CSIS Discussion Paper No. 144

A tale of two cities:
Urban spatial structure and mode of transportation

Takaaki Takahashi
Center for Spatial Information Science, the University of Tokyo,
5-1-5, Kashiwa-no-ha, Kashiwa, Chiba 277-8568, Japan

July 2016

Abstract
Paying special attention to the interdependence of the spatial structure of a city and the transportmode used there, we obtain two types of equilibria. One is the “auto city equilibrium” at which
workers, distributed over the city thinly, use automobile for their commutes. The other is the“rapid transit city equilibrium” at which a transport firm provides rapid transit services for thecommutes of workers, who are distributed densely. Examining the conditions for each type ofequilibrium, we discuss the possibility of multiple equilibria.
Keywords: auto city; multiple equilibria; population density; rapid transit city; spatial structureof a city, transport firm
JEL Classification Numbers: R14; R40
E-mail address: takaaki-t@csis.u-tokyo.ac.jp
I would like to thank Yasuhiro Sato and Jacques-F. Thisse for insightful discussions. I also appreciate the comments fromseminar participants at various institutions. This research is partly supported by the Grants in Aid for Research by the
Ministry of Education, Science and Culture in Japan (No. 25285070).

1 Introduction
The spatial distribution of economic activities within a city significantly varies across cities. Notably, some cities extend over a quite broad plain where population and employment spread out
fairly evenly, whereas others are compact with a large part of economic activities taking place ina quite small area. This variation is attributable to a number of factors such as histories, social
compositions, political environments and the policies toward urban planning. One of the mostimportant is probably a difference in the modes of transportation used in a city. It can be argued, for instance, that the spatial structure of Los Angeles has been shaped based on the use of
automobile while that of Paris on the use of horse-drawn wagons, buses, trams or subways.
The question of how the level of transport costs affects the spatial structure of a city is wellanswered by a standard economic theory on urban structure. The Alonso-Mills-Muth model ofland use patterns, for instance, demonstrates that the decline in transport costs leads to a spatial
expansion of a city.1 Indeed, a substantial number of studies argue that spatial decentralization,
one of the most pronounced changes that cities have undergone in recent centuries, occurs as aresult of the decline in transport costs, most importantly due to the spread of automobile (seeAnas et al. (1998), Brueckner (2001), Glaeser and Kahn (2004), and Baum-Snow (2007), among
others). In addition, the literature of new economic geography emphasizes the role of transport
costs in the determination of the spatial distribution of economic activities: In general, lower
transport costs work as a centripetal force (see Fujita et al. (1999) and Combes et al. (2008), among
others). However, most researches in the literature do not deal with the spatial distribution withina city.
For all that, this explanation tells only one side of a story. The immediate question is why sucha divergence in urban structure persists. Why did Paris, for example, not switch to an automobile
city like today’s Los Angels in the latter half of the 20th century? Obvious answer would be thatthe mode of transportation used in a city depends on the spatial structure of that city: In Paris, a highly
dense concentration of population and employment along with insufficient road capacities makes
the commutes by automobile infeasible. In Los Angels, contrastingly, the provision of rapid transit services is not profitable enough because of relatively thin population and employment in thecentral business district and/or suburban subcenters (it is only recently that a rapid transit system was constructed there, which carries merely a quite small portion of commuters comparedto automobile). The key observation is that not only is the spatial structure of a city determined by
the mode of transportation used there, but also the mode of transportation used in a city is determined by
the spatial structure of that city. In other words, the causality between the spatial structure and themode of transportation used goes both way. We pay attention to this interdependence, which has
seldom attracted the concern of researchers, nevertheless.
1 In that model, however, whether the decline raises population density or not depends on the nature of a city. When

it is “closed” (i.e., its population is fixed), it reduces the population density in its inner area. When it is “small and open”(i.e., intercity migration is allowed), to the contrary, it enhances the population density in its entire area.

1

Acknowledging this interdependence, we can understand the persistence of a certain spatial
structure as a result of a lock-in effect of multiple equilibria. That is, in one equilibrium, referred
to as an “auto city” equilibrium, population density is low and workers use automobile to go
to work. In the other equilibrium, a “rapid transit city” equilibrium, population density is highenough for a transport firm to earn a sufficient amount of revenue to cover the costs of construction and operation of a rapid transit system. Here, a particular urban structure is supported as
an equilibrium outcome when a particular mode of transportation is used; but it is no longer sowhen a different mode is used. At the same time, the use of a particular mode of transportation
is supported as an equilibrium outcome when a city has a particular spatial structure; but it is nolonger so when a city has a different structure. It is this property that generates the lock-in effect.
The aim of this paper is to examine such a possibility of multiple equilibria through a rigorous
analysis. For that purpose, we construct a model of urban land use of the Alonso-Miills-Muth
type incorporating a transport firm, and examine in the framework of a two-stage game wheneach type of equilibrium mentioned above emerges. Then, we analyze the case of multiple equilibria where both types emerge for the same set of parameters. In this case, even if two cities areendowed equally by nature, it may happen that one develops to an auto city like Los Angels andthe other to a rapid transit city like Paris by, for instance, a historical accident. This is our tale oftwo cities.
One significance of this study concerns the recent movement in urban policies toward a morecompact city. In many cities, especially those in developed countries, the attempts to concentrate
residences and work places in the narrower area of a traditional city center have becoming more
and more common. It is expected that they help us take advantage of the benefits of agglomeration economies further and reduce environmental burdens. What is important here is that suchattempts often accompany new construction or re-development of a public transportation systemrepresented by subways and trams (LRT’s). We can regard them as the attempts to switch theequilibrium from the auto city equilibrium to the rapid transit city equilibrium. In this context,
the newly developed mass transit system probably suffers losses, given the spatial structure of the
auto city, that is, sparse distributions of population and employment. Therefore, the transit systemcannot be run by a private company, and a government needs to continue to pay the losses until
the day when the city becomes concentrated enough to yield a sufficient amount of demand formass transit services. That day may or may not going to come, and even if it comes, it will be in aconsiderably distant future, because the spatial structure changes only gradually. In this way, ourperspective provides an adequate framework to understand and evaluate the attempts toward acompact city.
It is surprising that the relationship of mutual dependence between the spatial structure ofa city and the transport mode used there has seldom attracted the concern of researchers. In thefield of spatial economics, it is rare that different transport modes are explicitly considered.2 What
2 One exception is found in Takahashi (2006), who examines the adoption of a transport technology from “traditional”

2

is worse, transport costs are usually given exogenously, with noteworthy exceptions of the works
by Behrens et al. (2009) and Takahashi (2011), which study how transport costs are endogenously
determined and how it affects the spatial distribution of economic activities.
The rest of the paper consists of five sections. In Section 2, the model is presented. We first
explain a basic framework, then introduce transportation modes and finally discuss the structureof game. The next section discusses the decision makings of workers and a transport firm. In
Section 4, we derive two types of equilibria mentioned above, namely, the auto city equilibrium
and the rapid transit city equilibrium, and examine when each type emerges. In Section 5, we
explore the two types further by simulation analyses specifying functional forms and parameters.
Section 6 concludes.

