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The Effect of Transportati on Subsidies on Urban Sprawl by Qing Su A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Economics College of Business Administration University of South Florida Major Professor: Joseph S. DeSalvo, Ph.D. Kwabena Gyimah-Brempong, Ph.D. Gabriel A. Picone, Ph.D. Philip K. Porter, Ph.D. Kenneth F. Wieand, Ph.D. Date of Approval: May 2, 2006 Keywords: transit subsidies, highway subs idies, spatial expansion, census data, costs Copyright 2006, Qing Su
ACKNOWLEDGMENTS I am deeply indebted to those who assisted me in the various stages of this work and who made useful criticism and suggestions In particular, I wish to express sincere appreciation to my committee chairman, Prof essor Joseph S. DeSalvo, whose advice and generous contribution of time a nd talent made the completion of this thesis possible. I also want to thank the other members of my doctoral committee, Kenneth F. Wieand Kwabena Gyimah-Brempong, Philip K. Porter, a nd Gabriel A. Picone for their advice and support. Professor Wieand and Professor Pi cone's advice were especially helpful in the empirical analysis. I would also want to thank Mr. Steve P oyta from Census of Government for providing me access to data regarding transportation subsidies.
i TABLE OF CONTENTS LIST OF TABLES...........................................................................................................viii ABSTRACT....................................................................................................................... ix CHAPTER 1 INTRODUCTION.........................................................................................1 1.1 What is urban sprawl?..............................................................................................1 1.2 Consequences of Urban Sprawl...............................................................................3 1.2.1 Traffic Congestion..........................................................................................4 1.2.2 Loss of Open Space........................................................................................6 1.2.3 Income-Level and Racial Segregation............................................................7 1.3 What Causes Urban Sprawl?...................................................................................8 1.3.1 Local Public Finance Model...........................................................................8 1.3.2 Urban Economic Theories on Urban Sprawl................................................11 1.4 Summary................................................................................................................16 CHAPTER 2 SINGLE-MODE URBAN SPATIAL MODEL..........................................18 2.1 Definitions and Assumptions.................................................................................18 2.2 Demand Side of Housing Market..........................................................................21 2.3 Supply Side of Housing Market.............................................................................25 2.4 Overall Equilibrium of Urban Area.......................................................................28 2.5 Comparative Static Analysis of Single-Mode Urban Spatial Model.....................30 2.5.1 Effects of Change in Population...................................................................32
ii 220.127.116.11 Effect of Population on Utility..........................................................32 18.104.22.168 Effect of Populat ion on Urban Area Boundary.................................33 22.214.171.124 Effects of Population on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density.................34 126.96.36.199 Effect of Population on the Tax-Share Variable...............................35 2.5.2 Effects of Change in Agricultural Land Rent...............................................36 188.8.131.52 Effect of Agricultural Land Rent on Utility......................................36 184.108.40.206 Effect of Agricultural Land Rent on Urban Area Boundary.............36 220.127.116.11 Effects of Agricultural Land Rent on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density..............................................................................................38 18.104.22.168 Effect of Agricultural Land Rent on the Tax-Share Variable...........38 2.5.3 Effects of Change in Income........................................................................40 22.214.171.124 Effect of Income on Utility...............................................................40 126.96.36.199 Effect of Income on Urban Area Boundary......................................40 188.8.131.52 Effects of Income on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density.................42 184.108.40.206 Effect of Income on the Tax-Share Variable....................................44 2.5.4 Effects of Change in Commuting Cost.........................................................45 220.127.116.11 Effect of Commuting Cost on Utility................................................45 18.104.22.168 Effect of Commuting Cost on Urban Area Boundary......................46 22.214.171.124 Effects of Commuting Cost on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density.................48 126.96.36.199 Effect of Commuting Co st on the Tax-Share Variable.....................50 2.5.5 Effects of Change in Private Cost Share.......................................................51
iii 188.8.131.52 Effect of Private Cost Share on Utility.............................................51 184.108.40.206 Effect of Private Cost Share on Urban Area Boundary....................51 220.127.116.11 Effects of Private Cost Share on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density.................53 18.104.22.168 Effect of Private Cost Share on the Tax-Share Variable..................55 CHAPTER 3 TWO-MODE URBAN SPATIAL MODEL...............................................58 3.1 Modal Choices and Preliminary Results Used for Comparative Static Analysis.......................................................................................................58 3.2 Comparative Static Analysis..................................................................................61 3.2.1 Effects of Change in Population...................................................................63 22.214.171.124 Effect of Population on Utility..........................................................63 126.96.36.199 Effect of Populat ion on Urban Area Boundary.................................63 188.8.131.52 Effects of Population on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density.................64 184.108.40.206 Effect of Population on the Tax-Share Variable...............................65 3.2.2 Effects of Change in Agricultural Land Rent...............................................65 220.127.116.11 Effect of Agricultural Land Rent on Utility......................................65 18.104.22.168 Effect of Agricultural Land Rent on Urban Area Boundary.............66 22.214.171.124 Effects of Agricultural Land Rent on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density..............................................................................................67 126.96.36.199 Effect of Agricultural Land Rent on the Tax-Share Variable...........67 3.2.3 Effects of Change in Income........................................................................68 188.8.131.52 Effect of Income on Utility...............................................................68 184.108.40.206 Effect of Income on Urban Area Boundary......................................68
iv 220.127.116.11 Effects of Income on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density.................70 18.104.22.168 Effect of Income on the Tax-Share Variable....................................70 3.2.4 Effects of Change in Mode 1s Variable Commuting Cost..........................71 22.214.171.124 Effect of Mode 1s Variable Commuting Cost on Utility.................71 126.96.36.199 Effect of Mode 1s Variable Commuting Cost on Urban Area Boundary..................................................................................71 188.8.131.52 Effects of Mode 1s Variable Commuting Cost on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density.......................................................................74 184.108.40.206 Effect of Mode 1s Variable Commuting Cost on the Tax-Share Variable...........................................................................75 3.2.5 Effects of Change in Mo de 1s Private Cost Share......................................77 220.127.116.11 Effect of Mode 1s Private Cost Share on Utility.............................77 18.104.22.168 Effect of Mode 1s Pr ivate Cost Share on Urban Area Boundary...........................................................................................78 22.214.171.124 Effects of Mode 1s Private Cost Share on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density..............................................................................................79 126.96.36.199 Effect of Mode 1s Private Cost Share on the Tax-Share Variable...........................................................................80 3.2.6 Effects of Change in Mode 2s Variable Commuting Cost..........................81 188.8.131.52 Effect of Mode 2s Variable Commuting Cost on Utility.................81 184.108.40.206 Effect of Mode 2s Variable Commuting Cost on Urban Area Boundary..................................................................................82 220.127.116.11 Effects of Mode 2s Variable Commuting Cost on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density.......................................................................84
v 18.104.22.168 Effect of Mode 2s Variable Commuting Cost on the Tax-Share Variable...........................................................................85 3.2.7 Effects of Change in Mo de 2s Private Cost Share......................................86 22.214.171.124 Effect of Mode 2s Private Cost Share on Utility.............................86 126.96.36.199 Effect of Mode 2s Pr ivate Cost Share on Urban Area Boundary...........................................................................................87 188.8.131.52 Effects of Mode 2s Private Cost Share on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density..............................................................................................89 184.108.40.206 Effect of Mode 2s Private Cost Share on the Tax-Share Variable.............................................................................................90 3.2.8 Effects of Change in Mode 1s Fixed Cost...................................................91 220.127.116.11 Effect of Mode 1s Fixed Cost on Utility.........................................91 18.104.22.168 Effect of Mode 1s Fixed Cost on Urban Area Boundary................92 22.214.171.124 Effects of Mode 1s Fixed Cost on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density..............................................................................................93 126.96.36.199 Effect of Mode 1s Fixe d Cost on the Tax-Share Variable..............94 3.2.9 Effects of Change in Mode 2s Fixed Cost...................................................95 188.8.131.52 Effect of Mode 2s Fixed Cost on Utility.........................................95 184.108.40.206 Effect of Mode 2s Fixed Cost on Urban Area Boundary................96 220.127.116.11 Effects of Mode 2s Fixed Cost on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density..............................................................................................97 18.104.22.168 Effect of Mode 2s Fixe d Cost on the Tax-Share Variable..............98 3.3 Summary..............................................................................................................100 CHAPTER 4 EMPIRICAL ANALYSIS.........................................................................102
4.1 Objectives and Hypotheses..................................................................................103 4.2 Data Sources........................................................................................................104 4.2.1 United States Census 2000.........................................................................105 4.2.2 Census of Agriculture.................................................................................106 4.2.3 Census of Government................................................................................106 4.2.4 National Transit Database...........................................................................106 4.2.5 Highway Statistics......................................................................................107 4.2.6 Insurance Information Institute...................................................................107 4.3 Variables Used in the Analysis............................................................................107 4.3.1 The Urban Boundary, x .............................................................................108 4.3.2 Total Number of Households in Urban Area, L..........................................109 4.3.3 Mean Household Income, y........................................................................109 4.3.4 Agricultural Land Rent, r a ..........................................................................109 4.3.5 Fixed Transit Cost, f 1 ..................................................................................110 4.3.6 Variable Transit Cost, t 1 ..............................................................................110 4.3.7 Transit Subsidy, 1 .....................................................................................111 4.3.8 Fixed Auto Cost, f 2 .....................................................................................112 4.3.9 Variable Auto Cost, t 2 .................................................................................112 4.3.10 Highway Subsidy, 2 ................................................................................117 4.3.11 Local Property Tax Rate, .......................................................................122 4.3.12 Intergovernmental Grants, G....................................................................122 4.3.13 Missing Observations................................................................................123 vi
vii 4.4 Model Specification and Estimation....................................................................125 4.4.1 Functional Form..........................................................................................125 4.4.2 Potential Endogeneity Issue........................................................................126 4.4.3 Empirical Results........................................................................................131 4.4.4 Summary.....................................................................................................137 CHAPTER 5 POLICIES TO CONTROL SPRAWL......................................................139 5.1 The Sensitivity of Urban Spatial Size to Our Explanatory Variables..................139 5.2 Conclusion on Effect of Policy-Relevant Variables on Urban Sprawl................142 5.3 Future Research 5.3.1 Value of Time.............................................................................................144 5.3.2 Endogenizing Congestion...........................................................................145 5.3.3 Generalizing Mode Choice.........................................................................145 REFERENCES................................................................................................................146 About the Author...................................................................................................End Page
viii LIST OF TABLES Table 2.1 Comparative Static Re sults of Single-Mode Model.....................................57 Table 3.1 Comparative Static Results of Two-Mode Model......................................102 Table 4.1 Revenue Sources for Highways, 2000........................................................119 Table 4.2 List of Variables, Definitions, and Data Sources........................................125 Table 4.3 Summary Statistics.....................................................................................126 Table 4.4 OLS and 2SLS Regression Results.............................................................133 Table 4.5 First-Stage Results from Re gression of Auto Variable Costs on IVs........................................................................................................138 Table 5.1 Elasticities and Area Change......................................................................140
ix The Effect of Transportati on Subsidies on Urban Sprawl Qing Su ABSTRACT This dissertation investigates transporta tion subsidies as sources of urban sprawl. Apart from tolls, motorists do not pay highway user-fees, but they do pay gasoline taxes. Gasoline tax revenues are insufficient to c over the U.S. highway costs. Government, therefore, uses general tax revenues to cover highway expenditures. Since users do not pay the full cost of their travel, they have an incentive to travel longer commuting distances. Highway subsidies are, therefore, a po tential contributor to urban sprawl. A similar argument applies to public transit. To capture the effects of subsidized au tomobile and public transit travel, we extend the standard urban spatial single-mode model (Brueckner, 1987) to incorporate public subsidies for both one and two modes. Co mparative static anal ysis of both models produces empirically testable hypotheses. Our most important theoretical result is that transit subsidies are inversely related to urban sprawl while auto subsidies are directly related to urban sprawl. The empirical analysis focuses on tests of the two-mode model. For consistency with the monocentric assumpti on of our models, our sample consists of urbanized areas located within a single county and having only one central city. Spatial size of the urbanized area is the dependent variable. Following our theory, explanatory variables comprise the transit subsidy, the highway subsidy, number of households, agricultural land rent,
x mean household income, and fixed and variable costs for transit and auto. We find that the spatial size of the urbanized area shrinks with an increas e in the transit subsidy. The effect of highway subsidies, however, is ambiguous. We apply both ordinary least squares and two-stage least s quare regression analyses, and the results are qualitatively the same for both methods of estimation.
1 CHAPTER 1 INTRODUCTION Urban sprawl is a topic that has generated much debate in recent years and has become an important policy issue in the United States. The term urban sprawl was first used in 1937 by Earle Draper, one of the firs t city planners in the southeastern United States (Black 1996). In the years since Drap er introduced the term popular concern with urban sprawl has continued and grown. In 1998, more than 150 ballot measures were introduced to restrict ur ban sprawl in one way or another, and more than 85 percent of those measures were passed (Samuel 1998). By the year 2000, 14 states had adopted substantive land-use regulations to reduce ur ban sprawl (NARC 2000). The purpose of this chapter is to define urban sprawl and to discuss its consequences and causes. 1.1 What is urban sprawl? Osborn (Williams 2000), the British advocate of city planning, who summarized the major debate of urban sprawl in the 1940s, describes urban sprawl as a type of urban growth that is economically wasteful and so cially disadvantageous. It is economically wasteful because rising income and rapid transport induce people to move away from the central cities to suburbs wher e they find desirable residentia l surroundings at the expense of long and costly daily commutes. It is so cially disadvantageous because local community life is weakened or even destroyed, and access to the countryside is made more difficult for those people left in the central city (Nechyba and Walsh 2004). Brueckner (2000) defines urban sprawl as excessive spatia l expansion of cities. He notes that urban sprawl is a pejorative term in that it implie s the normative judgment that urban growth is
2 excessive. Mills (1999) prefers to use the non-pejorative term suburbanization to describe the phenomenon of continuous and unfe ttered urban growth in the United States. Nelson and Foster (1999) and Pendall (1999) define urban sprawl as unplanned, uncontrolled, and uncoordinated single-use development that does not provide a functional mix of uses and/or is not func tionally related to surrounding la nd uses and which variously appears as low-density, ribbon or strip, scattered, leapfrog, or isolated development. Generally, it is convenient to consider th ree different forms th at urban sprawl can take. It may involve relatively low-density residential and empl oyment areas coupled with low-density suburbanization at the ur ban fringe (Glaeser and Kahn 2003), planned communities that have their own downtowns perhaps located near a lake or park (Nechyba and Walsh 2004), or residences interspersed among rural areas. There is no doubt that U.S. population ha s increasingly urbanized. The first census in 1790 found 5 percent of U.S. populati on living in urban areas; by the 2000 census, nearly 80 percent did so ( Williams 2000 ) In addition, the urban population has decentralized, most notably since 1950, when nearly 70 percent of the populati on of urbanized areas lived in central cities while slightly over 30 percent lived in suburbs ( Nechyba and Walsh 2004) The last half of the twentieth century has witnessed an accelerated growth of suburbs within the urban areas as Americans moved to new homes and shopping centers developed on the fringes of metropolitan area s. Even in areas of the country that have seen less dramatic population growth, urban sp rawl has proceeded at a tremendous pace without letup. For instance, from 1990 to 1996, Little Rock, Arkansas, almost doubled its urban area from 109 to 199 square miles, while its population remained unchanged. Akron, Ohio, which had a population increase of only 3.5 percent, increased its urban area
3 by 65 percent in the same period. Austin, Texa s, experienced a population increase of 50 percent between 1990 and 1996 and expanded its urban area by 160 percent (Williams 2000). To most Americans, there is nothing pa rticularly unusual about the sight of new housing developments with business and pub lic facilities quickly going up around them. In the eyes of critics, however, this kind of real estate de velopment represents an unrecognized silent crisis in which Americas fin ite resources are wast ed at unsustainable rates (Williams 2000). In recent years, an increasingly heated debate has been under way between advocates of private urban development in a free market and advocates of government regulations against urban sprawl. Before turning to the debate itself, we need to consider the consequences a nd causes of urban sprawl. 1.2 Consequences of Urban Sprawl Urban sprawl has several consequences that are widely criticized by economists. Generally speaking, so-called scattered, unt imely, and unplanned urban development often occurs in urban fringe and rural areas, invading lands important to the environment for its open space and rich in natural re sources (Nelson and Foster 1999, Brueckner 2000). Other consequences of urban sprawl are the dominance of the private automobiles for transportation and the large income disp arities between city and suburbs (Downs 1999, Brueckner 2000). Additionally, unfettered urba n growth contributes to the decay of downtown areas, for it lowers developers incentive to redeve lop land closer to the central cities and lowers the ta x base of central cities. On the other hand, Gordon and Richardson (1997) argue that urban sprawl is a result of the fulfillmen t of consumer preferences. People choose to live in suburban areas,
4 thereby enjoying relatively lower land prices and larger amounts of living space, better schools for their children, a safer community, a nd better access to recr eational facilities. Although there are many alleged problems a ssociated with urban sprawl, we focus on three that are widely criticized by both urban and public finan ce economists: traffic congestion, loss of open space, and inco me-level and racial segregation. 1.2.1 Traffic Congestion Empirical studies show that traffic conge stion in the United States generates huge costs to urban residents. Shrank and Lomax ( 2003) find that the total increase in commuting costs from congestion in 75 urban ar eas is $69.5 million, of which $60 million are time costs associated with l onger commute time a nd $9.5 million are sp ent on additional fuel consumption resulting from congestioninduced decreases in transport efficiency. These figures translate into $520 per year for each resident in those urban areas. Urban economists argue that peak-load ta xes or toll roads are appropriate methods to internalize congestion externalities. Downs (1999) notes, however, that Americans may not be willing to incur the levels of congestion taxes that would be required to make a meaningful decrease in peak-hour traffic. Furthermore, the impact of congestion taxes on urban structure is not clear. Yinger ( 1993) argues that the im position of congestion tolls or peak-load taxes would increase populat ion density, thus re ducing urban sprawl. On the other hand, Rufolo and Bianco (1998) argue that the interna lization of congestion costs will lead to an increase in sprawl in the long run. They beli eve that the location decisions of both businesses and households are endogenous in the long run. Congestion pricing would raise commuting costs from s uburbs to the CBD but not commuting costs
5 between suburbs. This disparity, they cont end, will induce a decentralization of activity and further increase urban sprawl. Another issue associated with congesti on is air pollution. According to the U.S. Environmental Protection Agencys National Emission Inventory, in 2001, on-road vehicles in the United States accounted for 37 pe rcent of total nitrogen oxides (a major contributor to the formation of ground-level ozone particulate matter, haze, and acid rain), 27 percent of volatile organi c compounds (which react with nitrogen oxides to form ground-level ozone), and 62 per cent of total emissions of carbon monoxide (a particular threat for individuals who suffer from cardiovascular disease). Between 1970 and 2001, total vehicle miles traveled increased 151 pe rcent from 1.1 trillion miles to 2.8 trillion miles. Over the same period, miles traveled by passenger cars and motorcycles increased by over 75 percent from 920 billion miles to 1. 63 trillion (Federal Highway Administration, 2004). Empirical studies yield no strong results on the link between air pollution and urban sprawl. The majority of the empirical studies claiming to document a clear link between density and pollution fail to account for other important variab les such as income and household demographics (Nechyba and Walsh, 2003). Those studies that use microlevel data and attempt to control for these other factors generally co nclude that the relationship between density and travel behavior is weaker and less certain than is often claimed in the popular press (C rane, 2001). Glaeser and Kahn (2003) also point out that U.S. urban air pollution has been on the decline since the 1970s, with more newer, cleaner cars replacing older polluting cars.
6 1.2.2 Loss of Open Space Loss of open space is often identified as one of the key problems associated with sprawl. According to Burchfield, et al (2003), land area in urban areas increased by 47.7 percent between 1976 and 1992, an annualized rate of 2.48 percent. As the authors point out, however, these numbers can be misleadi ng because, although urbanization is increasing rapidly, only 1.92 percent of the entire U.S. land area as of 1992 is urbanized. Given this fact, why is the loss of open space rece iving such prominence in the debate over sprawl? Empirical studies by public finance economists suggest that open space within suburbs is significantly more important to hous eholds than open space at the urban fringe. Their work shows that housing prices are positively related to open space, such as public parks, privately owned open space, and th e natural land cover immediately surrounding household locations. Geoghegan, et al (1997) find that housing prices increase with the percentage of open space within a 0.1-kilo meter ring surrounding a house and decrease with the percentage of open space within a 1-kilometer ring surrounding the house. Acharya and Bennett (2001) find that housing pri ces increase with the percentage of open space at a decreasing rate within both a quart er-mile and a 1-mile radius. The evidence seems to suggest that households prefer ope n space in the immediate vicinity of their residence but they do not place great value on open space on the urban fringe. Nechyba and Walsh (2003) argue that open space is a local public good. The more local the nature of the open space amenities and the larger the sc ale of development, the stronger will be the incentives for developers to provide efficient levels of open space.
