Property Taxation and Urban Sprawl
by
Jan K. Brueckner
Department of Economics
and
Institute of Government and Public A®airs
University of Illinois at Urbana-Champaign
1206 South Sixth St.
Champaign, IL 61820
e-mail: jbrueckn@uiuc.edu
December 1999
For presentation at the Lincoln Institute Conference on
Property Taxation and Local Government Finance
Scottsdale, Arizona, January 16-18, 2000
Abstract
This chapter explores the connection between property taxation and urban sprawl. It is shown
that the property tax's tendency to reduce the intensity of land development, which follows
from its taxation of improvements, causes a city to expand spatially in order to accommodate
its population. Since the resulting expansion is socially ine±cient, the analysis suggests that
property taxation may contribute to undesirable urban sprawl. Using a di®erent model, the
second part of the paper shows that, by generating tax liabilities that lie well below the
marginal cost of urban infrastructure, the property tax makes urban development appear
arti¯cially cheap. Excessive development then occurs, with city populations and land areas
expanding beyond socially desirable levels.
0
Property Taxation and Urban Sprawl
by
Jan K. Brueckner*
1. Introduction
Urban sprawl has become a important policy issue in the U.S. over the last few years. Long
commutes, tra±c congestion, and the rapid disappearance of agricultural land on the urban
fringe have led many observers to conclude that American cities have expanded excessively.
The issue has even risen to the national agenda, with the Clinton administration proposing to
use federal funds to preserve open space.
Many economists caution that urban spatial growth is a natural result of several fundamental forces. These include expansion of the U.S. population as well as the rise in household
incomes, which spurs the demand for land by encouraging consumption of larger houses. In
addition, urban spatial expansion is a natural consequence of heavy investment in transportation infrastructure (mainly freeways), which eases commuting and thus encourages suburban
living.
Economists argue that the process of urban growth cannot be faulted as ine±cient unless
the operation of the above fundamental forces is distorted by market failure. The literature has,
in fact, identi¯ed a number of market failures that cause such a distortion, potentially leading
to excessive spatial expansion of cities. These include a failure to account for the social value
of open space around cities, which may provide environmental bene¯ts that are not taken
into account in market transactions. In addition, the negative externalities associated with
freeway congestion make the social cost of commuting higher than the private cost. Commuting
thus appears arti¯cially cheap to urban residents, so that commute trips are excessively long
(and cities excessively spread out) relative to the social optimum. Finally, because housing
developers are not required to pay for the full cost of the public infrastructure necessitated
by their projects, the cost of development under current ¯scal arrangements is arti¯cially low.
Therefore, too much development occurs, and cities expand excessively.1
1
The last of these market failures is ¯scally generated, and the purpose of the present chapter to further explore the ¯scal distortions that may encourage urban sprawl. The analysis
focuses in particular on the role of the property tax. As will be seen, use of property taxation
to ¯nance infrastructure investment is partly responsible for tendency toward excessive development noted above. But before outlining this rather subtle conclusion in greater detail, it is
useful to consider a di®erent and more-elementary connection between property taxation and
urban spatial expansion. This connection, which is analyzed in section 2 below, arises out of
the long debate on the virtues of land taxes versus property taxes.
As is well known, the property tax can be viewed as a tax levied at equal rates on both
the land and capital embodied in structures. While the land portion of the property tax has
no e®ect on resource allocation in a static setting, the tax on capital (i.e., improvements)
tends to lower the equilibrium level of improvements chosen by the developer. Thus, land is
developed less intensively under property taxation than under a pure land tax, where the rate
on improvements is set at zero. This conclusion has been derived and discussed many times in
the literature.2
In the case of residential structures, the lower level of improvements per acre under property
taxation means that developers construct shorter buildings, containing less housing °oor space
per acre of land. If the size of dwellings within each building remains roughly constant, the
shorter building height implies a decline in population density, with fewer households ¯tting
on each acre of land. But if the city must accommodate a ¯xed population, lower densities
mean that it must take up more space. Thus, by reducing the intensity of land development,
the property tax would appear to encourage urban sprawl.
Remarkably, the literature has not provided a clear analysis of this issue. Carlton (1981)
and LeRoy (1976) analyze certain e®ects of the property tax in a spatial context, without
considering its impact on urban expansion. Sullivan (1985) presents simulation analysis of the
spatial e®ect of the property tax, but his use of a complex model that includes both business
and residential property makes it hard to see the operation of the expansive force sketched
above. Mills (1998) investigates the spatial e®ect of the property tax in a business-oriented
model with no residential land, so that the above reasoning does not apply. Arnott and
2
MacKinnon (1977) present simulation results for a purely residential model, but their use of
an apparently nonstandard framework makes it di±cult to appraise the results.
The analysis presented in Section 2 is meant to ascertain whether the expansive force
outlined above indeed emerges in a standard urban model. The analysis shows that the basic
intuition from above, namely that the reduced intensity of land development under property
taxation encourages urban sprawl, is roughly correct. However, the argument is complex
because the property tax changes the price of housing at the same time that it raises the
cost of improvements. While the net e®ect is a reduction in development intensity in the
central part of the city, improvements per acre actually rise closer to the urban fringe because
the price of housing increases in this area. The overall e®ect, however, is to reduce average
population density in the original city, so that urban expansion is required to accommodate
the ¯xed population. In addition to showing that this spatial expansion occurs, the analysis
also establishes that the expansion is socially ine±cient. Therefore, the model suggests that
property taxation may belong on the above list of factors contributing to the excessive spatial
growth of cities.
It should be noted that the model used to derive these results embodies a key simpli¯cation, namely ¯xed dwelling sizes. While this allows clear identi¯cation of the expansive force
operating through development intensity, a more general analysis would allow both improvements per acre and dwelling sizes to change in response to the property tax. It is possible that
the present conclusion regarding the expansive e®ect of the property tax could be overturned
or quali¯ed by such an analysis.