2 Model
2.1

Basic framework

We consider a linear city with its width being unity in a homogenous plane. Because it is the setof the points within 0.5 miles from the straight line penetrating the middle of it, the city can beinterpreted as the area served by a rapid transit line running along that midway line when the
maximum walking distance for commuters is 0.5 mile (here, we are ignoring the fact that stations
are not continuously but discretely placed). With this interpretation, assuming a linear city israther more natural than assuming a disk-shaped city as in the standard model as long as only
one transit line is concerned.
The city is monocentric, that is, all the production activities are concentrated at the “center”
located at its endpoint. Locations within the city is identified by the distance from the city center,d ≥ 0.
In the city, there are workers who are endowed with 1 unit of time. Spending some portionof it, they work to earn a wage, whose rate is fixed at w. They consume land for housing, acomposite good and leisure, whose amounts are denoted by x, z and e, respectively. Here, wetake the approach of Train and McFadden (1978) in which workers, facing a trade-off between
consuming a greater amount of goods and enjoying longer time for leisure, freely choose thelength of leisure time and thus work hours, denoted by n.3 In order to work, furthermore, workers
need to commute to the city center, paying monetary and time costs. We assume that the numberof workdays is fixed although the work hours per workday are flexible and chosen by workers.
Thus, the costs of commute do not depend on the amounts of leisure consumed by workers, but
depend, it is assumed, only on the distance of commute. We denote the monetary and time costs
and “modern” ones, and its effects upon economic geography.
3 Another approach involves fixed work hours. The length of leisure time is automatically determined as a residualof the total available time after subtracting the work hours and commuting time. The reality is probably in between theworlds depicted by the two approaches. For the differences in their implications, see Jara-Diaz (2007), for instance.

3

for a worker living at d by m(d) and t(d), respectively, which are increasing functions.
Let us consider a representative worker living at d. Her time constraint is given by n(d) +e(d) + t(d) = 1. Furthermore, assuming that the composite good is a numeraire and that land
rent is taken by absentee landlords, we obtain their budget constraint as r (d) x (d) + z(d) + m(d) =wn(d), where r (d) is the land rent at d. It is convenient to combine these two constraints intor (d) x (d) + z(d) + we(d) = y(d) ≡ w − c(d),(1)

where c(d) ≡ wt(d) + m(d) is a generalized cost of commute inclusive of both monetary and time
costs. c(·) is an increasing function. Furthermore, y(d), a decreasing function of d, is the incomethat would be earned by a worker who worked for all the available time, w[1 − t(d)], subtracted
by the monetary cost of commute, m(d). That is, it is the measure of a potential disposable income.
Workers have the same preference represented by utility function U ( x, z, e), which they maximize subject to (1). Because the division of spendings between the two goods and the leisure is
not a main concern in this paper, we specify the utility function as U ( x, z, e) = β ln u( x, z) + (1 −β) ln e. Then, the amount of leisure consumption for the worker at d ise(d) =(1 − β ) y ( d )
.
w

(2)

Note that β becomes equal to the ratio of the actually earned disposable income to the potential
income, that is,β=wn(d) − m(d)
.
y(d)

(3)

Thus, we can interpret β as the relative measure of work hours.
Choosing x and z, workers maximize the sub-utility function, u( x, z), subject to rx + z = βy.
First order necessary condition for the worker at d is given by()()
u1 x (d), βy(d) − r (d) x (d) − r (d)u2 x (d), βy(d) − r (d) x (d) = 0,(4)

where ui (·) denotes a partial derivative of u(·) with respect to its i’th argument.
In a sufficiently long span, the city is small and open, that is, migration occurs between thiscity and the outside regions so that the level of utility for each worker in the city coincides withthat prevailing outside the city, denoted by ū. That is,()(1 − β ) y ( d )U x (d), βy(d) − r (d) x (d),
= ū
w

(5)

for any d at a locational equilibrium.
Solving (4) and (5) simultaneously, we can obtain the demand for land by each worker andthe land rent for each d, which we denote as xe(d) and er (d) (their arguments may be dropped if
doing so causes no confusion). Note that the variables associated with a a particular value of d aredetermined independently of those associated with a different value of d. In other words, they
do not depend on macro variables such as a population distribution and a geographical size of4the city. This independence property, which is a consequence of the small open city approach, will
greatly help us simplify the analysis.
Moreover, outside the city extends agricultural land, which is rented out at a fixed rent, r a .
Then, the location of city’s boundary, b, is given by the solution toer (b) = r a .

(6)

(
)
The resulting triplet of xe(d), er (d), b describes a spatial structure of the city.
One qualification needs to be satisfied. The sum of the spendings on land and on the composite
good must be nonnegative. Because (1) and (2) imply r (d) x + z = β [w − c(d)], this qualification
is expressed as c(d) ≤ w. We concentrate on the case where r a is so high and therefore, the city isso small that any d ∈ [0, b] satisfies this qualification, that is,c(b) ≤ w.

(7)

In this place, it is useful to review the impacts of changes in parameters upon the land consumption because it will play a key role in the determination of the transport mode used in thecity. For that purpose, let us define Υ as Υ ≡ u11 − 2u12e
r + u22e
r2 and Ψ as Ψ ≡ u12 − u22er,
where uij (·) denotes the second-order partial derivative of u(·) with respect to the i’th and j’tharguments (its arguments are omitted for the clarity). Note that Υ < 0 as long as the utility
function is strictly quasi-concave and the first order condition, (4), is satisfied. Furthermore, we/
assume that the land is a normal good, that is, ∂ex ∂ [ βy(d)] ≥ 0. Because differentiating (4) yields//∂ex ∂ [ βy(d)] = −Ψ Υ, the assumption is equivalent toΨ ≥ 0.

(8)

Now, since xe(d) and er (d) are determined by equations (4) and (5), we can derive the impacts ofchanges in parameters on xe(d) by totally differentiating them and eliminating a change in er ( d ).
From (8) and the condition that the marginal rates of substitution become equal to relative prices,//that is, Ux ( x, z, e) Uz ( x, z, e) = er (d) and Ux ( x, z, e) Ue ( x, z, e) = er (d)/w, the next results follow:

′

[
]

 xe′ (d) = − c (d) u2 + (1 − β)Ψex > 0,



Υe
x




[
]
∂e
x
(
d
)
1


= −
u2 + (1 − β)Ψex > 0,



∂c(d)
Υe
x



e
r
∂e
x (d)
(9)
= −
x ) > 0,
(u2 + Ψe

∂ū
Υe
x





∂e
x (d)
Ψy


= −
> 0,


∂β
Υ



[
]

}

∂e
x (d)
1 {


=
βy + m(d) u2 + (1 − β)Υe
x m(d) < 0.