7 1.2.3 Income-Level and Racial Segregation Generally speaking, peoples ability to move out of central cities into suburbs and their ability to choose among suburbs are not uniform across races and income groups. Within U.S. urbanized areas, the poor generally live in central cities while middle-income and high-income individuals live in suburbs (Mieszkowski and Mill s, 1993; Glaeser, et al., 2000). Glaeser et al., (2000) suggest that for those usi ng public transit, the autocentered suburbs may simply not be a choice. In addition, politically created distortions through zoning leads to further segregation based on income and race (Glaeser et al., 2000). The geographic segregation and lower mobility of poorer households are very likely to result in a variety of social pr oblems in poor areas (G laeser and Kahn, 2003). In the case of education, for example, it is well known that public school quality differs across neighborhoods and districts ev en when observable school inputs such as per pupil spending are e qualized (Vigdor and Nechyba, 2003). Analogously equalized spending on public safety does not lead to equa l levels of protection from crime, nor does equal public investment in basic infrastructu re result in uniforml y functional neighborhoods (Katz et al., 2001). In all these cases, the level of the public good depends critically on the characteristics of the local population that is being served by public expenditures on the good. Bayer, McMillan, and Reuben ( 2002) argue that housi ng markets, employment centers, and preferences for residential hom ogeneity rather than differences in tastes for quality of education repres ent the crucial explanatory forces for the racial school segregation patterns observed in the United States This result highlights the fact that urban economies arise from a blend of decisions about housing, employments, schooling, and neighborhood and of public and private institu tions that shape each of these decisions.
8 Further research on the impact of urban sprawl on neighborhoods with in cities is therefore necessary. Given the alleged consequences of urban sprawl, public officials and others seek ways of controlling it; knowle dge of what causes urban sprawl is therefore important in designing measures to control it. 1.3 What Causes Urban Sprawl? Why has urban development in the United States followed such patterns of suburbanization? Two groups of econom ists attempt to explain th is trend. Public finance economists focus on the role of households pr eferences for local taxes and amenities on households location decisions. Urban econo mists emphasize the role of population growth, rising income, and lower transportatio n costs with contributing forces from government regulations and taxation, including federal tax and local government zoning policies. 1.3.1 Local Public Finance Model The local public finance model stems from Tiebouts voting w ith ones feet theory in 1956 (Tiebout 1956). Tiebout shows th at local public goods could be provided without resort to poli ticians. Instead of voting for politic ians who would then set expenditures and taxes, households vote with their feet to select th e community that happens to offer the right level of public goods. In other words, people sort themselves into different local jurisdictions based on their preferences for local public goods. These sorting effects are divided into two broad categories for purpose of our study of urban sprawl: those that pull people ou t of central cities because of attractive
9 features of suburban areas and those that push people out of central cities because of inner city decay (Nechyba and Stra uss, 1998, Nechyba and Walsh, 2004). The pull side of the public finance mode l emphasizes a group of households with similar preferences willing to pay for the pr ovision of local public goods in a suburb. These like-minded households form a new co mmunity. People are attracted to the new community from the central city because they can enjoy some features they desire, such as a high-quality school system, low crime rate a reasonable tax rate, and clean streets. When combined with zoning policies enacted by local government, people in the new community could exclude those who are considered to create negative fiscal externalities, thus avoiding the free-ride r problem (Fischel 2001) Carruthers and Ulfarsson (2002) identify another possi ble contributor to urban sprawl, namely, governmental fragmentation, which falls under the category of the pull side of the public finance model. They argue that political fragmentation contributes to urban sprawl through Tiebout mechanisms. That is, in metropolitan areas, households choose locations that offer a desirable combin ation of amenities at a price they can afford. The price of entry is given by the cost of housing and the prope rty tax. This plays an important role in forcing residents to balance their preferences and their budget constraints. Since real estate markets in mo st metropolitan areas are highly segmented, potential residents can easily be priced out. C onsequently, growth at the urban fringe increases because land prices there are compara tively low. Empirical evidence reveals that fragmentation is associated with lower dens ities and higher property values but has no direct effect on public service expenditures. This suggests that fragmentation may be a possible cause of urban sprawl.
10 The push side of the public finance m odel focuses on inner city problems. Relatively high-income people are forced to leave the central city because of inner city problems such as a high crime rate, a high tax ra te, lower quality schools, and general fiscal distress within the central city. With more people leaving the central city, those remaining face higher and higher tax rates. With the decrease in tax revenues, the local government experiences fiscal distre ss. As a result, public goods pr ovision deteriorates, e.g., the crime rate increases and school quality decreases. Thus, people leave the central cities not because of attractive f eatures in suburbs but to avoid i nner city problems. There is some empirical support for this view (Cullen and Levitt 1999). The empirical studies on sorting effects and urban sprawl mainly involve racial segregation. The results are, however, inconc lusive. Some researchers believe that minorities are segregated in centr al cities because of their re latively lower income and exclusionary suburban policies. Yinger (1993) fi nds that urban development and racial segregation are positively related. Bayer et al., (2002) find that households tend to reside close to others with similar race and ethnicity, which suggests that urban sprawl may contribute to racial segregation. Some research ers argue, however, that with the spatial expansion of urban areas and increase in in come, an emerging minority middle class has appeared that can afford to move to suburbs thus decreasing racial segregation. Glaeser and Kahn (2003) find that fast growing urbanized areas have experi enced a sharper decline in racial segregatio n than have other areas. In conclusion, public finance economist s have explored the connection between urban sprawl and Tiebout sorting. Both push and pull sorting play an important role in determining household residential location decisions. The public finance economists,
11 however, fail to answer a key question, that is, whether Tiebou t sorting is the cause or the consequence of urban sprawl. 1.3.2 Urban Economic Theories on Urban Sprawl Much of our understanding of urban gr owth can be derive d from the monocentric urban model, early developers of wh ich are Alonso (1964), Mills (1967), and Muth (1969). This model finds three fundamental for ces responsible for sp atial growth of cities: population growth, rising real income, an d lower real transport costs. Population growth and rising real income increase th e demand for housing. Since land prices decrease with distance from the city center, people choose ho using in the suburbs, which causes the city to expand spatially. Lower real transport costs are largely due to highways that enable people to travel faster and cheaper, while f acilitating suburban living. Glaeser and Kahn (2003) identify automobiles and their accompanying lower transportation costs as the prim ary catalyst of sprawling citi es through much of the twentieth century. By 1910, car registrations in the United States had exceeded 500,000 (Nechyba and Walsh 2004), and in 1920, it reac hed eight million. By 1952, a majority of households in America owned at least one car (Glaeser and Kahn (2003). In 1960, 64 percent of commuters drove to work, growi ng to 78 percent in 1978, 84 percent in 1980, and 88 percent in 2000 (Center of Urban Transportation Re search, 2003). It is hard to imagine urban sprawl without the rise of the automobile. Although Mills (1969) finds suburbanization occurring as far back as 1890, th e concern with urban sprawl is a fairly recent phenomenon. The monocentric model also predicts th at rising incomes contribute to decreasing population densities in cities if the income elasticity of demand for housing is greater
12 than the income elasticity of marginal comm uting costs. Margo (199 2) suggests that as much as a 50 percent increas e in suburbanization between 1950 and 1980 could be explained by rising incomes. Brueckner and Fansle r (1983) find that a 1 percent increase in income results in a 1.5 percent increase in the spatial si ze of urban areas. The real question is whether or not the sp atial expansion of cities is excessive. In other words, do free markets allocate resource s efficiently in urban areas or do market failures result in inefficient resources allocation? Brueckner (2000) identifies three market failures that may lead to excessive spatial growth of c ities. The first involves the social value of open space. During the proce ss of suburbanization, open space near the city fringe is converted to urban use. People living in suburban areas do not, however, shoulder the full cost of loss of the social value of the open space. This market failure results in over-conversion of rural land into urban uses The second market failure concerns commuting. Commuters bear only the cost of vehi cle operation and their time as well as the average cost of congestion and pollution, but not the costs of conge stion and air pollution they impose on others. Consequently, the cost of commuting is und erpaid. As a result, households choose to live farther away from CB D than socially desirable. The third market failure concerns public infrastructure co st generated by new development at the urban fringe. When new living areas are built, public infrastructure, such as roads, sewer systems, schools, and recreation centers, are need ed. In the U.S., most infrastructure costs are paid through the property tax, which results in homeowners with equal assessed values in lowand high-density areas paying the same amount. That is, households private costs are only the average infrastructure costs, which may be less than the marginal costs they generate. This also means that resident s living in high-density areas subsidize those
13 living in low-density areas, ge nerally found in suburbs. Perver sely, in most urban areas, this means poor people subsidize rich people who live in suburban areas. As a result of the failure to fully charge for suburban infr astructure costs, developers can bid higher prices for the undeveloped land than would otherwise be the case, thus converting more rural land into urban use, contributing to urban sprawl (Mills 1978, Brueckner 2000). The effects of these thre e market failures are exacerbated by public policies such as land-use planning, which often creates situat ions where regulations are more restrictive in some areas than in others. As a result, new development is built in less restrictive areas generally located near the urban fringe. Many public policies have been suggested as potential contri butors to urban sprawl. For example, Jackson (1985) argues th at urban developments have been greatly affected by the New Deals creation of the Home Owners Loan Corporation and the Federal Housing Administration (FHA) to provide decent housing for poorer Americans. Williams (2000) identifies generous mortgage insurance and loan programs through the FHA and the Veterans Administration (VA) as potential contributors to urban sprawl. The FHA provides federal guarantees to priv ate mortgage lenders, lowering the minimum down payment to just 10 percent for homebuyers, and payba ck periods are extended to 20 or 30 years. The VA offers lowinterest mortgages without down payment to all qualified veterans. The federal income tax allows deductions for mort gage interest and property tax. All of these public policies worked to generate a massive e xpansion of suburban growth after 1945. Brueckner and Kim (2003) explore the connection between urban spatial expansion and the property tax. The pr operty tax is levied at eq ual rates on land and improve-
14 ments. However, the portion levied on land has quite differe nt effects from the portion levied on improvements. The property tax on land, as generally acknowledged, cannot be shifted and therefore is a neutral tax in that it has no effect on resource allocation. The underlying reason is that land is in fixed supply. A property tax on improvements, on the other hand, is not generally regarded as neutra l. Such a tax tends to lower the equilibrium level of improvements chosen by the developer. Thus, land is developed less intensively under the property tax, which may contribute to the spatial expansion of cities. Brueckner and Kim show that while the taxs depressive effect on improvements reduces population density, spurring the spatial expansion of cities, a countervailing effect from lower dwelling sizes may dominate raising densities and making cities shrink. Numerical examples suggest that the propert y tax may encourage urban sprawl. Based on their findings, Brueckner and Kim suggest that the distortions gene rated by the property tax may include inefficient sp atial expansion of cities. Other papers dealing with th e issue generate mixed results. Sullivan (1985) analyzes the spatial effect of the property ta x using a model including both business and residential property. In view of the comple xity of the model, however, the linkage between the property tax and urban growth is not clearly identified. Arnott and MacKinnon (1977a) undertake a general equilibrium simula tion of the spatial e ffects of the property tax and find that if there is an increase in the property tax, the city shrinks as a response. Pasha and Ghaus (1995) note, how ever, that this result might not hold in a more general model. Persky and Kurban (2003) identify federa l spending as a possible contributor to urban sprawl. They consider the influence on urban expansion of recent federal subsidies
15 in Chicago. They believe that federal spe nding influences metropolitan expansion in at least three ways. First, subsid ies can change the relative pric e of land and other goods at the urban fringe, which induces consumers to substitute land for other goods. Second, federal programs or subsidies can increase pe oples disposable income by reducing their need for local government or private spending, which increases the demand for living space. The last factor Persky and Kurban discuss concerns the employment opportunities generated by federal spending, which in turn influences employees residential locations. Specifically, if federal spending is dispersed rather than concentrated, it may encourage urban sprawl. They find that in Chicago federal spending aimed at alleviating poverty and supporting the elderly strongly stimulates la nd absorption in suburban areas. They estimate that residential land use in the outer ring of the Chicago area has been increased by 20 percent as a result of federal spending. They also believe that th e income tax subsidy for housing has a larger effect on urban sprawl in Chicago than federal spending. This is consistent with the findings of James and Lin (1991), who suggest that the federal tax subsidy for housing may be a possible cause of urban sprawl. Brueckner (2004) identifies transportation subsidies as another contributor to urban sprawl. According to economic theory, in dividuals transportati on and location decisions would be efficient if th e price paid for transportation closely matched the costs incurred by the user. Although users do not pay a direct fee every time they use a highway, they do pay a user-fee in the fo rm of the gasoline tax. Revenues from gasoline taxes have, however, been insufficient to cover the construction, maintenance, and administration of highways. Various levels of government must use general tax revenues to augment highway expenditures, which indicates that govern ments provide highway subsidies to users.
16 Given the fact that users do not pay the full cost of their travel, they have an incentive to travel longer and more often. Highway subsid ies not only foster increased travel and congestion, they also contribute to urban sprawl. Suburban ar eas traversed by new highways tend to attract residents because transportati on is improved while the full costs of the construction are not borne by those who benefit. User-fees only covered 68 percent of th e highway systems capital and maintenance costs during the period 1956-1986 (Voith 1989). The share of total highway expenditure covered by user-fees has fluctuated considerably over time and across regions. Voith (1989) examines thirteen major highway construction projects ra nging in cost from $97 million to $581 million. The cost of constr uction varies widely, from a low of $6.8 million per mile to a high of $133.3 million per mile. None of the projects generates sufficient user-fees to cover the infrastructure investment. As a matter of fact, user-fees covers 54 percent of the investment at best and 2.5 percent at wo rst. On a per-car basis, the subsidy ranges from $0.16 to $4.50. On a vehicl e-mile-traveled basis, the subsidy ranges from less than 1 cent per vehi cle to 41 cents (Voith 1989). 1.4 Summary This chapter has defined urban sprawl and discussed some of its consequences and causes. Although there are many potential caus es of urban sprawl, in this dissertation, we concentrate on one: subsidized transporta tion. In Ch. 2, we modify the monocentric model to suit our purpose by introducing transportation subs idies into a single-mode model; in Ch. 3, we do the same for a two-mode model. Ch. 4 presents empirical analysis, including research hypotheses, data sources, estimation methods, and estimation re-
17 sults. Ch. 5 contains a discussion of the research findings, focusing on policy recommendations.
CHAPTER 2 SINGLE-MODE URBAN SPATIAL MODEL We now present our theory of the relationship between urban sprawl and govern-mental transportation subsidies. We begin with some definitions and assumptions fol-lowed by a detailed discussion of the behavioral equations of the model. Finally, overall equilibrium conditions of the urban area are provided. Following standard practice in ur-ban economics since Mills (1972), we use the term urban area to refer to the legal city and its suburbs which may or may not be part of the legal city. 2.1 Definitions and Assumptions The basic framework employed is the Muth-Mills urban residential land use model exposited by Brueckner (1987). The following notation is used: y = an urban households money income, including both wage and non-wage in-come c = quantity of an urban households expenditures on all commodities except housing and transportation q = quantity of housing service consumed by an urban household p = price per unit of housing service t = money travel cost per round-trip mile = proportion of money travel cost per round-trip mile paid by household (01) x = distance from home to CBD 18
u = utility indicator = local tax rate N = capital, an input to housing production l = land, an input to housing production r = rent per acre of land, the price (rental rate) of the land input i = interest rate, the price (rental rate) of the capital input L= population of the urban area r a = agricultural land rent at the urban areas boundary x = urban area boundary = proportion of urban-area tax revenue used for transportation subsidies, which we refer to as the tax-share variable G= intergovernmental grants used for transportation The urban area is located on a featureless plain containing a central business dis-trict (CBD) to which every urban resident travels for work, shopping, and leisure activi-ties. The urban area has two sectors: consumption and production. Consumers choose the quantity of housing service and other expenditures based on their preferences and budget constraints. Housing service is produced using two inputs, capital and land. All urban residents earn the same exogenously determined income and possess the same utility function defined over housing and the composite good, whose price we normalize. An urban residents utility function is strictly quasi-concave, and housing ser-vice is assumed to be a normal good. Although local governments generally do not levy income tax, we assume local tax revenues from property taxes and sales taxes are propor-tional to income, therefore, local tax revenues are y. 19
20 Housing service, q, as originally defined by Muth (1969), captures all characteristics of a dwelling unit and the land on which it is located except location. We recognize that housing is a multi-attribute commodity with such characteristics as floor space, number of bedrooms, etc., and that households ma y have preferences for these attributes. The assumption of a single variable representing service from a dwelli ng unit, however, considerably simplifies the analysis and has b een used since it was introduced by Muth. Housing price, p, is the price per unit of housing servic e. For simplicity, we often refer to q as square feet of floor space and to p as the price (rental rate) per square foot. There is no distinction between ownership and rental of housing. If a household owns its house, housing price is the implicit cost of a unit of housing service. The quantity pq, therefore, is the observable or imputed rental value of a dwelling unit. We assume single-person households, so that the number of households equals the population of the urban area. Assumptions concerning transportation deal with the transportation system, travel cost, and the travel subsidy. Urban resident s only travel between home and the CBD. The transportation system is sufficiently develope d to allow straight-line travel. We begin by assuming a single transportation mode, or a lternatively, a single mode-combination, such as auto travel to a transit station. Later, we incorporate mode choice. The urban area residents private travel cost per round trip mile ,t, is less than the total travel cost per round trip mile, t because urban area government spending on transportation is higher than fees collected from residents. In other words, we assume government gives commuters a subsidy of (1 ) t per round trip mile of travel. This assumption has empirical support. In the United Stat es, only around 38 percent of transit operating costs and 25 percent of co mbined operating and capital costs of transit systems are
21 covered by the total fares collected from us ers. For highway systems, around 60 percent of highway disbursements are covered by gas taxes and other user-f ees (Brueckner 2003). In this model, households private travel cost is proportional to distance traveled, which implies that the urban transport system exhibits constant returns to scale. Although some empirical evidence suggest s that transportation systems, such as bus or rail, exhibit increasing return to scale, the increasing retu rns in those systems ma y stem from the existence of underutilized capacity. Passenger-miles could be increas ed with little cost by simply filling half-empty buses or rail cars wi th additional riders. But this possibility implies that the original capacity of the system was too large. If system capacity is instead adjusted to match demand, then costs should increase roughly in st ep with passengermiles. Adding a passenger-mile means adding a seat mile, which involves a constant additional cost (Brueckner 2004). 2.2 Demand Side of Housing Market We start our analysis from the dema nd side of the housing market. The households utility function is V ( c,q ), where c is expenditure on non-housing, nontransportation goods and q is housing consumption. The pr ice of the composite good (normalized to unity) is the same everywhere within the urban area, but housing price varies with location. Given the assumption that all urban residents have identical preferences, urban spatial equilibrium implies the same level of satisfaction, u, for all residents. The equilibrating factor is housing price so th at those living far from the CBD and therefore paying a large tr ansportation cost are compensated with a lower housi ng price, while those living close to the CBD pay a higher housing price. Households face a budget constraint represented by y = c + pq + y + tx This budget constraint tells us that urban
residents income is spent on the composite good, housing, urban-area taxes, and trans-portation. Solving the budget constraint for c and substituting that into the utility function, the households problem is to maximize 22 uqpqtxyV ,)1( (2.1) with respect to q. This maximization problem produces the following first-order condi-tion 21(1),(1),VytxpqqpVytxpqq (2.2) where subscripts denote partial differentiation with respect to the first and second argu-ments of the utility function. Equation (2.2) is the familiar condition that the marginal rate of substitution equals the price ratio where MRSVV12 and, because of normaliza-tion, the price ratio is p. All households must have the same utility for spatial equilibrium, so we have the equation: (1),Vytxpqq u (2.3) where u is the urban-area-wide utility level in spatial equilibrium. The simultaneous system composed of (2.2) and (2.3) yields solutions for the en-dogenous variables p and q as functions of exogenous variables x, y, t, u and Totally differentiating (2.3), we have
1212(1)(1)VdytxpqVdqduVdyydtdxxdttxdpdqqdpVdqdu Given V 1 p =V 2 from (2.2), we have txdVxdtVtdxVydVdyVduqdpV111111)1( (2.4) From (2.4), we derive 0qtxp (2.5) Housing price is a decreasing function of distance from the CBD. Given that u is constant for change in all other exogenous variables, we derive the effect of x on q as fol-lows 0xppqxqu where /uqp denotes the Hicksian demand slope. The explanation of the spatial behavior of p and q is simple. Urban residents liv-ing farther away from the CBD are compensated by lower housing price for their higher travel cost. Since housing service is a normal good, the decrease of housing price with distance from the CBD leads to more consumption of housing service, or larger dwell-ings. The other parameters influences on housing price and housing service are de-rived below because they play a very important role in the comparative static analysis presented in the next section. 23
24The influence (2.4) as follows of income on housing price and housing service is determined from 10pyq (2.6) 0 pqq ypyu The influence of money cost of travel on p and q is determined from (2.4) as fol-lows 0pxtq (2.7) 0 uqqptpt The influenceconsumption is derived from (2.4) as follows of the private transport cost share on housing price and housing 0qtxp (2.8) 0 ppqqu The influence of the urban-area tax rate on housing price and housing constion is derived from (2.4) as follows umpqyp < 0 (2.9) 0 ppqqu
25The influence of the utility level on derived from (2.4) as follows housing price and housing consumption is 0 11qVup (2.10) We cannot determine /qy by the method previously used, however, because u is now varying. Brueckner (1987, p. 825, n.6) provides the result as 0pcuu uqMRSpq In this model, /qycannot be used to define a normal good because utility is held fixed. Instead, following Wheaton (1974) and Brueckner (1987), we assume /qc M RSc > 0 as the definition of a normal good. This implies that indifference curves become steeper as c increases holding q constant. Coupled with the concavity of indifference curves (which follow s from the assumption of strict quasi-concavity of the utility function), this as-ptihe resulting decrease in p produces an increase q. ca sumon ensures that when u increases, t in2.3 Supply Side of Housing Market We now turn to the supply side of the housing market. Housing is produced with two inputs land, l, andpital, N. The production function, ),(lNHH is strictly con-cave and exhibits constant returns to scale. We assume positive marginal products, namely, 0lH, 0NH. Strict concavity implies that the marginal products of capital and land diminish, which is supp orted by the fact that as the height of a building increases, more and more non-land input is consumed in non-productive uses such as foun-dations, elevators, and stairways.
