Working Paper No June, 1999 DEPARTMENT OF ECONOMICS TULANE UNIVERSITY NEW ORLEANS, LA 70118

Transcription

1 TULANE UNIVERSITY Private Schools and the Demand and Supply of Public School Quality In Metropolitan Areas by David M. Brasington Working Paper No June, 1999 DEPARTMENT OF ECONOMICS TULANE UNIVERSITY NEW ORLEANS, LA 70118

2 Private Schools and the Demand and Supply of Public School Quality in Metropolitan Areas David M. Brasington Department of Economics Tulane University New Orleans, LA (Ofc) (Fax) ( ) June 30, 1999 Submitted:???

3 2 Abstract The role of private schools in the housing market and in the supply and demand for public schooling is investigated. The following are new to demand for public schooling studies: accounting for spatial autocorrelation in the hedonic house price estimation and the supply and demand, a theoretical model of education supply, an empirical estimation of supply, and addressing hedonic sample selection bias. Public school quality supply seems responsive to private school competition but not to competition from other public schools. The percentage of school-aged children attending private schools seems positively associated with house values, holding public schooling outcomes constant. JEL Classification codes: H41, R21, R22

4 I. Introduction 3 Do public schools respond to competition from private schools? Does the demand for public school quality depend on the attractiveness of the private school alternative? Public school quality is a well-known and important determinant of house prices. However, particularly in metropolitan areas, homeowners with children have a choice between public and private schools. The current study investigates the effect of the private school alternative on house prices and on the demand and supply of public school quality. This paper advances economic literature in four primary ways. First, with all the demand for school quality papers, it is amazing that no study has yet been published that estimates the supply of public school quality. Second, the calculation of prices from the hedonic model is the most careful and most extensive of any demand for school quality study; it is the only one that addresses either sample selection bias or spatial autocorrelation. In addition to the hedonic house price function, the demand for and supply of public schooling are estimated using spatial statistics; this is also an innovation. Third, the role of competition from private schools and other public schools on the supply of public school quality is investigated. Fourth, the first theoretical model of the supply of public school quality is formulated. The paper proceeds as follows. A hedonic house price equation is estimated. From the regression results, the implicit prices of public school quality and private school attractiveness are calculated. These implicit prices are then used in estimations of the demand and supply of public school quality. II. The Hedonic House Price Method The hedonic method expresses the price of a complex good as the summation of the value of its constituent parts. For housing, these characteristics include physical attributes such as the number of rooms and the square footage of the house and the yard. Other less tangible attributes also affect house price. These include crime in the community, pollution levels, noise, and the quality of the local public schools.

5 4 The current study estimates the supply and demand for public schooling. Supply and demand are functions of price, but because there is no explicit market for public school quality, there is no readily observable unit price. Sherwin Rosen (1974) explains how the implicit price of public school quality may be calculated using the hedonic approach. In addition, he suggests using a simultaneous equation technique such as two-stage least squares to estimate the supply and demand system. The main shortcoming of Rosen s method is that the estimated marginal attribute prices may not provide any real information over what the original hedonic provided. The only new information is the functional form restriction placed on the demand and supply equations. If there is no new information, the coefficients of the demand and supply system can be derived from the initial hedonic (Tinbergen, 1956; Brown and H. Rosen, 1982). The literature labels this a special type of identification problem. Therefore, this study derives marginal prices following Brown and Rosen s suggested method of segmenting the sample to deal with the identification problem. A separate hedonic house price function is estimated for each of the five major metropolitan areas in Ohio: the Cleveland CMSA, which includes the Akron and Cleveland MSAs; and the Cincinnati, Columbus, Dayton, and Toledo MSAs. Estimating each metro area separately will result in five different school quality coefficients. The question arises: What is the theoretical justification for using a separate hedonic for each metropolitan area? Palmquist (1984) assumes there is no segmentation within a metro area but there is between them because of moving costs. However, moving costs between suburbs of a single metro area are likely to be almost as high as those between metro areas. So instead, this study reasons that there is market segmentation between metro areas but not within a metro area because of job availability. It is relatively easy to commute from one part of a metro area to the workplace as it is from another part of the same MSA. In contrast, it is much more difficult to commute from a different metro area to the workplace in the original MSA. It is even more difficult in the short run to find a job in a new metro area, move to that new metro area,