∂w
Υwe
x
We can explain these results intuitively. First, the piece of land demanded by each worker expands as
we move farther away from the city center. As transport costs increase, a disposable income decreases.
To accomplish the same utility level, therefore, workers need to increase the consumption of either
5

land or composite good, or both. Second, the land lot size at a given location shrinks as the transport
costs from that location to the city center decline. The intuition behind this result is the same as before:
for the locational equilibrium, it is necessary to allocate a larger land lot to workers who pay
higher transport costs. In addition, the finding implies that population density rises as transport
costs decline.4 Third, as the target utility level rises, a worker needs to raise the amount of landconsumption. Fourth, the rise in the relative measure of work hours, β, needs to be accompaniedwith a change that will raise a utility level, namely, the increase in the consumption of land.
Finally, the impact of a change in wage rate is twofold. First, its rise brings about an increasein wage income. To keep the utility level constant, therefore, each worker needs to reduce theamount of land consumption other things being equal. This is a standard channel of effect that
one can see in the traditional land use theory. What is unique in this paper is, however, thatthe rise in wage rate implies the increase in the time cost of commute, because a higher value is
now attached to a given length of time. As the second line in (9) indicates, this indirect effect ispositive. The last line in (9) shows that the former direct effect more than offset the latter indirect
effect.
In addition, it is possible to derive by similar analyses the effects of changes in parameters
upon the land rent at each location and the location of city boundary. In particular, the effect of achange in transport costs on the land rent is negative, which is relevant for a later discussion:
1
∂e
r (d)
= − < 0.
xe
∂c(d)

2.2

(10)

Transport modes

In the economy, there is a transport “firm”, which may be a private firm or a government sector. It
constructs and operates a rapid transit system if it decides to do so. To make the analysis simple,
we confine our discussion to the case where the firm can construct only one rapid transit line
running from the city center to a terminal station located somewhere in the city.
The costs of construction and operation of the rapid transit system depend on the length of
the transit line and are expressed by Γ(l ). The function is increasing and Γ(0) > 0 because there
are some fixed inputs. It is true that in the reality, the costs depend also on the number of passengers, but we disregard it for two reasons. First, the change in variable costs arising from higher
ridership is usually much smaller than the change in fixed costs arising from the extension of
the line. To a railroad company, in other words, constructing and maintaining infrastructure is amuch heavier burden than daily operations. Second, it makes the analysis considerably simple
without affecting main results.
4 One might wonder if this result is inconsistent with the stylized fact that the decline in transport costs, mainly brought

about by the use of automobile, is a main reason for the decentralization of a city. This is partly true but partly false. Ourmodel demonstrates that the decline causes not only the increase in population density but also a spatial expansion of acity. Although the former effect may not be observed in the actual decentralization process, the latter effect is indeed oneof the most salient features of the process.

6In order to make the analysis tractable, we assume that the firm charges for its transport services a price proportional to the distance of a trip. (Although it is not difficult to assume otherfunctional forms for the price, the analysis would become much more complicated without yielding additional insights.) The price of the round trip between the location at d and the city centeris m R d, where m R is determined by the firm.
Workers possibly choose the transport mode for commute from the following two.
First, every worker can drive a car to go to her office. We refer to this mode as mode “auto,”
or mode A. The time necessary for a commute is proportional to its distance: t(d) = t A d fort A > 0. The monetary cost is, in contrast, not proportional to the distance but contains a fixed
term: m(d) = m A d + µ for m A > 0. The fixed term, µ > 0, includes various costs arising fromowning an automobile, such as the costs of purchasing a car, most importantly, an insurance fee, atax payment and a fee for a parking lot. In what follows, we will call m A d a “variable cost” and µ a“fixed cost” of automobile use, although in the standard theory, these terms are used with respect
to the amount of products and not the distance of trips. Furthermore, the money paid by workers
for their commutes is taken by someone outside the city, a petroleum industry in Texas, say, or
simply burned up as iceberg transport costs, as is often assumed in the literature of new economic
geography. Finally, the generalized cost is equal to c A (d) ≡ η A d + µ where η A ≡ m A + wt A .
Second, workers may use a rapid transit, a mode “rapid transit,” or mode R. We assume thatwhen a rapid transit is available, the worker living at d can take it from that point. To put it
another way, “stations” are placed continuously along the transit line. The time taken for thatmode is assumed to be proportional to the distance of a commute: t(d) = t R d. The monetary costis also proportional to the distance: there is no fixed term for mode R. Since the price of the rapidtransit services from d is m R d, m(d) = m R d. The generalized cost is equal to c R (d) ≡ ηR d whereηR ≡ m R + wt R .

2.3

Structure of the game

Analysis is made for the following two-stage game. In the first stage, workers decide whether tolive in the city or not; and if they decide to do so, they determine the location of residence andchoose the amount of land to consume. At the same time, the transport firm makes a decisionon whether to construct and operate a rapid transit system or not. When it decides to enter theindustry, it chooses the length of the rapid transit line, l, and the coefficient of its price, m R . In thesecond stage, workers decide the transport mode to use.
Recall that the aim of this paper is to discuss the relationship between the spatial structure ofa city and the transport mode used there. In this respect, what matters is whether a rapid transit
system can be successfully introduced, that is, whether the transport firm can raise a sufficientamount of revenue to cover the necessary costs. However, it will turn out that we can discuss
it without going through at what level of price, m R , and what length of rapid transit line, l, thetransport firm actually chooses. Thus, we do not formulate nor solve that part of transport firm’s
7decision problem in the first stage. Therefore, the model is consistent with any pricing scheme,
including the average cost pricing that prevails in the world.5

3

Decision makings of workers and the transport firm

In this section, we first examine workers’ choices in the second stage of the transport mode to use,
then, transport firm’s decision in the first stage upon whether to enter the industry or not, andfinally, workers’ decision on the location of residence.

3.1

Workers’ choices of transport mode

In the second-stage subgame, the spatial structure of the city has been already determined. Therefore, when the amount of potential disposable income of a worker changes according to her choiceof a transport mode, she adjusts the consumptions of composite good and leisure. In other words,
a worker who pays c(d) for transport services consumes β [w − c(d)] − r (d) x (d) units of composite good and (1 − β) [w − c(d)] /w units of leisure, where x (d) and r (d) here are the given valuesof the size of land lot and the land rent at each location.
First of all, if the transport firm chose not to enter the industry in the first stage, no rapidtransit service is provided and all the workers use automobile. We call such a city an auto city.
What we are going to discuss in the rest of this subsection is the case where the transport firm
did choose to enter in the first stage, and the city is provided with a rapid transit line up to theterminal at l, which is referred to as an l-mile rapid transit city.
Suppose that both mode A and mode R are available for a worker at d. Then, she prefers mode
R to mode A if
)
(
]
[
]
(1 − β ) [w − c R (d)U x ( d ), β w − c R ( d ) − r ( d ) x ( d ),
w
(
)
[
]
]
(1 − β ) [> U x ( d ), β w − c A ( d ) − r ( d ) x ( d ),w − c A (d) .
w(11)Since the utility function is increasing in each argument, (11) is equivalent to c R (d) < c A (d).
Furthermore, the worker is indifferent between the two modes if c R (d) = c A (d). Here, we assume
ad hoc that workers take mode R when indifferent between the two modes.6 Finally, she prefersmode A to mode R if c R (d) > c A (d). By definition, therefore, when both modes are available, aworker at d takes mode R if and only ifb (d) ≡ λ +mR ≤ m

µ
,
d

(12)

5 Our assumption that the production cost does not depend on the number of passengers implies that the rapid transit

industry exhibits decreasing costs. It is well-known that some regulations are necessary for decreasing cost industries inorder to reduce the loss from natural monopoly. The regulation of average cost pricing is one of such regulations, not only
actually observed in a number of cities in the world, but also well justified theoretically: To maximize social welfare under
the constraint that a transport firm not suffer losses, the price must be equal to the average cost because the consumer
surplus decreases with the price.
6 This assumption is made only for a technical reason. We can obtain similar results assuming otherwise.