26r, we refer to S as structural density. It is an index of the height of a building. tion exhibits constant returns to scale, we rewrite it as follows: To simplify notation, let S denote the capital-land ratio, N/l, and, following Brueckne Since the housing production func )1,(1,SHl NH Hlor )(SlhH A housing producers profit is ))((riSSphl (2.11) here The produ (2.12) wr is urban land rent per acre and i is the interest rate or rental rate per unit of capi-tal. cer determines S to maximize profit per acre of land, and the first-order condition for profit maximization is given as iShp)( In long-run competitive equilibrium, land rent must absorb profit, so that () p hSiSr (2.13) re functions of these same variables as well as the rental price per unit f capiant role in our model. This constitutes the spatial equilibrium condition for housing production, i.e., housing producers earn the same profit everywhere in the area. By (2.4), we know that housing price is a function of distance, x; income, y; money travel cost, t; private share of commuting cost, ; tax rate ; and the spatial equi-librium utility level, u. From (2.12) and (2.13), it is clear that the capital-land ratio, S, and urban land rent, r, a otal, i. In what follows, we suppress the variables and i as they play no import
27Totally differentiat ing (2.12) and (2.13) with respect to = x, y, t, u, yields 0pShph (2.14) ()phih Spr (2.15) rom (2.12), we kn Fow thatiShp )(, so (2.15) becomes h (2.16) pr Equation (2.14), upon rearrangement, becomes hp (2.17) phSFrom (2.16) and (2.17), we have the following results 0 0 01 0)10 0 xhSxhtrqhphyS qh(yrqthphxS qthxr qhpt q 0 011 10 00 0 qVhpu qVu From (2.18), we see that both urban land rent and the capital-land ratio are de-creasing function of distance from the CBD. These results are consistent with observa-tions that in the CBD land rents are higher and buildings are taller than in suburban areas. In our model, lower land rent compensates housing producers for lower housing price farther away from the CBD. As a result, housing produced farther from hShrqyhphS qhyrqxthphS qhxtr (2.18) the CBD uses less capital per unit of land, leading to shorter buildings in suburban areas. Another variable of importance in the comparative static analysis is urban popula-tion density. Under our assumption of single-person households, urban population density
28onsumer choice over housing service and producer decisions over hous-.4 the city boundary, is D=h(S)/q. Population density is a decreasing function of distance from the CBD be-cause S/x < 0 and q/x > 0. Much empirical evidence supports this result (e.g., Muth (1969)). Households living farther from the CBD live in larger dwelling units and shorter buildings than those living close to the CBD. Thus fewer and fewer people fit on each acre of land the farther from the CBD they live. Declining population density is the com-bined result of c ing production. 2Overall Equilibrium of Urban Area To complete the model, we need three equilibrium conditions. The first one is the boundary condition, which requires housing producers to outbid agricultural users for all urban land used in housing production. At x urban land rent equals gricultural land rent, ra. T ahis condition is a) (r, y, t, u,xr (2.19) The second condition states that the urban population must exactly fit inside the urban area. Let represent the number of radians of land available for housing at distance where 0 < < 2. The populat xion condition is 0 ) x x D(x,y,t,u,dxL (2.20) Transportation spending by government exceeds the fees collected from commut-ers. In addition, the part of transportation investment not covered by user-fees is covered om urban residents and by intergovernmental grants. he balanced budget condition is by part of the local tax collected fr T 20(1)()xyLGtxDx,y,t,u,dx (2.21)
where the left-hand side is urban-area taxes and intergovernmental grants used to cover the subsidized portion of urban-area transportation costs, which is the right-hand side. The interpretation of the urban equilibrium conditions depends on whether or not the city is a closed city or open city. A closed city is one in which population migration does not occur, so population, L, is exogenous but utility, u, is endogenous. An open city is one in which population migration occurs, so population is endogenous and utility is exogenous. We adopt the closed-city formulation. In summary, for a closed city, the single mode urban spatial model is composed of the following equations. Consumption sector 21(1),(1),VytxpqqpVytxpqq (2.22) (1),Vytxpqq u (2.23) Equations (2.22) and (2.23) imply that ),,,,( tuyxpp ),,,,( tuyxqq Production sector (2.24) iShp)( () p hSiSr (2.25) Equations (2.24), together with (2.22), (2.23), and (2.25) imply that ),,,,(tuyxSS r ),,,,(tuyxr 29
Urban boundary ar,y,t,u,xr() (2.26) Urban population 0)xxD(x,y,t,u,dxL (2.27) Balanced budget condition 20(1))xyLGtxD(x, y, t, u, dx (2.28) In a closed city model, the equilibrium conditions (2.26), (2.27), and (2.28) con-stitute simultaneous equations that determine equilibrium values for utility, the urban boundary, and the tax-share variable, or ),,,(, artyLxx ),,,(, artyLuu ),,,(, artyL 2.5 Comparative Static Analysis of Single-Mode Urban Spatial Model This section provides a comparative static analysis of the single-mode urban spa-tial model presented above. We begin by totally differentiating the three urban equilib-rium conditions. This enables us to derive the effect of L, y, r a, t, on the urban area boundary, x spatial equilibrium utility level, u, and the tax-share variable. We then de-rive the effect of the exogenous variables on the remaining endogenous variables, namely, p, q, r, and S. Some variables, such y, t, and, have both a direct effect on p, q, r, and S as well as an indirect effect through the endogenous utility. Other variables have only an indirect effects. The comparative static analysis provided in this section follows 30
Brueckner (1987) except for adding the variable and a balanced-budget equation for transportation subsidies. Many of our results are identical to those obtained by Brueckner. Since our main goal is to explore the relationship between transportation subsidies and urban sprawl, our comparative static analysis focuses on the effects of the relevant ex-ogenous variables L, y, r a, t, while neglecting the variables G, and i. Before presenting our comparative static analysis, we summarize those results obtained earlier that are used in this section. qthxrqtxp (2.29) 11 hpryqyq (2.30) qxhtrqxtp (2.31) qhxtrqxtp (2.32) 0 qhyrqyp (2.33) 11qVup 10rhuqV (2.34) phphS (where ,,,,,tyux ) (2.35) arxr() (2.36) xLdxqShx0)( (2.37) 31
xdxqShtxGyL02)()1( (2.38) The goal of the analysis is to deduce the impact of changes in L, r a, y, t, and espe-cially on the size of the urban area, x housing price, p, land rent, r, dwelling size, q, structural density, S, utility level, u, and the tax-share variables, 2.5.1 Effects of Change in Population 22.214.171.124 Effect of Population on Utility Recall that population density is defined as / D hq From (2.29) and (2.32) we have txrqhD Given this result, (2.37) is rewritten as 0xrxdxx tL (2.39) Integrating by parts, where u = x and dv = 0xrdx x we have 00xxtLxrrdx or (2.40) 0xatLxrrdx (since arxr )( ) (2.41) Totally differentiating (2.41) with respect to = L, r a y, t, we have 0xaaaLttLtLrxrruxrxdxru (2.42) After rearranging (2.42), we obtain 0xaLttLtLrrrudxxu (2.43) 32
or, since u is not a function of x 00xxaLttLtLrrurdxdxxu Then solving for /u we have 00xaxrtLLttLr x dxurdxu (2.44) When = L, (2.44) becomes 00xxtrdxuLrLdxu (2.45) From (2.34) 0. 1qVhur We see that 00xrdxu and because 0 L r 0 0xturLdxu (2.46) 126.96.36.199 Effect of Population on Urban Area Boundary Totally differentiating (2.36), we have arruurxxr (2.47) where bars mean the function is evaluated at x By rearranging the terms of (2.47), we have 33
34 xrruurr xa (2.48) ecall from (2.29) that xr < 0. Since ar = 0, therefore, when = L R LLu rurrLuurLrLuurLrLxaSign Sign SignSign L 0 and ,0 ,0 LLu, we have Because rur 0 x L (2.49) 188.8.131.52 Effects of Population on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density These effects op erate through u, so 0LuupdLdp (2.50) 0LuurdLdr (2.51) 0LuuqdLdq (2.52) 0LuuSdLdS (2.53)
184.108.40.206 Effect of Population on the Tax-Share Variable Totally differentiating (2.38) with respect to = L, r a y, t, we have 2222000(1) (1)(1)xxxLyxyLyLyLtxDtD Du x DdxtxDdxtxdxu (2.54) When = L, (2.54) becomes 220(1)(1)xxDutxDtxdxyLuLLyL 0 (2.55) We showed earlier that the urban area boundary, housing price, urban land rent, and structural density were increasing functions of population, while utility and housing consumption were decreasing function of population. This means that an increase in population expands the urban area spatially and leads to a lower urban utility. Population increase also results in a higher housing price, higher land rent, and taller buildings eve-rywhere within the urban area. As a result, urban residents consume less housing service. However, the effect of population on the tax-share variable is ambiguous. The logic behind these results is that an increase in population creates an excess demand for housing, which causes housing price to rise throughout the urban area, induc-ing urban residents to consume less housing. This expands the urban area because urban-area residents can now outbid farmers for agricultural land at the old urban area bound-ary. On the supply side of the market, rising housing price causes housing producers to bid up urban land for housing production. Higher urban land rent in turn forces housing producers to build taller buildings. When buildings are taller and housing consumption is 35
lower, population density increases everywhere. The increase in urban land rent every-where also means that at the old urban-area boundary, urban land rent exceeds agricul-tural land rent, the urban areas boundary therefore expands until urban land equals agri-cultural land rent again. Spatial expansion of the urban area and increase in urban popula-tion density both contribute to the elimination of excess housing demand, bringing the urban area to a new equilibrium with a larger urban area, higher population density, higher housing price, higher land rent, taller buildings, lower housing consumption, and lower utility level. The effect of an increase in population on the tax-share variable is ambiguous because when population increases, more revenue is collected, but more miles are driven because of the urban-areas spatial expansion. 2.5.2 Effects of Changes in Agricultural Land Rent 220.127.116.11 Effect of Agricultural Rent on Utility When = r a (2.44) becomes 0000 xaxxarxdxrurrrdxdxuu x (2.56) since because r is not a function of r /0arr a 18.104.22.168 Effect of Agricultural Rent on Urban Area Boundary When = r a (2.48) becomes xrruurrxaa1 (2.57) 36
Because 0xr aaruurrx1SignSign Recall that 0xauxrrdxu so 00011xxxarrdxxrurxuurrurudxdxuu From (2.34), we know that 1qVhur < 0, therefore, x0Sign Signa x rrdxxru uu Integrating by parts, where /ur and dv = dx, then xxx00001xrrrrrrdxxxxdxxxdxuuuxuuxuhxdxxqV Upon differentiating the integrand, we have DVxVxDqVhx11111 (since D=h/q) To determine the sign of this expression requires determining the sign of its partial de-rivative, since > 0 and D > 0 1V 02qxqhqxShxD 37
since > 0, h 0xphphxS and x q > 0 0111211xVVVx since x V1 > 0 for x a normal good (see Wheaton (1974) p. 227). Therefore, we have arx < 0 (2.58) 22.214.171.124 Effects of Agricultural Rent on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density 0aaruupdrdp (2.59) 0aaruuqdrdq (2.60) aarphdrdr > 0 (2.61) 0aaruuSdrdS (2.62) (Note we use total derivatives here since the corresponding partial derivatives are all zero.) 126.96.36.199 Effect of Agricultural Rent on the Tax-Share Variable When = r a (2.54) becomes 2201 (1)(1)xaaxDtxDtxdxryLrur au (2.63) 38
Upon totally differentiating (2.37) with respect to r a we have 00xaaxDuxDxdxrur or 0xaaDu x dxurxrxD (2.64) Substituting (2.64) into (2.63), we have 201(1)()0xaaDutxxxdxryLur (2.65) An increase in agricultural rent reduces the urban area spatially and lowers urbanarea residents utility. It also increases housing price, urban land rent, and structural den-sity everywhere. As a consequence, urban population density increases, but housing con-sumption and the tax-share variable decrease. The logic behind these results is similar to that of an increase in population. When agricultural rent increases, urban land rent at the old urban boundary is lower than agri-cultural land rent, which reduces the urban area spatially. This change, however, creates excess demand for housing because the unchanged population must fit within the new urban boundary. Excess demand for housing increases housing price and decreases hous-ing consumption everywhere within the urban area. Both rising housing price and rising agricultural rent push urban land rent up; as a consequence, housing producers build higher buildings. Since the urban area shrinks spatially and population density increases everywhere, people drive fewer miles from home to work, which lowers the proportion of local tax revenues used for the transportation subsidy. 39
2.5.3 Effects of Change in Income 188.8.131.52 Effect of Income on Utility When = y, (2.44) becomes 00xaxrtLLttLr x dxuyyyyyrydxu (2.66) Because 0yryytyLa and, from (2.30) and (2.34), 0and 0 uryr then 00xxrdxuyrydxu > 0 (2.67) 184.108.40.206 Effect of Income on Urban Area Boundary When = y, (2.48) becomes xryryuuryryxa (2.68) Because xr < 0 and yra = 0 SignSign yryuuryx 40
00000xxxxxrdxrurrryruyyuydxurrrrdxdxyuuy rdxu (2.69) Because 1qVhur < 0 00SignSignxxrurrrrrdxdxuyyyuuy From (2.30) and (2.34), we have q)h(yr qVhur 1and 1 so, upon evaluating variables at x we have q)(hyr Vqhur1and 1 Therefore, 00001101111111xxxxxrrrrh()hhh()dxdxdxdxyuuyqqVqVqhh()dxqqVV Recall that V 1 is the first partial derivative of the utility function with respect to the com-posite good, so 1111111111and 00VV ,VV, tVxV, V Given these results, we deri ve 41
011111 0xhh()dxqqVV Therefore, 011x11Signignxhh()SdyqqV 1xV so 0yx (2.70) 220.127.116.11 Effects of Income on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density The effect of an increase in income on housing price is complicated because in-come not only directly affects housing price, but also indirectly affects it through the util-ity level. Therefore, the relationship between income and housing price can be written as ypyuupdydp (2.71) At a given distance, let housing price be where 0 < < x p p p i.e., p evalu-ated at 0<< x x From (2.30), (2.34), and (2.67), we have xxdxurdxyryuqypVqup001 and ,1 ,1 Substituting these into (2.71), we obtain 01100010111111xxxxxrdxdpuyrdyyqqqVqVdxurrdxdxyquqVrdxu (2.72) 42
Because 0xrdxu < 0 00111SignSignxxdprrdxdxdyquyqV (2.73) Substituting qhyrqVhur)1( and1 into the above equation, we have 00001101111(1)(1(1)11xxxxxrrhhdxdxdxdxquyqqVqVqqVhdxqqVV 1) (2.74) At the urban area boundary ( x x ), 1111VV > 0 since 01 x V Hence, 0 yp at the urban area boundary. At the CBD (x = 0), 0 Hence .011011dydpVV at the CBD, where the subscript means evaluation at x = 0. Thus, the housing price function pivots counterclockwise. Because of the dependence of urban land rent and structural density on housing price, these functions pivot counterclockwise. The effect of income on housing consumption is more complicated, however. Breuckner (1987, p. 835) shows that q falls with y for, but is ambiguous for pp pp 43
18.104.22.168 Effect of Income on the Tax-Share Variable Since 0'02qyqhqyShqhyyDyx yu > 0 122221221 1 VcMRSpqqhupqpqhqhphqVcMRSpqhuqpqhquphphquqhquShqhuuDuuuu (2.75) Following Brueckner (1987), let uq p 22qhqhph and 121VcMRSqhqc Then upuD < 0 since > 0, up < 0, and < 0 (2.76) Therefore, we have 220(1)(1)0xxDDutxDtxdxLyyuyyyL (2.77) 44
45 The effects of an increase in income on endogenous variables are a little complicated. As we see from the above results, an increase in income will expand the city spatially and increase urban residents utility le vel. It will also increase housing service consumption and population density when p falls, but the effects on q and D are ambiguous when p rises. Incomes effect on housing price, ur ban land rent, and st ructural density is to pivot these functions counterclockwise. It s effect on the proportion of local tax revenues used for the transportation subsidy is ambiguous. Thus, according to our model, as income increases, people want to consume more housing. Since housing is cheap er farther out, the demand fo r housing farther from the CBD rises, causing price to rise there and to fall closer to the CBD. Because urban land rent and structural density are directly related to housing price, an increase in income will have the same effects on urban land rent and struct ural density as on ho using price. When housing price falls close to the CBD, housing consumption per household increases there. In the area close to the urba n boundary, the effect of an increase in income on housing consumption is ambiguous. The possible explan ation is that rising housing price, urban land rent and structural density there, by themselves, reduc e housing consumption. This, however, may be offset by rising utility becaus e of increase in income. The effect of income on the tax-share variable is ambiguous b ecause, on the one hand, an increase in income raises the tax base, while, on the other hand, as people move fa rther away from the CBD, they drive more miles and require more subsidy. 2.5.4 Effects of Change in Commuting Cost 22.214.171.124 Effect of Commuting Cost on Utility When =t (2.44) becomes
00xxLrdxutrtdxu (2.78) Because 0qhxtr and ur < 0 tu < 0 (2.79) 126.96.36.199 Effect of Commuting Cost on Urban Area Boundary From (2.48), substituting t for we have urtxxrtrtu (since tra = 0) Totally differentiating (2.37) respect to t and substituting tu into it, we have 00xxDDuxDxdxttut (since 0 ttL ) (2.80) 00xrrxxDDtxtxDxxdxrttuu 00xDrDrDrxxxxxtuutuxtxDdxrtu After rearranging terms, we have 46
00xxDrDr x dxxutturDrt x Dxduux x (2.81) We need to know tD to determine the sign of tx since upuD < 0 from (2.76) Recall that qhD uphphuS 11VcMRSuppquqqcu tppqtqtphphtSu Then 22220 (since 0)hpqSqhqhhqhDh p httttttqqqhhpppphqqttt (2.82) Since 0ur 0uD and 0xr then, from (2.81) we know that 0SignSignxxDrD r x dxtutt u Recall that 11, , ,DpDppxprhttuutquqVtq x and 1rhuqV 47
Furthermore, since .for and then ,011111xxVxVxVVxV Consequently, 0 1 01110100xxxxdxtrxVxVxqqhxdxtrxVqhqxxqxhqVxdxurtptrupxdxurtDtruDx Hence 0tx (2.83) 188.8.131.52 Effects of Commuting Cost on Housing Price, Housing Consumption, Urban Land Rent, and Structural Density Recall 11, , andrhxppxtquqVtq 00xxLrdxutrtdxu At a given location, housing price is affected by commuting cost both directly and indirectly through the utility function. Consequently, 01110 (where is some between 0 and )11111 xxdppupppxdtuttLrdxuxut x xrtqqtqqVVVdxu 48
0011011 xxxLrrdxxdxtuVVrqdxu Because 0ur 00111SignSignxxpLrrdxxdxttVV u (2.84) Substitute the following into the parenthetical expression in the right-hand side of (2.84) xxdxqxhxDdxLqhxtrqVhur001 and ,, Then we have 000001111101112xxxxxx L rrxhhxxhdxxdxdxdxdxtuqVVVqVqhxxdxqVV V (2.85) At (2.85) becomes ,0x 012xhxdxqV > 0, but at ,xx (2.85) becomes 01120xhxxdxqVV because both 12Vx and 1Vx are positive. Brueckner (1987) argues that the boundary value of p is invariant to t. That, coupled with 0tx implies that 0dtpd Since p decreases monotonically in x, then, p rises with t to the left of and falls to the right, i.e., the p function pivots clockwise for an increase in t. x 49
Recall 111, , and rhprprhxphuqVuqVuutqtq x Then xxx xxtqtuuqdtdq00 0 (2.86) 0 00 x xdSSuShpup xxxdtuttphutt (2.87) 0 00 x xdrrurph xxxdtuttt (2.88) 184.108.40.206 Effect of Commuting Cost on the Tax-Share Variable When = t, (2.54) becomes 22200(1)(1)(1)xxxDtxDxDdxtxdxtttyL Duut (2.89) Since ,0 and ,0 0 ,0 tuuDtDtx 0 t From the above results, we conclude that an increase in commuting cost reduces urban residents utility and shrinks the urban area spatially. The effects of an increase in commuting cost on housing price, urban population, land rent, and structural density vary with location, raising these variables near the CBD and lowering them farther out. The effect of commuting cost on housing consumption and population density are more com-plicated. An increase in commuting cost lowers housing consumption and raises popula50
tion density near the CBD, but its effect is ambiguous farther out. Finally, the effect of commuting cost on the tax-share variable is ambiguous. Our explanation for the results is that as transport cost per round-trip mile in-creases, people seek locations closer to the CBD to minimize transport costs. This movement reduces the spatial area of the city and lower residents utility. Also, as people move closer to the CBD, they drive up land rent and housing price there while driving those variables down farther out. This, in turn, causes households to consume less hous-ing in taller buildings near the CBD and raises the population density there, while lower-ing structural density farther out. The effects on housing consumption and population density farther from the CBD are, however, ambiguous. This is so because the low hous-ing price and land rent farther out, by themselves, would tend to raise housing consump-tion and lower population density. But the lower utility may offset these results. The ef-fect of commuting cost on the tax-share variable is ambiguous because, on the one hand, an increase in commuting cost per round-trip mile induces more transportation subsidy, while, on the other hand, the urban area shrinks spatially and people drive fewer miles, which reduces the transportation subsidy. 2.5.5 Effects of Change in Private Cost Share 220.127.116.11 Effect of Private Cost Share on Utility Recall that 0qtxhr and 10rhuqV Then, for = (2.44) becomes 51
000xxtLrdxurdxx (2.90) 18.104.22.168 Effect of Private Cost Share on Urban Area Boundary Equation (2.47), with = becomes 0 ruurxxr Solving for x we have xruurrx (2.91) From (2.32), (2.34), and (2.29), we have txDqtxhr 111VDVqhur tDqthxr Substituting these into (2.91), we get 01111utVxtDuVDtxDx (2.92) 52
22.214.171.124 Effects of Private Cost Share on Housing Price, Housing Consumption, Ur-ban Land Rent, and Structural Density Given the fact that affects housing price both directly and indirectly through the utility function, we have xx, xp ppuupdpd0at evaluated is where, (2.93) Then substituting from (2.32), (2.34), and (2.90), we get 00011100111xxxxxtLrrtLrdxtxdxdxudptxVVrrdqqqVdxdxuu Because ur < 0 00111SignSignxxdptLrrdxtxdxduVV (2.94) From (2.37) dxqxhdxxDLxx00 From (2.32), qhxtr and from (2.34), 1qVhur Substitute these into (2.94), get-ting 0000011111011112xxxxxxtLrrtxhtxhhdxtxdxdxdxtxdxuqqqVVVVthxxdxqVV V (2.95) 53
