The Optimal Population Distribution across Cities and the Private-Social Wedge David Albouy and Nathan Seegert ∗ July 23, 2010 ∗ The authors would like to thank seminar participants at the Michigan Public Finance Lunch and Urban Economics Association provided valuable feedback, and in particular Richard Arnott, Bernard Salanie, William Strange, Wouter Vermeulen, and Dave Wildasin for their help and advice. The Michigan Center for Local, State, and Urban Policy (CLOSUP) provided research funding and Noah Smith provided valuable research assistance. Any mistakes are our own. Please contact the author by e-mail at [email protected]or by mail at University of Michigan, Department of Economics, 611 Tappan St. Ann Arbor, MI.
Transcript
The Optimal Population Distribution across Cities and
the Private-Social Wedge
David Albouy and Nathan Seegert∗
July 23, 2010
∗The authors would like to thank seminar participants at the Michigan Public Finance Lunch and UrbanEconomics Association provided valuable feedback, and in particular Richard Arnott, Bernard Salanie,William Strange, Wouter Vermeulen, and Dave Wildasin for their help and advice. The Michigan Center forLocal, State, and Urban Policy (CLOSUP) provided research funding and Noah Smith provided valuableresearch assistance. Any mistakes are our own. Please contact the author by e-mail at [email protected] by mail at University of Michigan, Department of Economics, 611 Tappan St. Ann Arbor, MI.
Abstract
The conventional wisdom is that market forces cause cities to be inefficiently large, and public pol-
icy should limit city sizes. This wisdom assumes, unrealistically, that city sites are homogeneous,
that land is given freely to incoming migrants, and that federal taxes are neutral. In a general
model with city heterogeneity and cross-city externalities, we show that cities may be inefficiently
small. This is illustrated in a system of monocentric cities with agglomeration economies in pro-
duction, where cross-city externalities arise from land ownership and federal taxes. A calibrated
model accounting for heterogeneous suggests that in equilibrium, cities may be too numerous and
underpopulated.
JEL Numbers: H24, H5, H77, J61, R1
1 Introduction
Cities define civilization and yet are often perceived as too large. Positive urban externalities
from human capital spillovers – seen by Lucas (1988) as the key to economic growth – and from
greater matching and sharing opportunities (Duranton and Puga 2004), provide the agglomeration
economies that bind firms and workers together in cities. These centripetal forces are countered
by centrifugal forces that keep the entire population from agglomerating into one giant megacity.
Such centrifugal forces include the urban disamenities of congestion, crime, pollution, and con-
tagious disease, all thought to increase with population size. Many economists, including Tolley
(1974), Arnott (1979), Upton (1981), Abdel-Rahman (1987), and Fenge and Meier (2002), have
argued that because migrants to cities do not pay for the negative externalities that they cause, free
migration will cause cities to become inefficiently large from a social point of view. This view is
presented as fact in O’Sullivan’s (2009) Urban Economics textbook, and is easily accepted as it
reinforces ancient (e.g. Biblical) negative stereotypes of cities. Ultimately, this view provides sup-
port for policies to limit urban growth, such as land-use restrictions, and disproportionate federal
transfers towards rural areas.
The canonical argument explaining why cities are too large is analogous to the argument ex-
plaining why free-access highways become overly congested, first presented in Knight (1924). The
cost migrants pay to enter a city is equal to the social average cost rather than the social marginal
cost. This is illustrated in Figure 1, except that costs are translated to benefits using a minus sign.
The social marginal benefit curve, drawn in terms of population, crosses the average benefit curve
at its maximum, , and thus the marginal benefit curve is lower than the average benefit curve
beyond this size. Migrants, who ignore externalities and thus respond to the average benefit, will
continue to enter a city until the average benefit of migration equals the outside option at . This
population level is only stable when benefits are falling with city size, and thus cities can never be
too small.
The analogy of a city to a simple highway, which obviously appeals to urban economists, is
misleading for three fundamental reasons. First, the land sites that cities occupy may differ in
1
the natural advantages they offer to households and firms, such as a mild climate or proximity to
water. Thus, in a multi-site economy it may be efficient to add population to an advantageous
site beyond its isolated optimum at when the alternative is to add population to an inferior site.
Analogously, it makes sense to over-congest a highway when the alternative is a dirt road. Thus,
the outside marginal benefit from residing in another city may be below the peak benefit at , so
that the social optimum is at a point such as , where the social marginal benefit is equal to the
lower outside benefit.
Second, access to a city and its employment or consumption advantages is not free: migrants
must purchase land services and bear commuting costs to access these advantages. Thus, unlike
a free-access highway, migrants must pay a toll to access a city’s opportunities, and this toll is
highest in cities offering the best opportunities. Thus many of the benefits of urbanization are
appropriated by pre-existing land owners rather than by incoming migrants, whose incentive to
move may be below the social average benefit.
Third, workers must pay federal taxes on their wage incomes, which increase with a city’s
advantages to firms but decrease with a city’s advantages to households (Haurin 1980, Roback
1982). Thus, federal taxes create a toll that is highest in areas offering the most to firms and
the least to households, slowing migration to these areas. These effects are modeled by Albouy
(2009b) with exogenous amenities, but are modeled here with amenities that are endogenous to
city population. If urban size benefits firms but harms households, then federal taxes impose tolls
that are highest in the largest cities, strongly discouraging migration to them.
Land income and federal taxes together drive a wedge between the private and social gains that
accrue when a migrant enters a city. Migrants respond to the private average benefit, illustrated by
the dotted line in figure 1, putting the city population at point with free migration, or point
if migrants manage to maximize private benefits in the city. In this example, cities can be vastly
undersized, producing a welfare loss seen as large as the shaded area.
To the extent that individuals pay for land services and federal taxes, payments to land and
labor may be viewed as common resources. Because both rents and wages increase with city size,
2
cities can be too small in a stable market equilibrium as migrants have no incentive to contribute to
these common resources: migrants will artificially prefer less advantageous sites to avoid paying
higher land rents and federal taxes. In essence, inter-city migration decisions involve cross-city
fiscal externalities, which un-internalized, lead to inefficiently small cities. This may be amplified
if big-city residents have greater positive net externalities than small-city residents for non-fiscal
reasons, e.g. if big-city residents have lower greenhouse-gas emissions than small-city residents
(Glaeser and Kahn 2010).
We begin our argument in section 2 using a basic representation of cities, which may be viewed
as clubs with external spillovers. In section 3 we provide a microeconomic foundation to this
representation with a system of cities based on the canonical monocentric-city model of Alonso
(1964), Muth (1969), and Mills (1967) to give form to our functions and concreteness to our
simulations. Urban economies of scale are modeled through inter-firm productivity spillovers
that lead to increasing returns at the city level, while urban diseconomies are modeled through
commuting costs.1 In addition, city sites are heterogeneous in the natural advantages they provide
to firms in productivity or to households in quality of life. This model is calibrated as realistically
as possible in Appendix A to demonstrate the theoretical results concretely and to illustrate their
plausibility in the reality. Section 4 improves on existing work by allowing the number of cities to
vary, analyzing differences on the "extensive" margin, i.e. the number of sites occupied, as well
as on the "intensive" margin, i.e. on how the population is distributed across a fixed number of
occupied sites. The distribution of natural advantages across sites is modeled using Zipf’s Law.
Throughout the analysis we consider four types of population allocations. We begin with the
standard problem of how a city planner maximizes the average welfare of the inhabitants of a single
city, ignoring the effects on the outside population and internalizing any cross-city externalities.
Second, we consider the welfare optimum for an entire population, whereby a federal planner
allocates individuals across heterogenous sites, determining the number and size of cities. We
put particular emphasis on the case where individuals are equally well off in all cities, as would
1This model can be expanded to incorporate other realistic features of cities, e.g. non-central firm placement inLucas and Rossi-Hansberg (2002), without losing the main point.
