 103-127 125

general, we found that, where our tests were powerful enough to be meaningful, single demographic traits are not well correlated with preferences. As these are the most commonly used controls for preference variation, our results indicate the need to search for better controls, or alternative estimation procedures.

When multiple traits are used, it is possible to identify some groups of consumers who seem to have similar preferences. Our strongest results were for Democra t -Protes tants . Consumers who share these traits consistently exhibited homogeneous preferences. This result was confirmed in every case and was robust with respect to the method used for employing qualitative responses. To a lesser extent, the presence of school-age children, in combination with at least one other trait, tended to select consumers with similar tastes.

The largest degree of heterogeneity in tastes is that associated with Republican affiliation. This same result showed up no matter how the qualitative responses were handled and no matter how Republicans were fur ther filtered.

These results illustrate methods for testing for the presence of consumer heterogenei ty in a local public goods context. They are of equal value for testing for difference in preferences in a private goods setting, so long as the data exhibit adequate price variation. The implicit assumptions inherent in all demand estimation, that either heterogeneity is not a problem or that it can be controlled through the use of demographic traits as proxies for tastes, are called into question by these results. To the extent that we can control for heterogeneity, multiple characteristics seem to be a better bet, though it is by no means clear that, apart from Democra t -Pro tes tan t , even multiple characteristics are adequate.

Finally, our results point to the need for better data. Though our conclusions are somewhat negative, the results for the satisfied-only consumers are encouraging. The majority of these subsets are either exactly or almost exactly consistent. Perhaps with larger sample sizes the same pattern of consistency would emerge and be statistically significant. If so, more correlates with preferences might be suggested. Better data might have enabled us to say more about Afr ican-American preferences, preferences of households with other religious affiliations, and the effect of a college education on preference heterogeneity.

Though a somewhat different issue, several authors, including Hamilton (1983), Schwab and Zampelli (1987), Schwab and Oates (1991), and Schwarts (1993), have investigated the possibility that characteristics of the residents of various communities may make some communities more efficient at converting tax dollars into local public goods. Applying construc- tions given in Varian (1984), it should be possible, with appropriate data, to


cons t ruc t hypo thes i s tes t ing m e t h o d s s imi lar to those e m p l o y e d here to inves t iga te the hypo thes i s tha t c o m m u n i t y publ ic goods p roduc t i on func t ions vary with r e s iden t charac ter i s t ics .

W e have d e m o n s t r a t e d that r evea led p r e f e r e nc e t echn iques m a y be a p p l i e d to large da ta sets to inves t iga te the deg ree of local publ ic goods p r e f e r e n c e h e t e r o g e n e i t y by tes t ing for a r e l a t ionsh ip b e t w e e n c o n s u m e r cha rac te r i s t i c s and tas tes . O u r f indings call in to ques t ion the accuracy of p r i o r pub l i c goods d e m a n d e s t ima t ion and po in t to the need for fu r the r inves t iga t ion into p r e f e r e n c e s he t e rogene i t y .

Acknowledgements

T w o a n o n y m o u s re fe rees p o i n t e d ou t n u m e r o u s sho r t comings in an ea r l i e r ve r s ion and gene rous ly sugges ted s t ra teg ies for i m p r o v e m e n t . I am d e e p l y g ra te fu l for the i r ass is tance. A t severa l crucial s tages Ted B e r g s t r o m of fe red ins ight fu l and i m m e n s e l y he lpfu l sugges t ions , as d id Ha l Var ian and Paul C o u r a n t . D e b o r a h A n t k o v i a k , S teven Brona r s , Mel i ssa F a m u l a r i , and Bill H o l a h a n all p r o v i d e d m a n y va luab le obse rva t ions . Specia l t hanks to D a n K a i s e r for he lp in t r ans la t ing the tests into w o r k a b l e , eff icient c o m p u t e r code .

References

Afriat, S.N., 1967, The construction of utility functions from expenditure data, International Economic Review 8, 67-77.