At the CBD where x = 0, (2.95) is positive, which means that 00ddp At the urban area boundary, where = x x (2.95) is of ambiguous sign. Nevertheless, it can be shown that dpd < 0 and that the price function pivots clockwise about some 0 < x < x First, recall that the boundary value of p is invariant to exogenous variables other than r a and i. Thus, p is invariant to But, since x < 0, then p < 0. Also, since p decreases monotonically with x, the p function must pivot clockwise about some Thus, x 0 00 0 x xdpxxd x xx (2.96) From (2.16) and (2.17), we know that S and r functions exhibit the same behavior as the p function, then 0 00 0 x xdShdpxxdphd x xx (2.97) and 0 00 0 x xdrdphxdd x x xx (2.98) As we have seen in other contexts, the effect on q is more complicated. Since q falls when p rises but is ambiguous when p falls, then 0 00 x xdqd x xx (2.99) 54
126.96.36.199 Effect of Private Cost Share on the Tax-Share Variable When = (2.54) becomes 222001(1)(1)xxxD Du x DttxDdxtxdxyLu (2.100) Totally differentiating (2.37) with respect to and solving for x we have 0xDDu x dxxu x D Substituting the above equation into (2.100), getting 2001(1)())xxdDtxDdxtxxxdxdyLu Du < 0 (2.101) From the above results, we see that an increase in reduces the urban area spa-tially, lowers utility, and lowers the tax-share variable. An increase in increases hous-ing price, urban land rent, and structural density in the areas close to CBD while lowering these variables in areas close to the urban-area boundary. Finally, an increase in lowers housing consumption near the CBD but may raise, lower, or leave housing consumption unchanged near the urban-area boundary. Our explanation for these results is that as households unsubsidized transport cost increases, people seek locations nearer the CBD. This movement reduces the urban area spatially and lowers residents utility level. As people move closer to the CBD, they drive up land rents and housing prices there while driving down those variables father from the CBD. This, in turn, causes households to consume less housing in taller build-ings near the CBD, while lowering structural density farther out. The effects on housing 55
consumption and population density farther from the CBD area are, however, ambiguous. This is so because the lower housing price and land rent farther out, by themselves, would tend to raise housing consumption and lower population density. But the lower utility level may offset these results. Finally, because the private share of commuting cost increases, the tax-share variable is reduced. Table 2.1 summarizes the comparative static analysis of the single-mode urban spatial model. We find that an increase in population and income results in urban expan-sion, while an increase in agricultural land and commuting cost shrinks the urban area spatially. Since our goal is to explain how transport subsidies affect urban sprawl, we emphasize /x We find that in a single-mode model, when subsidies increase, the urban area expands spatially. This is so because an increase in transport subsidies raises peoples income net of transport cost, which results in a higher demand for housing in suburban areas because of lower housing price there. Consequently, housing price, urban land rent, and structural density in the suburbs increase, which leads to spatial expansion of the urban area. This finding suggests that subsidized transport is a contributor to urban sprawl in a single-mode model. 56
Table 2.1 Comparative Static Results of Single-Mode Model Endogenous Variable Exogenous Variable U x S r p q L + + + + ? xx + + + ? t xx ? ? xx + + + ? y xx + + + + + ? ? r a + + + x NC NC + NC xx + + + + 1xx + + + + + ? + 57
CHAPTER 3 TWO-MODE URBAN SPATIAL MODEL We now relax the assumption that only one transport mode prevails within a city. We begin with some new definitions and assumptions, proceed to a detailed discussion of the behavioral equations of the model, and then provide a comparative static analysis. 3.1 Modal Choice and Preliminary Results Used for Comparative Static Analysis All definitions and assumptions from the single-mode model remain unchanged except the following: Two different transport modes, 1 and 2, are available in the urban area and cost of commuting by each mode incurred by a resident at location x is repre-sented as follows: 582 1, )( ixtfxMiiii (3.1) where f represents fixed cost, which includes all travel cost independent of distance trav-eled, and xtii represents private variable cost, which include all costs that vary with distance traveled. We assume that variable cost is proportional to distance. As in Ch. 2, the s represent the proportion of total variable cost directly paid by travelers. We also assume that f 1 < f 2 while 1 t 1 > 2 t 2. As in Ch. 2, (1 i ) represents the governmental subsidy per round-trip mile. These assumptions imply that Mode 2 has a higher fixed cost but a lower variable cost per round-trip mile because it runs faster, more conveniently, more efficiently or re-ceives a larger governmental subsidy. It is necessary to assume that Mode 1s fixed cost is less than Mode 2s but that its variable cost per mile is greater so that one mode will not dominate the other for all trips. In our model, a resident chooses between two modes.
He will be indifferent between the two modes if he lives at where x )/()(221112ttffx will choose Mode 1 if he lives between CBD and and will choose Mode 2 if he lives betweenx and the urban area boundary. Therefore we refer to Mode 1 as the central city mode and Mode 2 as the suburban mode x Based on the new assumptions, the single-mode urban spatial model is modified as follows. Each individual chooses the amount of a composite good, c, and the amount of residential housing, q, so as to maximize utility u = u(c, q) subject to a budget constraint xMypqcy)( where 12()min()() M xMx,M x Substituting c from the budget constraint into the utility function, the requirement that the maximized utility equals u can be written as (1)(), M axVuqypqMxq (3.2) Since (3.2) represents the maximized utility, two conditions must hold. First, since consumers choose q optimally conditional on p, the first-order condition must hold, that is pqMpqyVqMpqyVii],)1[(],)1[(12 (3.3) Also, the resulting consumption bundle must generate the spatial equilibrium utility, u, so that ],)1[(uqMpgyVi 59
Given the assumption of mode choice, the exogenous variables are x, y, t i, i, i, G and f i (i = 1, 2). We suppress the variables i and G because they play no important role in our analysis. In presenting the results for p, q, S and r, we suppress notation identifying the mode, except for t i and i because these results hold for either mode. Later, it will become necessary to distinguish these variables by mode. Using the same method as in Ch. 2, we get 0 0iiutpqqxqxpx p (3.4) 10 0upqqyqypy p (3.5) 0 0iuiiiiitppqtqqxtp (3.6) i0 0iiiutxpqqqp p (3.7) qyp < 0 0ppqqu (3.8) qfpi1 < 0 iuifppqfq > 0 (3.9) 01 0111uqcpqVcMRSupuqqVup (3.10) Except for the addition of fixed cost, nothing is changed in the housing production sector of the model, so, from Ch. 2, we have phphS (3.11) 60
phr (3.12) where ,,,,,iii x ytfu Under our assumption, Mode 1 is used from the CBD to and Mode 2 is used from to x x x Hence, the boundary condition is more complicated than that of the single-mode model, namely, 11112222(,,,,,,)(,,,,,,)rxytufrxytuf (3.13) 222(,,,,,,)arxytufr (3.14) The population and balanced budget condition must also reflect mode choice. The popu-lation condition becomes 11,1,1222,20(,,,,)(,,,,)xxx x DxytfidxxDxytfidxL (3.15) where D i represent population density. The balanced budget condition becomes 21112220(1)(1)xxxyLGtxDdxtxDdx 2 (3.16) 3.2 Comparative Static Analysis The goal of the analysis is to deduce the impact of changes in L, r a y, t i i and f i on the spatial size of the urban area, x housing price, p i land rent, r i housing consump-tion, q i structural density, S i utility level, u, and the tax-share variable, Given that the purpose of this thesis is to study the relationship between urban sprawl and transportation subsidies, we ignore the effects of the exogenous variables G, and i. 61
Recall iiiiiitxrqhD (note we often use h i as shorthand for h(S i )) Substitute this into (3.15) to get 120112211xxxrrxdxxdxtxtx L (3.17) Integrating by parts, where u = x and dv = 120 or xxxr r x dxxdx x x we get Ldxrtxrtdxrtxrtxxxxxx 2 0 2222211101111111 (3.18) or 1210112211221111()[(()]xxaxLxrxxrxrxrdxrdxtttt 2 (3.19) Because we have )()(21xrxr 110221122112211111()xxax 2 L xrxxrrdxrdxttttt (3.20) Totally differentiating (3.20) with respect to 121212,,,,,,,,aLryttff we have 112211221112211112211221122()()()ttxttrxtt x rxrxxtttttt 1022221122111xaaarx 1 x rxrdxttt rt 1112011112211()xxxrruxdxrxrdxtutt 1 62
22222222211()xaxrruxxdxrxrtut 1t 21LL (3.21) 3.2.1 Effects of Change in Population 188.8.131.52 Effect of Population on Utility If L (3.21) becomes 11122110112211()1xrxttrru x dxLtttLuL 22221xxrrudxtLuL 1 (3.22) Since are not functions of L, (3.22) becomes )( and ),(),(211xrxrxr 11220112211xxxrrurrudxdxtLuLtLuL 1 Then, since u is not a function of x, we have 1201122111xxxuLrrdxdxtutu (3.23) Because 0)(1111VqShur and 0)(1222 VqShur 0Lu (3.24) 184.108.40.206 Effect of Population on Urban Area Boundary Totally differentiating (3.14) with respect to = L, r a t 1 t 2 1 2 f 1 and f 2 63 arruurx x r222 (3.25)
Then xrruurrxa 222 (3.26) Since 02 x r then we have Sign Lx = 2SignarrurLuLL 2 (3.27) Since 0Lra this becomes Sign L x = Sign LrLuur22 (3.28) From (3.24), we know 0Lu and 20arrLL because is not a function of L except through u. Also 2r 02ur from (3.10) and (3.12), so Sign Luur2 > 0, which means Lx > 0 (3.29) 220.127.116.11 Effects of Population on Housing Price, Housing Consumption, Structural Density and Urban Land Rent 0LuupdLdpii i =1,2 (from (3.10) and (3.24)) (3.30) 0LuuqdLdqii i =1,2 (from (3.10) and (3.24)) (3.31) 64
0)()(dLdpShpShdLdSiiii i =1,2 (from (3.11) and (3.30) (3.32) 0)(dLdpShLuurdLdriiii i =1,2 (from (3.12) and (3.30)) (3.33) 18.104.22.168 Effect of Population on the Tax-Share Variable dxDxtyLdxLuuDxtyLdxDxtyLdxLuuDxtyLyLGdLdxxxxxx222222222120112120112)1(1)1(1 )1(1)1(1 (3.34) Since we do not know the sign of the second and third terms of (3.34), 0dLd We have shown that, in a two-mode urban spatial structure model, the results are the same as in the single-mode urban spatial model. When population increases, the ur-ban area boundary, housing price, urban land rent, and structural density also increase, while the utility level and housing consumption fall. The effect of population on the tax-share variable is ambiguous. 3.2.2 Effects of Change in Agricultural Land Rent 22.214.171.124 Effect of on Agricultural Land Rent on Utility When ar (3.21) becomes 110221111xaarru x dxttrur 222210xxaarrudxtrur (3.35) Since are not functions of except through u, then 2 1 and rr ar 021 aarrrr and since u is not a function of x, (3.34) becomes 65
12011222211xxxaurrdxdxrtutut x After rearranging terms, we have 22120112211xxaxxutrrrdxdxtutu Because 01ur and ur2 <0 from (3.12), then 0aru (3.36) 126.96.36.199 Effect of Agricultural Land Rent on Urban Area Boundary Differentiate (3.15) with respect to getting ar 12120()()()0xxxaaaaaxDxDxxDxxdxxDxxdxxDxrrrrr 2 (3.37) The first and last term on the left side of (3.37) cancel because Solving for )()(21xDxD arx we have 1202()xxxaaDD a x dxxdxrxrxDx r (3.38) Since )0)()( (because 0 then 2 aiiaiaiiaiiaiiiaiiiirpShpShrSqrqhrShqrDqhD 66
and 0aiairuuqrq Thus, we know that (3.38) is negative; that is arx < 0 (3.39) 188.8.131.52 Effects of Agricultural Land Rent on Housing Price, Housing Consumption, Structural Density, and Urban Land Rent 0aiairuupdrdp i =1,2 (3.40) 0aiairuuqdrdq i =1,2 (3.41) 0)()(aiiaiairpShpShdrdS i =1,2 (3.42) 0aiairphdrdr i =1,2 (3.43) 184.108.40.206 Effect of Agricultural Land Rent on the Tax-Share Variable 221211220222(1)(1)10(1)()xxxaaaDuDutxdxtxdxururxryLtxDxr a (3.44) because 0airuuD while arx < 0. The results of an increase in rural land rent are the same as in the one-mode urban spatial model. When agricultural land rent increases, the utility level attained by urban area residents falls and the urban area shrinks spatially. An increase in agricultural rent 67
68 also raises housing price, urban land rent, and structural density everywhere but reduces urban residents housing consumption. Its e ffect on the tax-share variable is ambiguous.
3.2.3 Effects of Change in Income 220.127.116.11 Effect of Income on Utility When y (3.21) becomes 1211122011221122()110xxxrxffrrurrudxdxytttyuytyuy (3.45) since xMMrrrraat and,,21221 Then, since u is not a function of x, we have dxurtdxurtttffyxrdxyrtdxyrtyuxxxxxx 22201112211121222011111)(11 (3.46) Since f 2 > f 1 0)1(iiiqhyr and 0 uri so we have 0yu (3.47) 18.104.22.168 Effect of Income on Urban Area Boundary From (3.26), and the fact that 02 x r we have Sign xy Sign 2ruruyy 2 = Sign 22ruruyy yryuur22 = 1212102211221122120112211()11xxxxxxrrrxffdxdxrrtytyyttrruydxdxtutu = 69
dxurtdxurtdxuryrtdxuryrturttffyxrdxyrurtdxyrurtxxxxxxxxx 2220111 222201211222111212222012111111)(11 (3.48) The sign of yx is the opposite of the sign of the numerator of (3.48), the numerator of (3.48) is dxuryrtdxuryrturttffyxrdxyrurtdxyrurtxxxxxx 222011122111212220 1211211)(11 (3.49) Substitute the following equations into (3.49) 222)1(qhyr 1111Vqhur 2221rhuqV 111)1(qhyr 222)1(qhyr and note that 110Vx implies 1111VV then (3.49) becomes 21210112112111(1)(1)xhhhhdxtqVqqqV 2222222122211(1)(1)xxhhhhdxtqVqqqV + urttffyxr2211121)( 212201111122221111(1)111(1)11xxxhhhhdxdxtqqVVtqqVVV 0)(22211121urttffyxr (3.50) 70
22.214.171.124 Effects of Income on Housing Price, Housing Consumption, Structural Den-sity, and Urban Land Rent The effect of income on price includes direct and indirect effects ypyuupdypdiii (3.51) Because 0 ,0ypupii and 0 yu then 0dypdi (3.52) 0yqyuuqdydqiii (3.53) iiiiidShdpdydyph 0 (3.54) iiidrdphdydy 0 (3.55) 126.96.36.199 Effect of Income on the Tax-Share Variable 2212112202221112220(1)(1)10(1)(1)xxxxxxDuDuyLtxdxtxdxuyuyyyLLtxDdxtxDdx (3.56)since 0yx 71
In the two-mode urban spatial model, an increase in income increases urban resi-dents utility. Its effects on the urban area boundary, housing price, urban land rent, struc-tural density, and the tax-share variable are ambiguous. 3.2.4 Effects of Change in Mode 1s Variable Commuting Cost 188.8.131.52 Effect of Mode 1s Variable Commuting Cost on Utility Since is not a function of and 2r 1t arr 2 then when = t 1 (3.21) becomes 1112211222111111222211111112001111111112222122111()()()()111()11()axxxxr x rxttxxrxrxxttttttttrruxrdxdxrxttuttttruxdxrxtutt t 12210arxttt (3.57) Upon integrating by parts where 10xrdx dxdvru and ,1 we have 11122200111111111()xxrrdxxrxxdx x ttt Substitute this into (3.57) and rearrange terms, noting that 21rr then 1112000111111221111122111221111()()0xxxxxrrur 2r x dxdxdxdxxttttututrxttxttt (3.58) Then 1tu = 11111222001111112211120112211()()011xxxxxrrrxttxdxdxxxttttttrrdxdxtutu (3.59) 184.108.40.206 Effects of Mode 1s Variable Commuting Cost on Urban Area Boundary 72
Recall (3.15), 120()()xxx L xDxdxxDxdx Totally differentiating (3.15) with respect to t 1 yields 112120111121()()()0xxx 1 x DDuxDu x DxxdxxDxxdxttuttuxxDxt t (3.60) Because =, and after rearranging terms, (3.61) becomes )(1xD )(2xD 112011112()xxxDDuDu x dxxdxtututxtxDx (3.61) Recall from (3.25), since is not a function of and 2r 1t arr 2 1/art 0 we have 2221110rxrurxtutt (3.62) Then urtxxrurtxxrtrtu 212212121 (3.63) because 12tr =0. Substituting (3.63) into (3.62), we have 2211211022112()xxxrxrxDDDxtxt x dxxdxrrtuuuuxtxDx (3.64) 73
After rearranging terms of (3.64), we have 120112222120()xxxxDrxdxxtuDrDrrt x dxxdxxDxuxuxu (3.65) Because 12tr = 0, the sign of (3.65) depends on 11tD and uDi ()iiiDhSuuq = 2iiiiquqhuShq Recall that uphphuSiii 011Vqupii 11VcMRSuppquqiuiii Then 2211iiiuiiiuiiqhpDqqhMRShuqupcVphq iqu where 2211,and iiuiiuiiqhpqhhqpphq MRScV as originally defined in Ch. 2. Hence 74
0uDi We also need to obtain the sign of 11tD Note that 11111)(qShttD = 0)(2111111 qtqShqtSh (3.67) Because 111110Shptpht and 1111110uqqptpt given (3.66) and (3.67), the de-nominator and numerator in (3.65) are both positive, so we have 01tx (3.68) 220.127.116.11 Effects of Mode 1s Variable Commuting Cost on Housing Price, Housing Consumption, Structural Density, and Urban Land Rent The total effect of Mode 1s transport cost on housing price in the area of Mode 1 is 111111tptuupdtdp (3.69) Because 01up 01tu and 011tp the sign of (3.69) is ambiguous. Therefore, we have hdtdr11 11dtdp 0 (3.70) 11dtdS = 11111dtdphph 0 (3.71) 75
111111tqtuuqdtdq 0 (3.72) 1212tuupdtdp > 0 (3.73) hdtdr12 12tp > 0 (3.74) 12dtdS = 12222tphph > 0 (3.75) 01212tuuqdtdq (3.76) 18.104.22.168 Effect of Mode 1s Variable Commuting Cost on the Tax-Share Variable 12222121222212222 0 111211111211)()1()1()()1( )1()()1(1txxDtxdxtDtuuDxttxxDtxdxtDtuuDxttxxDtxyLdtdxxx (3.77) Because 01tx 011tD 01tx and 012 tD then 01dtd (3.78) In a two-mode urban spatial model, an increase in Mode 1s variable commuting cost results in a lower utility level, which is the same as in the a single-mode model. The above results show that an increase in Mode 1s commuting cost expands the urban area spatially, however, while in a single-mode model, an increase in commuting cost shrinks the urban area spatially. The effects of this variable on housing price, housing consump76
tion, urban land rent, and structural density in the area between the CBD and are am-biguous, while it increases housing price, land rent, and structural density while decreas-ing housing consumption in the area between and x. The effect of an increase in Mode 1s variable commuting cost on the tax-share variable is ambiguous. x x An intuitive explanation is that when Mode 1s commuting cost increases, all other things equal, the area using Mode 1 shrinks. (Recall that 22112ttffx so 01tx ) For those residents located in the area between the new and the original Mode 1s commuting cost exceeds that of Mode 2. Those residents switch from Mode 1 to Mode 2. Mode 1s higher t induces households between the old and the new to move farther from the CBD because their income net of transportation cost increases. Their move farther from CBD bids up housing price in the areas between and the ur-ban area boundary, which in turn raises urban land rent and structural density but lowers per-household housing consumption. Because of the increase in t x x x x 1 people originally lo-cated in the area between the CBD and new have lower income net of transportation costs, which lowers the demand for housing. At the same time, however, an increase in Mode 1s commuting cost makes living closer to the CBD more desirable, which bids up housing price close to the CBD. This in turn raises urban land rent and structural density in the area close to the CBD. As a result, when there is an increase in Mode 1s commut-ing cost, its effects on housing price, urban land rent, and structural density in the area between the CBD and are ambiguous. An increase in Mode 1s commuting cost re-duces population density in the area between the CBD and the new but its effect on x x x 77
population density in the area between and the urban area boundary is ambiguous. Thus, its effect on total transportation subsidies is ambiguous. As a result, the effect of an increase in Mode 1s variable cost on the tax-share variable is ambiguous. x 3.2.5 Effects of Change in Mode 1s Private Cost Share 22.214.171.124 Effect of Mode 1s Private Cost Share on Utility When = 1 (3.21) becomes 11122112112211111122221111()()()()ar x rxttxxrxrxxtttttt 1111200111111111111()xxrruxrdxdxrxttut 0)(1112211221222xtrxxrtdxuurtaxx (3.79) Substitute 11122200111111111()xrrdxxrxxdxttt xx into (3.79) and rearrange terms; then we have 111112220011111111221210112211()()011xxxxxrrrxttxdxdxxtxttturrdxdxtutu (3.79) 126.96.36.199 Effect of Mode 1s Private Cost Share on Urban Area Boundary Recall (3.15) 120xxx L xDdxxDdx Totally differentiating (3.15) with respect to 1 yields 78
1112011122110xxxDx 1 x DuxxDxxdxxDxuDuxxdxxDxu Since at we have 21DD x 112201111()()0xxDxDuxDuxdxxDxxuu xdx (3.80) Recall from (3.25) 0121212ruurxxr urxxrurxxrru 212212121 (since 012r ) (3.81) Substitute (3.81) into (3.80) and rearrange terms, getting 120112222120()xxxxDrxdxxuDrDrrxdxxdxxDxuxuxu (3.82) Since 0uDi and 02 x r the denominator of (3.82) is positive, so the sign of (3.82) depends on the sign of 11D 11111()DhSq = 2111111qqhqSh < 0 since 011S and 011q Therefore, we have 79
1x > 0 188.8.131.52 Effects of Mode 1s Private Cost Share on Housing Price, Housing Service Consumption, Urban Land Rent, and Structural Density 111111puupddp (3.83) Because up1 < 0, 1u < 0, and 11p < 0, the sign of (3.83) is ambiguous, so hddr11 11p 0 (3.84) 11ddS = 11111hpph 0 (3.85) 0111111quuqddq (3.86) 1212uupddp > 0 (3.87) 212hddr 12p > 0 (3.88) 12ddS = 12222phph > 0 (3.89) 01212uuqddq (3.90) 80
184.108.40.206 Effects of Mode 1s Private Cost Share on the Tax-Share Variable 12222121222212222011211111211)()1()1( )()1()1()()1(1xxDtxdxDuuDxtxxDtxdxuuDxtxxDtxyLddxxx (3.91) Since 01x 01x 011uuD and 012 uuD then 10dd These results show that when the private share of Mode 1s variable commuting cost increases, urban residents utility falls and the urban area expands spatially. Its ef-fects on housing price, housing consumption, urban land rent, and structural density in the area of Mode 1s use are ambiguous. An increase in the private share of Mode 1s variable costs raises housing price, urban land rent, and structural density while lowering housing consumption in the area of Mode 2s use. The effect on the tax-share revenue is ambiguous. The intuitive explanation is similar to that of an increase in Mode 1s variable commuting cost. When the private share of Mode 1s variable commuting cost increases, the area for which Mode 1 is cheaper shrinks (recall that 22112ttffx so 01tx ). The people who originally located between the new smaller and old now use Mode 2. Since they now use Mode 2, which has a lower variable cost than Mode 1, they are moti-vated to relocate farther from the CBD because their income net of transport cost inx x 81
creases. Their move farther from the CBD bids up housing price in the area between and x x which in turn increases urban land rent and structural density while reducing housing consumption. For the people originally located in the area between the CBD and the new an increase in x 1 reduces their income net of travel cost. As a result, demand for housing falls, which results in a lower housing price, urban land rent, and structural density. An increase in 1 also makes living close to the CBD more desirable, which bids up housing price close to the CBD, and increases urban land rent and structural density there. Over-all, an increase in 1 has ambiguous effects on housing price, urban land rent, and struc-tural density in the area between the CBD and Its effect on the tax-share variable is ambiguous because, on the one hand, an increase in x 1 decreases the subsidies needed for Mode 1, while, on the other hand, it expands the urban area boundary, which in-creases the total miles driven by the residents, thus increasing the subsidies needed for Mode 2. 3.2.6 Effects of Change in Mode 2s Variable Commuting Cost 220.127.116.11 Effect of Mode 2s Variable Commuting Cost on Utility When 2t (3.21) becomes 11222222112222221111()()aar x xxrxrxxrttttt tt 1122011211222111()xxxruxdxrxrdxtutttt 22212222222222111()()0xxrruxxdxrxrxttuttttt (3.92) 82
