The Effect of Transportation Subsidies on Urban Sprawl
by
Qing Su
March 30, 2005
Department of Economics
College of Business Administration
University of South Florida
4202 E. Fowler Ave. BSN3403
Tampa, FL 33620-5500
Telephone: 813-974-6526
Fax: 813-974-6510
E-Mail:
Key Words: Transportation Subsidies, Urban Sprawl
Introduction
Popular concern with urban sprawl has continued unabated since the term was coined in 1937. Professional concern has followed popular concern, as economists have documented decentralization of urban population and employment and provided explanations for it. The literature has grown to such an extent that the Journal of Economic Perspectives deems it worthy of a survey article (Nechyba and Walsh, 2004).
Although urban sprawl has several definitions, for economists it means excessive decentralization of urban population and employment (Brueckner, 2001; Mills, 1999). The emphasis on "excessive" means that mere decentralization of urban population and employment is not sprawl. It may simply be the efficient allocation of resources in urban areas.
Among urban economists, the fundamental factors causing decentralization of urban population and employment are real-income growth, real-transport cost reduction, and population growth. This view has both theoretical and empirical support. The theoretical support is found in the monocentric urban model, which grew out of the work of Alonso (1964), Muth (1969), and Mills (1972), and reached its culmination in Brueckner's (1986) synthesis. The empirical support is found in Muth's (1969) and Mills' (1972) studies of the determinants of population and employment density gradients as well as the more direct estimation by Brueckner and Fansler (1983).
That a perfectly competitive urban economy is efficient has been proved (Fujita, 1989), so sprawl, as seen by economists, must be due to deviations from the competitive assumptions. Brueckner has done more work in this area than any other economist and has identified several potential sources of excessive decentralization. These are (1) failure to account for the amenity value of open space (Brueckner, 2001), (2) failure to charge developers the full cost of urban infrastructure (Brueckner, 2001; Brueckner, 1997), (3) unpriced road congestion (Brueckner, 2001), (4) the property tax (Brueckner and Kim, 2003), and (5) transportation subsidies (Brueckner, 2003). For each of these, Brueckner has made the case that, in the presence of these factors, the urban area is larger than it would be under perfect competition; hence, these factors are potential sources of urban sprawl.
The word potential is used advisedly because there is no empirical evidence that these five factors are actual sources of urban sprawl. Brueckner has provided the theoretical rationale for these sources of sprawl and has in some cases also provided simulations that buttress the purely theoretical analysis. Also lacking is comparative static analyses of models containing these potential sources of sprawl. The purpose of the research reported here is to fill this gap, both theoretically and empirically.
I investigate transportation subsidies as potential sources of urban sprawl. I develop modifications of the monocentric model to include transportation subsidies and report results of comparative static analysis for both a single-mode and two-mode model. I test empirically the hypotheses generated by the comparative statics, so I proceed by discussing data and estimation issues. Finally, I briefly discuss proposed extensions of the models to include value of time, endogenous congestion, and a generalization of mode choice.
Monocentric Urban Model
Partial equilibrium models of urban areas grew out of Alonso's (1964) adaptation of von Thünen's model of agricultural land use to an urban area. Further work, both theoretical and empirical was provided by Muth (1969). General equilibrium analysis began with Muth's (1975) and Mills' (1972) urban simulation models. The first comparative static analysis of an analytical, as opposed to simulation, model was provided by Wheaton (1974). As noted earlier, Brueckner (1986) synthesized the work of Muth and Mills in an analytical general equilibrium model. I confine my discussion to the latter model.
The monocentric model has a predetermined center, the central business district (CBD), to which all travel is made for work and other activities. Travel is along radial and dense transportation routes between the household's residential location and the CBD. A household's quasi-concave utility function is defined over housing consumption, q, which is a normal good, and non-housing, non-transportation expenditures, c. The household spends its exogenous income, y, on housing; non-housing, non-transportation goods; and transportation. Round-trip transportation cost is determined by distance between home and CBD, x, and the round-trip cost per mile of travel, t. Thus, the problem of the household is to maximize v(c,q) subject to y = c + pq + tx, where p is the price per unit of housing. Upon eliminating c, this problem gives rise to the familiar first-order condition,
.
All urban households are assumed to be identical with respect to utility function and income. Consequently, for them to be in spatial equilibrium, in which no one wants to move, it is necessary that the following condition hold:
where u is the urban-area-wide spatial equilibrium utility level. The catchall variable, c, plays no role in the analysis and is therefore ignored.