The analysis sketched above shows how the property tax can cause a ¯xed population to
spread out spatially, with the housing needed to accommodate it taking up more space. This
conclusion is important because it draws a link between urban sprawl and key a institution of
public ¯nance, namely the property tax system. However, a di®erent avenue through which the
¯scal system can contribute to urban sprawl can be identi¯ed. In particular, ¯scal arrangements
may cause city populations to grow excessively, leading to excessive spatial expansion. Thus,
sprawl may be caused by unwarranted, ¯scally-induced population growth, independently of
any e®ects on development intensity. The argument supporting this point, which was formally
3
presented by Brueckner (1997) in a model without property taxation, focuses on infrastructure
costs, and it underlies the third market failure identi¯ed above.
To see the argument, consider ¯rst the rule for socially optimal conversion of land from
rural to urban use. Conversion should occur when the cost of development, which includes
forgone agricultural rent, construction cost, and the marginal infrastructure costs necessitated
by the development, equals the rent earned by the land in urban use. Suppose, however,
that developer/landowners are charged a tax equal to the average cost of infrastructure rather
than marginal cost. This would result from equal sharing of the cost of new infrastructure
built at the urban fringe by all landowners in the city. When the city operates in a range of
decreasing returns to scale in infrastructure provision, average cost will fall short of marginal
cost. The result is that the average-cost charge levied on developers understates the social cost
of the necessary infrastructure (i.e., marginal cost), so that the costs of development appear
arti¯cially low. The upshot is that land on the urban fringe is developed prematurely, before
it is socially desirable to do so. Thus, at each point in time, the city has a larger population
and takes up more space than it should.
When property taxation is realistically added to this model, the outcome is even worse.
The reason is that, since property values decline as distance to the center increases, fringe
landowners pay less than the average cost of new infrastructure under property-tax ¯nancing.
Landowners close to the center pay more than average, while the burden on landowners midway
to the fringe is roughly average. The result is that the infrastructure tax burden on fringe
landowners is even farther below the infrastructure's marginal cost than it would be under a
system of equal charges. The upshot is that at each point in time, the divergence between the
city's equilibrium and optimal population is bigger than before, implying a worsening of the
upward bias to spatial growth.
The remedy for this distortion is to change the method of infrastructure ¯nance, replacing
the portion of the property taxes used for this purpose with impact fees. Such fees require a
one-time payment for infrastructure by developers rather than ongoing payments ¯nanced by
property taxes, with the magnitude of the one-time payment equal to marginal cost. When
impact fees are levied, urban growth occurs in a socially optimal fashion, and cities have smaller
4
populations, and are more spatially compact, at each point in time.
The analysis supporting the above discussion is presented in Section 3 of the chapter.
Section 4 o®ers conclusions.
2. Sprawl Arising from a Tax-Induced Reduction in Development Intensity
2.1. Basic model
To begin the discussion of tax-induced urban sprawl resulting from the lower intensity
of land development, an analytical framework must be speci¯ed. It is convenient to adopt
the framework used by Brueckner (1986) to analyze land value taxation, a framework that
is common to many models in urban economics. Under this approach, housing is produced
by combining capital and land. Since the production function exhibits constant returns to
scale, the analysis can focus on the choice of capital per acre of land, or \improvements" per
acre. Letting S denote improvements per acre, housing output per acre of land is given by
the function h(S), which is increasing and strictly concave, re°ecting diminishing returns to
improvements (thus, h0 > 0 and h00 < 0). Note that improvements (per acre) is an index
of building height, which registers the intensity of land development, while housing output is
measured in units of °oor space. Thus, an increase in improvements leads to a taller building,
which contains more housing °oor space than the original building.
Since the cost per unit of improvements equals i, the housing developer's improvement
cost per acre equals iS. With land cost per acre equal to land rent, denoted r, the developer's
total cost per acre is given by iS + r in the absence of any taxes. To introduce property
taxation, it is convenient to represent the property tax as a tax levied at equal rates on both
land and improvements. An algebraically equivalent formulation portrays the property tax as
a tax on the developer's housing output. Adopting the ¯rst approach, and letting µ denote
the property-tax rate, the developer's total cost inclusive of the tax is given by (1 + µ)(iS + r).
Letting p denote the rental price per unit of housing °oor space, the developer's pro¯t per acre
is equal to ph(S) ¡ (1 + µ)(iS + r).
2.2. Accepted wisdom on property taxation and improvements
The property tax's e®ect on the intensity of development can be derived by computing the
5
impact of an increase in the tax rate µ on the level of improvements S chosen by the developer.
In the previous literature, this exercise is usually carried out in a partial-equilibrium setting
where the e®ect of the tax on the spatial size of the city (and thus on urban sprawl) is not
considered.
One approach is to assume that the housing price p is ¯xed exogenously in analyzing the
impact of a higher µ. To carry out the analysis, the developer's ¯rst-order condition for choice
of S is derived. Using the above pro¯t expression, this condition is ph0 (S) = (1+µ)i, which says
that the marginal revenue from increasing the level of improvements should equal marginal
cost. When the tax rate µ increases, the marginal-cost term rises, so that marginal revenue
must increase to satisfy the ¯rst-order condition. This increase is achieved via a reduction in S,
which raises h0 (S) under diminishing returns. Formally, @S=@µ = (1 + µ)=ph00 < 0, indicating
that an increase in the property-tax rate reduces improvements.
The equilibrium level of land rent is determined by the developer's zero-pro¯t condition,
which is written ph(S) ¡ (1 + µ)iS = (1 + µ)r. Di®erentiating this condition with respect to µ
shows that land rent falls as the property-tax rate increases, with @r=@µ < 0.