Upon substituting 2212222222222221111xxaxxrrdxxrxrxdxtttt x into (3.92) and rear-ranging terms, we have 1202112211xxxurrdxdxttutu 222222211xxr 2xxr x dxdxtxtt (3.93) Then 222222221220112211011xxxxxxxrrxdxdxutxttrrtdxdxtutu (since 02222tphtr ) (3.94) 18.104.22.168 Effect of Mode 2s Variable Commuting Cost on Urban Area Boundary Totally differentiating (3.15) with respect to t 2 yields 12120222222()()()0xxxxDuxDuD 22 x DxxdxxDxxdxtuttuttxxDxt (3.95) Then 12 22 022xxxxxDuDuD x dxxdxxdxututt 22)(txxDx (3.96) Totally differentiating boundary condition (3.14) with respect to we have 2t urtxxrtrtu222222 Substitute this into (3.96) and rearrange terms; then we have 83
12222202221222220()xxxxxxxxDrDrDr x dxxdxxdxxututtuDrDrrtxdxxdxxDxuxuxu (3.97) The sign of (3.97) is determined by the numerator because the denominator is positive. 22222()DhSttq = 222222qtqhqtSh = 22222222/0uhqphpphqqt since 22tS = 222hppht and 22tq = upq22 22tp This seems to imply that the numerator of (3.97) is ambiguous, but this is not so. Recall that 2222 p xtq 1221Vqup 22222)(qxShtr 1222)(VqShur where 0 and 0iiDquu 01 x V 01111VV 84
Upon substituting the above along with the previously derived expression for 2 2D tinto (3.97), we have 222222xxDrDr x dxtuut = 222222xxprpr x dxtuut = 22222222212122()()()()xxxhShShSxhSx x dxqqVqVqq = 222222112()11()0xxxxhSxhSxxdxxqqVVq dx (3.98) Since 12020xDrxdxut then the numerator of (3.97) is negative, so 2tx < 0 22.214.171.124 Effects of Mode 2s Variable Commuting Cost on Housing Price, Housing Service Consumption, Urban Land Rent, and Structural Density 2121tuupdtdp > 0 (3.99) 121hdtdr 21tp > 0 (3.100) 21dtdS = 1112hppht 1 > 0 (3.101) 2121tuuqdtdq 0 (3.102) 222222tptuupdtdp (3.103) 85
Because up2 < 0, 2tu < 0, and 22tp < 0, the sign of (3.103) is ambiguous. Therefore, we have 222hdtdr 22tp 0 (3.104) 22dtdS = 22222hptph 0 (3.105) 222222tqtuuqdtdq 0 (3.106) 126.96.36.199 Effect of Mode 2s Variable Commuting Cost on the Tax-Share Variable 2211111102222222222222222222222(1)()(1)1 (1)()(1) 0(1)()(1)xxxxxxDuxtDxtxdxtutdxDDxtDxtxdxdtyLttutxxtDxxDdxt 2u because 02tx 021tuuD 02 tx but 02222tuuDtD From the above results, we conclude that an increase in Mode 2s variable com-muting cost reduces utility and shrinks the urban area spatially. Its effects on housing price, housing consumption, urban land rent, and structural density are ambiguous in the urban area between x and x For the area between the CBD and x, however, an in-crease in Mode 2s variable cost increases housing price, urban land rent, and structural density while decreasing housing consumption. The effect of Mode 2s variable cost on the tax-share variable is ambi guous. 86
The intuitive explanation is that as Mode 2s variable cost increases, the area that uses Mode 1 expands from the old to new (recall that x x 22112ttffx so 02 tx ). For the people originally located in the area between the old and new Mode 1s com-muting cost is cheaper than that of Mode 2; therefore, they replace Mode 2 with Mode 1, which makes living closer to the CBD desirable. As a result, housing price rises in the area between the old and new which in turn bids up urban land rent and structural density while causing per-household housing consumption to fall. x x For those people living between the new and x x their income net of transport cost is reduced. As a result, housing price falls, which in turn reduces urban land rent and structural density while increasing housing consumption. Because urban land rent falls, the urban area shrinks spatially. On the other hand, an increase in Mode 2s variable cost makes living closer to the CBD desirable, which bids up housing price and land rent, in-creasing structural density and reducing housing consumption. Therefore, the overall ef-fects of an increase in Mode 2s variable cost on housing price, housing consumption, and structural density in this area are ambiguous. Its effect on the tax-share variable is ambiguous because, on the one hand, an increase in Mode 2s commuting cost results in higher subsidies, while, on the other hand, such an increase shrinks the urban area, which reduces transport subsidies within the urban area. 3.2.7 Effects of Change in Mode 2s Private Cost Share 188.8.131.52 Effect of Mode 2s Private Cost Share on Utility Let = 2 in (3.21), getting 87
22222222122011221111xxxxxxxrr x dxdxutxtrrdxdxtutu (3.107) Since 02 x r 022r and 0 uri we have 2u < 0 184.108.40.206 Effect of Mode 2s Private Cost Share on Urban Area Boundary Totally differentiating (3.15) with respect to 2 and canceling terms involving we have )(1xDx 12022()xDuxxdxxDxu 22220xxDuDxu dx (3.108) Let = 2 in (3.25) and solve for 2 u ; then we have urxxrru222222 Upon substituting this into (3.108) and solving for 2 x we get 122222022212222220()xxxxxxxxDrDrDr x dxxdxxdxxuuDrDrrxdxxdxxDxuxuxu u (3.109) 88
We know that the denominator of (3.109) is positive; therefore the sign of (3.109) is determined by the numerator. We need to know the following to determine the sign of (3.109) 2222222222222222222222()0uSqhqhDhSqqqhphpphqq 2p 2222 p txq 1221Vqup 2222rhtq 2x 1222Vqhur 01 x V which implies 01111VV Substitute the above into the second and third terms of the numerator of (3.109); then we have 222222xxDrDr x dxuu = 222222xxprpr x dxuu = dxqxthdxqxthVqVqhqxtxxxxx 2222 22212122221 89
= 0)11( 2222 112222xxxxdxqxthdxVVqqxthx Because the first term of the numerator of (3.109) is negative, we conclude that 02x 220.127.116.11 Effects of Mode 2s Private Cost Share on Housing Price, Housing Consump-tion, Urban Land Rent, and Structural Density 2121uupddp > 0 (3.110) 121hddr 21p > 0 (3.111) 21ddS = 1112hpph 1 > 0 (3.112) 02121uuqddq (3.113) 222222puupddp (3.114) Because up2 < 0, 2u < 0, and 22p < 0, the sign of (3.114) is ambiguous. Therefore, we have 222hddr 022ddp (3.115) 22ddS = 022222ddphph (3.116) 90
0222222quuqddq (3.117) 18.104.22.168 Effect of Mode 2s Private Cost Share on the Tax-Share Variable 2211111102222222222222222222222(1)()(1)1(1)()(1)(1)() xxxxxxDuxtDxtxdxudxD 2Du x tDxtxdxdyLuxxtDxtxDdx Since 02x 021uuD 02 x and 02222uuDD then 02 dd The above results show that an increase in the private share of Mode 2s variable cost reduces urban residents utility and shrinks the urban area spatially. For the area be-tween the CBD and housing price, urban land rent, and structural density increase while housing consumption per household decreases. For the area between and the ur-ban area boundary, the effects of the private share of Mode 2s variable cost are ambigu-ous. x x The intuitive explanation is that when the private share of Mode 2s variable commuting cost increases, the area in which Mode 1s transportation cost is lower than Mode 2s expands (recall that 22112ttffx so 02 x ). Those people located be-tween the old and new use Mode 1 rather than Mode 2, which makes living closer to the CBD desirable. As a result, housing price, urban land rent, and structural density in x x 91
the area increase, while per-household housing consumption decreases. For the people located between the new and urban area boundary, an increase in the private share of Mode 2s variable cost reduces income net of transportation costs, which reduces housing price, urban land rent, and structural density in the area. Because of the decrease in urban land rent, the urban area shrinks spatially. On the other hand, because of the increase in transportation costs, people are motivated to move closer to which bids up housing price there. Therefore, the overall effects on housing price, urban land rent, and structural density in the area beyond are ambiguous. The effect of the private share of Mode 2s variable cost on the tax-share variable is ambiguous because an increase in the private share of Mode 2s variable cost reduces subsidies for Mode 2s use while increasing population density in the area between the CBD and, which in turn raises subsidies for Mode 1s use. x x x x 3.2.8 Effects of Change in Mode 1s Fixed Cost 22.214.171.124 Effect of Mode 1s Fixed Cost on Utility When 1f (3.21) becomes 1211110112211122111111xrxffrrurxdxttftttfuf 222110xxrudxtuf (3.118) Rearranging (3.118), we have 112011111122121011221()11xxxxrrxfdxutffttrrfdxdxtutu 1f (3.119) 92
Recall that 01ur 02ur 011fr Therefore, we have 1fu < 0 126.96.36.199 Effect of Mode 1s Fixed Cost on Urban Area Boundary Totally differentiating (3.15) with respect to we have 1f 112011 1xx ()xDDuxdxxD f uff 210xxDuxdxuf (3.120) Substitute 1f into (3.26) and solve for 1fu ; then we have urfxxrfrfu212121 Substitute this into (3.120) to obtain 12121222000111()xxxDrDrxDrrxxdxxdxxdxxDx 1 f uuffuxu f 2222110xxxxDrxDrxdxxdxuffux (3.121) Rearranging the terms of (3.121), we have 93
121222001122122120()xxxxxxxDrDrDr 1 x dxxdxxdxx f uufuDrDrrfxdxxdxxDxuxuxu f (3.122) Because 12fr = 0, we have 120122122120()xxxxDrxdxxfuDrDrrfxdxxdxxDxuxuxu Since 111111112111()/0SqqhhhSqDffffq then 1fx > 0 188.8.131.52 Effects of Mode 1s Fixed Cost on Housing Price, Housing Consumption, Ur-ban Land Rent, and Structural Density 111111fpfuupdfdp (3.123) Because up1 < 0, 1fu < 0, and 11fp < 0, the sign of (3.123) is ambiguous. Therefore, we have hdfdr11 011dfdp (3.124) 11dfdS = 011111dfdphph (3.125) 94
0111111fqfuuqdfdq (3.126) 01212fuupdfdp (3.127) hdfdr12 012fp (3.128) 12dfdS = 012222fphph (3.129) 01212fuuqdfdq (3.130) 184.108.40.206 Effect of Mode 1s Fixed Cost on the Tax-Share Variable 221111111011222222221122221(1)()(1)1(1)()(1)(1)()xxxxDDtxDxtxdxffudxtxDxtxdxdfyLfufxtxDxf 11ufDu Since 01fx 011fdD 01fx and 012fuuD then 01dfd The above results show that an increase in Mode 1s fixed cost leads to lower util-ity but expands the urban area spatially. For the area between the CBD and the effects of Mode 1s fixed cost on housing price, housing consumption, urban land rent, and structural density are ambiguous. For the area between and x x x however, an increase in Mode 1s fixed cost increases housing price, urban land rent, and structural density while reducing housing consumption. The effect of Mode 1s fixed cost on the tax-share vari-able is ambiguous. 95
The intuitive explanation is that an increase in Mode 1s fixed commuting cost shrinks the area where Mode 1 is cheaper than Mode 2. Those people located between the new and old switch to Mode 2. The resulting increase in their income net of transport cost motivates them to move farther from the CBD to enjoy lower housing prices and larger dwelling sizes. As a result, housing price beyond the new increases, which in turn increases urban land rent and structural density. Rising urban land rent ex-pands the urban area spatially. x x x For the people located in the area between the CBD and the new, an increase in Mode 1s fixed cost reduces their income net of transport cost. Their demand for housing falls, leading to a decrease in housing price, lower urban land rent, and lower structural density. On the other hand, an increase in Mode 1s fixed cost makes living closer to the CBD more desirable, which counteracts the income effect. Therefore, the overall effects on housing price, urban land rent, housing consumption, and structural density are am-biguous in the area between the CBD and the new The effect of an increase in Mode 1s fixed cost on the tax-share variable is ambiguous. An increase in Mode 1s fixed cost expands the urban area, which increases subsidies for Mode 2. The effect on the subsidies for Mode 1 is ambiguous because the effect of Mode 1s fixed cost on population density between the CBD and the new is ambiguous. x x x 3.2.9 Effects of Change in Mode 2s Fixed Cost 220.127.116.11 Effect of Mode 2s Fixed Cost on Utility When 2f (3.21) becomes 1122111011221122112111122111()()xttrurxdxrxtttttufttt 1 96
2212222221122111()0xxrrudxrxtfufttt (3.131) After rearranging terms, we have 2111122221111222222120221111111()()011xxxxxrrxrxdxtttttttfurrfdxdxtutu (3.132) 18.104.22.168 Effect of Mode 2s Fixed Cost on Urban Area Boundary Totally differentiating (3.15) with respect to we have 2f 2202121)()(fxxDxdxfuuDxfxxDxx dxfDfuuDxxx2222 0)(22fxxDx (3.133) Substituting 2f into (3.25) and solving for 2fu we get urfxxrfrfu222222 Substitute this into (3.133) and rearrange terms; then, we have 122222022212222220()xxxxxxxxDrDrDr x dxxdxxdxxufuffuDrrDrf x dxxDxxdxuxuux (3.134) Since we know the denominator of (3.134) is positive, the sign of (3.134) is determined by the numerator. 97
Upon substituting the following into the second and third terms of the numerator of (3.134), 2222fpfD 2221qfp upuD22 1221Vqup 1222Vqhur 2222rh f q we have 222222222221212222211211110xxxxxxxxDrDrhhh 2 x dxxdxxdxuffuqqVqVqqhhxdqqVVq x Given that the first term of the numerator is negative, we have 02fx (3.135) 22.214.171.124 Effects of Mode 2s Fixed Cost on Housing Price, Housing Consumption, Ur-ban Land Rent, and Structural Density 02121fuupdfdp (3.136) 98
121hdfdr 021fp (3.137) 21dfdS = 021111fphph (3.138) 02121fuuqdfdq (3.139) 222222fpfuupdfdp (3.140) Because up2 < 0, 2fu < 0, and 22fp < 0, the sign of (3.114) is ambiguous. Therefore, we have 222hdfdr 022dfdp (3.141) 22dfdS = 022222dfdphph (3.142) 0222222fqfuuqdfdq (3.143) 126.96.36.199 Effect of Mode 2s Fixed Cost on the Tax-Share Variable 2211111102222222222222222222(1)()(1)1(1)()(1)(1)()xxxxDutxDxtxdxfufdxDtxDxtxdxdfyLffufxtxDxf 2Du Since 02fx 021fdD 02fx and 012dfdD then 01dfd 99
We have shown that an increase in Mode 2s fixed cost reduces urban residents utility and shrinks the urban area spatially. It increases housing price, urban land rent, and structural density in the area between the CBD and while decreasing per-household housing consumption. The effects on these variables in the area between and urban area boundary are ambiguous. The effect of an increase in Mode 2s fixed cost on the tax-share variable is ambiguous. x x The intuitive explanation is very similar to that of an increase in Mode 2s vari-able commuting cost. An increase in Mode 2s fixed cost increases the area using Mode 1. Those people who originally located in the area between the new and old switch to Mode 1. Given the resulting decline in their income net of transport cost, living close to the CBD becomes more desirable. The increased demand for housing bids up housing price there, which in turn increases urban land rent and structural density while decreas-ing housing consumption. Those people located beyond the new find their income net of transport cost fall, which lowers the demand for housing. This reduces housing price, urban land rent, and structural density. The decrease in urban land rent shrinks the urban area spatially. It also makes living closer to more desirable, which increases housing price closer to Therefore, its overall effects on housing price, housing consumption, urban land rent, and structural density are ambiguous. An increase in Mode 2s fixed cost shrinks the urban area spatially, which reduces the subsidies needed for Mode 2; how-ever, it raises population density in the area between the CBD and the new thus in-creasing the subsidies needed for Mode 1. Therefore, its overall effect on the tax-share variable is ambiguous. x x x x x x 100
3.3 Summary Table 3.1 summarizes the comparative static analysis of the two-mode model. Rather than discuss all of these results, we emphasize those that are important for the em-pirical analysis to follow. Since our main goal is to explain how transport subsidies affect urban sprawl, we concentrate on 1 x and 2 x Recall that i is the ith modes private cost share. Since ,01x an increase in the subsidized share of Mode 1s transport cost causes the urban area to contract. This result may seem counterintuitive, but when the private share of Mode 1s variable cost decreases, the area for which Mode 1 is cheaper expands (recall that 22112ttffx so 01x ). The people who originally located between the old, smaller and new, larger now use Mode 1. Since Mode 1 has a higher variable cost than Mode 2, they are moti-vated to relocate closer to the CBD because their income net of transport cost increases more rapidly with Mode 1 than with Mode 2. Their move closer to the CBD reduces housing price in the area between and x x x x which in turn decreases urban land rent and structural density while increasing housing consumption. As a result, the urban area shrinks. Since 02x the urban area expands if the subsidized portion of Mode 2s cost increases. This result is more appealing intuitively and happens because when the private share of Mode 2s variable cost decreases, the area in which Mode 1s transporta101
tion cost is lower than Mode 2s shrinks. Those people located between the old, larger and new, smaller use Mode 2 rather than Mode 1, which makes living farther from the CBD desirable since the increase in transport costs associated with a move farther out is now less. As a result, housing price, urban land rent and structural density in the area de-crease, while per-household housing consumption increases. For the people located be-tween the new and urban area boundary, a decrease in the private share of Mode 2s variable commuting cost increases income net of transportation costs, which raises hous-ing price, urban land rent, and structural density in the area. Because of the increase in urban land rent, the urban area expands spatially. x x x Table 3.1 Comparative Static Results of Two-Mode Model Endogenous Variable Exogenous Variable u x S i r i p i q i L + + + + ? )(1xxf + + + + ? )(2xxf ? ? ? ? ? 1txx + + + + ? 2txx ? ? ? ? ? ()yxx + ? ? ? ? + ? Ar + + + x NC NC + NC 1 x x + + + + ? 2 x x ? ? ? ? ? 102
CHAPTER 4 EMPIRICAL ANALYSIS Although there have been many empirical studies of urban sprawl (see, for exam-ple, the citations in Nechyba and Walsh (2004)), the one closest in spirit and approach to ours is Brueckner and Fansler (1983). Consequently, we provide a brief review of that paper. Brueckner and Fansler (1983) base their empirical analysis on the theoretical model exposited by Brueckner (1987). That model is similar to, but simpler than, ours: it has only one transportation mode, no variables representing transportation subsidies, and no balanced budget equation. They estimate the following general relationship 12(,,,,)A A reafLrytt (4.1) where Area is the land area in square miles of forty urbanized areas in 1970, L is 1970 urbanized-area population, is the 1969 median agricultural land value per acre for the county in which the urbanized area is located, y is 1970 urbanized-area mean household income, is the percentage of urbanized-area commuters using bus transit in 1970, and is the percentage of 1970 urbanized-area households owning one or more autos. Ar 1t 2t Equation (4.1) is essentially the reduced-form urban-rural boundary equation from the theoretical model, (,,,,,)AxxLtyir where x is the urban-area radius, which is of course directly related to Area. The two transportation variables are proxies for round-trip transport cost per mile. Brueckner and Fansler believe that and transport cost should be 1t 103
positively correlated because bus travel has a high time cost relative to auto travel, whereas and transport cost should be negatively correlated because auto travel involves less time cost than bus travel. The variables for structure cost, i, and radians of urban land used for residences, are suppressed. 2t Since functional forms are not specified by the theory, the authors estimate a lin-ear equation by ordinary least squares (OLS) and a flexible form using the Box-Cox transformation. The results of both estimation techniques are qualitatively the same. The estimated coefficients on population and income are statistically significantly positive, while the estimated coefficient on agricultural land rent is statistically significantly nega-tive. The transportation cost proxies have the correct signs but are not statistically signifi-cant. The coefficient of determination is approximately 0.80 for both estimated equations. Brueckner and Fansler do not discuss potential endogeneity problems arising from the transportation-cost variables; we shall return to this topic in our empirical analysis. The authors conclude that sprawl is the result of an orderly market process, that transferring farmland to urban use is not a waste, and that the empirical robustness of the theory increases confidence in the closed-city model as a policy tool. We turn now to our own empirical work. 4.1 Objectives and Hypotheses The objective of this thesis is to explore the effect of transportation subsidies on urban sprawl. The theoretical model of Ch. 3 predicts that highway subsidies promote urban sprawl, while public transit subsidies retard it. Although these are the primary hy-potheses of interest, we also test other hypotheses of the theory. 104
105 We start with OLS and flexible-form procedures, as did Brueckner and Fansler. We also investigate endogeneity, which require s us to estimate the model using two-stage least square (2SLS). Some of the variables in our model are the same as those used by Brueckner and Fansler, and some are differe nt. In addition, we sometimes measure the same variables in different ways from those of Brueckner and Fansle r. We use what we think are more accurate measures of modal commuting costs, and we add new variables for highway subsidies, transit subsidie s, and the local income tax rate. 4.2 Data Sources We draw all of the basic data used in our empirical work from the web sites of various organizations. From the U.S. Bureau of the Census, we have drawn data from the population census of 2000; the censuses of government of 1992, 1997, and 2002; and the census of agriculture of 1997 and 2002. From the Federal Highw ay Administration (FHWA), we use Highway Statistics 2000. From the Federal Transit Administration (FTA), we use the National Transit Database (NTD). From the Insurance Information Institute, we use data on average expenditures for auto insurance. 4.2.1. United States Census 2000 United States Census 2000 Summary File 3 (SF3) (U.S. Census Bureau 2000) provides number of households, population, household income, and spatial size of urbanized areas. SF3 contains some of the richest and most complete statistical data available on U.S. residents, collected from one in si x households in the United States. There are 813 detailed tables available in the SF3 data products. The total number of household by urbanized area is from Table H1. Mean hous ehold income by urbanized area is derived from Table P54 and Table H1.