Housing is produced via a constant-returns-to-scale concave production function defined over land, l, and non-land inputs, N, as follows:
,
but, because of constant returns to scale, this may be rewritten as
where S is the nonland-to-land ratio, called structural density.
Profit per unit of land is given by
where p is housing price, as before, i is the rental rate of the nonland input, and r is the rental rate of the land input. Maximizing rent per unit of land produces the following first-order condition:
which is the familiar result that marginal revenue product equals factor price at the profit-maximizing S. Finally, the spatial equilibrium condition for housing producers is that land rent absorb profit, so all housing producers are equally well off at any location:
.
To complete the model requires an urban-area boundary condition and an urban population condition. The urban boundary condition is:
,
whereis the distance from the CBD at which the urban area ends and the rural area begins and is the rural land rent (or opportunity cost of land). Urban households outbid rural land users between the CBD and , while rural land users outbid urban land users beyond .
The urban population condition is:
where is the number of radians in a circle available for urban residential use and L is the urban population, which is assumed to be the same as the number of urban households. The quotient is population density since it is the total quantity of housing per unit of land at any given x divided by the per-household consumption of housing at that x. This condition ensures that the population of the urban area exactly fits inside the boundary of the urban area. To simplify notation, I will sometimes use the relation D = h/q, where D is population density.
At this point in the development of the model, it is necessary to distinguish between the "open-city" and "closed-city" versions of the model. In an open city, utility is exogenous while population is endogenous. The idea behind this is that if utility is higher in one urban area than another, people will migrate to the first urban area from the second. This raises population and lowers utility in the first urban area while lowering population and raising utility in the second. Eventually, the utility level is the same in both urban areas and migration stops. In the closed-city model, the reverse is the case. An exogenous increase in population lowers utility in the urban area but no out-migration occurs. It is sometimes said that the closed-city model is a "short-run" model while the open city model is a "long-run" model. Most applications of the monocentric model have used the closed-city version. I use the closed-city version in my work as well.
Table 1 summarizes the comparative statics of the model (I have suppressed the variables for the price of the nonland input and the radians of urban land available for residential use). An increase in urban-area population increases the demand for housing, thereby driving up the price of housing, causing households to consume less housing and become worse off. The increased demand for housing encourages housing producers to produce more housing per unit of land in taller buildings, driving up land rent which, in turn, allows the urban area to expand outward.
Table 1
ExogenousVariable / Endogenous Variable
u / / S / r / p / q
L / - / + / + / + / + / -
t / / - / - / + / + / + / -
/ - / - / - / - / - / ?
y / / + / + / - / - / - / +
/ + / + / + / + / + / ?
/ - / - / + / + / + / -
x / NC / NC / - / - / - / +
NC = no change
An urban-area-wide increase in transport cost will induce households to decrease housing consumption and move closer to the CBD. The attempt to do so will drive up housing price and land rent between the CBD andand lower them farther out, i.e., the housing price and land rent functions pivot clockwise about . Thus, households located closer in will consume less housing, pay higher housing prices, and have lower utility. Although housing price falls farther out, utility also falls (or there would not be a spatial equilibrium), and the resulting effect on housing consumption is ambiguous. Because of the fall in urban land rent farther out, the urban area contracts.
An increase in urban-area-wide income has the opposite effects as that of an increase in transport cost for the same reasons, except for the effect of income on housing consumption farther out. In this case, the housing price function pivots counter-clockwise so housing price rises farther out, but utility also rises, so the effect on housing consumption is ambiguous.
An increase in rural land rent causes the urban area to shrink. Squeezing the given population into a smaller land area causes land rent and housing price to rise, thereby reducing utility, increasing structural density, and reducing housing consumption. Finally, neither utility nor the urban boundary are functions of distance, so there is no change in these variables with x. Housing price, land rent, and structural density fall with distance, while housing consumption rises with distance, for reasons stated earlier.
Transportation Subsidies in a Monocentric Urban Model with a Single Mode
The Magnitude of Transportation Subsidies in the U.S.
Individuals' transportation and location decisions would be socially efficient if the 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 form of gasoline taxes. Revenues from gasoline taxes have, however, been insufficient to cover the construction, maintenance, and administration of highways in the U.S. Various levels of government must use general tax revenues to augment highway expenditures, a fact that indicates that governments provide highway subsidies to users. Given that users do not pay the full cost of their travel, they have an incentive to travel longer and more often. Highway subsidies are, therefore, a potential contributor to urban sprawl. A similar argument applies to public transit.