To avoid distorting the choice of improvements, the local government could abandon the
property tax and switch to a land tax. Under such a tax, improvements are not taxed at all,
with land again taxed at a rate µ. The developer's pro¯t per acre in this case is ph(S) ¡ iS ¡
(1 + µ)r, and the ¯rst-order condition for choice of S is ph0 (S) = i. Since this condition does
not involve µ, it follows that the level of improvements is una®ected by the tax rate on land.
Moreover, the higher land tax is fully capitalized in a lower land rent. To see this, note that
the zero-pro¯t condition requires (1+µ)r = ph(S)¡iS. With S una®ected by µ, it then follows
that (1 + µ)r is itself una®ected by µ. Thus, the land tax is fully capitalized, with r falling
enough as µ rises to keep the gross-of-tax rent (1 + µ)r constant.
The above analysis of the property tax is limited by its partial-equilibrium nature, which
is re°ected in the assumption of a ¯xed housing price p. One way of endogenizing p is to
assume that the city has a ¯xed land area, and that aggregate demand for housing is given by
a demand function F (p). Brueckner (1986) shows that an increase in the property-tax rate in
this modi¯ed model once again reduces the level of improvements.
6
While this modi¯cation improves the realism of the model, the assumption of a ¯xed urban
land area rules out any connection between the property tax and urban sprawl. However, there
is strong reason to suspect that such a connection exists, as explained in the introduction. The
argument is that, by reducing improvements per acre, the property tax is also likely to reduce
population density. But if density declines, then a larger land area is required to accommodate a
¯xed population. The upshot is that the city grows spatially as the property-tax rate increases,
leading to urban sprawl.
To see the argument more clearly, observe that population density is given by housing
°oor space per acre of land, h(S), divided by housing °oor space per person, denoted q (this is
dwelling size). The resulting ratio, h(S)=q, gives dwellings (and hence people) per acre of land,
or population density. Given the above discussion, the decline in improvements as µ increases
leads to a reduction in housing output h(S) per acre. Unless this e®ect is o®set by a lower
dwelling size q, population density falls with µ. As explained above, this lower density should
lead to a spatial expansion of the city.
In order to explore this conjecture, the model must be modi¯ed to allow the city's land
area to be endogenouly determined. The easiest way to achieve this endogeneity is to suppose
that the city's residents commute to work in a central business district (CBD). With CBD
access valued, land rent then falls as distance to the center increases, and this decline imposes
a natural limit on the size of the city. When property taxation is added to this type of model,
the e®ects on improvements and population density are more complex than those outlined
above. But the ultimate conjecture, namely that property taxation leads to urban sprawl, is
con¯rmed, as shown in the next section.
2.3. A spatial model
Let x denote the distance from a consumer's residence to the CBD, and suppose that
commuting cost per period from distance x is given by kx, where k > 0. Letting y denote the
common income per period earned at the CBD by all urban residents, disposable income for
an individual living at distance x is then equal to y ¡ kx.
To focus on the sprawl e®ect arising through changes in development intensity, dwelling
sizes in the model are assumed to be ¯xed. In particular, each resident is assumed to consume
7
one unit of housing °oor space, with q above set equal to unity. Housing expenditure, which
equals pq, is thus given by p, where p again denotes the price per unit of housing °oor space.
Since residents also consume a nonhousing good, denoted u, the budget constraint for an
individual living at distance x is given by u + p = y ¡ kx (u is numeraire).
Because all urban residents are identical, they must achieve the same utility in equilibrium.
But, since housing consumption is ¯xed at one unit, a resident's utility can be measured
directly by the level of nonhousing consumption u. Therefore, equalization of utilities means
that u must be the same regardless of the individual's residential location. In order for this to
happen, the housing price p must vary so as to exactly o®set the commuting cost di®erences
across locations. Using the budget constraint, p must then satisfy
p = y ¡ kx ¡ u ´ p(x; u):
(1)
where u now represents the uniform nonhousing consumption (utility) level in the city. This
utility level is determined endogenously, as explained below. Di®erentiating (1) yields
@p @p
;
@x @u
< 0;
(2)
so that p falls as x increases, o®setting the higher commuting cost associated with a moredistant residence. Also, a higher utility level u requires a lower p, which frees up more funds
to be spent on the nonhousing good.
Rewriting the pro¯t expression from the previous analysis, the developer's pro¯t is given
by p(x; u)h(S) ¡ (1 + µ)(iS + r). As above, the ¯rst-order condition for choice of S is then
p(x; u)h0 (S) = (1 + µ)i:
(3)
This equation determines the optimal S as a function of x, u, and µ, which is written
S = S(x; u; µ):
8
(4)
Di®erentiating (3) yields
@S @S @S
;
;
@x @u @µ
< 0:
(5)
Since @S=@x < 0, the level of improvements varies with location. At more distant locations,
where poor CBD access reduces the housing price p, the incentive to improve the land is weak,
leading to a low S. In addition, an increase in the property-tax rate reduces improvements
per acre, with @S=@µ < 0 holding as before. This e®ect on S occurs at each location x within
the city. Finally, by lowering p at each location, an increase in the utility level u reduces S
throughout the city, yielding the same e®ect as an increase in µ.
The developer's zero-pro¯t condition, which can be written r = p(x; u)h(S)=(1 + µ) ¡ iS,
determines land rent. Substituting the optimal solution from (3) in place of S, r is given by
r(x; u; µ) ´
p(x; u)
h[S(x; u; µ)] ¡ iS(x; u; µ):
1+µ
(6)
Because S is chosen optimally, application of the envelope theorem shows that the derivatives
of r with respect to x, u, and µ have the same signs as the corresponding derivatives of
p(x; u)=(1 + µ). Using (2) to sign the latter derivatives, it follows that
@r @r @r
;
;
@x @u @µ
< 0:
(7)
Thus, land rent decreases with distance, mirroring the decline in p, and it is also decreasing in
µ and u.