8b (d) gives the highest price that induces the workers living at d towhere λ ≡ m A + w(t A − t R ): mb (·) is a decreasing function, however, any d < d′ satisfies (12) if d′ satisfies it.
take mode R. Since mb (d) is actually the highest price that induces all the workers living within d miles
Consequently, mfrom the city center to take mode R. In this sense, we call it an upper limit price for d. Moreover,those workers are referred to as d-mile inner city workers.
We can view this result from the opposite perspective, using an inverse function, db(m R ):µmR − λdb(m R ) ≡∞ifmR > λotherwise.

There are two cases to consider. First, suppose that m R > λ. Then, db(m R ) represents the locationof the marginal worker who is indifferent between using automobile and using rapid transit, that()()
is, c A db(m R ) ≡ c R db(m R ) . Thus, the workers living at d ≤ db(m R ) take mode R if that mode isavailable at d (also note that (12) is rewritten as d ≤ db(m R )). This implies that the workers living[]b (l, m R ) ≡ min db(m R ), l take mode R since the rapid transit line is constructed up to
at d ≤ D(]
the terminal at l (see Figure 1). In contrast, the workers living at d ∈ db(m R ), l , if any, and thoseliving at d > l take mode A because the former do not want to use it even though it is availableb (l, m R )-mile inner city workers take modeand the latter have no access to it. Consequently, the D
R. Second, suppose that m R ≤ λ. In that case, the workers at d ≤ l have an access to mode R
and take that mode because (12) is satisfied. The workers at d > l do not have an access to it andb (l, m R )-mile innertherefore take mode A. For this case, too, we can repeat the conclusion that the Dcity workers take mode R and the rest take mode A.
Figure 1: Workers’ choices of transport mode
In the above discussion, we have not considered the possibility of “park-and-ride”, or the
possibility that workers living farther from the city center than the terminal station drive a carfrom home to a certain station of the rapid transit and then take it to the city center. It turns
out, however, that the transport firm can always earn a higher profit by charging the price that
does not induce the park-and-ride, as long as the fixed monetary cost of the automobile use, µ, is
sufficiently high. Therefore, we do not consider this possibility in this paper. For more detail, see
the appendix.

3.2

Entry decision of the transport firm

Let us turn to the decision of the transport firm in the first stage on whether to go into the businessor not.
It is now clear from the above discussion on the second-stage subgame that the demand for theb (l, m R )-mile inner city workers. Since the populationtransport firm’s services comes from the D9density at d is equal to 1/x (d), therefore, the revenue of the firm is equal to()Λ l, m R ; x (d) = m R∫ Db (l,mR )
0

d
dd
x (d)

(the argument x (d) of function Λ(·) may be omitted when doing so causes no confusion). The()
firm enters the industry if and only if its profit, Π(l, m R ) ≡ Λ l, m R ; x (d) − Γ(l ), is nonnegative.7

3.3

Workers’ decisions on land consmption

Workers’ decision in the first stage on the amount of land to consume has been already explained.
To recapitulate briefly, solving (4) and (5) simultaneously yields the size of land lot each worker
consumes, xe(d), as well as the land rent at each location, er (d). The solution further gives thelocation of a city boundary, b, through (6).

4Equilibrium

Now we are ready to discuss equilibria. It immediately follows from the above discussion that
there are two kinds of equilibria, each of which is associated with a particular type of city, namely,
the auto city or the l-mile rapid transit city. We call them an auto city equilibrium and a rapid transit
city equilibrium.

4.1

Auto city equilibrium

In the auto city equilibrium, no rapid transit service is provided and all the workers use automobile. Therefore, c(d) = c A (d) for any d ∈ [0, b A ] where b A is the equilibrium value of b. The sizeof land lots at each location is determined based on c(d) = c A (d). It is denoted by xeA (d), and thecorresponding land rent is similarly denoted by er A ( d ).
The necessary and sufficient condition for the auto city equilibrium is that the transport firm
suffers a loss by entering the industry. That is, the auto city is supported as an equilibrium outcome if and only if
Λ A (l, m R ) < Γ(l )

(13)

(
)
for any l ∈ [0, b A ] and any m R ≥ 0. Here, Λ A (l, m R ) is the value of Λ l, m R ; x (d) in the auto city,()
that is, Λ A (l, m R ) ≡ Λ l, m R ; xeA (d) .
It is obvious that (13) holds for any l ∈ [0, b A ] and any m R ≥ 0 if and only ifmax

l ∈[0,b A ], m R ≥0

Π A (l, m R ) < 0,(14)

where Π A (l, m R ) ≡ Λ A (l, m R ) − Γ(l ). Two observations are important. First, suppose thatthe price is so high that db(m R ) < l. Because the db(m R )-mile inner city workers use the rapid
7 Here, we assume that the firm enters when the cost equals the revenue.

changed.

10

This assumption is arbitrary and can betransit, the profit increases as the length of the rapid transit line is curtailed from l miles todb(m R ) miles. Second, suppose that the price is so low that db(m R ) > l. Then, we can raise theb (l ), without changing the number of users, which yields
price up to the upper limit price, mb (l ) at the maximum ofa higher profit. It follows from these two observations that m R = m
Π A (l, m R ) (for more rigorous derivation, see the proof of the subsequent proposition). Let us
∫ l
(
)
A
A
b (l ) = mb (l )
define Ω (l ) ≡ Λ l, md/ex A (d) dd. We have the following result:0

Proposition 1

The auto city is supported as an equilibrium outcome if and only ifΩ A (l ) < Γ(l )(15)holds for any l ∈ [0, b A ].
(For the proof, see the appendix.)
The impact of l on Ω A (l ) is ambiguous:b (l )lmµ− 2Ω (l ) = Alxe (l )
A′

∫ l
0

d
xeA (d)dd.