In the U.S. user fees covered only 68 percent of the highway system's capital and maintenance costs during the period 1956 to 1986 (Voith, 1989). The share of user fees has fluctuated considerably over time and across regions. Voith (1989) examines thirteen major highway construction projects ranging in cost from $97 million to $581 million. The cost per mile of construction varies widely, from a low of $6.8 million to a high of $133.3 million. None of the projects generates sufficient user fees to cover the infrastructure costs. As a matter of fact, user fees cover 54 percent of the investment at best and 2.5 percent at worst. On a per-car basis, the subsidy ranges from $0.16 to $4.50. On a vehicle-mile basis, the subsidy ranges from less than $0.01 to $0.41.
User fees fail to cover the costs of public transit. Brueckner (2003, pp. 3-4) reports that "data on 600 public-transit agencies compiled by the American Public Transit Association show that total fares collected in 2001, which amounted to $8.89 billion, covered only 38 percent of total system operating costs and 25 percent of combined operating and capital costs."
Brueckner's Work on Transportation Subsidies and Urban Sprawl
In an urban spatial model similar to the one described above, Brueckner (2003) shows that transportation subsidies lead to spatial expansion of urban areas, even though the subsidies are offset by a higher general tax burden on urban residents, which would, by itself, reduce the demand for space. He also argues that if the transportation system exhibits constant returns to scale, then the transportation subsidies are inefficient and, hence, that the spatial expansion of the urban area they engender is excessive. This analysis is done under the assumption of a given transportation system. Brueckner extends the analysis to allow the urban area to select a transportation system from a continuum of transportation systems that vary in terms of their money and time costs.
Brueckner's paper is the only one of which I am aware that deals with transportation subsidies as a potential source of urban sprawl, and it is this paper that motivated my interest in the subject. It may be useful therefore to note the differences between Brueckner's and my approaches. In so doing, I ignore Brueckner's analysis of system choice. (Although I confine my attention to Brueckner because of his work on transportation subsidies, I should note that others have worked with urban spatial models in which transportation figured prominently, e.g., Arnott and MacKinnon (1977) and Arnott, Pines, and Sadka (1983).)
Brueckner assumes that the urban area has only one type of transportation system, such as automobile travel on local streets and highways. I develop both a singe-mode and two-mode model. Brueckner's urban area is linear, whereas mine is circular (or any part of a circle that may be desired). Brueckner's households consume only land and a numéraire good, whereas my households consume housing, which is produced by land and nonland inputs, and a numéraire good. This means that I model a housing production sector as described in the previous section. Brueckner assumes that households pay a head tax to cover the subsidy-induced deficit in the operation of the transport system. I assume households pay an "income" tax, which is meant to capture property and sales taxes paid by urban residents. Brueckner introduces a balanced-budget equation, which ensures that the transportation deficit is covered by the head tax. I also introduce a balanced budget equation, but mine includes the "income tax," as noted above, and intergovernmental grants, which are a common source of funds to urban-area governments.
Brueckner devotes his attention to finding the qualitative effect of a transportation subsidy on the spatial size of the urban area. He is unable to do this for the general case "given the complexity of this equation system (p. 9)." Instead, he provides a "more limited result (p. 9)" that shows that the urban area's size will increase under a transportation subsidy. I derive this result for the general case in my single-mode model. I also derive boundary effects for subsidies to each mode in my two-mode model. Brueckner contends (p. 10) that the effect of transportation subsidies on the utility of urban residents is ambiguous, while in my model the effect is positive; in other words, the greater the subsidy, ceteris paribus, the better off are urban residents. Brueckner provides no other comparative static results, devoting the rest of the section containing this model to a discussion of the constant-returns assumption in transportation production. I provide comparative static results for almost all exogenous variables.
The Model
Consumption Sector. In this subsection, I modify the basic urban spatial model, described above, to incorporate transportation subsidies. The utility function stays the same, but the budget constraint changes as follows:
,
where represents the "income tax rate" and is the subsidized share of transportation cost. Of course, few cities in the U.S. impose the income tax. Rather, represents the combined property and sales tax rate paid by an urban-area resident, andy reflects the fact that an individual's property and sales taxes tend to rise with income. If is the subsidized share of an individual's transport cost, thenis the share paid privately. Both of these new variables are considered exogenous.
For the consumption sector of the model, these modifications generate the following structural equations:
and
.
Production Sector. The production sector of the model changes only because the housing price function changes. The structural equations become:
and
.
Urban Boundary, Population, and Balanced Budget Conditions. The structural equations for the urban boundary and population conditions are modified as follows:
and
Since municipalities and counties are required by state law to balance their budgets, I add a balanced-budget equation, as follows:
,