6 5 and then commute to work. Also see Epple (1987) and Des Rosiers and Theriault (1996) for further discussion of segmented markets in hedonic estimations. III. Hedonic Estimation Technique A hedonic house price estimation must be executed for each metropolitan area so that the implicit price of schooling may be calculated. The hedonic in the current study contends with two estimation problems that are not commonly addressed: sample selection bias and spatial autocorrelation. Haurin and Hendershott (1991) were the first to suggest that hedonic coefficient estimates may be biased due to certain types of homes being over-represented in a housing sales data set. Overrepresentation may come from two sources: 1) starter homes that sell more frequently than other homes, and 2) economic conditions in the neighborhood. Jud and Seaks (1994) were the first to correct housing price estimates for sample selection bias. Suppose there is an unobserved variable w* that determines whether or not a property is sold in a census block group during a certain time period. If a house is sold, assume w* is positive and w takes the value 1. If there are no housing sales, w* is negative and w=0. The probit that governs the sale is thus 1) w i * = αz i + u i, u i ~N(0,1) 2) w i = 1 if w i * > 0; w i = 0 otherwise 3) prob (w i = 1) = Φ(αz i ); prob (w i = 0) = 1 - Φ(αz i ) where i represents house i, z is a vector of observable characteristics determining whether a property is sold, and Φ is the cumulative normal function. Let the traditional hedonic house price estimation be the following: 4) v i = βx i + ε i, ε i ~N(0, σ 2 ) where house value v i is a function of characteristics x i. However, this housing sales price is only observed when w i = 1. This implies 5) E[v i w i = 1] = βx i + ρσλ(αz i )

7 6 where ρ is the correlation between v and w, and λ is the inverse Mills ratio. The bias in 5) may be ameliorated, but not solved (Kennedy, 1998, p. 252), by employing the familiar Heckmann (1979) procedure. All the counties that constitute an MSA are chosen. A probit is then run in which the dependent variable is a dummy variable for whether there are no housing sales in each census block group during 1991 (EMPTY = 1) or there are (EMPTY = 0). This process is repeated for each MSA. Means of the variables used in the probits are shown in Table 1, while Table 2 shows variable definitions and sources. The results of the sample selection probits are found in Table 3. (Insert Table 1) (Insert Table 2) (Insert Table 3) The results of the probit regressions are generally consistent with expectations. There tends to be a lack of housing sales in census block groups that have a large percentage of vacant buildings, persons living in poverty and houses using well water, this last of which indicates a lack of city sewage service. On the other hand, census block groups with a large percentage of white residents and high income levels tend to have housing sales in 1991, possibly indicating a vibrant housing market. Although it is possible to criticize the probits for a paucity of explanatory variables, data is only available for a select number of characteristics for those census block groups that have no housing sales. In fact, only one usable variable is excluded from the probits: the percentage of persons holding graduate degrees. From experimentation, it seems that it is somewhat collinear with %WHITE. It is also possible to criticize the probits on the grounds that an overwhelming proportion of the census block groups have housing transactions. However, the slope coefficients are not greatly affected by unequal sampling rates in a probit and are completely unaffected in a logit model. Furthermore, weighting the probit because of unequal sampling rates is not only unnecessary but improper (Maddala, 1992, p.331). The inverse Mills ratios are calculated from the probits and are included in the hedonic regressions.

8 7 The estimation of the hedonic house price regressions is now discussed. Abstracting from sample selection bias for the moment, consider the traditional hedonic estimation given by equation 4) above. LeSage (1997) explains that spatial autocorrelation may exist if each house price influences other nearby house prices. Ordinary least squares does not account for this interplay between observations, which may lead to biased, inefficient and inconsistent parameter estimates (Anselin, 1988, p.58-59). A study by He and Winder (1999) demonstrates bi-directional price causality between three adjacent housing markets in Virginia, suggesting that there may indeed be spatial effects in housing markets. An instrumental variables technique may be used, but most attempts to adjust for spatial autocorrelation have been based on maximum likelihood (Anselin and Hudak, 1992). Epple (1987) discusses the use of instrumental variables to solve the identification problem of Brown and Rosen (1982) and Tinbergen (1956), but Epple also advocates the use of maximum likelihood and specifically mentions its usefulness in housing market estimation. Therefore using maximum likelihood in the manner that follows will not only address spatial effects but may also help with identification. Neighboring houses affect each other more than houses far away; consequently, a spatial weights matrix must be constructed to summarize the spatial configuration of the data. See LeSage (1997) for an excellent, intuitive discussion of the spatial weights matrix. The spatial autoregressive model takes the following form: 6) v i = ρwv i + βx i + ε i where ρ is the spatial autoregressive parameter and W is the spatial weights matrix that tells how much influence neighboring observations have on the observation in question. Griffith (1988) provides actual SAS programming code with which to perform the spatial autoregressive estimation in 6), but let researchers be warned: even the regression of the smallest MSA in this paper exceeded the mainframe s capacity by at least a factor of six. Instead, a mixed regressive spatially autoregressive model with common factor specification (Pace and Barry, 1997a; Anselin, 1988, p. 35) is proposed: 7) v i = ρwv i + βx i + Wx i + ε i