(16)

The first term of the right hand side represents a marginal effect: As the rapid transit line isb (l )l.
extended, additional workers with their mass equal to 1/ex A (l ) come to use it, each paying m
The second term, furthermore, shows the effect through the change in price: In order to tempt
workers to use a longer transit line, the transport firm needs to lower its price by the amountb ′ (l )| = µ/l 2 .
equal to |m
It is worthwhile examining the impacts of changes in parameters upon the condition.
First, the auto city becomes more likely to be supported by an equilibrium as the time cost of
the use of rapid transit, t R , rises because∫l∂Ω A (l )ddd < 0.
= −w
A (d)
∂t R
e
x
0

b (l ), which reThe rise in t R , making rapid transit less attractive, lowers the upper limit price, mduces the revenue of the transport firm.
Second, the directions of the impacts of changes in the costs of automobile use are ambiguous.
As an example, consider the change in m A :∂Ω A (l )
=
∂m A

∫ l
0

d
b (l )
dd − mAxe (d)∫ l0∂ex A (d)dd.
2[ xeA (d)] ∂c A (d)d2

For one thing, as a result of the rise in m A , the upper limit price rises, which directly raises therevenue of the transport firm. This is represented by the first term of the right hand side. At thesame time, the higher m A brings about a more sparse population distribution, which reduces the
profitability of rapid transit (remember that ∂ex A (d)/∂c A (d) > 0 by (9)). This is captured by the11second term. If the latter indirect effect dominates the former direct effect, the overall impact isnegative. In that case, the auto city becomes more likely to be supported by an equilibrium as m A
rises. If the indirect effect is dominated by the direct effect, instead, the opposite result holds. Thechanges in t A and µ have similar impacts.
Third, recall that the decline of the prevailing utility level (ū) and that of the relative measure ofwork hours (β) bring about a denser population distribution at each location (see (9)). Therefore,
these changes result in a higher revenue of the transport firm, which makes it less likely for the
auto city to be supported by an equilibrium.8
Fourth and last, the effect of a change in the wage rate (w) is ambiguous. It affects the revenueof the transport firm through two channels. First, it alters a spatial structure: As the wage rate
rises, the population density goes up (remember (9)). Second, it changes the upper limit price.
Recall that the rise in the wage rate implies the increases in time costs. If the time cost of theuse of rapid transit is relatively lower compared to that of automobile, the former mode becomes
further attractive as a result of the rise in the wage rate, which raises the upper limit price. If the
opposite holds, the upper limit price falls as the wage rate rises. Indeed, we have∂Ω A (l )= (t A − t R )
∂w

∫ l
0

d
b (l )
dd − mxeA (d)∫ l0∂ex A (d)dd.
∂w[ xeA (d)]d2

The two terms in the right hand side represent the ambiguous effect through the change in theupper limit price and the positive effect through the change in the spatial structure (recall that∂ex A (d)/∂w < 0), respectively. As long as the time cost of rapid transit use is not too high compared to that of automobile, the first effect is dominated by the second, and total effect becomes
positive.

4.2

Rapid transit city equilibrium

In the rapid transit city equilibrium, rapid transit services are provided. Therefore, c(d) = c R (d)[](]b (l, m R ) and c(d) = c A (d) for d ∈ Db (l, m R ), b R , where b R denotes the equilibrium
for d ∈ 0, D
value of b. Workers decide the amount of land consumption based on this scheme of transport
costs, which is denoted by xeR (d). The corresponding land rent is similarly denoted by er R ( d ).
In the first place, two properties of the rapid transit city equilibrium are worth mentioning.
First, when db(m R ) ≤ l, the amount of the land consumed by each worker and the land rent are()()
b (l, m R ) since c A db(m R ) = c R db(m R ) . They, however, have a kink at that
continuous at d = D
point, that is, their derivatives with respect to the distance exhibit discontinuity. When db(m R ) < l,b (l, m R ).
in contrast, they are discontinuous at d = DSecond, the variables at the suburban part of the rapid transit city where workers use automobile coincide with those at the same location in the auto city. In other words, xeR (d) = xeA (d) andb (l, m R ). It is a consequence of the independence property mentioneder R (d) = er A (d) for any d > D
8 For χ ∈ { ū, β }, (9) implies that ∂ΩA (l )∂χb (l )
= −m

∫ l

d

0

2
[ xeA (d)]12∂ex A (d)dd < 0.
∂χearlier. One implication is that the locations of the boundaries of the two types of cities coincidewith each other, b R = b A . Another implication is that the city population is greater at the rapidtransit city equilibrium than at the auto city equilibrium. The reason is as follows: Recall that thesize of land consumed by each worker increases with transport costs (see the second equation in[)b (l, m R ) since c R (d) < c A (d) for d < db(m R ). Fur(9)). Therefore, xeR (d) < xeA (d) for d ∈ 0, Db (l, m R ) since c R (d) ≤ c A (d) for d ≤ db(m R ). Therefore, thethermore, xeR (d) ≤ xeA (d) for d = D[]b (l, m R ) in the rapid transit city is greater than that in the
population of the interval d ∈ 0, D(]b (l, m R ), b are equal for the two
auto city. The populations of the interval outside of it, i.e., d ∈ Dtypes of cities because of the property derived above.
Now, let us consider the conditions for the rapid transit city equilibrium. It immediately follows from the analysis of transport firm’s entry decision that the l-mile rapid transit city is supported as an equilibrium outcome if there exists m∗ ≥ 0 such that
Λ R (l, m∗ ) ≥ Γ(l ).

(17)

(
)
Here, Λ R (l, m R ) is the value of Λ l, m R ; x (d) at the rapid transit city equilibirum, that is, Λ R (l, m R ) ≡()Λ l, m R ; xeR (d) . Remember that we do not discuss the choice of a price by the transport firm. It
means that we are not to obtain m∗ that is “optimal” from a certain viewpoint (i.e. maximizing
profit) for the transport firm. Furthermore, note that there are two channels through which theprice affects the revenue. One is a direct channel, which involves a change in the payment by
each worker, and the other is an indirect channel, which involves a change in the spatial structure of the city. For the latter channel, one caution is necessary. The price does not “affect” thespatial structure in the sense of usual chronological causality: The spatial structure and the priceare determined simultaneously in our game. What is intended here is that the equilibrium spatial
structure and the equilibrium price are mutually dependent on each other.
It is obvious that there exists m∗ ≥ 0 that satisfies (17) if and only ifΩ R (l ) ≡ max Λ R (l, m R ) ≥ Γ(l ),m R ≥0(18)which establishes the following result:

Proposition 2

The l -mile rapid transit city is supported as an equilibrium outcome if and only if (18) holds.

It is straightforward to see Ω R′ (l ) ≥ 0 for almost all l, because the envelope theorem implies/
that it is equal to m R l xeR (l ) if l < db(m R ) and 0 if l > db(m R ).9
Two observations follow.
First, the solution to the maximization problem in (18) can be lower than the upper limit price,b (l ), that is, db(m R ) can exceed l at the solution. This exhibits a sharp contrast to the finding for them
9 Ω R′ ( l ) is not defined at l = db( m

R ).