3
be implied by free mobility. Third, we look at the equilibrium that occurs when populations are
freely mobile, but in a private ownership economy where they must rent land and pay federal taxes.
Fourth, we consider political equilibria in a private ownership economy that could arise when local
governments restrict population flows into their city, ignoring the effects on other cities. These
four cases share a symmetry illustrated below:
Multiple Authority Single Authority
Planned Economy City Planner Federal Planner
Private Ownership Political Equilibrium Competitive Equilibrium
We find that the efficient population distribution tends to concentrate the population in the fewest
number of cities, fewer than would be allocated by isolated city planners. Meanwhile, equilibrium
forces disperse the population inefficiently, causing inferior sites to be inhabited, with local polit-
ical control potentially exacerbating this problem. Examples throughout the paper are illustrated
graphically using the calibrated model from section 3. The simulation in section 4, which endoge-
nizes the number of cities, demonstrates that there may be (roughly) 40 percent too many occupied
sites, with welfare costs equal to 1 percent of GDP.
2 Basic Model
2.1 Planned Economy
A homogenous population, numbering , must be allocated across a set of sites, J = {0 1 2 }, indexed by , with the population at each site given by , so that
X=0
= ≥ 0 for all (1)
The non-negativty part of the condition reflects that sites may be uninhabited. The population
allocation in vector form is written as N = (0 1 ). Assume that the social welfare
4
function can be written as an additively separable function
(N) =
X=0
() (2)
where () is the social net benefit of having members living on site ; it is normalized so
that an uninhabited a site produces no benefit (0) = 0. The social benefit, now always meant
to be net of costs, includes the value of goods produced by residents and the amenities they enjoy
net of the disamenities they endure. Some benefits only affect residents inside the city — such
as climate amenities, transportation costs, or congestion – while others – such as global pollution,
technological innovations, and federal tax payments – may affect residents of other cities. Region
= 0 is assumed to be a non-urban area with 0() = 0 , where 0 is a constant.
By definition, the social average benefit of residing in city , () ≡ () .
Assume that () is twice continuously differentiable, strictly quasi-concave, and
( )
= 0 for some finite
0, for all (3)
making the function single-peaked: urban economies of scale dominate diseconomies of
scale for small populations, while the opposite holds true of large populations. This single peak
at designates the choice of a city planner (hence "cp") maximizing the social average benefit
welfare of residents, assuming that all city benefits are internalized.
The social marginal benefit of residing in city is given by the following identity
() ≡ ()
= () +
()
(4)
The second term, (()) ≡ (), or within-city wedge, captures the effect
of an additional migrant on infra-marginal inhabitants of city through scale economies. It makes
higher than when is increasing, and lower when is decreasing, and
5
therefore at :
( ) = (
) (cp)
Because there is no coordination between city planners, it is unclear which sites, J ⊆ J , are
inhabited under a city planner allocation, and hence how many cities = |J | this solution
implies. It is possible to imagine mechanisms that allow cities with the highest maximum ’s
to be inhabited first. An integer problem arises if the optima do not add up to the total population,
i.e.P
∈J 6= , for J ⊆ J .
The federal optimum, which determines the efficient population distribution, maximizes the
social welfare given in (2) subject to the constraints in (1), implying the necessary condition
( ) = (
) = (fp)
across any two sites and that are inhabited, where ≥ 0 is the multiplier on the population
constraint, and refers to the population chosen by the federal planner. The sites efficiently
inhabited, J – which gives the efficient number of cities, =¯J
¯– can be determined
through backwards induction: for every J ⊆ J , the efficient population allocation N(J ) can
be determined using (fp) and the associated second-order conditions; then the J that maximizes
[N(J )] over (J ) determines the solution J andN= N(J ). Since the federal plan-
ner can allocate populations below the peak of , it does not face an integer problem.
When cities are heterogeneous N will generally differ N because the federal planner al-
locates a greater number of individuals to more advantageous sites. Only if all sites have the
same maximum ( ) and if integer problems are absent, do the two solutions coincide.2
Consider the case where sites are ordered so that, if
() () for all 0
2To show this, let satisfy (3) for all, then by homogeneity, all cities will have the same (), and
through the absence of an integer problem = ∗, the optimal number of cities. With homogeneity, andequal allocation of will satisfy (fp), however the global optimum also maximizes each individual
6
which defines to be uniformly more advantageous than site . When such an ordering exists,
consider the city planner solutions, which imply ( ) ≥ (
) (
),
where the first inequality holds by definition of , and the second from being uniformly more
advantageous. Since ( ) = (
), this implies that (
) (
),
and hence city planners do not achieve a federal optimum. Furthermore, the federal optimum
allocates more population to the superior site: when 0 welfare can always be improved by
moving a resident away from city to city , since ( ) (
), as the first-order
effects on welfare are zero, and so
. When
= 0, then
≥
trivially.
Figure 1 illustrates this difference: here 1 is given by the solid curve and 2, by the
outside option, where point gives the city planner solution, and point , the federal planner.
Figure 2 illustrates an example with 2 cities where city 1 is uniformly more advantage than city 2,
and where 1
1 = 2 =
2
2 . The city planner solution is at points and ,
and the federal planner, at point . In both figures, the deadweight loss of the city planner solution
is equal to the area between the curves, from the efficient to the inefficient population levels.
2.2 Private Ownership and Individual Incentives
Because of cross-city spillovers, residency in a city may affect the income or amenities of residents
in other cities. This drives a wedge between the average social and private benefit of residing in a
city, which we refer to as the across-city wedge:
() ≡ ()− () (5)
where () is the private average benefit, which like the is assumed twice continuously
differentiable, strictly quasi-concave, and is single-peaked as in (3) We normalize private average
benefits so that they equal the sum of social average benefits, so the the across-city wedges sum to
zero. X
() = 0 (6)
7
Note that this normalization holds for inhabited sites; as different sites are inhabited, these wedges
may change.3
In a competitive equilibrium, all individuals are mobile across cities, and receive the same
private average benefit within or across cities. In this case, the population distribution is determined
by the population and non-negativity constraints in (1), and a mobility condition under which no
resident can be made better off by moving, and therefore
( ) = (
) (ce)
for all inhabited cities and . In addition, the population of cities should be stable, requiring
( )
+
( )
≤ 0 (7)
for all inhabited cities and . Otherwise the population size will be unstable, because increasing
the size of one of the cities will make it more beneficial than the others, attracting further migrants.
The competitive equilibrium concept determines the population levels across sites that are inhab-
ited, but does not determine what sites are inhabited. When the non-urban area is inhabited N
can be determined by ( ) = 0, as with point , in figure 1
The competitive equilibrium may not maximize the welfare of city residents, and so it plausible
that existing residents or city developers may limit the size of a city to boost local welfare levels.
A political equilibrium, denoted with "pe," refers here to the situation where mobility is restricted
to maximize the of a city. This solution, given by point in figure 1, is analogous to the city
planner optimum, except that cross-city externalities are not internalized, implying
( ) = (
) (pe)
3This is justified in a closed economy where all external effects are fiscal. It is possible that residence anywherecould contribute, say, negative externalities, e.g. some level of greenhouse gases, but in this case it is formally innocu-ous to subtract off some base level, so that the sum still averages to zero. This comes from the fact that the populationlevel is fixed. In an optimal population problem, the benefit of an additional life would be weighed against the ex-ternal effect that life imposes on others, which would correspond to the case where the multiplier on the populationconstraint, = 0.
8
As with the city-planner problem, the political equilibrium is subject to integer problems and does
not determine which sites will be inhabited. In the micro-foundation discussed in section 3.6,
we provide a mechanism by which the sites with the highest maximum are inhabited first.