9 Equation 7) has the following log-likelihood function (Anselin, 1988, p. 181): 8 8) L = -(n/2)lnπ - (n/2)lnσ 2 + ln I-ρW -(1/2σ 2 )[(I-ρW)v i x i β] [(I-ρW)v i x i β] where n is the number of observations and is the determinant of the matrix. Equation 8) may be simplified to the following concentrated log-likelihood function (Pace and Barry, 1997a): 9) L = C + ln I-ρW - (n/2)ln(sse(ρ)) where C is a constant and SSE(ρ) is the sum of squared errors associated with a given value of the spatial autoregressive parameter. In this format the sparsity of the spatial weights matrix W may be exploited (Pace, 1997; Pace and Barry, 1997a, 1997b) so that a personal computer can handle the large data set estimations with computational ease 1. This procedure has been demonstrated to greatly improve cross-sectional regression estimates that are spatial in nature (Pace 1998a, 1998b; Pace and Barry, 1997c). IV. Choice of Hedonic Variables The house characteristics used are an air conditioning dummy, a fireplace dummy, the number of outbuildings, square feet of lot, age of the house, size of the house, a garage dummy, the number of full and partial bathrooms, and dummy variables for the presence of porches, patios, decks and pools. The squares of lot size, house size, and age are included because these variables may influence a house s value in a nonlinear fashion. The following community variables are hypothesized to be capitalized into house value: the percentage of census block group residents that are white, commuting time, the percentage of residents who have a graduate degree, income levels, poverty and tax rates, a pollution measure, the percentage of schoolaged children who attend private schools, and public school proficiency test passage rates. Following standard urban economic theory, commuting time should influence house price. In this theory, households all work downtown and are therefore expected to value proximity to the workplace to lower commute costs. The amenity literature inspires the income, racial composition, educational attainment, poverty and pollution

10 9 variables. Higher income, percentage of white residents and education levels are expected to raise constantquality house price, while a higher poverty rate and danger from polluting sources are expected to lower house price. Local public economic theory suggests that taxes and school quality will be capitalized into house value. In accordance with Yinger, Bloom, Borsch-Supan, and Ladd (YBBL, 1988), the measure of taxes employed is the community s property tax rate. All else constant, higher tax rates are expected to lower house value while improved school outcomes are expected to increase house value. Public school quality is one of the most important determinants of house price (Goodman and Thibodeau, 1998; Haurin and Brasington, 1996; Black, 1999); even today, though, many researchers include no such measure to help explain house price (Benjamin and Sirmans, 1996; Des Rosiers and Theriault, 1996; Jud and Seaks, 1994). Those who do account for school quality use many different measures. These include expenditure per pupil (Edel and Sclar, 1974; Gustely, 1976; Bradbury, Case and Mayer, 1995), the pupil/teacher ratio (Harrison and Rubinfeld, 1978; Grether and Mieszkowski, 1974), and a dummy variable for whether residents believe school quality is bad (Linneman, 1980). Rosen and Fullerton (1977) criticize the use of expenditures as proxies for the quantity or quality of public education. They claim that it is not certain that the output of schools can be measured by the expenditure on inputs. Accordingly, they use reading and math achievement scores of 4th graders as school quality measures. They find that using an output measure yields more theoretically consistent capitalization results than using an input measure. Mozayeni (1995) compares the use of expenditures and outcomes of public services in a hedonic house price study and finds less consistent results when outcomes are used, but most of the recent literature has followed Rosen and Fullerton s advice of using outcomes to measure public school quality. However, even within the outcomes category of school quality measures there are choices. There are two primary competing alternatives: value-added measures and proficiency tests. The value-added approach is rising in popularity, and the author is currently aware of two studies that use value-added in a house price

11 10 hedonic (Haurin and Brasington, 1998; Hayes and Taylor, 1996). Nevertheless, the school outcome variable chosen here is the percentage of students who pass all four sections of the Ohio 9th-grade proficiency test administered in The sections are reading, writing, math, and citizenship. It is not a perfect measure of school quality, but this outcome measure is very visible to parents (Jud, 1985), it avoids the sample selection bias of SAT scores (Hanushek and Taylor, 1990) 2, and using proficiency test scores follows the precedent of the majority of the recent literature (Black, 1999; Haurin and Brasington, 1996; Walden, 1990; Goodman and Thibodeau, 1998). Furthermore, Hayes and Taylor (1996) do not find that value-added is more valued by the housing market than proficiency tests; rather, both measures are found to work well. Future work will address a comparison of expenditures, value-added, and proficiency test scores as school quality measures. Finally, it is possible that the attractiveness of private schools affects house value. Hayes and Taylor (1996) find a positive relationship between house prices and the percentage of students attending private school. Using a linear structural relations model, Seiler (1996) finds that the percentage of students enrolled in private schools is not significantly related to home values. The current study, too, uses the percentage of students enrolled in private school instead of private school tuition to capture the attractiveness of the private school alternative for whatever reason: cost, quality, or inclusion of religious instruction. V. Data The primary source of data for the hedonic regression is a tape of housing purchases that occurred during 1991 in Ohio (Amerestate, 1991). The data are cleaned to eliminate all but single-family detached dwellings. To keep the sample urban, any house with lot size greater than two acres is suspected of being a farm and deleted. Houses that transact at prices above $400,000 are deleted for being unrepresentative, and houses that transact for less than $10,000 are deleted for suspicion of being either uninhabitable or a gift between family members. In addition, outliers in square feet of housing and garage size are deleted. Any district with less than 17 observations is also omitted. There are 44,255 houses left in 140 communities, and