13

auto city equilibrium that raising the price up to the upper limit price results in the increase in therevenue. The reason is that the change in the price invokes a change in the spatial structure of therapid transit city to be determined simultaneously. To see this, suppose that the price falls fromthe upper limit price. Because the users of rapid transit are still the l-mile inner city workers, therevenue declines other things being equal. However, the other things are not equal: Corresponding
to the fall in the price, the population distribution realized becomes denser, which gives a positive
impact on the revenue.
Second, the solution to the maximization problem can be greater than the upper limit price.
This finding is also different from that obtained for the auto city equilibrium. Here, our concern
is to maximize the revenue for a given length of the rapid transit line. Consequently, there is apossibility that a part of the line is left unused at the maximum.
These two observations are easily verified by computing the following derivative:dΛ R (l, m R )
=
dm R

∫ Db
0

[
]
b2
b
d
∂ ln xeR (d)D∂ ln D·1−
dd + R·R∂ ln m R
xe (d)xe (d) ∂ ln m R(19)b (l, m R ) are dropped in some cases). The(for the sake of clarity, the arguments of db(m R ) and D
right hand side shows three components of the effects of the rise in a price. The first, represented
by the first term in the square brackets (“1”), is a positive direct effect. The second component,
captured by the second term in the brackets, is a negative effect of the decrease in the population
density. The last component, appearing at the end, is a nonpositive effect that the range of theb ( l ),
locations of users is shortened. The first observation above concerns the case with m R < mb (l, m R ) = l and therefore, the last term of (19) disapor equivalently, l < db(m R ). In that case, D
pears. The overall derivative can be negative because we have the negative density effect. If itb (l ). Instead, the second observation aboveis negative, the maximum may be attained at m R < mb (l, m R ) = db(m R )b (l ), or equivalently, l > db(m R ). In this case, Dconcerns the case with m R > m
and therefore, the negative last term remains. Nonetheless, the overall derivative can be positive.
b (l ), or equivalently, l > db(m R ) can give the maximum.
Then, m R > m
Furthermore, it is straightforward to examine the impacts of changes in parameters upon
Ω R ( l ).
First, Ω R (l ) increases or remains unchanged as the costs of automobile use rise, that is, as m A ,µ and/or t A rises. This is because the change is transmitted only through db(m R ), and

mR


>0
if
db(m R ) < lR∂Ω (l )R ( db)
e
x
=

∂db
 0
if
db(m R ) > l.
(The derivatives of Ω R (l ) are not defined at db(m R ) = l because Γ R (l, m R ) has a kink at that point.)
Second, Ω R (l ) decreases as the time cost of rapid transit use, t R , rises. For this change, we

14

obtain


[





 −w

µm R

∫ db{

d

+
∂Ω R (l )(m R − λ)2 xeR (db)xeR (db)
0
=
∫ l

∂t R

d
∂e
x R (db)
dd < 0 −wm ReR (db) ∂c R (d)
0 x

}2

]
∂e
x R (db)
dd < 0∂c R (d)ifdb(m R ) < lifdb(m R ) > l.

The first term of the right hand side in the first line represents the effect of the fall in the numberb The second term captures the effect of the change in the spatial structure toward aof users (d).
lower population density (recall that we have seen in (9) that ∂ex R (db)/∂c R (d) > 0).
Third, the changes that lead to a denser population distribution, namely, the decrease in theprevailing utility level and that in the relative work hours, give a favorable effect on the rapidtransit city equilibrium through a rise in Ω R (l ).10
Fourth and finally, the effect of a change in the wage rate is ambiguous.

∫ db{
]
[
}2


∂e
x R (db)dµdb(t A − t R )



dd
if
−
 mR
∂w
∂Ω R (l )xeR (db)(m R − λ)2 xeR (db)
0
=
∫ l

∂w

d
∂e
x R (db)
dd > 0if −m ReR (db) ∂w
0 x

db(m R ) < ldb(m R ) > l.

The two terms of the right hand side in the first line represent the ambiguous effect of the change
in the number of users and the positive effect of the change in the spatial structure (recall that∂ex R (d)/∂w < 0), respectively. As long as the time cost of rapid transit use is not too high compared to that of automobile use, the first effect is dominated by the second, and the total effectbecomes positive.

4.3

Discussions

In this subsection, we answer two questions related to the two types of equilibria obtained above:
One is the question of when multiple equilibria emerge, and the other is that of which type ofequilibrium yields a higher welfare level.
4.3.1

Multiple equilibria

One of the interesting consequences of our model is a possibility that both the auto city equilibrium
and the rapid transit city equilibrium are realized for the same set of parameters. In this case ofmultiple equilibria, an identical city becomes the auto city in some circumstances and the rapidtransit city in others, depending on the factors outside economic considerations such as historical
accidents and expectations.
A necessary and sufficient condition for the multiple equilibria immediately follows from
Proposition 1 and Proposition 2:
10 Since ∂Ω R ( l ) /∂χ = ∂Λ R ( l, m

∫ db

where Ψχ ≡ −m R0d2
[ xeR (d)]·R ) /∂χ for χ = { ū, β }, (9) implies that ∂Ω∂ex R (d)dd.
∂χ15

R ( l ) /∂ ū = Ψū < 0 and ∂Ω R ( l ) /∂β = Ψ β < 0,

Corollary 1

Both the auto city and the l ∗ -mile rapid transit city are supported as equilibrium outcomes for the[]same parameter set if both the following conditions are met: (15) is satisfied for any l ∈ 0, b A ,andΩ R (l ∗ ) ≥ Γ(l ∗ )(20)holds.

We have seen above that the directions of the effects of a change in a parameter on Ω A (l )and Ω R (l ) may be ambiguous. Even if they are unambiguously determined, however, the effecton Ω A (l ) and that on Ω R (l ) go in the same direction as in the case of changes in t R , ū and β.
Therefore, it is hard to decide analytically the directions of the impacts of changes in parameterson the multiple equilibria.
4.3.2 Comparison of welfare levels
The next task is to compare the levels of social welfare realized at the two types of equilibria.
Recall that the utility level of each consumer is pegged at a fixed level at the equilibria because
we are considering a small open city. What we need to take into account for the computationof social welfare are, therefore, the land rents earned by absentee landlords and the profit of thetransport firm when rapid transit is introduced.
First, when we consider the welfare of not only workers in the city but also that of the landlords who live outside it, it is necessary to count land rents as a part of social welfare. Here,remember the independence property: The land rent in the rapid transit city at any location farther away from the city center than the terminal station coincides the counterpart in the auto
city. Therefore, the difference in the total land rents in the two types of equilibria is equal to∫ Dber R (d) − er A (d) dd. However, we know that the land rent is negatively related to the trans[)b (l, m R ) since c R (d) < c A (d) for
port costs (see (10)). Consequently, er R (d) > er A (d) for d ∈ 0, D
0b (l, m R ) since c R (d) ≤ c A (d) for d ≤ db(m R ). Thesed < db(m R ), whereas er R (d) ≥ er A (d) for d = D∫ Dbobservations imply that0er R (d) − er A (d) dd ≥ 0: Social welfare is higher at the rapid transit city

equilibrium than at the auto city equilibrium when one takes into account the absentee landlords.
Second, when we consider both the land rents earned by absentee landlords and the profit of
the transport firm, social welfare is still higher at the rapid transit city equilibrium than at the
auto city equilibrium, because the profit cannot be negative.
Thus, we have established the following result:

Proposition 3

Social welfare is higher at the rapid transit city equilibrium than at the auto city equilibrium.