106 4.2.2. Census of Agriculture The United States Department of Agri culture conducts a census of agriculture every five year. We obtain rural land rent from the censuses of agriculture for 1997 and 2002 (National Agricultural Statistics Service 1999, 2004). 4.2.3. Census of Government The U.S. Census Bureau conducts censuses of government every five years. These censuses cover three majo r subjects: government organization, public employment, and government finances. We calculate highway subsidies from information on highway expenditures and revenues of counties in wh ich urbanized areas are located. We detail this procedure in Section 4.3 be low. We also derive the prop erty tax rate using data provided by the census of government. 4.2.4. National Transit Database We calculate two variables from the NT D (Federal Transit Administration 2004): transit cost per passenger-mile and trans it subsidy per passenger-mile. The FTA collects and disseminates data on the state of mass transportation via the NTD program. Over 600 of the nation's transportation providers submit data to the NTD annually. Although the NTD reports data on twelve mass transporta tion modes, we focus on bus because most urbanized areas provide bus serv ice, while very few provide ot her kinds of transit service. We define the transit subsidy as the difference between transit operating and capital expenses and transit fare revenues, expressed per passenger-mile. We explain how we estimate this variable later.
4.2.5 Highway Statistics The FHWA annually publishes Highway Statistics, which contains data on motor fuel, motor vehicles, driver licensing, highway-user taxation, state and local government highway finance, highway mileage, and federal aid for highways. States submit the highway data, and the FHA reports the data at the state level. We use Highway Statistics 2000 (Federal Highway Administration 2003) to calculate motor fuel taxes per vehicle-mile traveled (VMT) as a proxy for highway cost. We provide details of this procedure in section 4.3.7. 4.2.6. Insurance Information Institute The Insurance Information Institute publishes data on average expenditures for auto insurance in 2000 by state (Insurance Information Institute 2005). We use these data as part of fixed auto cost. 4.3 Variables Used in the Analysis This section discusses each variable used in the analysis. Our two-mode theoreti-cal model produces the following reduced form equation 1122,,,,,,,AxfLyrtt (4.2) where x is the urban boundary, measured by distance from the CBD; L is population; y is income per household; is rural land rent at the urban boundary; is transit cost per round-trip mile; ar 1t 1 is transit subsidy per round-trip mile (please note that in Ch. 3, 1 is used to represent the private share of transit cost per round-trip mile); is highway cost per roundtrip mile; 2t 2 is highway subsidy per round-trip mile (in Ch. 3, 2 is used to represent the private share of auto cost per round-trip mile); and is the local income 107
tax rate. Following Brueckner and Fansler, we suppress the structural input rental rate, i, in (4.2) because the most likely proxy for it, the interest rate, does not vary much nation-wide. Again, following Brueckner and Fansler, we also suppress the variable for radians of urban land used for residences. This variable theoretically affects the size of the urban area, but it is very difficult to get data on it. Of the various kinds of urban areas the census delineatesincluding legal cities, census designated places, urban clusters, urbanized areas, metropolitan statistical areas, primary metropolitan statistical areas, and consolidated metropolitan statistical areasit is the opinion of urban economists that urbanized areas best conform to the generic urban area used for theoretical analysis. Many urbanized area have more than one central city. For example, the Tampa Bay area has three: Tampa, St. Petersburg, and Clearwater. Since our model is monocen-tric, we eliminate those urbanized areas with more than one central city. In addition, there are sometimes more than one urbanized areas in a single county. In these cases, we are concerned that our proxy for rural land value at the urban fringe may be inflated by the presence of another nearby urbanized area. There are a total of 465 urbanized areas de-lineated by the 2000 census. Using GIS software, we identify 201 urbanized areas that are located within a single county and have only one central city. This set of urbanized areas constitutes our sample. We turn now to a detailed discussion of our variables. 4.3.1 The Urban Boundary, x Theoretically, the dependent variable in our estimating equation should be the ra-dius of an urban area with the CBD as the center. To avoid having to calculate an implied radius for each of our sample urbanized area, we use instead the spatial size of the urban108
109 ized area, which is, of course, directly re lated to the implied radius. The 2000 census of population provides the spatial sizes of urba nized areas, measured in square meters, which we convert to square miles. This va riable is found in Cens us 2000(U.S. Census Bureau 2002) 4.3.2. Total Number of Households in Urban Area, L In our theoretical models, we assume single-person households, which is, of course, not the case in reality. Since the house hold is the basic unit of analysis in our models, we use households, rather than population, to measure L in our model. This variable is found in U.S. Census 2000, Table P15 (U.S. Census Bureau 2002). 4.3.3. Mean Household Income, y The U.S. Census 2000 reports median household income in 1999 dollars by urbanized area, but not mean household income. The census does, however, report aggregate household income in 1999 dollars in Ta ble P54. We calculate mean income per household by dividing aggregate household income by the total number of households residing in the urbanized area. 4.3.4. Agricultural Land Rent, r a In our theoretical models, agricultural land rent is the land rent immediately adjacent to the built-up part of the urban area. This theoretical construct is not available empirically in any published source of which we ar e aware. As an alternative, we use mean estimated market value of farmland per acre fo r the county in which the urbanized area is located. This variable is available from the Census of Agriculture (National Agricultural Statistics Service, 1999, 2004). Since the Census of Agriculture is conducted every five years and in different years from the decennial census, our variable is the mean of the
110 means reported for 1997 and 2002. We perform th is calculation to ob tain a value for the rural land variable as consistent as possible with other variables, most of which are for the year 2000. 4.3.5 Fixed Transit Cost, f 1 Fixed transit cost includes travel time between home and bus stop and waiting time at the bus stop. Since there are no data by urbanized area on travel time to and waiting time at bus stops, we use as a proxy the percentage of the working-age population taking transit to work. We expect this proxy to be negatively related to fixed transit cost. Brueckner and Fansler (1983) use the pe rcentage of urbanized-area commuters using bus transit as a proxy for their single-mode variable transport cost. They reason that the percentage of commuters using transit s hould be positively correlated with variable transport cost because bus has a high time co st. This argument is no t supported by travelbehavior studies, however. Studies of travel behavior find that tran sit passengers find waiting time to be more onerous than in-vehicle travel time (Fed eral Transit Administration 2004). We argue therefore that the longer the waiting time, the less attractive transit becomes as a means of transportation, and the fewer users it will attract. Our view is further supported by a Florida Department of Transportation report on f actors affecting trans it use (Hinebaugh and Zhao 1999)). The report says that transit use deteriorates exponentia lly with walking distance to transit stops. In addition, Nationwide Personal Transportation Studies (U.S. Department of Transportation 1986) reports th at the maximum convenient walking distance is about 450 meters. We conclude that the longer the walking time, the fewer the people who use transit. Data on the percentage of the working age population taking transit to
111 work are from U.S Census 2000 Table P32. Another potential proxy we may use is the transit density, defined as the land area covered by each bus route mile. It is expected to be positively related to the fixed transit cost The larger the land area is served by each bus route mile, the longer the waiting time, thus higher fixed transit cost. We use data from National Transit Database to derive th e transit density. Its effect, however, is not statistically significant. 4.3.6. Variable Transit Cost, t 1 In our two-mode model, pr ivate transit cost is the households money cost of transit per round-trip mile. Since no direct measure of this cost exists, we use a close substitute: private transit cost per passenger-mile. This variable is derived by dividing annual bus fare revenue by the annual total passenger-miles reported in the National Transit Database. The National Transit Databa se 2002 data tables (Federal Transit Administration, 2004) provide detailed summaries of financial and operating data submitted to the FTA by the nations mass transit agencies. For the 2002 report year, 613 transit agencies submitted data. The FTA has established a uniform system of accounts and records, as well as a reporting system for the collection and dissemination of public mass transportation financial and operating data by uniform categories. Additionally, according to federal law, all applicants and direct beneficiaries of federal assistance are subject to the reporting system and the uniform sy stem of accounts and records. Therefore, data reported in the NTD are uniform, consistent comparative, and reliable. We use these data for the year 2002 ra ther than 2000 for a couple of reasons. First, since 2002 the FTA has re quired reporting of fare revenues by mode. Before 2002,
112 transit agencies reported only an aggregate fare revenue for all the modes they operated, which makes it impossible to identify the exact amount of fare revenue generated by a particular mode and makes the data less co mparable if the modes operated by different transit agencies are di fferent. Second, 2002 was the first ye ar that transit agencies were required to report performance indicators, su ch as unlinked passenger trips and passenger-miles by mode. These data enable us to obtain fare revenue per passenger-mile by mode. Third, we compare the transit reve nues and transit expenses from 2000 to 2002 and find that the amounts reported are very close, which justifies using 2002 data. The NTD program collects data on eighteen modes, including railroad, cable car, demand response (i.e., taxi), commuter rail, ferr yboat, heavy rail, inclined plane, jitney, light rail, bus, monorail, and trolleybus. Of the 536 transit ag encies that report fare revenues, 456 provide bus service, 432 provide dema nd-response service, forty-five provide vanpools, twenty-four provide lig ht rail service, and sevent een provide ferryboat service. Other modes have less than fi fteen providers. Since bus serv ice is widely provided and demand response is generally not considered public transit, we use bus to represent transit in our analysis. Based on the transit agency address, we match the transit agencies and the urbanized area for which they provide servic e. We then derive private transit cost per passenger-mile from fare revenue and passenger-miles. 4.3.7 Transit Subsidy, 1 In our theoretical model, 1 is used to represent the priv ate share of transit cost. In this empirical analysis, we use 1 to represent the transit subs idy. We derive transit subsidy per passenger-mile as follows. First, we sum bus transit opera ting cost (from Table 1, NTD 2002) and transit capital cost (from Table 7, NTD 2002) to get total cost. Second,
113 we subtract fare revenue from total cost to ge t the subsidy. Finally, we divide the subsidy by total passenger miles to get tr ansit subsidy per passenger-mile. 4.3.8. Fixed Auto Cost, f 2 Fixed auto cost includes all costs invariant to distance traveled. For highway users, fixed costs generally incl ude registration fees, license fees, motor vehicle taxes, auto insurance premiums, maintenan ce, and depreciation. Since data on maintenance and depreciation are not available, we use the su m of average annual motor vehicle fees and taxes and annual auto insurance premiums per household to represent fixed auto cost. Data on average annual motor vehicle fees a nd taxes are from Highw ay Statistics 2000, and data on annual auto insuranc e premiums are from the Insu rance Information Institute. 4.3.9 Variable Auto Cost, t 2 In the two-mode model, variable auto co st is the households auto expenditure per round-trip mile. No direct measur e of this cost exists, and th is is the most difficult variable for which to develop a proxy. After c onsulting many sources, we conclude there is no way to get an accurate estimate of this cost by urbanized area. Consequently, we use highway fuel tax per VMT to represent private variable highway cost. To help justify the use of this variable, we summarize problems with alternative measures. The Bureau of Labor Statistics annual consumer expenditure survey reports consumer units annual expenditures on trans portation, including vehicle purchase, gasoline and oil purchase, other related expenditu res, and expenditures on public transportation. (A consumer unit comprises: (1) all me mbers of a particular household who are related by blood, marriage, adoption, or othe r legal arrangement; (2 ) a person living alone or sharing a household with othe rs or living as a roomer in a private home or lodging
114 house or in permanent living quarters in a hote l or motel, but who is financially independent; or (3) two or more persons living to gether who use their income to make joint expenditure decisions.) Financial independen ce is determined by the three major expense categories: housing, food, and other living expenses (Bureau of Labor Statistics 2000). This seems an ideal source of data with whic h to construct variable auto cost. Unfortunately, the consumer expenditure survey re ports only the average transportation expenditures for four major regions (Midwestern, Northeastern, South, and Western) and the twenty-eight largest Metropolitan Statistical Areas. We c onsidered using the public use micro-data files from the Bureau of Labor Statistics to calculate average highway expenditures by urbanized area or by county or MS A in which our sample urbanized areas are contained. This approach was precluded by the fact that the geographical location of the consumer units is suppressed or recoded to preserve confidentiality. Thus, it is impossible for us to obtain the desired transport cost. The only available proxy for the variable we want to measure is average highway motor fuel taxes per VMT. Other things equa l, the higher the motor fuel taxes paid, the higher are variable auto costs. Before we explain how we obtain motor fuel taxes per VMT, which is the largest component of highway user-fees by urbanized area (Federal Highw ay Administration, 2003), we explain what highway user-fees ar e and how they are collected and apportioned. These fees play a key role in highw ay financing in the United States. Highway user-fees consist of highwaybased exercise taxes; federa l use tax and taxes on tires, trucks, and trailers; and vehicl e-related license and operation taxes. The three major categories of highway taxes are federal motor fuel taxes, state motor fuel taxes, and local mo-
115 tor fuel taxes. Vehicle-related license and operation taxes are genera lly collected at the state and local level (Federal Highway Administration 2003). Federal motor fuel taxes are specific taxes that differ by type of fuel. These taxes are not paid directly by highway users to the Internal Revenue Service (IRS). Rather, most taxes are collected by the IRS from several large corporations, typically large oil companies or distributors with storage facilities. This makes it impossible to measure fuel usage within each state. Therefore, there is no direct measure of highway taxes paid by the residents in each state. After the IRS collects highway-based excise taxes, the Office of Tax Analysis estimates monthly highway receipts from each state. The Financial Management Service and the Bureau of Public Debt then credit the Highway Trust Fund (HTF) with the estimated revenue. Each quarter, the IRS certifies total receipts to reflect actual tax revenue collected by type of tax. The Financial Ma nagement Service and the Bureau of Public Debt then adjust the amount initially estimat ed based on the IRSs quarterly verification. The final information from the Department of Treasury, which documents receipts for the previous fiscal year, determines the overall le vel of highway user ta x receipts attributed to the HTF by tax type. This amount repres ents the total highway program funds available for apportionment to each state. Among ma ny criteria, state-reported motor fuel data are a critical component of the process that distributes HTF to the states (Federal Highway Administration 2003, 2005). States, however, have very different tax le gislation, tax forms, and administrative procedures. There are significant variations among the states in exemptions or refunds for off-highway use or government us e. Therefore, it is not su fficient for the FHWA to sim-
116 ply sum up the total gallons of fuel reported by states and use those data for attribution. The FHWA has to adjust the data reported by states and separate the data into use (onhighway versus off-highway), fuel type (gaso line, gasohol, special fuels), and tax statutes based on public, private, and commercial categories (exempted, refunded, taxed-at-otherrate). As a final step, the FHWA sums up the on-highway gallons to derive a national total for each of the three motor fuel types and divides each states gallons by the total gallons to derive each states share (F ederal Highway Administration 2003). Highway users not only pay federal motor fuel taxes but also state motor fuel taxes. State motor fuel tax rates on gasoline va ry from 7.5 cents per gallon in Georgia to 29.3 cents per gallon in New York State, w ith a national average of 20.17 cents in 2000 (Federal Highway Administra tion 2003). Based on state laws, state governments have authority to collect motor license fees, regist ration fees, and other motor-related taxes. Revenues from state motor fuel taxes and moto r-related fees and taxes are generally used by state governments for trans portation purposes and are therefore counted as a part of user-fees. In some states, due to population growth and inflation in the early 1960s and 1970s, new demands placed on local governments we re greater than thei r ability to raise capital for local transportation projects. Theref ore, in the early 1970s, several states allowed counties to piggyback, or add to, the states tax on highway fuels. For example, the Florida legislature authorizes counties to collect up to 12 cents per gallon gasoline tax to be used on local transportation projects (Florida Department of Transportation 2005). The majority of states, however, do not al low local governments to collect motor fuel taxes.