In standard fashion, the urban equilibrium is determined by two requirements. First, urban
land rent must equal the agricultural rent ra at the edge of the city, denoted x.3 Second, the
city must ¯t its ¯xed population, N . The ¯rst condition is written
r(x; u; µ) = ra ;
(8)
and it is illustrated in Figure 1, which shows several land-rent curves (the Figure is discussed
further below). To write the second condition, assume for simplicity that the city is linear with
9
unit width. Then, the population ¯tting in a slice of land between x and x + dx is equal to
population density times dx, or h[S(x; u; µ)]dx (recall that q = 1). Integrating this population
expression from the CBD to the edge of the city and setting the result equal to N yields4
Z
x
h[S(x; u; µ)]dx = N:
(9)
0
Equations (8) and (9) determine equilibrium values of the variables x and u as functions of
the parameters µ, N, and ra . The e®ect of property taxation on urban sprawl is revealed by
the sign of the derivative @x=@µ.
Analysis presented in the appendix shows that
@x
@µ
> 0:
(10)
Thus, an increase in the property-tax rate does indeed enlarge the spatial size of the city,
leading to urban sprawl. Although the underlying analysis is somewhat complex, this important conclusion can be explained intuitively in a transparent fashion. First, suppose that the
property-tax rate µ increases, but that the utility level u, and thus the housing price p(x; u)
in each location, remains ¯xed. As seen in (5), the higher µ reduces improvements S at each
location in the city, and this in turn reduces population density at each location. From (7), the
increase in µ also reduces land rent r throughout the city. But from the boundary condition
(8), the reduction in urban rent implies that the city contracts, with x falling. The city is now
spatially smaller and has lower population densities, and this means that it no longer ¯ts its
population. Thus, there is excess demand for housing, with the available housing stock unable
to accommodate the population.
In response to this excess demand, the housing price p rises throughout the city. Formally,
this price increase is driven by a decline in utility u, which raises p at each location given (1).
With u falling and p rising everywhere, the level of improvements S rises at each location,
o®setting the tax-induced decline (from (5), S rises as u falls). In addition, higher housing
prices pull up land rents, with r rising at each location as u falls (see (7)). With higher improvements raising population density at each location, and with rising land rents causing land
10
to be reclaimed from agriculture, the city's ability to accommodate its population improves,
so that the excess demand for housing is reduced.
To see why this process ultimately raises x, it is helpful to focus on the nature of the shifts
in the land rent function. By di®erentiating (6), it can be seen that, when µ rises, the decline in
land rent is ampli¯ed by di®erences in housing prices. In particular, rent declines substantially
near the CBD, where p is highest, while declining more moderately near x, where p is lower. By
contrast, the o®setting increase in land rent caused by the decline in u is more nearly parallel
across locations.5 The upshot is that the combined increase in µ and decrease in u lead to a
counterclockwise rotation of the land-rent curve, with the point of rotation being a distance
x that lies between the CBD and x, as shown in Figure 1. As can be seen from the Figure,
b
such a rotation necessarily leads to an increase in the distance x at which the land-rent curve
intersects the horizontal agricultural-rent line. Thus, rotation of the land-rent curve provides
the key that links property taxation to the extent of urban sprawl.
Additional insight comes from noting that the two curves representing improvements S and
p=(1 + µ), which are not shown in Figure 1, rotate in step with the land-rent curve. Therefore,
improvements decline at locations inside of b
x, mirroring the decline in land rent, while rising
at locations outside of b
x, mirroring the rent increase far from the CBD. This outcome shows
that an increase in the property-tax rate actually raises the level of improvements at some
locations in the city, contrary to what a partial-equilibrium analysis would predict. However,
improvements fall over a su±ciently broad range that the city must expand spatially in order
to ¯t its population. Thus, despite the added complexity of the present analysis, the intuition
suggested by the partial-equilibrium approach is valid: by tending to reduce the intensity of
land development, the property tax causes a city to expand spatially in order to accommodate
its population.
It is important to realize that this conclusion is conditional on the underlying assumption
that dwelling sizes are ¯xed, an assumption that is meant to isolate the e®ect of changes in
development intensity. When dwelling sizes are free to adjust in response to the tax change,
then the complexity of the analysis escalates, and derivation of results may require a simulation
approach. To see the source of the complexity, recall that the price p(x; u) per unit of housing
11
rises as u declines (see (1)). When q is °exible, this price increase would tend to reduce
dwelling sizes, raising population density at each location and putting downward pressure on
the spatial size of the city. This pressure would tend to o®set the spatial expansion due to
lower improvements per acre, making the net impact on x ambiguous. In fact, Arnott and
MacKinnon (1977) present simulation results showing a reduction in city size in response to a
property-tax increase. Their model, however, appears to di®er in structure from the standard
urban model, as presented above, making it hard to appraise their results. Nevertheless,
Arnott and MacKinnon's analysis suggests that, once dwelling-size e®ects are incorporated,
the present conclusions regarding urban sprawl and the property tax may be overturned or
quali¯ed. Clearly, further exploration of such a model would be a useful avenue for future
research.
A ¯nal point concerns the use of a land tax in the present model. Following Sullivan (1985),
it is useful to model the land tax as a tax on di®erential land rent, which equals r ¡ ra . Land
costs for the developer are then r + µ(r ¡ ra ). Because improvements are untaxed under a
land-tax regime, S is independent of µ, as in the earlier discussion. Since it can be shown that
µ is also absent from the boundary condition determining x, it follows that the equilibrium
values of x and u are independent of the land-tax rate.
This conclusion implies that substitution of the land tax in place of the property tax would
lead to a spatial shrinkage of the city. To see this, observe that eliminating the property tax by
setting the rate at zero would reduce the spatial size of the city, while imposing a land tax and
raising the rate su±ciently to recover the lost property-tax revenue would have no e®ect on
city size. Thus, switching between the two tax regimes would shrink the city, limiting urban
sprawl.