16

5 Numerical simulations
The key functions, Ω A (l ) and Ω R (l ), are too complicated to characterize the two types of equilibria analytically. In this section, we, specifying functional forms and parameters, explore themthrough numerical simulations.

5.1

Specification of functional forms and parameters

5.1.1

Utility function and cost function

Suppose that the preference over the consumptions of land and composite good is represented bya log linear utility function, u( x, z) ≡ α ln x + (1 − α) ln z. The demand for land and the land rent
that simultaneously solve (4) and(5) become[]1θθ
w − c (0)xe(d) =
·
1
−
θ
e
r (0)
{w − c(d)}[w − c(d)and er (d) = er (0)w − c (0)

]1
θ

,

[
] 1− α
1− β
1− β
1
1
respectively, where θ ≡ αβ and er (0) ≡ θ (1 − α) β α (1 − β) θ ū− θ w− θ [w − c(0)] θ are positive
constants. Moreover, we can derive from (6) the location of the city boundary as a solution to[c ( b ) = w − w − c (0)

]

[

]
ra θ
,
e
r (0)

which implies that the boundary qualification (7) is always satisfied for this preference.
For the sake of exposition, furthermore, we focus on the benchmark case of linear Γ(l ), i,e.,Γ(l ) = γ̄ + γl for some γ̄ > 0 and γ > 0, which are respectively referred to as a “fixed cost” anda “marginal cost” not with respect to the amount of products but with respect to the length of therapid transit line.
In the rest of the paper, we evaluate the revenue of the transport firm in terms of the total land
rent at the city center in the auto city, i.e., er A (0):

∫ Db

1− θ
Λi (l, m R )mR=d · [w − ci (d)] θ dd
1
A
e
r (0)
θ [w − c A (0)] θ 0

[
}]

1+ θ
1{
mR

b
b

θ
θ
·
θ
(
w
−
µ
)
−
(
w
−
η
D
−
µ
)
θ
(
w
−
µ
)
+
η
D

A
A
1
 2
η A (1 + θ )(w − µ) θ=
[ 1+ θ
]mRb ) 1θ (θw + ηR Db)
· θw θ − (w − ηR D
for i = R.
 2
1
ηR (1 + θ )(w − µ) θfori = A,

Here, the assumption that the transport costs are linear in d, that is, c A (d) = η A d + µ with
η A ≡ m A + wt A and c R (d) = ηR d with ηR ≡ m R + wt R , enables us to calculate the integral.
Furthermore, the boundary qualification (7), which we have shown to be satisfied, guarantees
that w − ci (d) in the integral is greater than 0 for i ∈ { A, R}.

17

5.1.2

Parameters

For the time cost coefficients, Brueckner’s model case furnishes a clue (Brueckner (2001)). He considers a worker who commutes 250 times a year by driving a car at 30 miles per hour.11 However,one may argue that the driving speed of 30 miles per hour is rather too high in many cities in
the world: In most of European and Japanese big cities, for example, the average driving speed
is much lower. Thus, we use the speed of 25 miles per hour instead of 30 miles. Computing thecoefficient for such a worker, we obtain t A = et A ≡ 0.002283.12 Furthermore, we use et R ≡ 0.8 et A
as a benchmark value.
For the monetary costs of automobile use, furthermore, we can use the estimates by the American Automobile Association (AAA) and the Automobile Association in the United Kingdom(AA).13 The AAA reports that the variable costs are equal to $0.1454 for a small sedan and $0.1718
for a medium sedan, both per vehicle-mile.14 The figures of the AA are much higher: $0.3466 fora car whose price ranges from £13,000 to £18,000, and $0.3743 for a car whose price ranges from
£18,000 to £25,000, both per vehicle-mile. In this paper, we adopt a figure in between those in
the two countries, $0.3.15 Then, assuming that a typical worker works 250 days a year, we cane A ≡ 150 (= 0.3 · 2 · 250). For theobtain the parameter for the yearly variable costs as m A = m
fixed cost, the AAA gives $4,548 for a small sedan and $6,139 for a large sedan, both per year.
The figures in AA’s are not too different: $4,936 for a car from £13,000 to £18,000, and $6,030 fore ≡ 5000. In thea car from £18,000 to £25,000, both per year. Here, we take $5,000, that is, µ = µe A, µ = µe and t R = et R are usedfollowing simulations, benchmark figures of t A = et A , m A = m
unless otherwise mentioned.
In the United States, household median income is equal to $53,657 (the US Census) and the
11 The figure of 250 for the yearly work days is not far from that indicated by Japanese data (Monthly Labor Survey),which is 248.4 in 2000, 253.2 in 2005, and 228 in 2010.
12 The worker spends 250 · 2/25 = 20 hours on a 1-mile commute (2 miles for a round trip) in one year, which occupies
20/(365 · 24) = 0.002283 of all the available time.
13 Several economists have also estimated the costs. Small and Verhoef (2007) estimate that the private cost of operationand maintenance, the vehicle capital cost, roadway cost and parking cost are $0.141, $0.170, $0.016 and $0.007 per vehiclemile, respectively, in the United States, which totals $0.334. Moreover, Roy (2000) cites figures of e0.462 and e0.449 per
vehicle-km (or, equivalently $0.752 and $0.731 per vehicle-mile) at peak hours and off-peak hours, respectively, for asmall gasoline car in the urban areas of the United Kingdom excluding London area. Also he estimates the cost at e0.497
and e0.310 per vehicle-km (or, equivalently $0.810 and $0.505 per vehicle-mile) for a small gasoline car and diesel car,
respectively, in the urban areas of France. Here, the exchange rate of $1 = e0.987 in January of the year 2000, which istaken from Shams (2005), is used.
14 The figures presented here are those in 2015 for the AAA and those in 2014 for the AA. They are obtainable at their
web sites. Furthermore, the variable costs in this paper correspond to the “operating costs” in AAA’s report and the“running costs” in AA’s, whereas the fixed costs correspond to the “ownership costs” in AAA’s and the “standing charge”
in AA’s. Their definitions are, however, quite similar. The variable costs include the costs of fuel, tires and maintenance;
and the fixed costs include insurance, taxes, depreciation and finance charge. In addition, AA’s original data are in termsof pounds, which are converted to the American dollars based on the exchange rate in 2015 provided by the IMF.
15 Here, we take into account the fact that the price of gasoline is exceptionally low in the United States compared tothose in other advanced countries.