12 11 the mean deflated sale price is $72, The Ohio Environmental Protection Agency (1994), the Ohio Department of Education, and the U.S. Bureau of the Census (1990) provide the remainder of the explanatory variables. Table 4 contains variable means, and Table 2 shows definitions and data sources. (Insert Table 4) VI. Hedonic House Price Equation Results Table 5 shows the results of the five hedonic regressions. (Insert Table 5) The hedonic regressions in Table 5 explain a remarkable percentage of the variation in LOG HOUSE PRICE; adjusted R-square ranges from 0.90 in Cleveland to 0.94 in Toledo. The optimal lag coefficient ranges from 0.37 to Ordinarily this would suggest mild spatially autoregressive effects. However, with fifty-one explanatory variables, the regressions are already capturing almost all of the effects in the model except spatial effects. In the presence of so many independent variables, 0.40 is a fairly copious spatial autocorrelation parameter (Pace, 1999). The lagged variables in Table 5 essentially indicate how far off estimates would be if the spatial lag were omitted. For example, consider PATIO for Cincinnati. It has a coefficient of LAG PATIO has a coefficient of PATIO is somewhat correlated with its spatial lag. Therefore, if only PATIO were included without its spatial lag in the regression, the coefficient of PATIO might have been negative ( < 0). The parameter estimates are almost exactly as expected for house characteristics. It is noteworthy that the age of a house is negatively related to its price, while its square is always positive. Together this implies that a graph of a house s value to its age would be concave to the origin: older houses command a lower price, all else constant, and for every year older a house becomes, the greater is the marginal discount. The community characteristics are also generally consistent with theory. Community education and poverty levels

13 12 have particularly consistent relationships with constant-quality house price. On the other hand, COMMUTE TIME has sporadic signs and significance levels. Perhaps the more decentralized nature of employment has eroded the empirical relationship between house value and proximity to work (Palmquist, 1984). Contrary to Seiler (1996) but consistent with Hayes and Taylor (1996), the percentage of high schoolaged students attending private school has a positive association with house prices, all else constant. In each of the five regressions this variable has a positive coefficient, and it is statistically significant in four of the five. Next, the tax rate is insignificant in four of the five MSAs. The tax rate may be capturing the effect of additional services that are omitted from the regression; it could also be capturing the cost of services or economies of scale in service provision. The focus variable, proficiency test score, works very well. It is positive and significant in every hedonic. Toledo has the highest coefficient estimate: In contrast, Columbus has the lowest: Because Toledo s coefficient is much greater than Columbus, it seems that there is sufficient variation in the estimates to achieve identification in the manner Palmquist (1984), Epple (1987), and Brown and Rosen (1982) suggest. VII. Calculating the Implicit Prices The results of the hedonic regressions are now used to calculate the implicit price of public and private school quality. Whereas the unit of observation was the 44,255 individual housing transactions for the hedonic regression, the unit of observation now becomes the 140 school districts. There are 66 school districts in Cleveland, 26 in Cincinnati, 18 in Columbus, 18 in Dayton, and 12 in Toledo. The implicit prices of public and private schooling are the partial derivative of each school quality variable with respect to house value. Therefore, for the implicit price of public school quality, if there are n school districts in a metropolitan area, each of the n median voters house values is multiplied by that market s coefficient estimate of LOG PROFICIENCY TEST SCORE and divided by the proficiency test score. The

14 13 same technique is applied to every metro area. Each of the resulting mathematical products is an implicit price of public educational quality. The implicit price of private school quality is calculated in the same way, except that, due to functional form, there is no division by the proficiency test score. For example, Madeira City School District is one of the Cincinnati MSA s school districts. To calculate the implicit price of public school quality, take the median home value in Madeira and multiply it by 0.064, the coefficient of public school quality in the Cincinnati MSA s hedonic in Table 5. The product of these two numbers is then divided by Madeira s proficiency test score. The implicit prices are adjusted for the tax rate because the property tax affects the capitalization value of education on house prices: households in communities with high tax rates will pay more property taxes and therefore will receive lower capitalization benefits of school quality (Crane, 1990). communities: The following simultaneous system of equations for school quality is then estimated for the ) demand: Q j = f(m j, d j, t j, s j ) 11) supply: Q j = f(m j, r j ) where Q j is school quality in district j, m j are the marginal implicit prices calculated from the first-stage hedonics, d j are the socioeconomic factors that influence demand, t j are the public finance variables that influence demand, s j are the demand variables related to sorting, and r j are supply shifters for education. The demand and supply are now discussed. VIII. The Demand for Public Schooling The following theoretical model of the demand for public school quality is based on Murdoch, Rahmatian and Thayer s (1993) model of recreation expenditures. Let the median voter in school district j have the following utility function: 12) U j = U j (n j, x j, Ω j )