117 Since highway users pay three layers of fu el taxes, we add them to obtain the total fuel taxes paid in each state. We derive the urbanized areas share of fuel taxes by multiplying the amount of total state fuel taxes by the urbanized areas propo rtion of its states vehicles, calculated as the number of vehicl es owned by urbanized-a rea residents divided by the total number of vehicles owned by stat e residents. Since states vary significantly regarding the tax exemption for public use or government use, we use the total number of privately owned vehicles in each state. After obtaining a figure for annual fuel ta xes by urbanized area, we divide that figure by annual total VMT, thus deriving high way fuel taxes per VMT, which represents variable auto cost in our empirical work. The annual VMT by urbanized area is derived by multiplying average daily VMT by 365. To ge t a per-vehicle figure for each urbanized area, we divide the sum of each states share of federal motor fuel tax receipts, state motor fuel tax receipts, and local motor fuel tax receipts by the annual VMT. Federal motor fuel taxes attributed to each state are from Hi ghway Statistics 2000, Table FE-9. Data on state motor fuels taxes an d related receipts and state motor vehicle and motor carrier tax receipts are from Table MF-1 and Table MV-2. Data on local motor fuels tax receipts are from Table LDF. Data on the total number of vehicles by state and urbanized area are from U.S. Census 2000, Table H46. Data on VMT by urbanized area are from Highway Statistics 2000 Table HM72. 4.3.10 Highway Subsidy, 2 In our theoretical model, we use 2 to represent the private share of auto cost. In this empirical analysis, we use the same not ation to represent the highway subsidy. We present our method for measuring the highway subsidy by urbanized area in this section.
118 One question must be answered before we di scuss how we measure the highway subsidy: What are highway subsidies or what constitute highway subsidies? There is no official defi nition of highway subsidies. If user-fees collected for highway purposes are not sufficient to cove r highway expenditures, governments must use other revenues to make up the difference, which amounts to subsidization. According to Highway Statistics 2000 total U.S. highway user-fees were $106 billion in 2000, of which $81.0 billion was used for highway purposes (Federal Highway Administration 2003). Total U.S. highway expenditures in 2000 were $127.5 billion. Therefore, the subsidy was $46.5 billion, which was contributed by all levels of government from other revenue sources. Revenues contributing to the subsidy are from a number of sources, including local property taxes and assessments, other dedicated taxes, general funds, bond issues, and other investment income. As shown in Table 4.1, the degree to which highway programs are funded by highway-user-fees differs widely among differe nt levels of government. At the federal level, 95.6 percent of highway revenues came from motor-fuel and motor vehicle taxes in 2000. The remainder came from general fund appropriations, timber sa les, lease of Federal lands, oil and mineral royalties, and mo tor carrier fines and penalties. Highway userfees provided 75.5 percent of highway revenu es at the state level in 2000. Bond issue proceeds were another significant source of funding, providing 12.3 percent of highway funds at the state level. The remaining 14 percent of state highway funding came from general fund appropriations, ot her state taxes and fees, in vestment income, and other miscellaneous revenue sources (Fed eral Highway Administration 2003).
119 Many states do not permit local governme nts to impose motor-fuel and motor vehicle taxes, or they cap them at relatively low levels. At the local government level, only 7.5 percent of highway funding was provided by highway user-fees in 2000. Local general funds, property taxes, and other taxes a nd fees were the sources of 67.5 percent of local highway funding. Bond issue proceeds provided 9.8 percent of local highway funding, while investment income and miscellaneous receipts provided the remaining 14 percent (Federal Highway Administration 2003). Since only 62.9 percent of revenue sources for highways come from highway user-fees, highway use is clearly subsidized. The question is how to use this information to form the highway subsidy variable in our model. For consistency with our theoretical model, highway subsidies should be measured per round-trip mile per household for tr avel within an urban area. (Although the model assumes single-person households, we us e per-household figures, rather than perperson figures, because the household is assume d to be the decision maker and the bearer of travel costs.) Highway subsidies are the difference between highway expenditures paid by various level of governments in a given urban area and the total user-fees collected from highway users in that same area. Such a measure of highway subsidies is unavailable because of complicated highway owne rship and financing systems in the United States.
120 Table 4.1 Revenue Sources for Highw ays, 2000 (billions of dollars) Governments Revenues and Expenditures Federal State Local All Percent of Total Revenue User Charges Motor-Fuel Taxes $25.1 $28.7 $1.0 $54.8 42.5 Motor-Vehicle Taxes and Fees $4.6 $15.5 $0.7 $20.8 16.2 Tolls $0.0 $4.7 $0.7 $5.4 4.2 Subtotal $29.7 $49.0 $2.3 $81.0 62.9 Other Sources of Revenue Property Taxes and Assessments $0.0 $0.0 $6.4 $6.4 4.9 General Fund Appropriations $1.2 $4.1 $11.9 $17.2 13.4 Other Taxes and Fees $0.1 $2.4 $2.8 $5.4 4.2 Investment Income & Other Receipts $0.0 $2.7 $4.8 $7.5 5.8 Bond Issue Proceeds $0.0 $8.2 $3.1 $11.2 8.7 Subtotal $1.4 $17.5 $28.9 $47.7 37.1 Total Revenues $31.1 $66.4 $31.3 $128.7 100.0 Total Expenditures $27.7 $67.0 $32.7 $127.5 99.0 Source: Federal Highway Admi nistration, 2003, Table HF-10 Highways are typically classified by eith er ownership or purpose. Ownership is determined by the jurisdiction that has primar y responsibility for a particular structure, while purpose (as well as level of service) is identified from the structures function. Ownership is divided among federal, state, and local governments. According to the De-
121 partment of Transportation, roads and br idges owned by these governments are considered public, while structures owned privately are commonly considered nonpublic. States own almost 20 percent of the nations road system. The federal government has control over about 3 percent of the network. Over 77 percent of Ameri can roads are locally owned although some intergovernmental agreem ents may authorize states to construct and maintain locally owned highways. Highway financing is not solely determ ined by highway ownership, however. Highways owned by the federal government (inc luding interstate highways; the strategic highway network, which is generally highway s important to military mobilization; other rural and urban principal arterials; strategi c highway network connectors; and intermodal connectors) are 90-percent funded by the federal government. For highways owned by states, capital outlays are largely funded by the federal government through intergovernmental grants. Even local roads are not funded solely by local governments. State and federal governments fund the majority of cap ital outlays for local roads (Federal Highway Administration 2002). According to Highway Statistics 2000, although the federal government funded $27.7 billion (21.7 percent) of total highway expenditures, the majority of the federal governments contribution to highways consisted of grants to state and local governments. Direct federal spendi ng on capital outlay, maintena nce, administration, and research amounted to only $2. 3 billion (1.8 percent). The remaining $25.4 billion was in the form of transfers to st ate and local governments. State governments combined $24.4 billion of federal funds with $52.1 billion of state funds and $1.3 billion of local funds to make direct expenditures of $77.9 billion (61.1 percent). Local governments combined
122 $1.0 billion of federal funds with $14.9 billi on of state funds and $31.4 billion of local funds to make direct expenditures of $47.3 billion. Given the complicated highway ownershi p and funding systems, it is almost impossible to obtain an accurate estimate of highway subsidies by urbanized area. For example, in a particular urbanized area, urban residents may have access to interstate highways owned and mainly funded by the federa l government but maintained by the state government, state-owned highways funded by th e state, highways owned by counties or cities maintained and funded by the state, a nd county or city roads owned and funded by them. For the various roads within an urbanized area, local governments report only revenues and expenditures on the roads owne d and funded by them. No financing data are available for those roads not owned or funded by local governments. We are therefore restricted to use county and c ity data to calculate highway subsidies for a given urban area. Unfortunately, city data are not reliable because most highway revenue data are not reported. We are therefore reduced to using only county data to calculate highway subsidies. The highway subsidy variable used in our analysis is derived as follows. First, we obtain county highway subsidie s as the difference between total county highway expenditures and total county highway revenues. The urbanized areas share of highway subsidies is calculated as the product of total county subsidies and the number of vehicles owned by urbanized-area resi dents divided by the number of vehicles owned by county residents. The urbanized areas share of hi ghway subsidies then is divided by urbanizedarea VMT to derive the highway subsidies per VMT. If major construction occurred in a particular year, highway expenditures and rela ted subsidies in that year may have risen
123 substantially. To smooth this kind of effect, we use a thre e-year average for 1992, 1997, and 2002. All the three-year data are adjusted to 2000 dollars based on the CPI. Our data are provided by censuses of government. The census of government collects detailed data on government financing every five years. We use their data for the survey year 1997 and 2002. Highway-related ex penditures and revenues are reported by county. According to the census of governme nt, highway expenditu res include expenditures on maintenance, operation, repair, and construction of non-toll highways, streets, roads, alleys, sidewalks, bridges, and rela ted structures, while hi ghway revenues include revenues from motor fuel taxes, motor vehicl e license fees, motor vehicle operation fees, transfers from state government and federal governments, and regular charges on highways (U.S Census Bureau 2004). 4.3.11 Local Property Tax Rate, The local property tax rate is derived from information provided by the census of government. We deflate 1997 and 2002 county and city property tax revenues to 2000 dollars by the CPI. To derive the per-househ old property tax payment, we average the property tax figures an d divide by the number of urbani zed-area households. Then the per-household property tax payment is divided by mean household income to derive the local property tax rate. Although the urbanized areas in our model are smaller than the counties in which they are located, this s hould not introduce much error since the high value property is located predominantly in the urbanized areas. 4.3.12 Intergovernmental Grants, G The variable intergovernme ntal grants is derived fr om information provided by
124 the census of government. We deflate 1997 an d 2002 county and city intergovernmental grants for transportation purposes from state government and derive the average intergovernmental grants by urbanized area. 4.3.13 Missing Observations In this empirical analysis, 201 urbanized ar eas meet our criteria that the urbanized area be located within a single county and have only one central city Because of missing observations, however, we end up with 93 urbanized areas. Al most all the incomplete observations are due to the abse nce of two key vari ables, the transit subsidy and the highway subsidy. Although it is fairly common for a data set to have gaps for a variety of reasons, it is nevertheless impor tant to ensure as well as possible that these missing observations do not cause problems in the empirical analysis. In general, there are two cases to co nsider, depending on why the data are missing. One is that the data are simply unavailable and not systematically related to other observations in the sample. This case is called ignorable (Greene 2000) because the analyst may simply ignore the problem without reduc ing the statistical relia bility of the estimated results. Were the missing data availabl e and not used, there w ould of course be a loss of efficiency. The second case involves missing data th at are systematically related to other variables in the model. This case occurs most often in surveys in which the data are selfselected. This case causes a more serious problem because the gaps in the data set represent, not only missing information, but also in formation that, if available, could change the estimated results. All our incomplete observations are due to the absence of data on two key variables, the transit subsidy and the highway subsidy. The transit subsidy is
125 missing for some urbanized areas because they simply have no public transit. This situation is not related to self-selection or non-response error. Because of the non-existence of public transit in 96 urbanized areas, our sa mple size falls from 201 to 107. Because of non-existence of VMT by urbanized area, our final sample is 93. A list of variables, definition of the variables in the empirical analys is, and their data sources are presented in Table 4.2. Table 4.3 provides summary statistics. Table 4.2 List of Variables, Definitions, and Data Sources Variable Definition Source Area Spatial size of the urbanized area in square miles Census 2000 L Number of households by urbanized area Census 2000 r a Estimated market value of farmland per acre for the county in which the urbanized area is located Census of Agricultural 1997, 2002 y Mean household income by urbanized area Census 2000 f 1 Percentage of working age population using transit Census 2000 f 2 Sum of auto annual insurance premium, registration fee, license fee, and motor vehicle tax per household by urbanized area Highway Statistics 2000, Insurance Information Institute t 1 Bus fare cost per passenger-mile National Transit Database 2002 t 2 Fuel tax payment per vehicle-mile traveled Highway Statistics 2000 1 Subsides to bus service per passe nger mile National Transit Database 2002 2 County subsidies to auto use per vehicle mile traveled Census of Government 1997, 2002 G Intergovernmental Transfer from State to local governments for transportation purposes Census of Government 1997, 2002 Local income tax rate: percentage of average household income paid as property tax payment Census of Government 1997, 2002
Table 4.3 Summary Statistics Variable Unit Mean Std. Error Minimum Maximum Area Sq. mi. 85.7978 75.01 13.6256 323.556 L 1000 79.839 92.85 17.87 518.61 r a $1000 2.176 1.143 0.28 5.87 y $1000 49.43 8.036 31.38 92.86 f 1 Percentage 1.83 1.54 0.35 7.61 f 2 $1000 1.275 0.203 0.9909 1.747 t 1 Cent 15.755 10.406 1.65 75.05 t 2 Cent 3.449 1.006 1.73 6.64 1 Cent 112.6 80.69 17.26 430.6 2 Cent 0.4004 0.3327 0.0071 1.7215 G $1000 20379.5 58908.34 0.9572 500000 Percentage 1.642 0.6516 0.43 3.28 4.4 Model Specification and Estimation 4.4.1 Functional Form We estimate the following equation: 20123451617282921021121213lnAAreaLryyftftG 1 (4.3) We adopt this particular form for the following reasons. Following the method presented in Wooldridge (2003, pp. 208) in order to obtain a goodness-of-fit measure for the log model that can be compared with the R-squared from the level model, we find that the semi-log model explains more of the variation in the spatial size of urbanized areas than does the linear model. We therefore prefer the semi-log model on goodness-of-fit 126
127 grounds. We also choose the semi-log model b ecause, except for the intercept, each estimated coefficient is the proportional change in the dependent variable per unit change in the independent variable and is, therefore, i ndependent of the units in which the variables are measured. We then conduct specificatio n tests that are embedded in STATA, the software we use to run the regression. An added-value plot reveals that there are significant nonlinearities in the data for income and the auto subsidy. We therefore, add two quadratic terms for these variables. This is further justified by the Lagrange multiplier test for adding variables. The chi-square value obtained by running regression of the residuals of the restricted model on all explanatory variables is 27.44, exceeding the 1 percent critical value of 9.21 for 2 degrees of fr eedom, which indicates that we should reject the restricted model. 4.4.2 Potential Endogeneity Issue In our two-mode model, al l the independent variables in (4.3) are theoretically exogenous because the equation is a reduced form. Econometrically, however, a variable on the right-hand side of (4.3) may be endogenous if it is correlated w ith the error term. If such a case or cases exist, OLS will pr oduce inconsistent estimates. Our empirical model, following from the theoretical model, fo cuses on what we regard as the most important forces determining the spatial size of an urban area. We have not included all possible factors that may affect urban sprawl, and one of these omitted variables may give rise to endogeneity in one or more of the included variables. A pot ential candidate for endogeneity is variable highway cost, which m easures the per-mile cost of driving to the CBD. Endogeneity could result from unobserve d or unmeasured heterogeneity, which would occur if highway variable cost were correlated w ith any of the unmeasured spatial
128 size determinants that are buried in the error term. One of the unmeasured spatial size determinants could be congestion. Although we do not believe congestion plays an important role in determining the spatial size of an urban area (Altmann and De Salvo, 1981, find that it does not in their urban simulation model), it may be related to variable highway cost. For example, when congestion is severe, stop-and-go driving results in inefficient fuel usage, which causes higher variable highway cost. If this were th e case, the resulting coefficient estimates in (4.3) would suffer from omitted-variable bias. This bias cannot be eliminated because the omitted variable or variables representing congestion are, by definition, not in our data set. If congestion is important and is not held constant in the regression, the coefficient on highway variable cost would be biased down, and we would therefore underestimate its effect on urban sprawl. When potential endogeneity is involved, researchers generally use two-stage least squares (2SLS) to correct the problem and obtain consistent and unbiased estimates. To serve this purpose, researchers introduce instru mental variables (IVs) into the regression. The IV for an endogenous explanatory variable must satisfy two propert ies: (1) it must be uncorrelated with the error term of the st ructural equation (although in theory our estimated equation is a reduced form equation, here, because of the potential endogeneity, we refer to it as a structural equation); and (2) it must be correlated with the endogenous variable. When used properly, 2SLS allows us to estimate ceteris paribus effects in the presence of endogenous explanatory variables. When the instrumental variable and the endogenous explanatory variable are only weakly or even moderately correlated, how-
129 ever, IV estimates can have large standard e rrors or even a large asymptotic bias, which means 2SLS can be worse than OLS. In our empirical analysis, we use two in strumental variables for the potential endogenous highway variable cost, state gas tax per gallon and ur banized-area freeway lane-miles. To satisfy the condi tions for suitable instruments, state gas tax per gallon and freeway lane-miles must not affect the spatia l size of the urbanized area in any way other than through the impact of these variables on highway variable cost per mile. In other words, these variables must be correlated w ith highway variable cost but not with the second-stage residuals. To determine if thes e conditions hold, we must investigate the processes that determine interstate differences in gas taxes and inter-urbanized-area differences in freeway lane-miles and evaluate whether the determinants of these variable have direct effects on the spat ial size of an urbanized area. State gas taxes are set within a complex system of institutional relationships designed to plan, finance, and operate the nations highways. According to Dunn (1998), the system has two important components, in tergovernmental transf ers and the highway trust fund. Intergovernmental transfers ensure federal funds are funneled to the states that are responsible for construction and maintena nce of highways. The highway trust fund ensures that the majority of revenues from federal and state gas taxes are dedicated to highway infrastructure needs. As discussed in 4.3.7, the federal government pays for 90 percent of interstate highway construction costs and over 50 percent of capital outlays for non-interstate highway construc tion. States are responsible for the construction and maintenance of highways owned by the federal gove rnment. The states ra ise funds for their share of highway costs through state gas taxes an d other fuel taxes. The majority of states
130 have exclusive dedication provisions that prot ect gas tax revenues from general use other than highway financing, as shown by the fact that twenty-eight st ates have gas-tax earmarking provisions in either th e state constitution or state la w, while eleven states commit receipts to special-purpose f unds to finance highway projects. A states gas tax is determined by the states matching obligation for the construction of federally funded highway projects and the financing of state-initiated projects. In most states, revenues collected from state gas taxes are separated from state general funds. Combining that fact with the exclusive dedication provisions existing in most states, it seems reasonable to argue that the principal determinants of state taxes are highway project financing and the availability of federal funds. It is therefore reasonable to believe that the process determining the stat e gas tax is independent of the spatial size of the urbanized area. Thus, we may argue that state gas tax per gallon is a suitable instrument for highway variable cost. To the best of our knowledge, there is no literature directly analyzing interurbanized-area differences in freeway lane -miles. Among roadway types, freeways provide the highest level of mobility at the hi ghest speed for long, uninterrupted travel. They generally have higher design standards than other highways, often with multiple lanes and a high degree of access control. Freeways comprise interstate highways and other expressways with fully controlled access. Highway functional classification is the grouping of roads, streets, and highways into systems of similar characteristics based primarily on the length of trips served. Additionally, functional classification defines the role that a particular road or street plays in serving the flow of trips through a highway ne twork and analyzes the services provided
131 or that should be provided by each highway facility in servin g the two principal functions of a highway, mobility and access. Two nationwide studies of highway functional classification were conducted duri ng the period 1969. The firs t study required the functional classification of existi ng (1968) highways, while the second study used the same functional classes and basic functional criteria as the first study, but provided for the classification to be based on projected 1990 fac ilities and usage. The Federal-Aid Highway Act of 1973 required th e use of functional highway classi fication to update and modify the federal-aid highway systems by July 1, 1976 (Federal Highway Administration 2003). This legislative requirement is still in eff ect and the National Highw ay Functional Classification System has been in use since that time. From the perspective of history, the most important use of functional classification has been to identify those streets and ro ads that are eligible for federal funds. The Federal-Aid Highway Acts specifically manda ted nationwide studies in cooperation with states and local governments to bring greater consistency to highway classifications and to classify all public streets and highways according to their function. According to the Federal-Aid Highway Act of 1973, interstate highways and other expressways with access fully controlled are in hi gh priority to receive federal funds. Therefore it is reasonable to believe that the location and lane -miles of freeways are determined by federal fund availability. From an engineering perspe ctive, freeway capac ity is generally set equivalent to engineering capacity because the through movement is the only concern for those highways while other road s capacity is not only base d on engineering capacity but also other factors such as population density and accessibility to major activities centers. Combined with the fact that the highway func tional identification was done in 1970s, it is
132 reasonable to believe that freeway lane-miles are unlikely to be corr elated with the unobservable determinants of the spatial size of urbanized area. This is confirmed by the change in freeway lane-miles over time. We an alyze data from Highway Statistics on the freeway lane-miles for urbanized area. We find that from 1992, at the national level, 33.7 percent of urbanized areas freew ay lane-miles decreased, 41.3 percent increased at an annual rate of less than 1 percen t, and 26 percent increased at an annual rate of 2 percent or higher. For the urbanized areas in our data set, 73 percent of freeway lane-miles either decreased or remained unchanged, while 37 percent increased, most at an annual rate of less than 3 percent. These facts support the argument that freeway lane-miles within an urbanized area are pure functi onal classification and unlikel y to correlate with the unobserved determinants of the ur banized areas spatial size. The data for these instrumental variable s are from Highway Statistics 2000. Data on total freeway miles and state gas taxes ar e from also from Highway Statistics 2000, Table HW71 and Table MF-121T, respectively. 4.4.3 Empirical Results In this subsection, we discuss the results of our estimation of (4.3). We confine our discussion to the statistical significance of the estimated coefficients and to the conformity of our estimated results with our theore tical expectations. In Ch. 5, we discuss the economic significance of our findings for urban sprawl. Table 4.4 presents the OLS results a nd the second-stage pa rameters from the 2SLS estimation. The first-stage variable highw ay cost regression results are presented in Table 4.5. In the 2SLS estimation, we incor porate the explanatory variables discussed in
Section 4.2 and use the interstate variation in gas taxes and inter-urbanized-area freeway lane-miles as instruments. Comparing the estimated coefficients from the OLS and 2SLS regressions, we find the magnitudes of most are similar and the signs are the same. We conduct a Hausman specification test and find that we cannot reject the hypothesis that the differences in coefficients are not systematic. The results are therefore materially un-affected by the endogeneity correction, so we discuss only the OLS regression. Table 4.4 OLS and 2SLS Regression Results OLS 2SLS Variable Coefficient Std. Error p Coefficient Std. Error p L 0.0077 0.0006 0.000 0.0075 0.0006 0.000 r a -0.0041 0.0338 0.905 -0.0115 0.0351 0.743 y 0.1187 0.0271 0.000 0.1153 0.0277 0.000 y 2 -0.0011 0.0002 0.000 -0.0011 0.0002 0.000 f 1 -0.0956 0.0266 0.001 -0.0927 0.0271 0.002 t 1 0.0068 0.0039 0.084 0.0058 0.0041 0.162 1 -0.0010 0.0005 0.052 -0.0009 0.0005 0.067 f 2 -0.2936 0.1953 0.137 -0.3375 0.2027 0.100 t 2 -0.0690 0.0381 0.074 -0.1278 0.0687 0.067 2 0.6431 0.3575 0.076 0.6968 0.3666 0.061 2 2 -0.6589 0.2893 0.025 -0.6783 0.2943 0.024 -0.1224 0.0588 0.041 -0.1156 0.0601 0.058 G -4.18e-7 1.00e-06 0.677 -1.88e-7 1.04e-6 0.857 Constant 1.3520 1.7300 R 2 0.8161 0.8105 133
The coefficient on L, the number of households, is positive, as predicted, and sta-tistically significant at the 1-percent level. The coefficient on agricultural land rent, r a is negative, as predicted by our model, and is statistically insignificant. The statistical insignificance may be due to two factors. The mean estimated market value of rural land in a county may in fact do a poor job of capturing actual rural land value at the urban fringe. Brueckner and Fansler (1983) allow for this possibility by restricting their sample to urbanized areas in small coun-ties, and their estimated coefficient is statistically significant. The statistical insignifi-cance of our variable may also result from its small range of variation in our sample, which prevents the emergence of a precise estimate. The estimated coefficients on y and y 2 income and income squared, are statisti-cally significant at the 1-percent level, and their signs indicate that the spatial size of the urbanized area increases at a decreasing rate with income. In the estimated equation, the marginal effect of income on urban size is given by 34ln2Areayy where 3 and 4 and are the estimated coefficients on y and y 2 respectively. For the es-timated coefficients and the mean urbanized-area income of $49,430, the result is posi-tive, which is in line with the prediction of the standard single-mode model, as well as ours. When income is $54,000 or higher, however, our result is negative. Since urban-area size can increase, remain unchanged, or decrease with urban-area mean income, the effect of income on urban spatial size is, in general, ambiguous, which is what is pre-dicted by our two-mode model. 134
135 The effect of income on urban size has generated considerable discussion in the urban economics literature, so we devote some time to how our findings comport with the findings in the literature. In partial equilibrium urban models that include the value of time spent commuting, the ambiguity of inco mes effect on location has been known since Muth (1969), and the empirical results fi nd the effect variously positive or negative (e.g., Muth, 1969, Wheaton, 1977, Hekman, 1980, DeSalvo, 1985). The result depends on the relative magnitude of the income (o r wage-rate) elasticity of housing demand and the income elasticity of marg inal (distance) commuting cost (first noted by Muth (1969)). If the former is greater than, equal to, or less than the latter then, as income increases, the household locates farther from, at the same distance from, or closer to the CBD, respectively. Since the value of time spent commuti ng rises with income (Hensher, 1976) it is possible that at incomes above $54,000 households choose to live closer to the CBD than they would at lower incomes. Thus, urba nized areas with mean household incomes greater than $54,000 might well be smaller than urban areas with smaller mean incomes. Partial equilibrium urban models and empi rical tests of them generally not do include mode choice. DeSalvo and Huq (2005) in troduce mode choice in to the partial equilibrium urban model and find that the wage-rate elasticity of marginal commuting cost varies with the wage rate in a complicated way. The effect of th e wage rate on location can generate a wide variety of possibilities, including our empirical result. Although a few general equilibrium models include m ode choice (e.g., Anas and Moses, 1979), we are unaware of general equili brium urban models that include the value of time spent commuting.