Since utility rises as the property-tax rate is lowered and is una®ected as the land-tax
rate increases, consumers would bene¯t from this change of regimes. More generally, it can be
shown that switching to the land tax raises an overall measure of social welfare, which equals
total consumption of the urban residents plus tax revenue plus the rental income accruing to
absentee landowners, who own the urban land. While a±rming the well-known virtues of the
land tax, this conclusion also establishes that the urban sprawl caused by the property tax is
12
socially ine±cient. Thus, the analysis suggests that property taxation may belong on the list
of factors causing ine±cient spatial expansion of cities, as discussed in the introduction.
3. Sprawl Arising from Tax-Induced Mispricing of Infrastructure
3.1. The model
The analysis in section 2 showed that under property taxation, a city expands excessively,
with its ¯xed population taking up more land than is socially desirable. The analysis in this
section uses a di®erent model to explore a related e®ect of property taxation: the encouragement of excessive population growth in cities (and thus excessive spatial expansion) through
mispricing of urban infrastructure.
The model used to analyze this question (drawn from Brueckner (1997)) is related to that
in Section 2, but it di®ers in a number of important respects. First, the model is explicitly
dynamic, focusing on urban growth.6 Second, the production of housing, as represented by the
developer's choice of improvements per acre, is suppressed, with urban residents consuming
land directly after its conversion from agricultural use at a ¯xed cost per acre. Third, urban
residents require public infrastructure, which was absent in the previous model. Fourth, rather
than having a ¯xed population, the city is \open," with its population expanding to occupy
newly converted land.
Each urban resident consumes one unit of (converted) land, paying rent r. At time t,
the residents earn income y(t) at the CBD, and commuting cost from distance x is again kx
(this cost is time-invariant). Because the city is open, its residents must enjoy whatever utility
prevails in the rest of the economy. This utility, which again measures consumption of the
nonland good, is given by u(t), an exogenous function of time. Thus, unlike in the previous
model, the utility of residents is exogenous rather than endogenously determined within the
city.
Combining the above assumptions, an individual's budget constraint is given by u(t) + r =
y(t) ¡ kx. In order for the individual to be able to a®ord u(t) worth of the nonland good, r
must adjust to satisfy this equation. Thus, r is given by r(t; x) ´ y(t) ¡ u(t) ¡ kx. As before,
land rent is decreasing in x, and it is also assumed that y 0 (t) ¡ u0 (t) > 0 holds for all t, so
13
that rent is increasing over time. Note that, in contrast to the previous model, this land rent
function is exogenous. Before, endogeneity of u meant that housing prices, as given by the
analogous expression in (1), were endogenous.
As seen below, rising land rents cause the city to grow over time. For simplicity, the model
does not specify the microfoundations of this growth, which is driven instead by the exogenous
increase in the city's income (and the resulting rise in land rents). As population growth
proceeds, spending by the local government must increase to maintain the level of public
services z, which is assumed to be ¯xed at z. Public services are generated by infrastructure,
and the cost of the infrastructure necessary to provide a service level z to a population of
size n is given by C(z; n). The partial derivative Cn is positive, indicating that infrastructure
expansion is required to maintain the service level at z as population grows. In addition, it is
assumed that per capita infrastructure cost, C(z; n)=n, is a U-shaped function of n, re°ecting
eventual diminishing returns to population.
A key additional assumption is that infrastructure is in¯nitely lived. Thus, previously-built
infrastructure continues to generate services, but augmentation of the infrastructure stock is
needed to maintain the service level as population grows. A relevant example would be a city's
road network, which must be expanded spatially as well as widened to accommodate population
growth without service degradation.7 It should be noted that the role of current inputs (e.g.,
snow removal from roads) in providing public services is ignored under this formulation, with
services generated entirely from infrastructure. However, current inputs can be added to the
model, as explained below.
3.2. Socially optimal urban growth
Consider now the question of socially-optimal conversion of land from agricultural to urban
use. Because consumer utilities are exogenous, social welfare is measured by the total value
of land in the region containing the city. Accordingly, the immediate bene¯ts from conversion
of a unit of land at location x are captured by the rental income generated by the land in
urban use, given by r(T (x); x), where T (x) denotes the land's conversion date. The costs of
conversion include the forgone agricultural rent, ra , along with the annualized construction
cost required to ready the land for occupation. This construction cost, which is incurred as a
14
lump sum, is denoted D, and the cost equals ½D on an annual basis, where ½ is the interest
rate. The ¯nal element of conversion cost is the cost of expanding the city's infrastructure to
accommodate a new resident at location x. For the moment, let this cost be denoted I(x).
Conversion of the land at x is optimal when bene¯ts equal costs, or when
r(T (x); x) = ra + ½D + ½I(x);
(11)
where annualized infrastructure costs are used. Once I(x) is speci¯ed, this equation determines
the optimal conversion date, T (x), for land at location x. Recognizing that conversion occurs
at x, the edge of the city, (11) can be rewritten so that it instead determines the city edge x(t)
at an arbitrary time t. The resulting condition is
r(t; x(t)) = ra + ½D + ½I(x(t)):
(12)
To derive the infrastructure-cost term I(x(t)), it is helpful to impose the previous assumption that the city is linear and one unit wide. Since individual land consumption is assumed to
equal one unit from above, it follows that the distance to the edge of the city, x(t), is the same
as the city's population n(t) at the given time. Therefore, when the city edge has expanded to
a given x(t), the marginal infrastructure cost from a new resident is Cn (z; n(t)) = Cn (z; x(t)),
which represents I(x(t)). Substituting, (12) can be rewritten as
r(t; n(t)) = ra + ½D + ½Cn (z; n(t));
(13)
where x(t) in the land-rent function has been replaced by n(t).