15 14 where n is consumption of a numeraire good, x is consumption of the pure private component of schooling, and Ω is total consumption of the public aspect of public schooling, and U is a twice continuously differentiable, strictly quasi-concave and strictly increasing function of its arguments. An unusual feature of this model is its incorporation of possible external benefits to the production of public education which may spill across jurisdictional boundaries. Formally, 13) Ω j = q j + ω j so that total consumption of the public aspect of schooling by members of school district j is determined by own provision of the public component of education (q) and spill-ins from provision in neighboring communities (ω). Each community undertakes an activity g j : providing public schooling. This activity jointly produces x and q, with the benefits of x staying strictly within the community but allowing q to spill over into neighboring school districts. The following joint product technology (Sandler, 1977) is assumed: 14) x j = θ j *g j 15) q j = φ j *g j so that a certain fraction θ and φ from education g goes toward generation of the private component and pure public component of schooling. Because equation 15) holds for all school districts, 16) ω j = (φ i g ) i j i where φ i is the fraction of activity in school district i that spills in to jurisdiction j as a public good. Spillovers from education are expected to be inversely related to the distance between the school districts in question. Reiter and Weichenrieder (1997) present one way to conceptualize spillovers. Another way to account for spillovers is to incorporate spatial autocorrelation in the statistical estimation. The mixed regressive spatially autoregressive model used for the hedonic estimation is therefore applicable to the demand for public school quality estimation as well. From equations 13), 14) and 15) the median voter s utility function may be rewritten as follows:

16 17) U j = U j (n j, θ j *g j, φ j *g j + ω j ). 15 Utility is maximized by choosing g j and n j subject to the following budget constraint: 18) y j = n j + τ j g j where y is the median voter s income and τ is the per-unit cost of education faced by the median voter. Schooling is chosen in a Nash equilibrium sense, so that the median voter in community j chooses g taking other districts schooling decisions as given. Utility maximization may proceed in the usual manner. In addition to spatial considerations of spillovers, there are other issues that must be addressed when performing a demand for schooling estimation. These issues include Tiebout bias and the choice of variables in the demand estimation. IX. Tiebout Bias and Other Demand Estimation Hazards Jud and Watts (1981) is one study that derives the implicit price of school quality from a housing market hedonic. In so doing, they simply average the implicit prices of each household in a community to arrive at a single price measure for each community. Although adequate, this technique is not based on any theoretical justification. In contrast, the current study relies on the median voter model (Bergstrom and Goodman, 1973) for the calculation of each community s implicit price. However, the median voter approach is not free from criticism. Goldstein and Pauly (1981) discuss the potential drawbacks in some detail, with particular emphasis on its relation to the Tiebout hypothesis (Tiebout, 1956; Hamilton, 1975). The Tiebout hypothesis presents a description of how households sort themselves among communities that have different levels of taxation and public good provision. Each household moves to the jurisdiction that meets its particular taste for local public goods. Goldstein and Pauly say a problem arises because the median voter model assumes proportionality of desired service level in income. In reality, the proportionality of income assumption is violated because there is imperfect Tiebout sorting, and choosing a median voter on the basis of income will bias regression results. Goldstein and Pauly

17 16 call this Tiebout bias. They say the use of individual micro data will more likely produce unbiased parameter estimates. However, Rubinfeld, Shapiro, and Roberts (RSR, 1987) point out that even if micro data is used, it is likely to suffer from the same Tiebout bias that Goldstein and Pauly say may occur when the median voter model is used. RSR suggest including sorting variables that will minimize the mismatch between actual and desired spending, thus reducing the magnitude of the error term and reducing omitted variable bias. X. Choice of Variables for Demand for Education The issue of the proper measure of school quality in a hedonic house price regression has already been addressed; however, an additional concern is the appropriate measure of school quality in a demand estimation. The traditional dependent variable is expenditures per pupil. One such example is Rubinfeld (1977). However, Rosen and Fullerton (1977) argue that in hedonic estimations, student test scores may better represent school outcomes than expenditures on an input to the education process. Similarly, using a school output is more appropriate than using expenditures on inputs to measure the demand for education quality. In addition, using expenditures to measure demand may trigger an apples-to-oranges comparison problem. Large school districts may offer specialized courses like Chinese and programs like swimming and diving that small school districts cannot offer, but the increased spending may not represent different demand for education per se (Reiter and Weichenrieder, 1997). The hedonic house price equations are estimated, the implicit price vector, m j, is calculated and is used endogenously (Palmquist, 1984; Bartik, 1987) in the school quality demand and supply estimations. No prior demand estimation has included the implicit prices of both public and private school quality, but both should be included. Quantity demanded is a function of the price paid. The implicit price of public education measures the price of the good from a capitalization framework. Furthermore, demand for a commodity should depend not only on the price of the commodity, but also on the price of available substitutes. Private