For a given number of cities the competitive equilibrium population levels will be larger than the
political equilibrium levels, with the exception of at most one city.4
2.3 Private versus Efficient Incentives
The difference between efficient and private incentives is equal to the sum of the across-city and
within-city wedges:
()− () =() +() (8)
These two wedges are illustrated in figure 3 to the right of where both wedges are positive. The
previous literature has emphasized how eliminating within-city wedges should eliminate locational
inefficiency, a point that already breaks down under heterogeneity. With homogeneity and ignoring
integer problems, this inefficiency arises as all points to the right of = , are potentially
stable competitive equilibria, so long as they exceed (0), while no points to the left are. This
leads to the textbook maxim that "cities are not too small" (O’Sullivan 2009) while they can be too
big. But when across-city wedges are present, stable competitive equilibria to the right of can
be inefficiently small, even when sites are homogenous.
To show this, assume city sites are homogenous and that , where both of these
numbers are small relative to . Then, the unique political equilibrium, which because of ho-
mogeneity is one of many competitive equilibria, produces an overabundance of inefficiently small
cities. This is illustrated in figure 4, where the number of cities = . The welfare max-
imum is achieved at point , since homogeneity implies = , and by single-peakedness,
() ( ). As cities are homogenous, then by (6), () = (), and
4The condition that ≤
for all in where is in the set J \{} for = 0 , without loss of generality.This condition is a direct implication of the stability condition (7).
9
thus the peak of private benefits at point is below that of social benefits because of cross-city
spillovers. If a single city is given the population , its will be lower at point , although
its is at point , with its given by the distance between and . If all cities coor-
dinate to lie at point , the cross-city spillovers will cause the curve to rise by the average
amount of the spillovers, which are additive. Yet the residents of a single city could be made better
off by shrinking the city to point to maximize its own private benefits, while still receiving the
spillover benefits of larger cities. Yet, if all cities do this, the equilibrium will return to point , as
the spillovers are lost and the curve shifts back down.
When city sites are heterogeneous, the following proposition proves to be helpful:
Proposition 1 A competitive equilibrium produces under-sized cities in the most advantageous
cities, i.e.
for all ∈ {1 } for some integer ≥ 1, if the wedges are such that
( )−(
) (
)−(
) (9)
The proof of this proposition, shown in the Appendix, rests on the fact that ( ) −
( ) (
)−(
) implies (
) (
) and the fact
that across-city wedges add up to zero by (6). There are two cases to consider in the proposition
regarding the wedge. In the first case the right-hand side of condition (9) is positive, in which case
is necessary for
for all ∈ {1 } for some integer ≥ 1.
In the second case, the right-hand side of condition (9) is negative and , is
sufficient for the competitive equilibrium to produce undersized cities for the most advantageous
cities. In both cases, the important relationship is that more advantageous cities have larger wedges,
.
10
3 Parametric System of Monocentric Cities
3.1 City Structure, Commuting, Production, and Natural Advantages
In each city, individuals reside around a central business district (CBD) where all urban production
takes place. The city expands radially from the CBD with the conventional assumptions that land
is homogenous, commuting costs are a function of distance from the CBD, and land at the fringe
is available at no cost. Each resident demands a lot size with a fixed area, normalized to one, so
that a city of radius contains a population = ()2. The average resident lives a distance of
= (23) = (23)()12.
Besides land, individuals consume a produced good, , which is tradable across cities, and has
a price normalized to one. Utility is given by (), which is strictly increasing and quasi-
concave in both arguments, with the second argument, , representing the quality of life in city .
is uniform within a city, and is independent of city size until section 3.8. We restrict attention
to where utilities are the same within a city, meaning that consumption levels, , do not depend on
, which is justified in the market case where individuals can freely choose their location.
Each individual supplies a single unit of labor, which is divided between working and commut-
ing. The net labor supply of a worker who lives at distance is () = 1 − where is the
time cost of commuting. In addition, workers face a material cost of commuting, , measured
in units of the tradable good. In a city, aggregate labor supply and material commuting costs are
() = − (23)32 and (23)32, where ≡
−12 and ≡ −12
Aggregate city production is ( ) = (), where is the natural advantage
of city in productivity, while the term , with ∈ (0 12), models local scale economies,
which are external to firms but internal to cities, so that firms exhibit constant returns but cities ex-
hibit increasing returns.5 The social average product of population, ( ) ≡ [ ( )−(23)
32] , initially increases with population through the production externality, but even-
tually decreases as commuting costs increase. The social average product reaches a maximum
5This parameterization follows Dixit (1973) and encompasses the micro-foundations of local information spilloversas well as search and matching economies, as reviewed in Duranton and Puga (2004)
11
where it meets the social marginal product meet, i.e. ( ) = ( ) ≡ ( )−
12.
The social marginal product is the sum of three terms, each functions of and :
= + − (10)
where = (), the marginal product that accrues to the firm from placing a worker at
; = (), the agglomeration externality, which goes to firms for which the household
does not work; = [(13) + ]
12, the increase in average commuting costs,
netting out the average commute time.6
3.2 System of Cities and Mobility
Each city site differs in its exogenous natural advantages to firms and residents ( ). In ad-
dition, we posit a hinterland region, indexed with = 0, which is free of commuting costs and
agglomeration economies, so that its production is 00 with 0 = 0 = 0. While
somewhat artificial, the hinterland may be used to model agricultural opportunities or a set of
small homogenous cities occupying inferior sites, much like a competitive fringe, so that the hin-
terland population is proportional to the number of fringe cities. When the hinterland is occupied,
it simplifies formulas considerably and aids intuition, without sacrificing much in content.
When the population is mobile across cities, individuals receive the same utility regardless of
where they live, otherwise they will move. Across any two inhabited sites, and , ( ) =
( ) = , a common utility level. Let ( ) be the implicit solution to ( ) =
with 0 and 0. Inverting this mobility condition, we have = ( ),
6The full mathematical forms for social and marginal average products are
( ) = n[1 + − [1 + (23)]12
o−
12 (11)
( ) = h1− (23)12
i− (23)12 (12)
12
which means that consumption in city equals the consumption needed to obtain in a city
with quality-of-life , so that cities with higher require lower . Mobility requires that
the difference in consumption in both cities must equal the compensating differential which makes
up for differences in quality of life across both cities
− = ( )− ( ) (13)
3.3 Planned Economies
An isolated city planner will simply maximize the consumption of residents, given by ( ),
since all resources are kept within the city and if fixed. Thus, is determined implicitly by
the equation ( ) = (
), which is an implicit function of , but not .
Furthermore, it increases with only through material commuting costs, as more productive cities
more easily bear these costs.7
A federal planner allocates population and consumption levels, (Nx) = {
}=0
across all city sites to maximize the generalized utilitarian welfare function
(Nx) =X≥0
1
[( )]
(14)
In addition to the population constraints in (1), we require that aggregate production is equal to the
sum of consumption and material commuting costs
00 +X≥1
() ≥
X≥0
+X≥1(23)
32 . (15)
Taking N parametrically, and solving out the consumption plan x(N) and substituting this into
(14), produces a value function £Nx(N)
¤like (2).
7With = 0, = [3((1 + 2))]2, which is independent of natural advantages.
13
Using the envelope condition for the general case
=1
h(
)i+
h(
)−
i− (16)
which at the social optimum equals zero if 0, where =
h(
)i−1
( )
is the marginal welfare of a consumption unit, and 0 is the multiplier on the population
constraint. We focus on the case of perfect mobility, equivalent to the Rawlsian case where →−∞. Taking this limit in (16), this condition may be rearranged in units of the numeraire, allowing
us to reintroduce the social marginal benefit notation:
( ) = (
)− (
) = if 0 (17)
where = . (17) equates the social marginal benefit with the social marginal product minus
the consumption needed to provide in a city with is the same for every inhabited city. Be-
tween any two inhabited cities, ( )− (
) = (
)−( ),
meaning that the social marginal productivity in city should be greater than in a nicer city, , by
the differential that compensates those in city for not living in city , accounting for their mobil-
ity. Condition (17) is a parametric form of () increasing in both and and is dependent on
a global level of utility .8 The social average benefit can be calculated through integration, by
definition () =1
R 0().