Equation (13) determines the socially optimal population size for the city at time t. At
this population size, land rent at the city edge (where x(t) = n(t)) is equal to the costs of
conversion, which also depend on n(t) through the infrastructure term. The determination of
n(t) at a given time is illustrated in Figure 2, which shows the downward sloping land-rent
curve along with the marginal infrastructure cost curve Cn (z; n(t)), translated upward by the
15
amount ra + ½D (referred to subsequently as the \planner's curve"). The intersection between
the rent curve and the planner's curve gives the city's optimal population size at a given time.
In the Figure, the optimal population size at t = t0 equals n00 . Note that as time progresses,
the land-rent curve shifts upward, so that the optimal population grows (the rent curve for a
later time t1 is shown). The other curves in Figure 2 are discussed below.
3.3. Urban growth under head-tax and property-tax regimes
The next step is to compare the urban growth paths generated by various infrastructure
¯nancing methods to the socially optimal path. Consider ¯rst a regime under which each unit
of converted land in the city pays an equal share of infrastructure costs. This arrangement is
denoted a \head-tax" regime because each resident generates a tax liability for the owner of
the land he occupies.
To compute the required head tax, suppose that the city sells in¯nite-maturity bonds
to ¯nance each incremental addition to the stock of infrastructure. As shown by Brueckner
(1997), this scheme generates a total interest payment in each period equal to the interest rate
times the cost of the entire existing stock of infrastructure, or ½C(z; n(t)). Since this cost is
divided among all the occupied parcels of land in the city, each unit of land bears a cost of
½C(z; n(t))=n(t).
Rather than being decided by a planner, land conversion is now decentralized, with each
landowner determining the conversion date for the land he owns. Bene¯ts and costs are
compared, as before, but the infrastructure costs are those generated by the head-tax regime,
rather than the actual marginal costs considered by the planner. As shown by Brueckner
(1997), the population growth path resulting from these decentralized decisions is characterized
by (13) with the marginal cost term replaced by the above average-cost expression, which arises
from the head-tax regime. This condition is written
r(t; n(t)) = ra + ½D + ½
C(z; n(t))
:
n(t)
(14)
Inspection of Figure 2 shows that the growth path characterized by (14) di®ers from the
socially optimal path. Once the city has reach the point of diminishing returns to population,
16
with C=n rising as n increases, the head-tax regime leads to excessive growth. To see, this
observe that the city's population under the head-tax regime is determined by the intersection
between the land-rent curve and the curve representing ra +½D +½C=n (the \head-tax curve").
Over the range of diminishing returns, this intersection lies to the right of the intersection
between the land-rent curve and the planner's curve, establishing that the city's population
is too large. In Figure 2, the intersection between the land-rent and head-tax curves yields
a population of n000 at time t0 , which exceeds the socially optimal population n00 . Thus, the
mispricing of infrastructure under the head-tax regime leads the city to expand excessively as
it enters the range of diminishing returns. Since this overexpansion applies to both population
and the city's spatial size, mispricing of infrastructure generates urban sprawl.
Two additional observations are useful. First, the head-tax regime has exactly the opposite
e®ect on urban expansion over the range of increasing returns, where C=n is falling. Inspection
of Figure 2 shows that if the land-rent curve intersects the head-tax curve on its downwardsloping range, the resulting population is smaller, not larger, than the socially optimal level.
Thus, urban overexpansion is a problem only when prior growth has pushed the city into
the range of diminishing returns. The second observation is that, whatever its direction, the
divergence between optimal and equilibrium city sizes can be corrected by switching to a system
of impact fees, which are lump-sum marginal-cost charges levied at the time of land conversion.
Before discussing this alternative, however, it is important to ask how the preceding conclusions
are altered in the presence of a property-tax regime.
Under the property tax, which is levied only on converted land, the value of a unit of land
is equal to the presented discounted value of rent minus property-tax payments. Let v(t; x)
give the value at time t of a unit of land at location x, an expression that implicitly depends
on the future time path of the property-tax rate. The property-tax payment at time t is then
µ(t)v(t; x) ´ P (t; x), where µ(t) gives the tax rate at t.8 Because v is decreasing in x, the
property-tax payment P is also lower at more distant locations.
To characterize land conversion under the property-tax regime, the annual property-tax
liability at the edge of the city, which equals P (t; x(t)), is substituted in place of ½C=n in (14).
The modi¯ed equation then governs decentralized conversion of land under the property-tax
17
regime. To investigate how this substitution changes the population path, a comparison of
the magnitudes of P (t; x(t)) and ½C=n is needed. The ¯rst step is to recognize that, since P
is decreasing in x, the property-tax payment at the city's edge is less than the average tax
payment in the city. The average payment, denoted Pavg (t; n(t)), is equal to total property
R x(t)
taxes 0 P (t; x)dx divided by n(t). Since the tax at the city's edge is less than Pavg , it
follows that P (t; x(t)) = Pavg (t; n(t)) ¡ Á(t; n(t)), where Á is a positive function that captures
the tax discrepancy. But in order for total taxes to cover interest on the existing structure, the
average tax payment Pavg must equal ½C=n. It then follows that P (t; x(t)) = ½C(z; n(t))=n(t)¡
Á(t; n(t)), so that the property-tax liability at the edge of the city equals the head-tax payment
minus the discrepancy term.