18 17 schooling may be a substitute for public schooling. The capitalized value of private school attractiveness is therefore included in the estimation of the demand for public school quality. There is an additional price of education present in the theoretical model that belongs in the demand estimation: taxes. The implicit price of public schooling and the tax rate are not duplicative; in fact, the correlation between them is only The implicit price measures the capitalization price; the tax rate measures the tax effort required to achieve a certain amount of spending on public schooling. For a fixed amount of public school expenditure, a community with a lower overall tax base requires a higher tax rate. A higher tax rate is expected to be associated with lower demand for school quality. Similarly, the implicit price of education should be negatively related to the quantity of school quality demanded just as price is negatively related to the quantity demanded for any typical commodity. The current demand estimation will therefore include the implicit price of public school quality, the implicit price of private school quality, and taxes. Previous studies, in contrast, use either the implicit price of public school quality or taxes, but omit the other two variables. There is some debate about the operation of taxes in the median voter framework (Reiter and Weichenrieder, 1997). Because of possible assessor bias, the nominal property tax rate is used as a long-term estimate of the tax price (YBBL, 1988). In addition, due to the decisiveness of the median voter, the tax rate is not an exogenous variable and may depend on such things as exported tax (Downes and Pogue, 1994). Furthermore, taxes should be nonlinear in demand: a higher budget leads to capital migration, which in turn means a larger share of the budget must be borne by the median voter (Reiter and Weichenrieder, 1997). The tax rate is therefore treated as a nonlinear and endogenous variable in the demand estimation, using instruments based on discussion by Reiter and Weichenrieder (1997), Bartik (1987) and Epple (1987). Demographic variables hypothesized to influence demand are average resident age, income levels, racial composition and educational attainment. These variables either appear directly in the theoretical model or influence the choice of g relative to n. RSR (1987) inspire the inclusion of the percentage of residents who have lived in a community for less than six years and a central city dummy to help mitigate Tiebout bias 4.

19 18 A list of variables used in the demand estimation along with means is presented in Table 6. Definitions and sources are found in Table 2. (Insert Table 6) XI. Demand for Education Results The results of the demand for public education regression are shown in Table 7. Before specific results are discussed, note first that spatial autocorrelation does not appear to be a factor in the demand estimation. The optimal spatial lag coefficient is zero; moreover, a likelihood ratio test accepts the null hypothesis that the spatially lagged variables as a group have no effect on the regression 5. Perhaps the idea of spillovers in education demand is only of theoretical, not empirical, concern. More study is required to support this assertion. Also note that the maximum likelihood estimation has an astounding adjusted R- squared of 0.97, so that much of the variance in proficiency test passage is explained by the included variables. (Insert Table 7) The signs of all coefficients are consistent with expectations. Few are statistically significant, but this may not be too much of a concern (McCloskey and Ziliak, 1996). Own price has the anticipated negative sign; the price elasticity of demand is 0.20, calculated at the mean. Taxes are negatively associated with the demand for schooling. The elasticity of school quality with respect to taxes is These elasticities are precisely in line with prior studies: the implicit price and tax elasticities in demand studies typically range from 0.20 to 0.40 (Reiter and Weichenrieder, 1997). Private schooling appears to be a substitute for public schooling; the first cross-price elasticity of demand is Communities with older residents tend to demand less public schooling, as do communities with low educational attainment levels. School districts with large proportions of white residents and those with high incomes are associated with high levels of school quality demand. The income-elasticity of demand is 0.20,

20 19 which is not much larger than the elasticities reported in RSR (1987). Attention now turns from the demand side to the supply of school quality. XII. Theoretical Model of Public School Quality Supply No paper has yet been published which formulates a theoretical model of the supply of public school quality. This paper introduces such a model, building off Khanna s (1992) model of agricultural research grants from national to state governments. As discussed in the demand section, education is a joint product with both pure private and pure public benefits, the latter of which may spill over into neighboring communities. A local school district may use subsidies from state and national government grants in addition to its own tax efforts to fund its public schools. In Ohio, local tax effort accounts for 50%, state grants contribute 35%, and national and other sources contribute the remaining 15% of the average school district s funding. Local effort consists almost entirely of property taxes in the Ohio metro area sample, although one suburban district has a local school income tax. The local government does not decide on the total amount of education quality supplied: such a decision is beyond its control and depends partially on the demographic composition of students and parents in the community. However, the local government does allocate spending toward education. Spending, parental factors and other factors together determine quantity supplied. The relationship between spending and supply shifters and the choice of supply factors will be discussed in the next sections. Total funding for public schooling may be represented as follows: 19) f j = f jj + f gj where f j is total funding for district j, f jj is funding from strictly local sources, and f gj is funding from state and national government grants that district j accrues. As in the demand model, local public schooling is a joint product that yields a pure private good x and a pure public good q according to some function of total spending:

21 20) x j = x j (f j ) 20 21) q j = q j (f j ) and the total public component of schooling consumption available is 22) Ω j = q j + ω j where 23) ω j = ω j ( ( f + ) ). i j ii f gi That is, the total public element of schooling consumption available is that derived from local effort and spillins, where the spill-ins themselves depend on local funding by neighboring jurisdictions and the grants that they receive. As with demand, the existence of spillovers and spill-ins in the theoretical supply model suggests the use of spatial statistics. Local governments preferences may be represented as 24) U j = U j (n j, x j, Ω j ) where U is strictly increasing, quasi-concave, and twice continuously differentiable, and n is the numeraire good. Substituting 19), 20), 21) and 23) into 24) yields 25) U j = U j [n j, x j (f jj + f gj ), q j (f jj + f gj ) + ω j ( ( f + ) )] i j ii f gi Equation 25) is maximized by choosing f jj and n j in a Nash framework; that is, holding the actions of other agents constant. There are two constraints on local government s utility maximization. First is the familiar budget constraint, which has the following form: 26) I j = n j + p*(f jj + f gj ) where I is total funds available to the local government, including those from intergovernmental grants; p is the price of funding a unit of public schooling expenditure, which may differ from the price of the numeraire good n.