3.4 Calibration
The shape of these curves is already given in figures 1 through 4 using parameter values whose
calibration is explained in Appendix A. Income is measured on an annual per-capita basis. City
population density is 2,800 per square mile. Half of the population goes to work for up to 8
hours a day, 250 days a year, at a wage of $22 per hour, and commutes an average distance of 8
8The main impact of letting vary is that it determines the marginal rate of substitution between consumption andquality-of-life through income effects.
14
miles at a material cost of 50 cents per mile and at a speed of 20 miles per hour. This implies
= 00001333 and = $1333, where distance is measured in individual lots, and a typical
city size of 1.27 million people. The agglomeration parameter is set to 006 (Ciccone and Hall,
1996) and produces a typical = $9 468, this produces.
If we consider again the case of homogenous cities (figure 4) where = and pretend
as if all cities are typical, then the calibration implies (127M) = −$414, and efficient size
= 721 000 million, meaning that the typical city is oversized. The shape of the curves is
very sensitive to , a parameter whose empirical value is uncertain, but thought to range between
0.03 and 0.08. The agglomeration concept here has all workers working in the CBD, providing a
stronger form of agglomeration than should occur in an actual metropolitan area, where production
is dispersed. It also depends on a commuting structure which implies that the marginal commut-
ing cost always exceeds the average by 50 percent: thus, the average time-cost of 10 percent of
working-time translates to a marginal cost of 15-percent. This translates to an external cost equal
to 5 percent of income, which along with external material commuting costs over 2 percent of
income, must be weighed against the external benefit, equal to percent of income.9 Overall, the
monocentric city model creates both strong economies and diseconomies of scale. Inference about
city-size efficiency becomes more complex with city-site heterogeneity, which requires a treatment
of private ownership.
3.5 Private Ownership and Individual Incentives
With private ownership, individuals receive income from labor and land, and pay for taxes, rent,
tradable consumption, and material commuting costs. Factor and output markets are competitive,
so that firms pay a wage = per labor unit, of which a worker at distance supplies ().
Labor income is taxed at the federal rate of ∈ [0 1], calibrated at = 033, leaving workers
with (1− )().10 Federal taxes are redistributed in the form of federal transfers , which
ª− (13)1210It is appropriate to use the marginal tax rate since we are considering marginal changes in labor income due to
migration decisions. See Albouy (2009b) for further discussion.
15
may be location dependent, and obey the constraintP
= P
(). When federal
transfers are not tied to local wage levels, federal taxes turn a fraction of labor income into a
common resource, reducing individuals’ incentive to move to areas with high wages.11
Within a city, rent fully compensates for distance from the CBD, causing all residents to pay
the same in location costs, leaving them with the same consumption, . Figure 5 depicts the cone-
shaped rent gradient () = ((1− ) + )( − ),which declines to () = 0 at the edge
of the city, where we assume the opportunity cost of land is zero.12 In terms of population, the
average rent in city is
=1
3
£(1− )
+
¤12 =
1
3(0) (18)
where (0) gives the full location cost, given by the height of the cylinder in figure 5. The rental
income of residents in city is
= (1− ) + (19)
where = 1
P
=0 is the average rent paid in all cities, and ∈ [0 1] is an exogenously
fixed parameter. Much of the previous literature has focused on the special case where = 0
implying individuals receive the average rental income in the city they live in. This assumption
appears unrealistic for modeling migration decisions as a new migrant to city inherits a free
plot of land at distance and gives up any other land holdings without payment. Consequently,
residents ignore the average cost of rent in a city. When = 1, migrants to a city have to pay
rent on any plot they occupy, but still receive income from land, albeit in an amount unrelated to
their location decision. This assumption treats individuals anonymously and appears to be the most
realistic situation because migrants pay to access the advantages of a city through rent. The larger
11Emprically, Albouy (2009b) finds that federal transfers are not strongly correlated with wage levels in the UnitedStates, however Albouy (2010) finds that they are negatively related in Canada, increasing the size of the across-cityfiscal spillovers.
12More generally, we discuss land rents that are differential land rents. Assuming that the opportunity cost of landis greater than zero adds little to the model unless the opportunity cost varies with or . For instance it may bepossible that sunnier land is more amenable to urban residents, but also contributes to agricultural productivity, raisingthe opportunity cost as well. Given the low value of agricultural land relative to residential land, these effects are likelyto be of small consequence.
16
, the more land income can be thought of as a common (federal) resource, as a higher share of
rent is redistributed within the economy at large, rather than only within the city.13 Further detail
on how affects the model is provided in Appendix C.
Adding income and expenditure items leads to the budget constraint for a resident in city at
distance of the form (1− )(1− ) + + = + + (). Consumption, or net
income, may be written in terms of the population size by substituting in from (18):
( ; ) =£(1− )
(1− 12)−
12¤+ [ + ] (20)
The first term indicates labor income net of all location costs and the second indicates income
redistributed from land and federal revenues, which may change with location. This net income
level minus the level of consumption needed in a city with is interpreted as the private average
benefit of living in city ,
() = ( ; )− ( ) (21)
Equation (21) is the parametric form of (pe), which is increasing in natural advantages and
as well as fiscal advantages and . This parametric shape of the with respect to has
been used in figures 1 through 4 in the case where = 033, = 1 and = across areas.
3.6 Competitive and Political Equilibria
In a competitive equilibrium, the mobility condition (13) is satisfied, which by (21) implies that
’s are equal across cities, as in (ce). An equilibrium consumption and population distrib-
ution (xN) will satisfy the constraints in (1) and (15), although this still allows for multiple
13If migrants owned plots of land in an origin city, they would still sell the land when moving to the destinationcity, since they can only live in one city at a time. This would unnecessarily complicate the analysis through incomeeffects, and require us to consider the origin as well as destination of migrants. The situation with = 1 may also becharacterized as one of a migrant from a typical city in the economy, as denotes the average rent on a plot of landanywhere. One could also assume that land is owned by the federal government or absentee landlords. In these casesrental earnings are the same and zero for all individuals.
17
equilibria. If in equation (21) is set to a constant, as in the case of a small city, then the popula-
tion, , becomes an implicit and increasing function of and as well as and . Figure
6 graphs this implicit function, ( + ) with = 1, against and . Note the
function accounts for endogenous agglomeration effects on productivity and that average rents,
increase with , and thus also increase with and ; without taxes, rents increase at the exact
same rate.14
A political equilibrium may be micro-founded in several different ways. We consider equilibria
established in two stages. In the first, coalitions of individuals are formed sequentially until either
all individuals are in a coalition or forming a coalition from the set of individuals not in a coalition
is not advantageous. In the second, coalitions inhabit sites and are able to limit mobility to varying
degrees. At one extreme, coalitions are able to perfectly limit migration, so that city populations
attain the maximum of , leading to the highest level of consumption. At the other
extreme, coalitions are unable to limit migration, causing utility to be the same within and across
all cities, leading to a particular competitive equilibrium we label ∗. 15 This conceptualization
allows for the fact that coalitions are forward-looking and able to predict how mobility will alter
welfare in a city.16
14Because land is not used in production, wages do not negatively capitalize consumption amenities as in Roback(1982) – see Albouy (2009b) for details. However, when not all sites are inhabited, individuals may choose to reside inareas with high but low which can produce a negative correlation between wages and consumption amenities.
15These political equilibria resemble the equilibria in Helsley and Strange (2004), which models how developerscan influence city size, according to the level of competition between them.