Substituting the last expression in place of P (t; x(t)) in the modi¯ed version of (14), the
condition that determines the city's population path under the property-tax regime is then
r(t; n(t)) = ra + ½D + ½
C(z; n(t))
¡ Á(t; n(t))
n(t)
(15)
In Figure 2, the solution to (15) corresponds to the intersection between the land-rent curve
and the dotted curve representing ra + ½D + ½C=n ¡ Á (the \property-tax curve"). Since the
property-tax curve lies below the head-tax curve, it follows that, once the city has entered the
range of diminishing returns, its growth is even more excessive under the property-tax regime
than under the head-tax regime. At time t0 , for example, the city's population equals n000
0,
00
0
where n000
0 > n0 > n0 . Thus, while the average-cost infrastructure payment under the head-
tax regime understates marginal cost, the property-tax payment at the city's edge represents
an even more serious understatement. By generating arti¯cially low infrastructure costs, the
property-tax system therefore encourages excessive population growth, and thus urban sprawl,
once the city has entered the range of diminishing returns.
This conclusion is quali¯ed over the range of increasing returns, where the comparison
between the optimal and equilibrium population sizes is ambiguous.9 However, the conclusion
becomes stronger in the case of constant returns to population, where C=n is constant and
equal to Cn . In this case, the head-tax curve coincides with the planner's curve, with both
18
represented by a horizontal line. While the head-tax regime then generates the socially optimal
population path, the property-tax regime now generates excessive growth over the entire range
of populations, a consequence of the fact that the property-tax curve lies everywhere below
the horizontal planner's curve. Thus, in the constant-returns case, excessive urban expansion
is a ubiquitous feature of the property-tax regime.
As noted above, switching to an impact-fee regime can ensure that the city follows the optimal population growth path. Under this regime, landowners are charged a lump-sum fee equal
to marginal infrastructure cost when their land is converted to urban use. Thus, landowners converting their land at time t are charged an amount Cn (z; n(t)), and this marginal-cost
expression (multiplied by ½) then replaces the ½C=n term in the decentralized conversion condition (14). Since the resulting condition is the same as (13), decentralized land conversion is
socially optimal. Encouragingly, the use of impact fees to ¯nance infrastructure is becoming
increasingly common throughout the U.S. (see Brueckner (1997) for references describing current practice). This institutional change may limit the bias toward urban expansion created
by reliance on the property tax.
As a ¯nal point, it should be noted that the model can be generalized to include the use
of current inputs in the provision of public goods along with infrastructure. With the cost of
current inputs written K(z; n), the term Kn (z; n(t)) would be added to the right-hand side
of the condition (13) determining optimal population growth, and K=n would be added to
(14) (because these terms capture annualized costs, ½ is absent). As before, reliance on the
property-tax leads to excessive urban expansion. But unlike in the pure infrastructure case,
the revenue from marginal-cost charges for current inputs, which are required to support the
optimum, could be larger or smaller than the cost of those inputs.10
4. Conclusion
This chapter has explored the connection between property taxation and urban sprawl. It
has been shown that the property tax's tendency to reduce the intensity of land development,
which follows from its taxation of improvements, causes a city to expand spatially in order to
accommodate its population. Since the resulting expansion is socially ine±cient, the analysis
19
suggests that property taxation may contribute to undesirable urban sprawl. Switching to a
land-tax regime would remove this potential source of sprawl.
Using a di®erent model, the second part of the paper showed that, by generating tax
liabilities that lie well below the marginal cost of public infrastructure, the property tax makes
urban development appear arti¯cially cheap. Excessive development then occurs, with city
populations and land areas expanding beyond socially desirable levels. An impact-fee regime,
under which developers are charged the marginal infrastructure cost associated with their
developments, would tend to correct this overexpansion.
Both of these points are new to the literature on property taxation, and they may provide a
point of departure for additional research on this important institution of local public ¯nance.
20
Appendix
This appendix derives the results of Section 2. To begin, (1) is substituted into (3) to yield
h0 (S) =
Solving for S yields
(1 + µ)i
:
y ¡ kx ¡ u
(a1)
¸
(1 + µ)i
;
S(x; u; µ) = f
y ¡ kx ¡ u
·
(a2)
where f ´ (h0 )¡1 , with f 0 = 1=h00 < 0. Using (1) and (6), condition (8) may then be written
as
µ ·
¸¶
·
¸
y ¡ tx ¡ u
(1 + µ)i
(1 + µ)i
h f
¡ if
= ra :
1+µ
y ¡ kx ¡ u
y ¡ kx ¡ u
(a3)
This equation can be viewed as determining m ´ kx+u as a function of µ. Total di®erentiation
yields
@m
@µ
´ k
@x
@u
+
@µ
@µ
= ¡
y¡m
:
1+µ
(a4)
Next, condition (9) may be written as
Z
x
0
µ ·
¸¶
(1 + µ)i
h f
dx = N:
y ¡ kx ¡ u
(a5)
Total di®erentiation of (a5) yields
@u
@x
h +
@µ
@µ
Z
x
0
(1 + µ)ih0 f 0
dx +
(y ¡ kx ¡ u)2
Z
0
x
ih0 f 0
dx = 0;
y ¡ kx ¡ u
(a6)
where h denotes the integrand in (a5) evaluated at x = x. Solving (a4) and (a6) yields
@x
@µ
1
= ¡
A
Z
0
x
ik(x ¡ x)h0 f 0
dx > 0;
(y ¡ kx ¡ u)2
21
(a7)
where
A = h ¡
Z
0
x
(1 + µ)ikh0 f 0
dx > 0:
(y ¡ kx ¡ u)2
(a8)
Note that the positive sign of @x=@µ in (a7), which establishes the key result in (10), uses
h0 > 0, f 0 < 0, and the fact that x < x holds over the range of integration. A's positive sign
follows from the ¯rst two facts. In addition,
@u
@µ
=
"
#
Z x
1
(y ¡ m)h
ikh0 f 0
¡
+
dx
< 0;
A
1+µ
0 y ¡ kx ¡ u
(a9)
showing that utility falls as µ rises, as claimed in Section 2.