22 21 The second constraint on local governments is the joint-use constraint that results from the privatepublic nature of grants from the state and national government: 27) f g = f gj + f gi Equation 27) states that the total amount of grants available from state and national sources must be allocated across local governments in some manner, and these grants are rivalrous in consumption. Equation 27) is substituted into 25) and the Lagrangian is formulated: 28) L = U j [n j, x j (f jj + f gj ), q j (f jj + f gj ) + ω j ( ( f + f f ) )] + λ (I j - n j p*(f jj + f gj )) i j ii g where λ is the Lagrange multiplier. In addition to the budget constraint, the following first order conditions are derived: U j x j U j q j 29) + λ p = 0 for all j x f q f j jj j jj gj U j 30) λ = 0 n j for all j. After substitution, first-order conditions 26), 29) and 30) yield the following solution: U 31) (f jj + f gj ) x j j x f j jj U + q j j q f j jj = U n j j (I j n j ). However, choosing levels of funding alone will not determine supply of school quality. Funding will only affect the supply of schooling through the demographic and economic characteristics of the district. Explanatory variables to be used in the empirical model and the relationship between education supply and an education production function are now discussed. XIII. The Supply of Education

23 22 There is a critical distinction between the supply of education and an education production function. In a production function, an output measure is regressed upon a collection of inputs to production. For the production of education, these inputs may include expenditure per pupil, teacher salary, the pupil/teacher ratio, and teacher education and experience levels. In contrast, a supply function only contains prices and shift variables. Expenditure per pupil is an input to the production of schooling, but it is not a supply shifter. A supply shifter is a variable that shifts the supply curve holding spending constant. For example, in an estimation of hours of labor a person supplies, relevant factors include the wage rate and presence of children. However, technological equipment is only an input to the productivity of labor; it is not a shifter of the number of hours of work supplied by an individual. There is some confusion between the supply of education and an education production function. Jud and Watts (1981) present (but do not estimate) an education supply function; unfortunately, it is actually a cross between an education supply and an education production function. For example, it contains the implicit price of school quality, but it also contains the pupil/teacher ratio and expenditure per pupil. McNamara and Johnson (1985) claim to estimate a supply curve of education. Instead of calculating an implicit price, however, they substitute expenditure per pupil. They thus convert their supply curve into a production function. There is also some confusion between the supply of education and the demand for education. For instance, Rubinfeld (1977) says our model rules out the real possibility that individuals will vote for a millage proposal solely on the grounds that it is likely to raise the value of their own property, or to lower the cost of the public services by raising the value of taxable property in the community. There is nothing wrong with this statement if it is an estimation of the supply of education; however, Rubinfeld is modeling the demand. If there were a measure of the implicit price of education in Rubinfeld s demand estimation, his argument says that the price would be positively related to demand.

24 XIV. Choice of Variables for Supply of Education 23 Any supply estimation is incomplete if it excludes the price of the good. The implicit price of education therefore belongs in the supply of public school quality estimation, although it does not belong in an education production function. The higher the capitalization price of school quality into house value, the more incentive homeowners have to supply additional education quality (Jud and Watts, 1981). Shifters of the supply of public school quality are not confined to demographic influences. Murnane and Levy (1998) tell how a school district in Texas shifted its supply curve to the right through the efforts of an outstanding principal. At a school meeting, the principal let parents know that although the students were getting As and Bs, the school s proficiency test passage was worst in the state. Parents became outraged then more involved, teachers became scared then energized, and the school s proficiency test passage rate surged; all this with the same demographic mix of students. Typically, though, demographic and related influences are the primary shifters of educational supply. Supply shifters include student and parent attributes like the percentage of students living with both parents. Given expenditures, an increase in the proportion of students living with both parents is expected to shift the supply curve of education to the right. This is because a two-parent household can devote more time to nurturing its children than single-parent households. Student characteristics are more difficult to measure. Coleman et. al. (1966) suggests that student motivation levels are important determinants of school quality. In this spirit, the student attendance rate is included to capture student motivation levels. Holding spending constant, a higher attendance rate is expected to shift the supply curve of education to the right. The degree to which the community and the school are conducive to learning are also supply shifters. Psychologists argue that a primal need for humans is safety; therefore, a lack of safety may distract a student from his or her studies. The crime rate is one way to measure the safety of the community in which a student lives. A higher crime rate is expected to shift the supply of education to the left, holding spending constant. It is more difficult to measure the environment of the schools themselves. Good measures might include