16The first case is one possible formalization of the extensive margin decision for the competitive equilibrium. Theliterature typically assumes that new cities are created either by developers or when one resident is able to start a newcity and receive higher utility. The latter option means that utility will generally fall to (1) of the next-best city,which in this model can be very low, meaning cities can be excessively large. We find this option with no coordinationwhatsoever to be unrealistic.
18
3.7 Private versus Efficient Incentives
With the private economy described, we can write the difference between the social marginal and
private average benefit as the sum of two wedges, as in (8):
()− () = − ∗| {z }()
+ ( − ) + ( − )| {z }()
(22a)
= + [1− 12]− − (22b)
where ∗ = +(13) is the average rent with = 0. The across-city wedge,() =
( − ) + ( − ), is equal to the net average rental payment, subtracting off rental in-
come received federally, plus the net average federal tax payment, subtracting off any transfers.
The first wedge, , reflects the Henry George Theorem result (Arnott and Stiglitz, 1979)
that at , differential land rents – ignoring the impact of federal taxes – should equal the ag-
glomeration externality. A migrant confers a positive benefit on residents equal to but reduces
local rental income by ∗ , and at the these two effects offset exactly. When land is purchased,
= 1, a migrant does not reduce local rental income, and so the social marginal benefit is greater
than the private benefit, as reflected in the across-city wedge.17 In addition, with heterogenous
sites, 0 in more advantageous sites is federally efficient, as the values of locations depend
on differences in natural advantages as well as agglomeration rents.
The expression for the second wedge, , shows how land-rent and federal-tax payments,
proportional to and , turn land and labor income into common resources. As and
increase with city population, so does the size of this wedge.18 In the case where transfers are
17The same is true in the case where each city has a non-congestible public-good paid for equally by residents(Flatters et al. 1974, Albouy 2010). An incoming migrant can reduce the cost by 1 to each resident, withoutreducing the amount of the public good available to others, introducing another wedge. In the case where public-goods are set optimally, this would involve adding the public-good externality = (1 − ) , where is the expenditure on public goods, and ∈ [0 1] expresses the amount of congestion in the public good.
18Substituting in for , and the wedge is given by
() = (13)( (1− ) + )
12 +
h1− (23)12
i− (+ ) (22c)
19
homogenous, = , the tax wedge is directly proportional to labor income.
The second expression, (22b), expresses that the two wedges are driven fundamentally by
externalities from within-city agglomeration and across-city tax-payment inequalities. The tax-
term [1−12] reflects the full effect on taxable income from city , as marginal loss in labor
time is higher than the average cost. These externalities are juxtaposed against the differential fiscal
transfers from land and labor,+ . Interpreted differently, this equation says the efficient federal
transfer to residents of city , ∗ = − + [1 − 12], will subsidize agglomeration
economies, negate differences in rental incomes – which could result from differences in source-
based taxes – and refund federal differences in federal tax payments, adjusted for marginal time
costs.19
With the parametric form of (), and for a city uniformly more advantageous than ,
( ) (
) is a necessary condition for the most advantageous cities to be un-
dersized. By equation (22) the necessary condition that the wedge is increasing in the production
amenity holds. Furthermore, the sufficient condition from Proposition 1 can be rewritten as
(1− )(∗ − ∗) +
3(
−
) ( + )( − ) (23)
The right-hand side of (23) is always positive, while the first term on the left goes to zero as → 1,
which leaves only the second minor interaction term, which is relatively small.
In the situation where = 1, figure 6 compares populations allocated under the federal planner
and a competitive equilibrium that both produce a population level of = 721000 at a base
amenity level. With increasing natural productivity, , the planner’s population increases at a
higher rate than the equilibrium. This is understandable as the across-city wedge, which increases
at a high rate with the natural productivity level; in fact the slope is greater than one. This occurs
because has a large direct effect on , while around the calibrated values, increases in
19In a closed-city context, Wildasin (1985) notes that the time costs of commuting are implicitly deducted fromfederal taxes, although the material costs are not, and argues that taxes lead to excessive sprawl by reducing the time-cost of commuting. This mechanism does not work in a closed-city setting with fixed lot sizes, but it does matter inan open-city setting by leveling the slope of the wage gradient, causing it to hit zero at a further distance, implying alarger population.
20
have a relatively small effect. With natural quality-of-life differences, , increases and
then decreases with , causing the tax wedge to increase and eventually decline. The planner’s
population increases at a slightly lower rate than the equilibrium, as past the base amenity level,
and the size of the rent wedge grows at a smaller rate than the within-city wedge (not shown).
3.8 Endogenous Quality of Life
The model so far has taken quality of life, , as exogenous, although it may depend on the pop-
ulation level, . The relationship could be negative, if higher population levels bring about urban
disamenities such as pollution, crime, congestion, or disease. At the same time, a higher popula-
tion level should increase the availability of non-tradeable private goods, as well as public goods,
through greater inter-jurisdictional choices, as in Tiebout (1956). Theoretically, it is ambiguous
whether a higher city population reduces quality of life, although it is generally assumed: in this
case, higher population leads to a lower within-city-wedge.20
For now consider the simple case where = 0
, for some constant . In this case, the
social marginal benefit in (17) gains an additional term
() = ( )− ( )−
(24)
In the case where 0, the social marginal benefit is made lower by (− ) , which
should be added to the within-city wedge in (22a). Analyzing the quality of life of U.S. cities,
Albouy (2008) estimates that is close to zero, although it cannot control for whether populations
disproportionately inhabit sites with greater amenities not measured in the data. Nevertheless, it
seems unlikely that takes on a large negative values and so we calibrate the model in the third row
of figure 6 in the case where = −02. This adjustment actually increases the tax wedge slightly,
making the the disparity between the efficient and equilibrium population levels very small.
20It is also possible that a higher population could affect amenities in other cities, such as through lower averagelevels of global pollution, or through greater shopping externalities.
21
4 City Numbers and the Population Distribution
The goal of this section is to endogenize the number of cities determined by our solution concepts.
As the total population increases the federal planner and political coalitions will inhabit more cities.
As we will see, the private-social wedge may cause too many sites to be inhabited in equilibrium
rather than concentrating the population in the most advantageous sites where wages and rents are
highest.
4.1 Homogenous City Formation
We begin with the case of homogenous sites, illustrated in figure 4 when the total population is
large. As the total population grows, new sites are populated by the federal planner to maxi-
mize the social welfare. Eventually, as the integer problem disappears, city sizes converge non-
monotonically to the single-city maxima. Figure 7 graphically depicts the population the federal
planner allocates to the first city as a function of the total population (Warning: numbers based
on a previous calibration). As the total population increases and the federal planner distributes
population across more cities, as seen in figure 7b.
Residents able to form coalitions in a political equilibrium consider only their own potential
consumption in inhabiting a new site. When coalitions are unable to limit mobility, the population
inhabits more than one city when consumption levels increase in all cities. When coalitions are able
to perfectly limit mobility coalitions inhabit sites and limit population at the peak level , so that
j cities are inhabited when = . Figure 7 demonstrates that the political equilibrium
weakly populates more cities than the federal planner.
4.2 Small System of Heterogenous Cities
For simplicity, let cities differ only in natural production-amenity levels with a distribution of the
form = + where follows a uniform distribution, where +1 = + $120. Here we
consider five inhabited city-sites and a hinterland that offers a relatively high outside benefit and
22
contains an effectively infinite population, so that 0 = 0. Thus, the fiscal spillovers from
the five cities are spread across such a large portion of the population, that they do not affect the
hinterland benefit if the population is shifted.21
Putting the population on the horizontal axis, figure 8a illustrates how the federal planner
equates the across all sites to determine , where each new curve comes starts at the
population used cumulatively by cities with a higher productivity on the left. Figure 8b shows
an analogous graph with the competitive equilibrium, which equates the with the outside
option. In this, admittedly stylized, example, we see that the efficient city sizes are much larger
than those in the competitive equilibrium, especially for the most advantageous cities.