Finally, to investigate rotation of the rent and S curves, the derivative of p(x; u)=(1 + µ) is
evaluated at an arbitrary x = e
x. Using (a9), this yields
@ y ¡ ke
x¡u
@µ 1 + µ
=
"
#
Z x
t
(1 + µ)ik(x ¡ e
x)h0 f 0
¡h(x ¡ e
dx :
x) +
A
(y ¡ kx ¡ u)2
0
(a10)
Inspection shows that (a10) is negative for e
x = 0 and positive for e
x = x, and that the
expression is increasing in e
x. This implies that the curve representing p(x; u)=(1 + µ) rotates in
a counterclockwise fashion as µ increases, with the point of rotation being some intermediate
x value denoted b
x, which satis¯es 0 < b
x < x. From (4), the same rotation occurs in the
land-rent curve r(x; u; µ), and (2) generates identical rotation in the curve S(x; u; µ) showing
improvements per acre.
22
References
Anderson, J.E. 1986. \Property Taxes and the Timing of Urban Land Development." Regional Science and Urban Economics 16: 483-492.
Bentick, B.L. 1979. \The Impact of Taxation and Valuation Practices on the Timing and
E±ciency of Land Use." Journal of Political Economy 87: 859-868.
Arnott, R.J. and J.G. MacKinnon. 1977. \The E®ects of the Property Tax: A General
Equilibrium Simulation." Journal of Urban Economics 4: 389-407.
Brueckner, J.K. 1986. \A Modern Analysis of the E®ects of Site Value Taxation." National
Tax Journal 39: 49-58.
Brueckner, J.K. 1997. \Infrastructure Financing and Urban Development: The Economics
of Impact Fees." Journal of Public Economics 66: 383-407.
Brueckner, J.K. 1999. \Urban Sprawl: Diagnosis and Remedies." International Regional
Science Review, forthcoming.
Carlton, D.W. 1981. \The Spatial E®ects of a Tax on Housing and Land." Regional Science
and Urban Economics 11: 509-527.
LeRoy, S.F. 1976. \Urban Land Rent and the Incidence of Property Taxes." Journal of
Urban Economics 3: 167-179.
Mills, D.E. 1981. \The Non-Neutrality of Land Value Taxation." National Tax Journal 34:
125-129.
Mills, E.S. 1999. \The Economic Consequences of a Land Tax." In Land Value Taxation:
Can It and Will It Work Today? D. Netzer, ed. Cambridge, Mass.: Lincoln Institute of
Land Policy.
Nechyba, T.J. 1998. \Replacing Capital Taxes with Land Taxes: E±ciency and Distributional Implications with an Application to the United States Economy." In Land Value
Taxation: Can It and Will It Work Today? D. Netzer, ed. Cambridge, Mass.: Lincoln
Institute of Land Policy.
Oates, W.E., and R.M. Schwab. 1997. \The Impact of Urban Land Taxation: The
Pittsburgh Experience." National Tax Journal 50: 1-21.
Sullivan, A.M. 1985. \The General Equilibrium E®ects of the Residential Property Tax:
23
Incidence and Excess Burden." Journal of Urban Economics 18: 235-250.
Wildasin, D.E. 1982. \More on the Neutrality of Land Taxation." National Tax Journal 35:
105-108.
24
Footnotes
¤ This
paper was funded by the Lincoln Institute of Land Policy, and a later version will
appear in a Lincoln Institute conference volume edited by Wallace Oates.
1 For
an informal elaboration of these points, as well references to the literature, see Brueckner
(1999).
2 See,
for example, Brueckner (1986), Oates and Schwab (1997), Mills (1998), and Nechyba
(1998).
3 It
is assumed that agricultural land is not subject to the property tax.
4 Since
the CBD extends from both directions on the line, (9) applies only to the right half
of the city. The equation gotten by multiplying both sides of (9) by 2 would state that the
entire city ¯ts its population.
5 Di®erentiating
(6), using the envelope theorem along with (1), yields
@r
@µ
@r
@u
p(x; u)
h[S(x; u; µ)]
(1 + µ)2
1
= ¡
h[S(x; u; µ)]:
1+µ
= ¡
(f1)
(f2)
Because the p(x; u) term is present in (f1) and absent in (f2), @r=@µ exhibits greater spatial
variation than @r=@u, with a larger e®ect closer to the CBD, where p(x; u) is large.
6A
related literature considers dynamic models of land development, asking whether the
optimal development time is a®ected by the magnitude of the property-tax rate. These
papers reach an a±rmative answer to this question, establishing the nonneutrality of the
property tax. See, for example, Bentick (1979), Mills (1981), Wildasin (1982), and Anderson
(1986). As will be seen below, the present analysis addresses a di®erent question, namely
the e®ect of a change in the tax regime on the timing of development, rather than the e®ect
of a change in the tax rate under a given regime.
7 An
important underlying assumption, which makes the analysis workable, is that infrastructure cost in°ation is absent.
25
8 It
can be shown that
v(t; x) =
Z
1
R¿
¡[
r(¿; x)e
t
t
µ(s)ds+i(¿ ¡t)]
d¿
(f3)
9 Figure
2 also shows that the property-tax curve can lie above or below the planner's curve
over the range of increasing returns. Since a comparison of the land-rent intersections is
then ambiguous, the city's population could be larger or smaller than optimal under the
property-tax regime.
10 To
support the optimum, a marginal-cost charge for current inputs should be levied on each
parcel of land at each point in time, re°ecting the ongoing nature of current costs. By
contrast, the marginal-cost charge for infrastructure is levied only once, when the land is
converted to urban use. Because this impact fee exactly pays for incremental infrastructure,
it follows that the cost of the existing infrastructure stock is fully covered by the sequence of
fees levied over time. By contrast, because the marginal cost of current inputs can be larger
or smaller than average cost, total yearly payments under a marginal-cost scheme could be
larger or smaller than current costs.
26