Afriat, S.N., 1976, Combinatorial theory of demand (Input-Output Publishing Co., London). Athreya, K.B., 1987, Bootstrap of the mean in the infinite variance case, The Annals of

Statistics 15, 724-731. Bergstrom, T.C. and R.P. Goodman, 1973, Private demands for public goods, American

Economic Review 63, 280-296. Bergstrom, T.C., D.L. Rubinfeld and P. Shapiro, 1982, Micro-based estimates of demand

functions for local school expenditures, Econometrica 50, 1183-1205. Bickel, P.J. and D.A. Freedman, 1981, Some asymptotic theory for the bootstrap, Annals of

Statistics 9, 1196-1217. Bronars, Stephen G., 1987, The power of nonparametric test of preference maximization,

Econometrica 55, 693-698. Courant, P., E. Gramlich and D. Rubinfeld, 1979 Tax limitation and the demand for public

service in Michigan, National Tax Journal 32 (Special Supplement), 147-158. Courant, P., E. Gramlich and D. Rubinfeld, 1980, Why people support tax limitation

amendments: The Michigan case, National Tax Journal 33, 1-20. Crane, R., 1990, Price specification and the demand for public goods, Journal of Public

Economics 43, 93-106. Diewert, W.E., 1973, Afriat and revealed preference theory, Review of Economic Studies 40,

419-426.


Diewert, W.E. and C. Parkan, 1985, Tests for the consistency of consumer data, Journal of Econometrics 30, 127-147.

Efron, B., 1979, Bootstrap methods, another look at the jackknife, Annals of Statistics 7, 1-26. Efron, B., 1982, The jackknife, the bootstrap, and other resampling plans, Society for

Industrial and Applied Mathematics, Philadelphia, PA. Freedman, D., 1981, Bootstrapping regression models, Annals of Statistics 9, 1218-1228. Gramlich, E.M. and D.L. Rubinfeld, 1982, Micro estimates of public spending demand

functions and tests of the Tieboat and median-voter hypotheses, Journal of Political Economy 90, 536-560.

Gross, J., 1991, On expenditure indices in revealed preferences tests, Journal of Political Economy 99, 416-419.

Gross, J., 1995, Testing data for consistency with revealed preference, Review of Economics and Statistics, to appear.

Hamilton, B., 1983, The flypaper effect and other anomalies, Journal of Public Economics 22, 347-361.

Rose, H., 1958, Consistency of preference: The two-commodity case, Review of Economic Studies 35, 124-125.

Rubinfeld, D.L., 1977, Voting in a local school election: A micro analysis, Review of Economics and Statistics 59, 30-42.

Rubinfeld, D,L., 1987, The economics of the local public sector, in: A. Auerbach and M. Feldstein, eds., Handbook of public economics, Vol. II, (Elsevier Science, Amsterdam) 571-645.

Rubinfeld, D.L. and P. Shapiro, 1989, Micro-estimation of the demand for schooling, Regional Science and Urban Economics 19, 381-389.

Rubinfeld, D.L., P. Shapiro and J. Roberts, 1987, Tiebout bias and the demand for local public spending, Review of Economics and Statistics 69, 426-437.

Schwab, R.M. and E.M. Zampelli, 1987, Disentangling the demand function from the production function for local public services, Journal of Public Economics 33,245-260.

Schwab, R.M. and W.E. Oates, 1991, Community composition and the provision of local public goods, Journal of Public Economics 44, 217-237.

Schwartz, A.E., 1993, Individual production, community characteristics and the provision of local public services, Journal of Public Economics 50, 277-289.

Singh, K., 1981, On the asymptotic accuracy of Efron's bootstrap, Annals of Statistics 9, 1187-1195.

Srivastava, M.S. and B.R. Singh, 1989, Bootstrapping in multiplicative models, Journal of Econometrics 42, 287-297.

Varian, H., 1982, The nonparametric approach to demand analysis, Econometrica 50,945-973. Varian, H., 1984, The nonparametric approach to production analysis, Econometrica 52,

579-597. Varian, H., 1988, Revealed preference with a subset of goods, Journal of Economic Theory 46,

179-185. Varian, H., 1990, Goodness-of-fit in demand analysis, Journal of Econometrics 46, 125-140. Wildasin, D., 1989, Demand estimation for public goods: distortionary taxation and other

sources of bias, Regional Science and Urban Economics 19, 353-380. Wyckoff, J.H., 1984, The nonexcludable publicness of primary and secondary public education,

Journal of Public Economics 1984, 331-351.

Heterogeneity of preferences for local public goods: The case of private expenditure on public...

Documents

Transcript of Heterogeneity of preferences for local public goods: The case of private expenditure on public...