4.3 Large System of Heterogenous Cities
In this example, we consider the total population needed to populate 135 heterogeneous cities.
We let follows a distribution resembling a Pareto distribution.22 To distribute population across
135 cities, the federal planner requires a total population close to 190 million while the political
equilibrium requires only 155 million (Warning: numbers based on a previous calibration). In
comparison, a total population of 155 million are efficiently allocated across only 94 cities im-
plying that the political equilibrium populates about 40 percent more cities than is efficient. The
welfare loss associated with allocating population across too many cities is approximately $250
per person per year. Figure 9 provides scatter plots of both the political and efficient allocation of
population across 135 cities.
Figure 10 is a scatter plot of the political allocation with log rank on the vertical axis and log
population on the horizontal axis with the following linear relationship
= 1053− 1005 (25)
21Thus the effective = 0 and = (13)($17500) = $5833.22This distribution is chosen to match empirical findings, including Zipf’s law. A complete explanation of the
distribution is given in Appendix A.
23
This linear relationship and scatter plot matches closely the empirical findings of George Zipf
(1949), Gabaix (1999), and others that city size approximately follows Zipf’s law. The exact
distribution of city sizes and the reasons cities follow them are still open questions in the literature.
One explanation given by Krugman (1996a) is that the distribution of natural advantages is the key
determinant in the resultant distribution of cities. Krugman supports this claim by noting that the
relationship between the log rank and flow of the 25 largest rivers in the United States is roughly
linear. Our model provides micro-foundations to Krugman’s conjecture by varying the level of
natural advantages to produce a distribution of cities that follows Zipf’s law.
5 Conclusion
The above analysis does not prove that cities are necessarily too small, but it does call into question
the necessity of cities being too large in an economy where federal taxes are paid and residential
land must be purchased. As a result, the ability of local governments to reduce city sizes by re-
stricting development through impact fees, green belts, and zoning may do much to reduce overall
welfare, as they will likely neglect across-city spillovers, fiscal and otherwise, and allow a small
minority to monopolize the best sites, forcing others to occupy less naturally advantageous sites.
Many other factors certainly play a role in determining efficient city sizes, among them, the
ability of governments to provide adequate regulation, public goods, and infrastructure to make a
large city function well. This may be a particular challenge in developing countries, where rapidly
growing cities suffer disproportionately from negative externalities such as dirty air, infectious
disease, and debilitating traffic. Moreover, in these cities the marginal resident, perhaps a poor
rural migrant, may not pay federal taxes or for their land costs by working in the informal sector
and squatting on land they have no property rights to. Thus, the problem of under-sized cities
may be a relatively new one historically, seen primarily in the developed world, but one that will
become increasingly important as property rights develop, federal governments tax increasingly,
and urbanization rises.
24
References
ABDEL-RAHMAN, H.M. (1988), "Product Differentiation, Monopolistic Competition and City
Size", Regional Science and Urban Economics 18 69-86.
ALBOUY, D. (2008), "Are Big Cities Really Bad Places to Live? Improving Quality-of-Life
Estimates across Cities", Working Paper no. 14472 (November), NBER, Cambridge, MA.
ALBOUY, D. (2009a), "What are Cities Worth? Land Rents, Local Productivity, and the Value of
Amenities", Working Paper no. 14981 (May), NBER, Cambridge, MA.
ALBOUY, D. (2009b), "The Unequal Geographic Burden of Federal Taxation", Journal of Political
Economy 117, 635-667
ALBOUY, D. (2010), "Assessing the Efficiency and Equity of Federal Fiscal Equalization" Work-
ing Paper no. 16144 (July), NBER, Cambridge, MA.
ALONSO, W. (1964), "Location and Land Use", Cambridge:Harvard University Press
ARNOTT, R. (1979), "Optimal City Size in a Spatial Economy", Journal of Urban Economics.
ARNOTT, R. and STIGLITZ, J. (1979), "Land Rents, Local Expenditures, and Optimal City Size",
Quarterly Journal of Economics, 93 471-500.
BEHRENS, K.; DURANTON, G. and ROBERT-NICOUD, F. (2010), "Productive Cities: Sorting,
Selection, and Agglomeration." University of Toronto, mimeo.
BUCHANAN, J., and GOETZ, C. (1972), Efficiency Limits of Fiscal Mobility: An Assessment of
the Tiebout Model", Journal of Public Economics I: 25-43.
BUREAU OF TRANSPORTATION STATISTICS (2003) "Commuting Expenses: Disparity for
the Working Poor." Department of Transportation.
CICCONE, A. and HALL, R.E.(1996) "Productivity and the Density of Economic Activity." Amer-
ican Economic Review, 86: 54-70.
25
DIXIT, A. (1973), "Optimum Factory Town", The Bell Journal of Economics and Management
Science, 4 No.2 637-651.
DURANTON, G. and PUGA, D. (2004), "Micro-foundations of urban agglomeration economies",
In: Henderson, V., Thisse, F. (Eds.), Handbook of Regional and Urban Ecnomics, vol. 4. North-
Holland, Amsterdam: 2063-2117.
FENGE, R. and MEIER, V. (2001), "Why Cities Should Not be Subsidized", Journal of Urban
Economics, 52 433-447.
FLATTERS, F; HENDERSON, V, and MIESZOWSKI, P. (1974) "Public Goods, Efficiency, and
Regional Fiscal Equalization." Journal of Public Economics, 3, 99-112.
FUJITA, M. (1989) Urban Economic Theory Cambridge: Cambridge University Press.
GABAIX, X. (1999), "Zipf’s Law and the Growth of Cities." The American Economic Review,
89(2), 129 - 132.
GLAESER, E, and KAHN, M(2010) "The Greenness of Cities: Carbon Dioxide Emissions and
Urban Development." Journal of Urban Economics.
HAURIN, D. R. (1980), "The Regional Distribution of Population, Migration, and Climate", Quar-
terly Journal of Economics, XCV, 293 -308.
HELSLEY, R. W. and STRANGE, W. C. (1998) "Private Government", Journal of Public Eco-
nomics, 69 (2), 281 -304.
HENDERSON, J. V. (1986), "Efficiency of Resource Usage and City Size", Journal of Urban
Economics, 19, 47-70.
HU, P. and REUSCHER, T. (2004) "Summary of Travel Trends: 2001 National Household Travel
Survey" U.S. Department of Transportation.
INTERNAL REVENUE SERVICE (2009) "Revenue Procedure 2009-54"
26
JACOBS, J. (1969), "The Economy of Cities", Random House.
KNIGHT, F. H. (1924), "Some Fallacies on the Interpretation of Social Cost." Quarterly Journal of
Econmics 38, 582-606.
KRUGMAN, P. (1996a), "Confronting the Mystery of Urban Hierarchy", Journal of the Japanese
and International Economies, 10, 399-418 0023.
LUCAS, R. E. Jr. (1988), "On the Mechanics of Economic Development", Journal of Monetary
Economics, 22, 3-42.
LUCAS, R. E. Jr. and ROSSI-HANSBERG, E. (2002), "On the Internal Structure of Cities",
Econometrica, 70,(4) 1445-1476.
MILLS, E. S. (1967), "An Aggregative Model of Resource Allocation in a Metropolitan Area",
American Ecnomic Review Proceedings. 57 197-210.
MIRRLEES, J.A. (1972), "The Optimum Town", Swedish Journal of Economics, IV: 114-35.
MOOMAW, R. L. (1981), "Productivity and City Size: A Critique of the Evidence", Quarterly
Journal of Economics, 95, 675-688.
MUTH, R. F. (1969), "Cities and Housing", Chicago: University of Chicago Press.
O’SULLIVAN, A. (2009), Urban Economics: 7th edition, New York: McGraw-Hill.