136 In our two-mode model, if public transits fixed cost, variable cost or private cost share increases, the urban area expands, whereas the opposite is the case for auto commuting. In our empirical work, the percenta ge of people using transit is the proxy for fixed transit cost, f 1 The estimated coefficient on f 1 is statistically significant at the 1percent level, and its negative sign is consistent with our theoretical expectation. The coefficient on transit variable cost, t 1 is positive, as expected, and statistically significant at the 8-percent level. The coefficien t on the transit subsidy variable, 1 is negative, as predicted, and statistically signi ficant at the 5-percent level, suggesting that the higher the transit subsidy, the smaller is the urbanized area. Another potential proxy variable for the fixed transit cost is transit density, which is derived as the land area of the urbanized areas divided by the total bus serv ices miles. When we use this proxy in our regression, we find it to be insignificant. The coefficient of fixed auto cost, f 2 is negative, as predicted, but not statistically significant. Our measure of fixed auto cost, wh ich is composed only of the auto insurance premium and motor vehicle taxes, may fail to capture the behavior of actual auto fixed cost because we do not have data on auto maintenance and depreciation. The coefficient on variable auto cost, t 2 is negative, as predicted, and statistically significant at better than the 1-percent level. This means that if auto commuting cost increases, people will move closer to the CBD, thereby redu cing the size of the urbanized area. We use a quadratic polynomial to captu re the nonlinearity of the highway subsidy, 2 The coefficient on highway subsidy and highway subsidy squared are statistically significant at the 8-percen t level and 3-percent level, respectively. The coefficient on highway subsidy is positive, while that on highway subsidy squared is negative, in-
dicting that the spatial size of the urbanized area is increasing at a decreasing rate with the highway subsidy. The marginal effect of the highway subsidy is: 101122ln2Area where 10 and 11 are the estimated coefficients on 2 and 22 respectively. If we insert the estimated coefficients and the mean value of the highway subsidy of 0.4 cents per ve-hicle-mile traveled, the result is positive, which is consistent with our prediction that the higher the highway subsidy, the larger is the urbanized area. When the highway subsidies exceed 0.4880 cent per VMT, however, our result is negative. The overall effect of high-way subsidies on the urban size is not consistent with our theoretical prediction. This may be due to the fact that our highway subsidy variable only includes subsidies from the county where the urbanized area is located, which fails to capture the subsidies from the cities. It may also be because our highway subsidy variable is not an accurate measure-ment because, given current data availability, there is no way to figure out exact highway subsidies by urbanized area. The coefficient on the property tax rate, is negative, as predicted, and statisti-cally significant at the 6-percent level. Song and Zenou (2006) also obtain this result in a paper dedicated to determining the effect of the property tax on urban sprawl. The coefficient on the intergovernmental grant for highway purposes, G, is nega-tive but not statistically significant. Our model predicts that the relationship between ur-ban size and the intergovernmental grant is ambiguous. Before turning to the 2SLS estimates, a brief discussion of the results from the first-stage regression reported in Table 4.4 is in order. It is clear that state gas taxes exert 137
138 a positive effect on variable auto cost, while freeway lane-miles exert a negative effect. Both instruments are independent ly significant at the 0.001-l evel. The F-statistic for the test of the joint significance of the two inst ruments is 18.56. Hence, the instruments are strongly correlated with highway variable cost. Returning to the results in Table 4.3, the 2SLS coefficient estimates on all variables except auto variable costs are quite cl ose to the estimates from OLS. OLS underestimates the effect of auto vari able costs: the coefficient on auto variable costs is .069, compared to .128 using 2SLS. Nevertheless both sets of results largely support our theoretical predictions. Since we use two instru mental variables, we are able to perform a test of the over-identifying rest riction. The test fails to reject the over-identification restriction, suggesting that the dependent variables effects on urban size are not sensitive to the choice of instruments. 4.4.4 Summary This chapter presents the data and econo metric issues in estimating our two-mode model. The discussion of our empirical findings focuses on th e statistical significance of estimated coefficients and on the conformity to our of our empirical results with the twomode theoretical model. Based on our theory, the variables explaining the spatial size of an urban area are the number of households, rural land va lue, household income, fixed and variable costs of public transit and auto travel, subsidies to public transit and auto travel, property taxes, and intergovernmental grants. We find that the spatial size of the urbanized area shrinks with an increase in th e transit subsidy. The effect of highway subsidies, however, is ambiguous. Both OLS and 2S LS are applied, and the results are qualitatively the same.
139 Table 4.5 First-Stage Results from Regression of Auto Variable Costs on IVs Variable Coefficient Std. Error p L 0.0086 0.003 0.010 r a -0.2002 0.0840 0.020 y -0.0359 0.0617 0.562 y 2 0.0003 0.0005 0.489 f 1 -0.0091 0.0665 0.892 f 2 -0.0006 0.0005 0.226 t 1 -0.0169 0.0095 0.079 1 0.0005 0.001 0.694 2 0.9379 0.8864 0.293 2 2 -0.3611 0.7177 0.616 Freeway Lane Miles -0.0067 0.0018 0.000 State Gas Tax Rate 0.1083 0.0246 0.000 G -2.37e-06 2.76e-06 0.393 -0.0175 0.1477 0.906 R 2 0.43 N 93 F ( Joint Significance of IV ) 18.56
140 CHAPTER 5 POLICIES TO CONTROL URBAN SPRAWL In Ch. 4, we presented the econometric results of estimating (4.3) and discussed how these results conform to our two-mode theoretical models predications. In this chapter, we asses the economic significance of our findings a nd relate them to the issue of urban sprawl. 5.1 The Sensitivity of Urban Spatial Size to Our Explanatory Variables To get a sense of the sensitivity of sprawl to our explanatory variables, we calculate elasticities of the urban areas spatial size w ith respect to the i ndividual explanatory variables, and we calculate their individual eff ects on the spatial size of the urban area due to a 1-percent change in the variable (all evaluated at the means of the data). These calculations are shown in Table 5.1. We show results only for those explanatory variables whose coefficients are statistically significant at least the 10-percent level. On this criterion, we omit rural land value, r a auto fixed cost, f 2 and intergovernmental grants, G Perhaps the first thing to notice in Table 5.1 is that the spatial size of the urban area is not very sensitive to any single explan atory variable: all elasticities are numerically less than one. The largest in magnitude is the elasticity of area with respect to number of households, L so that a 1-percent increase in the number of households, which is about 800 households, produces an approximately 0.6-percent increase in the spatial size of an urban area. This increases the size of the urban area by about 0.5 square miles, or about 0.4 acres per household, which is slightly larger than the averag e residential lot size of one-third acre. The next la rgest is the income elasticity of about 0.5. A 1-percent in-
crease in income, y, which is about $500, produces an expansion of the mean urban area of 0.4 square miles. Table 5.1 Elasticities and Area Change (Due to 1-Percent Change in Explanatory Variables) Variable Elasticity Area Change (sq. mi.) L 0.615 0.53 y 0.492 0.42 f 1 .175 .15 t 1 0.107 0.09 1 .113 .10 t 2 .238 .20 2 0.046 0.04 0.201 .17 Variables affecting public transit use have relatively small effects on the spatial size of urban areas. A 1-percent increase in the percentage of commuters using public transit, which is an increase of about 0.02 percent, reduces the urban area by about 0.2 percent, or by about 0.2 square miles. The percentage of commuters using public transit is a proxy for our theoretical variable, and we expect it to be inversely related to the the-orys transit fixed cost variable. Therefore, an increase in transit fixed cost increases ur-ban-area size, which is consistent with our theory. A 1-percent increase in public transit 141
cost per passenger-mile, which is about 0.01 cent and is our proxy for transit variable cost, increases urban-area size by about 0.1 percent, or by slightly less than 0.1 square miles. Finally, a 1-percent increase in the transit subsidy per passenger-mile, 1 which is about $0.01, reduces urban-area size by about 0.1 percent, or about 0.1 square mile. Variables affecting auto usage also have small effects on urban-area size. A 1-percent increase, an increase of about 0.03 cents, in highway fuel taxes per VMT, our proxy for variable auto cost, produces a reduction in urban-area size of about 0.2 percent, or about 0.2 square miles. A 1-percent increase in county highway subsidies per VMT, an increase of about 0.004 cents, increases urban-area size by about 0.05 percent, or about 0.04 square miles. Finally, a 1-percent increase in household property tax paid as a percentage of household income, our proxy for the urban-area income tax, which is about 0.02 per-cent, produces 0.2 percent decrease in urban-area size, or about 0.2 square miles. One conclusion from the above is that the most powerful policy variable is the number of urban households. Policy makers can control this variable directly, through urban population constraints, or indirectly through land-use controls. The other policy variables are those affecting the cost of transportation. As we have seen, their effects are relatively small. Although the individual effects of changes in policy variables are small, joint changes in those variables would produce a larger effect. To see how large an effect, we change all of the policy variables by 1 percent in the appropriate direction. Before pro-ceeding to this exercise, we note that the proxy for fixed transit cost, namely, the percent-age of working-age population using transit for commuting, cannot be changed directly 142
143 by policy action. Rather, it will change due to changes in other variables, such as transit variable cost and transit subs idy, so we omit that variable from our list. The remaining variables we treat as policy variables are tran sit variable cost and s ubsidy, auto fixed and variable costs and subsidy, and the propert y tax rate. Of these, policy should decrease transit variable cost and the auto subsidy while increasing transit subsidy, auto fixed cost, auto variable cost, and the property tax rate to produce a reduction in urban-area spatial size. If we change these variables by 1 percen t in the appropriate direction, the urban area decreases in size by about 0.8 percent, or 0.7 square mile. This is still a modest change, slightly larger than a 1 percent decrease in th e number of households in the urban area. 5.2 Conclusion on Effect of Policy-Relevant Variables on Urban Sprawl Although our intent was to determine the effect of transport subsidies on urban sprawl, our empirical an alysis suggests that the spatial si ze of the urbani zed area is not very responsive to any single policy-relevant variable in our model, which we have defined as variable transit cost, transit subsidy, fixed and variab le auto cost, auto subsidy, and property taxes. Even used together and for small changes, namely, 1-percernt increases or decreases, their effect on urban sprawl is small. Since this thesis focuses on the effect of transportation subs ides on urban sprawl, we emphasize the transportation policies or practices that may be effective in curbing sprawl. Our empirical analysis provides evidence that the spatial size of an urbanized area is negatively related to the transit subsidy and the percentage of the working-age popula-
144 tion using transit and positively related to tr ansit cost. This suggests that improving transit use may help curb urban sprawl but that the effect may be small. Transit ridership is influenced by a variety factors, both internal and external to the transit system. Internal factors refer to those under the control of transit agencies, such as the level of service provided, fare structures and levels, service frequency and schedules, route design, and size of service area. External fa ctors, on the other hand, are those outside the transit agencys control, such as population and employment growth, residential and workplace locati on, and factors that affect the relative attractiveness of transit to other modes, such as gas price and parking cost s (Mineta Transportation Institute 1991). One disadvantage of public transit use is its high time cost, including both collection and distribution cost and line-haul cost. Tr ansit agencies or transportation authorities may reduce transit collection and distribution costs by adding additional routes, expanding coverage, increasing service frequency, and operating longer hours. They may reduce line-haul costs by busways, queue-jumping lanes, and intersection signal controls that give priority to public transit. Our empirical analysis finds that transit cost and urban spatial size are positively related. This suggests that reducing transit cost may be another way to improve transit ridership, which may help curb urban sprawl, but again, the effect is likely to be small. A deep discount fare program is a good way to reduce transit cost. Such policies generally offer a per-ride discount for purchase of a multi-ride pass or transit card, aiming to attract potential riders of low usage or high price-sensitivity. Park ing price can be an effective means of increasing transit use for work trip s. Since increasing parking costs increases
145 the relative attractiveness of tr ansit as compared to auto, its effect on transit modal choice is significant. A higher gas price is expected to have the same impact on transit use. Our empirical analysis suggests that auto costs are negatively re lated to the spatial size of urban area. Increasing auto costs also helps reduce auto subsidies, which are expected to contribute to urban expansions although our em pirical analysis indicates that the spatial size of an urbanized areas increases at a decreasing rate with increase in auto subsidies. The federal government can indirectly infl uence the cost of auto use by removing subsidies and tax benefits to oil and gasolin e production. State governments may use road pricing to increase auto cost. Road pricing means that highway users pay directly for driving on a particular road or in a particul ar area. State governments may also change motor carrier regulations and vehicle insurance regulations to increase the cost of auto use. Local governments can offer local tr ansportation demand management programs and allow more flexible zoning requirements to reduce auto use (Victoria Transport Policy Institute 2005). 5.3 Further Research There are three extensions of the preceding models that we would like to pursue. Specifically, these are (1) including value of travel time, (2) endogenizing congestion, and (3) generalizing mode choice. Only very brief discussions of these extensions are provided. 5.3.1 Value of Time To deal satisfactorily with the value of transportation time, it is necessary to introduce time elements explicitly into the utility function, the budget constraint, and a time
146 constraint. A partial equilibrium model of this type has been worked out by DeSalvo (1985). We intend to develop a general equilibrium mode l of this sort. 5.3.2 Endogenizing Congestion Congestion is absent from the models discussed above, but congestion affects commuting cost. The first formulations of a general equilibrium urban spatial model incorporating endogenous congestion were those of Mills (1972) and Solow (1972, 1973). Muth (1975) also included congestion in his simulation model. In addition, simulation models (e.g., Muth (1975)) use a congestion functi on that expresses time cost of travel as a function of traffic density. Henderson (1985) discusses three ways to model congested systems: current demand dependent on past and future travel conditions, current travel speeds dependent on past and future conditions, and the traditional mode l (a static version of the first and second ways). Since most urban models with congestion use the Mills/Solow framework (Ross and Yinger, 2000 ), we intend to use the same framework for expositional simplicity. 5.3.3 Generalizing Mode Choice Mode choice in our approach is determin ed by where a traveler lives and is dependent on only two parameters, fixed and vari able transportation co st. We plan to use the more general approach proposed by DeSal vo and Huq (2005). This approach is similar to the "abstract mode" approach proposed many years ago by Quandt and Baumol (1966), in which a mode is defined by its char acteristics. This approach has the advantage of being able to deal with past, present, a nd future modes, as well as mode combinations, rather than being restricted to modes currently in use.
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About the Author Qing Su received her Bachelors degree in Economics with a major in International Trade from Dongbei University of Finance and Economics in 1993 and M.S. in Economics with a major field in Internationa l Trade Rules and Regulation from University of International Busine ss and Economics in 1996. She then worked as an Assistant Professor of Economics in Beijing University of Technology teaching international trade, international trade rules and re gulation, and foreign trade poli cy analysis. She entered the Ph.D. program in Business Administration with a major in Economics at the University of South Florida in 2000.
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The effect of transportation subsidies on urban sprawl
h [electronic resource] /
by Qing Su.
[Tampa, Fla] :
b University of South Florida,
ABSTRACT: This dissertation investigates transportation subsidies as sources of urban sprawl. Apart from tolls, motorists do not pay highway user-fees, but they do pay gasoline taxes. Gasoline tax revenues are insufficient to cover the U.S. highway costs. Government, therefore, uses general tax revenues to cover highway expenditures. Since users do not pay the full cost of their travel, they have an incentive to travel longer commuting distances. Highway subsidies are, therefore, a potential contributor to urban sprawl. A similar argument applies to public transit. To capture the effects of subsidized automobile and public transit travel, we ex-tend the standard urban spatial single-mode model (Brueckner, 1987) to incorporate public subsidies for both one and two modes. Comparative static analysis of both models produces empirically testable hypotheses. Our most important theoretical result is that transit subsidies are inversely related to urban sprawl while auto subsidies are directly related to urban sprawl. The empirical analysis focuses on tests of the two-mode model. For consistency with the monocentric assumption of our models, our sample consists of urbanized areas located within a single county and having only one central city. Spatial size of the urbanized area is the dependent variable. Following our theory, explanatory variables comprise the transit subsidy, the highway subsidy, number of households, agricultural land rent, mean household income, and fixed and variable costs for transit and auto. We find that the spatial size of the urbanized area shrinks with an increase in the transit subsidy. The effect of highway subsidies, however, is ambiguous. We apply both ordinary least squares and two-stage least square regression analyses, and the results are qualitatively the same for both methods of estimation.
Dissertation (Ph.D.)--University of South Florida, 2006.
Includes bibliographical references.
Text (Electronic dissertation) in PDF format.
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
Title from PDF of title page.
Document formatted into pages; contains 152 pages.
Adviser: Joseph S. DeSalvo, Ph.D.
x Business Administration
t USF Electronic Theses and Dissertations.