25 24 suspensions and expulsions and gang activity, but this data is unavailable. It may be possible to infer school environment by the quality of teachers the school district is able to attract. Holding spending constant, better teachers will tend to self-select to safer school environments (Hanushek, Kain and Rivkin, 1999). Average teacher education level is an available measure of teacher quality. Therefore, higher teacher education levels, as a proxy for school environment, are expected to shift the supply of school quality to the right. Because teacher education levels are a choice variable to the school district, this must be treated as endogenous (Akerhielm, 1995). Competitive pressures may also influence the supply of public school quality. Competition may come from private schools as well as from other public schools. Each of these possibilities is discussed in turn. Couch, Shughart and Williams (1993) find that the percentage of students in a public school district who attend private schools has a positive relationship with standardized algebra test scores in North Carolina s public schools, suggesting that public schools respond to competition from private schools. Borland and Howsen (1996) extend Couch, Shughart and Williams analysis to include the effect of both public and private school competition on public school performance. This competition from both sources is captured in a single Herfindahl index, which is found to be positively related to public school performance. Hoxby (1998) also finds competition from private schools is positively related to higher public school achievement. Dee (1998) finds competition from private schools has a positive and statistically significant impact on high school graduation rates of neighboring public schools. In contrast, Zanzig (1997) finds a negative relationship between public school math test scores and the percentage of students in the county who attend private schools. %PRIVATE is thus included in the supply function to capture the relationship between private school competition and public school performance. Following Hoxby (1998), this variable is treated as endogenous to public school quality. There may also be competition between public schools. Hoxby (1998) says such competition should exist because efficient and high-quality education providers are rewarded with higher budgets. An increase in

26 25 quality causes an increase in house prices, which increases the tax base, tax collections, and the size of the school budget. In addition, Hoxby suggests schools are more responsive to parent s desires as opposed to school staff desires when there are many public school districts in the area. Hoxby further asserts that when parents have more choice among districts, they are more involved in their children s schooling. One may expect a positive relationship between the number of school districts and public school outcomes, then. Indeed, Hoxby finds such a relationship. Using a Herfindahl index to measure competition, she finds an increase in competition is associated with a small but statistically significant increase in achievement. Zanzig (1997) generally finds that the number of public school districts in a county has a positive effect on public school performance until a threshold of three or four districts is reached, beyond which additional public schools depress performance. Based on the literature, different variables representing public school competition are included. One such variable is #DISTRICTS. The more public school districts there are in a metro area, the more Tiebout sorting will occur, and the more competition any one public school district will face. Another competition measure employed is FRAGMENTATION 1, the ratio of the number of public school districts in the metropolitan area to MSA population. Finally, FRAGMENTATION 2 is the ratio of the number of public school districts in the metropolitan area to the population of each school district. The greater the fragmentation of school districts in the metro area, the more competition is expected between public school districts. An additional regression will be run in which private school competition is included without any public school competition measure. Hoxby (1998) suggests that the number of districts in a metro area is endogenous to school quality, and that the number of streams in the area should be used as an instrument. However, Brasington (1999) empirically finds that consolidation is not a function of the quality of schools and therefore the number of districts may be exogenous to school quality. The public school competition variables will be treated as exogenous in this study. Table 6 shows the means of the variables used in the supply estimation, and definitions and sources appear in Table 2.

27 26 XV. Supply of Education Results The results of the supply of education estimations are reported in Table 8. As it was in the demand estimation, the optimal spatial lag coefficient is estimated at zero. However, a likelihood ratio test rejects the null hypothesis that the spatially lagged variables have no effect on the model at the 0.05 significance level. Therefore, there may be spillovers in the supply of public education. (Insert Table 8) The implicit price of public schooling has a coefficient of zero in all four regressions. Its likelihood ratios are higher than those of any other independent variable, so the parameter estimates appear to be measured with precision. The implication is that the supply of public school quality is perfectly inelastic. Although crime has the expected relationship with public school test outcomes, the relationship is neither economically significant nor consistently statistically significant. The school environment variable is not statistically significant, but it is economically significant, with an elasticity of Furthermore, increasing the percentage of school-aged children who live with both parents by ten percentage points would be expected to raise proficiency test passage by seven percent. The strongest supply shifter is the student attendance rate, which attempts to capture student motivation level. Its elasticity is This means a one percent increase in attendance rate is associated with an almost four percent increase in proficiency test passage. The focus variables are those related to competition. %PRIVATE has a positive effect on the supply of public school quality. The magnitude of the effect is not trivial: for every increase in %PRIVATE, public school supply increases by This result is all the more impressive in light of the assertion that this parameter estimate is biased toward finding a negative relationship (Hoxby, 1998). It seems that, holding expenditures constant, the more competition there is from private schools, the better public schools are at extracting good performance from their students. Alternatively, in the face of higher private school competition, it takes less expenditures to reach the same level of public school supply. The %PRIVATE