RAPPAPORT, J. (2008) "A Productivity Model of City Crowdedness." Journal of Urban Eco-
nomics, 63, 715-733.
ROBACK, J. (1982), "Wages, Rents, and the Quality of Life", Journal of Political Economy, 90,
(December): 1257-1278.
SEEGERT, N (2010) "A Dynamic Model of City Formation" University of Michigan, mimeo.
27
SHAPIRO, J. M. (2006), "Smart Cities: Quality of Life, Productivity, and the Growth Effects of
Human Capital", The Review of Economics and Statistics , 88, (May) 324-335.
STARRETT, D.A. (1974), "Principles of Optimal Location in a Large, Homogenous Area", Journal
of Ecnomic Theory, IX 419-48.
SVEIKAUSKAS, L. (1975), "Productivity of Cities", MIT Press The Quarterly Journal of Eco-
nomics, 89,(3) 393 - 413.
TIEBOUT, C. M. (1956), "A Pure Theory of Local Expenditures", Journal of Political Economy,
64, (May) 416-424.
TOLLEY, G. (1974), "The Welfare Economics of City Bigness", Journal of Urban Economics, 1,
324-345.
UPTON, C. (1981), "An Equilibrium Model of City Size", Journal of Urban Economics, 10, 15 -
36.
WILDASIN, D. E. (1980), "Locational Efficiency in a Federal System", Regional Science and
Urban Economics, 10, (November) 453-471.
WILDASIN, D. E. (1985), "Income Taxes and Urban Spatial Structure" Journal of Urban Eco-
nomics, 18, 313-33.
WILDASIN, D. E. (1986) Urban Public Finance. Chur, Switzerland: Harwood Academic Pub-
lishers.
VICKERY, W. (1977), "The City as a Firm", The Economics of Public Services M.S. Feldstein
and R. P. Inman, eds. (London:Macmillan).
ZIPF, G. (1949), Human Behavior and the principle of last effort. Cambridge, MA:Addison-
Wesley.
28
Appendix
A Calibration
The parameters in the model may be subdivided into three sets: agglomeration and production,commuting, and amenity levels. Each set of parameters is calibrated to match empirical observa-tions from the U.S. data and microeconomic studies in the literature. Given that the monocentriccity model provides only a very rough approximation to reality, the parameter calibration is meantto help illustrate the empirical relevance of the issues raised here, rather than provide an exactrepresentation of city structures.
Available time hours are set at 2000 per year, assuming workers have 40 hours per week and50 weeks per year available. According to the 2006-2008 American Community Survey 3-YearEstimates, average household size is 2.6 of which about half are employed, meaning there are 1.3workers per household: therefore this is one worker for every two individuals. Across metropolitanareas, the median individual lives in an area with a population density of 2,800 per square mile(Rappaport 2008) so that a mile crosses
√2800 ∼= 529 individual lots, of which 2.6 make a
standard household lot. In the model, is distance in individual lots, and is the time-cost perround trip across a lot. We take the average distance to work as 8 miles, or = 423.3 lots, whichis close to the distance reported by the Summary of Travel Trends (Hu and Reuscher 2004) fromdata in 2001 National Household Travel Survey. According the average one-way commute is about24 minutes long (ACS), giving an average speed of 20 miles per hour. As a fraction of the day orwork-year, the average time-cost of a two-way commute is = 088 = 01, meaning that =
−12 = 01(423312) = 00001333. As estimated by the Internal Revenue Service (2009),the monetary cost of driving is 50 cents per mile, $1 per round-trip mile: this amounts to $8 dailyfor an average commute, or $2000 per 250 workday-year. Per capita, this must divided by 2, so = $1000, and therefore =
−12 = $1000(423312) = $1333.The marginal federal tax rate is taken at 0.33, based on Albouy (2009b), taking into account
federal income taxes, payroll taxes, and a portion of state taxes, and averaging out deductions forhousing and increases in social security benefits.
At an average hourly wage of $22 (Bureau of Labor Statistics 2010) the average daily commutecost is $1833 in pre-tax time or $1222 in post-tax time, and $8 in material costs. Out of a 7.2hour workday, the commuting costs are equal to 8(72 ∗ 22) = 00505 percent of labor income.This is consistent with commuting data from the Survey of Income and Program Participation(SIPP) reported by the Bureau of Transportation Statistics of 3.9 percent of all personal income,or 4.9 percent for drivers. The implied city radius from the calibration is 12 miles (1.5 times theaverage commuting distance), implying a population of = (2800)(12)2 = 1267 million, or633 thousand workers, which is very close to the metropolitan size experienced by the averageindividual.
The agglomeration factor of production, = 006 is set on the higher side of estimates seenin the meta-analyses by Rosenthal and Strange (2004) and Melo et al. (2009). Melo et al. use asample of 729 elasticity estimates from 34 studies from 1965 to 2002 and find a range of estimateswith a mean of 0058. Controlling for selection, Behrens et al. (2010) estimate = 0051.23 These
23Using Moomaw’s (2001) adjustment with a conservative labor share of two-thirds, Sveikauskas’ (1975) estimates
i
estimates apply to an entire metropolitan area, although not individuals actually commute to theCBD: if they did agglomeration effects would likely be larger. Ciccone and Hall (1996) finds anelasticity of wages with respect to density of roughly 0.06.24
Through the implied agglomeration effects, wage levels in an average city should be (1267000) timeswages in a site with one inhabitant, which in the case of = 006 is 23236. We take the averageannual wage at $44 000 per worker for a 2,000 hour work-year, or $22 000 per individual. Dividingby the agglomeration effect this implies a productivity of = $9468 in the typical city. In termsof hourly-wage, this provides $947 at a population of one, or $1646 at 10,000, or $2490 at 10million. This latter number, which corresponds to 5 million workers, already exceeds the numberof workers in Manhattan. (175009468)106 = 27947
Empirically city sizes are heterogeneous and roughly follow Zipf’s law. In Section 4, heteroge-nous city sizes are caused by heterogeneous productivity levels given by = + where issome fixed level and comes from a distribution that resembles a Pareto distribution. The specificdistribution for is chosen to match the fact that empirical observations of both the smallest andbiggest cities generally lie below the linear relationship between city size and rank. The thoughtthat amenity levels should resemble a Pareto distribution is not new, in fact Krugman (1996a)plots the log of river rank and log river flow and finds the relationship to be roughly linear con-cluding this may explain the city size distribution. For the calibrated example = 21995 and = 4141( + 3).
B Mathematical Appendix
Federal Planner’s Problem
The full Lagrangian from section 3.3 is
L =X
=0
1
[( )]
+
"00 +
X=1
()−
X=0
−X=1
(23)32
#
+
X=0
+
" −
X=0
#
which implies the following first order conditions for each for consumption
L
= ()−1
− = 0
L
=1
()
+ ( − ) + − = 0
implies ∈ (00077 00569), which is not completely consistent with the estimates here.24Typically, agglomeration is estimated by using the relationship between wages and population explicit in a zero-
profit condition for firms, controlling for other factors such as education and industry mix. A second method of esti-mating agglomeration estimates the production function directly using value-added, employment, capital, population,and education attainment data. Both approaches interpret the coefficient on the population size as an approximation ofthe agglomeration economies and both produce similar results.
ii
where 0 = 0 for = 0, and is otherwise given by equation (10). Furthermore, the equationsobey non-negativity through
= 0
Dividing by , the social marginal benefit can be written:
( ) =
( ∗)
( ∗)
+ ( )− ∗ ≤ if
0
The first terms goes to zero as → −∞.
B.1 Parametric Sufficient Condition
In the case where federal transfers are equal, then from proposition 1 and equation (22a), then city will be undersized relative to city if
( − ) + ( − ) − + ∗ − ∗ (A.1)
Rewriting equation (1) substituting in the fact that = and rearrange.
(1− )( − ) ( + )( − ) (A.2)
C Model Implications of taxes, , and rental earnings
Modeling federal taxation, material costs of commuting, and rental earnings has important implica-tions on the competitive equilibrium. To understand these effects substitute the budget constraintand rental payment function into the free mobility condition and rearrange to get the followingrelationship
− =Φ( ) + ( − ) + (( − ∗ ()
12)− ( − ()12))
1− (A.3)
where = −()12. The numerator on right hand side of equation (3) consists of three
terms that correspond to three market distortions of federal taxation. The first distortion arisesbecause federal taxes distort utility between cities with heterogeneous quality of life amenities,distorting population toward nicer cities. Second, population will be distorted toward regions thatreceive more benefits from the federal government, here lump sum transfers, relative to the amountof taxes the city paid in. Finally, federal taxation distorts population by changing the relative costof living between cities whenever 6= 0. Therefore these three market distortions caused byfederal wage taxation can only be seen by modeling heterogeneity in quality of life, lump sumtransfers, and material costs of commuting.
Finally, consider the effects rental earnings have on the allocation of population between cities.Without loss of generality let the average rent in city j be higher than in city k, . Under theassumption that rental earnings are location dependent, rental earnings will be higher in city j than
iii
in city k, , leading to more population in city j than if rental earnings were independent oflocation. Despite the prevalence of location dependent rental earnings in the literature it seems thatit provides for perverse incentives. Consider the marginal person deciding between city j and cityk, with location dependent rental earnings the marginal person has the seemingly awry incentiveto move to the region with higher land rents.
D Intuition and Algebra Inefficiently Small Cities
A two-city example is used to formalize how the competitive equilibrium allocates populationinefficiently. Initially assume both cities are the same such that the competitive equilibrium andfederal planner’s allocation allocate half of the total population to each city.
Next, totally differentiate the federal planner and competitive equilibrium conditions allowingone parameter to increase, in this example production amenities . The competitive equilibriumcondition is given by
´holds the competitive equilibrium allocates less population to city 1
than the efficient allocation. Rewrite³11
´in the following way.
µ1
1
¶
= + Φ
+Ψ(A.8)
where is the numerator of³1
1
´, is the denominator and Φ and
Ψ are the wedge terms. Φ and Ψ consist of three parts; a base differ-
iv
ence, a rent difference, and a tax difference which are given below.
Φ = (1)| {z }
−
1|{z}
+ (1− 12)(1)| {z }
(A.9)
Ψ =
21(1)−1(1− (23)12) + (13)1(1)−12| {z }
+
1|{z}
− [(1− 12)(1)−1− 1
2(
1)−12]| {z }
Consider how federal taxes affect³11
´by analyzing the tax parts of Φ and Ψ. Including
federal taxes makes Φ bigger and Ψ smaller which makes the numerator of³11
´less negative
and the denominator more negative, decreasing the competitive equilibrium. In the calibratedexample federal taxes are large enough to decrease the competitive equilibrium below the sociallyefficient population.
When = 1 rental earnings are location independent
=
= 0. However when 16=
= 0 which will make Φ smaller and Ψ bigger, increasing the competitive equilibrium.
The formal necessary and sufficient condition for the competitive equilibrium to allocate asuboptimal level of population to large cities isµ
1
Θ
¶
Φ(Θ)
Ψ(A.10)
In our example Θ = 1 however Θ is allowed to be any parameter, including parameters such asagglomeration and commuting costs. 25.
E Political Equilibria
The formal description of the political equilibria consists of two stages. In the first stage, individ-uals form coalitions sequentially from the pool of individuals not yet associated with a coalition,the uncoordinated set. Coalitions cease to form when there are no possible coalitions from theuncoordinated set that will increase the welfare of those in the coalition in stage two. In stage two,coalitions build cities. If a coalition is unable to limit mobility as in case one, individuals from theuncoordinated set are able to move freely into the city. If the coalition is able to limit mobility, asin cases two and three, it can create a green belt around the city to limit participation in the local
25Note that Φ is parameter specific
v
labor market. Finally, if the coalition can restrict mobility and discriminate against migrants, as incase three, the coalition can charge a premium to individuals in the uncoordinated set to move intothe city.
Maximizing the indirect utility function () over will equate the marginal benefit andmarginal cost of an additional individual from the point of view of an individual in coalition j.
−1 (1− ) =
1
2(
(1− ) + )()
−12 + −1 (1− )
12 (A.11)
In the free-mobility case, coalitions are formed in the first stage until the utility provided byinhabiting a new site decreases the utility of everyone in the second stage. The distribution ofpopulation across inhabited cities is the same as in the competitive equilibrium26. In the restrictedmobility case coalitions of size 2
, which is characterized by the necessary condition 11, willbe formed until everyone is in a coalition or until the utility provided in the hinterland is greater thanthat provided by forming a coalition. In the case where coalitions can charge a premium to migrantsthe distribution of population will be weakly between the extremes because the willingness to payby a group of migrants may exceed the utility loss to the coalition from incorporating the newmigrants.
The comparison between the federal planner and political equilibrium is characterized by awedge similar to the comparison between the federal planner and the competitive equilibrium. Un-like the comparison between the federal and competitive equilibrium that focused on the intensivemargin this comparison is on the extensive margin which is inherently a discrete space. Thereforethe optimization over cities consists of comparing the objective functions, given optimal intensivemargins, of populating additional cities. For simplicity this comparison works with the politicalequilibrium in case one where coalitions are unable to limit migration.
The federal planner populates k regions when the social average benefit of k cities is largerthan that of − 1 and + 1 cities. 27Similarly, the political equilibrium inhabits k cities when thecompensated consumption, + , of all individuals is larger with k cities than that of − 1 and + 1.28.
Formally, the federal planner inhabits k regions when the following conditions holds.
( ) ( − 1) (A.12)
and( ) ( + 1) (A.13)
When the second set of conditions holds, the political equilibrium will inhabit k regions.
( ) + ( ) ( − 1) + ( − 1) (A.14)
and
( ) + ( ) ( + 1) + ( + 1) (A.15)
26The difference between the free-mobility case and the competitive equilibrium exists only on the extensive marginin stage one.
27The single peaked nature of the federal planner’s objective function allows for this simple formulation28The single peaked nature of private consumption with respect to population allows for this simple formulation
vi
Which can be written as
( )) + ( ) ( − 1) + ( − 1) (A.16)
and
( )) + ( ) ( + 1) + ( + 1) (A.17)
where ( ) is the wedge term between the federal and political objective functions. Thewedge term for the inframarginal city is increasing in the number of regions inhabited because thewedge is increasing in the disadvantageousness of a city and successive cities are more disadvanta-geous. Therefore ( −1) ( )⇒ ( −1)+( −1) ( )+( ). Intuitively, this relationship implies that the political equilibrium pop-ulates at least as many regions as the federal planner and in general the political equilibrium willpopulate more.
vii
Figure 2: Two cities with the same city-planner but different federal-planner populations.
Figure 1: Optimal and equilibrium city sizes with a constant outside option.
Figure 4: Representation of within-city and across-city wedges.
Figure 3: Optimal and equilibrium city sizes with a large number of homogenous sites.
Figure 5: Monocentric city location costs and differential land rents
CBD z
z
Figure 6: Population and across-city wedges as a function of natural advantages.
Natural productivity differences
Natural quality-of-life differences
Natural quality-of-life differences with endogenous quality-of-life
Figure7: Extensive margin with homogenous cities.
Figure 8a: Efficient sizes of more productive cities with hinterland option
Figure 8b: Competitive equilibrium sizes of more productive cities with hinterland option.
Figure 9: Efficient and political equilibria with a fixed number of large cities
Figure 10: City size distribution predicted by model and Zipf's Law