18

average annual hours actually worked per worker are 1,789 (the OECD), which occupies 20.42%
of all the time, both in 2014. From these numbers, we estimate w = 53, 657/0.2042 = 262, 767(which is the amount of money that one could earn in one year by working 24 hours a day).
Finally, we estimate preference parameters. First, the share of spending on housing is usually
10% to 20%.16 Thus, we assume that the share of spending on land, α, is equal to 15%. Second, forβ, we evaluate (3) for an “average” worker. Our computation on a hypothetical average worker
suggests that the reasonable size of β is 18% to 21%; and thus we assume that β = 0.2.17 Lastly,
α = 0.15 and β = 0.2 yield θ = 0.03.
Now, we are ready to conduct numerical simulations based on these benchmark values ofparameters.

5.2

Simulation analysis of the two types of equilibria

To begin with, let us examine function Ω A (l ). Figure 2 depicts it as an Ω A revenue curve. Eachof the five curves corresponds to a different value of t A . It is apparent that for sufficiently smallvalues of l, the curve has an upward slope. In the benchmark case with t A = et A , for instance, it is
upward sloping for l ∈ [0, 39.43). For such values of l, the positive effect of the extension of therapid transit line, which arises from the increase in the number of users (the first term of the righthand side in (16)), dominates the negative effect due to the decline in a price (the second term).
Figure 2: Ω A revenue curve
Next, let us turn our attention to function Ω R (l ). We have seen that the impact of a change
in m R on the revenue of the transport firm at the rapid transit city, Λ R (l, m R ), may be positiveor negative, and therefore, there is a possibility that the maximum of Λ R (l, m R ) is attained atb (l ). However, simulation analyses show that such a case is not likely to occur for plausiblem R ̸= mb (l )
values of parameters. In other words, it is likely that dΛ R (l, m R )/dm R is positive for m R < mb (l ), which implies that the maximum is attained at m R = mb (l ), the upperand negative for m R > mlimit price for l.
To see this, we can take a look at Figure 3, which describes the relationship between m R andb (l, m R ) isb (l ). Because D
Λ R (l, m R ). The upward sloping curves represent the revenues for m R < mequal to l in this case, the revenue depends on l and thus, each curve corresponds to a particular

16 In Japan, for example, it is 16.2% for owner-occupied housing and 13.5% for rented housing in 2014 (Family Income

and Expenditure Survey).
17 Suppose that the average worker lives at 10 miles away from the city center and commutes by a car. We can obtain thee A and µ = µ
e. Furthermore, wn is estimated at $53,657, the household
estimates of m(10) and t(10) using t = et A , m = m
median income. Using these figures, we obtain β = 0.1884 from (3). In some countries, a rapid transit is widely used forcommutes. If the marginal cost parameters for the rapid transit use are the same for the automobile use, that is, t R = et A
e A , β is equal to 0.20432. The estimate is not sensitive to the changes in cost parameters: If t R = 0.5et A andand m R = m
e A , the corresponding β only slightly changes to 0.20485, for instance.
m R = 0.5m

19

value of l. As l rises, the revenue increases for given m R and the curve shifts upward. Furtherb (l, m R ) is equal tob (l ). For such m R , D
more, the dashed curve represents the revenue for m R > mb ( l ).
db(m R ) and does not depend on l. This curve intersects each of the upward sloping curve at m
For instance, the intersection, point A, of that curve and the upward-sloping curve for l = 15 isb (15). Putting these two kinds of curves altogether, the revenue is described by a mountainat mb (l ). Again for l = 15, for example, it is described byshaped kinked curve with its summit at mb (l ), the maximum revenue iscurve OAB. As long as the dashed curve is downward sloping at m
attained at that point. For the parameters we are considering, it is the case for l ≤ 61.6, which islikely to hold in a real city. In the rest of the paper, therefore, we concentrate on this case with()
b (l ) .
Ω R (l ) = Λ R l, m

(21)

Figure 3: The relationship between the price and the revenue in the rapid transit city
When (21) is true, we haveb (l )Ω (l ) − Ω (l ) = m
R

A

∫ l [
0

]
1
1
d R − A dd.
xe
xe

That is, the difference between the values of Ω functions for the two types of equilibria dependsb ( l ), c R ( d ) −only on the difference in population densities. Furthermore, note that for m R = mc A (d) is equal to µ(d − l )/l, which is negative for d < l. Because higher c(d) implies higher xe(d)(see (9)), it follows that Ω R (l ) − Ω A (l ) > 0. In other words, the lower transport cost at the rapidtransit city than at the auto city brings about a denser spatial structure, which yields a higher
revenue. This result implies that for any l ≥ 0, there exists some Γ(l ) that satisfies both (15) and(20), that is, there is always a possibility of multiple equilibria.
Figure 4 depicts Ω R (l ) as an Ω R revenue curve as well as the Ω A revenue curve. Also described
are five dashed straight lines, which represent Γ(l )’s. We call them cost curves. Their intercepts
with the vertical axis are equal to γ̄’s, the fixed costs with respect to the length of the transit line,
and their slope represents γ, the marginal cost (in the figure, γ = 0.19).
Figure 4: The revenue curves and the cost curve
The figure illustrates how different sets of equilibria emerge depending on parameters, whichis the crux of the paper. Consider the situation where the fixed cost, γ̄, gradually declines.
First, when it is sufficiently high that the corresponding cost curve is given by Γ1 (l ) in Figure
4, the auto city is supported by an equilibrium because the cost curve lies above the Ω A revenue
curve for the entire range of l (see Proposition 1). At the same time, no rapid transit city can besupported by an equilibrium because the cost curve is situated also above the Ω R revenue curvefor the entire range of l (see Proposition 2).
Second, as the fixed cost gradually declines, the cost curve finally touches the Ω R revenue
curve. In the figure, it occurs when γ̄ = γ̄ Ru = 1.911. After it reaches that critical value, the rapid
20

transit city is supported as an equilibrium outcome for some l. For instance, when the cost curve
shifts to Γ2 (l ), any rapid transit city where the transit line is constructed up to the terminal locatedin between l3 and l5 is supported by an equilibrium. Thus, the critical value γ̄ Ru represents theupper limit of the fixed cost for which the rapid transit city is supported by an equilibrium. Note that the
auto city is still an equilibrium outcome because the cost curve remains to lie over the Ω A revenue
curve in the entire range. In this case with a medium size fixed cost, therefore, there are multiple
equilibria (see Corollary 1).
Third and finally, when the fixed cost declines to another critical value γ̄ Al = 0.823, the cost
curve becomes tangent to the Ω A revenue curve. Afterward, it becomes possible that we canintroduce rapid transit successfully into the auto city and therefore, that type of city is no longer
supported as an equilibrium outcome. When the cost curve is described by Γ3 (l ), for example, therapid transit line with length between l2 and l4 is feasible in the auto city. The critical value γ̄ Al ,thus, represents the lower limit of the fixed cost for which the auto city is supported by an equilibrium.
At the same time, a rapid transit city continues to be supported by an equilibrium. Indeed, forthe cost curve given by Γ3 (l ), any rapid transit city with the transit line extending to the terminal
between l1 and l6 is supported as an equilibrium outcome.
The two critical values of the fixed cost, γ̄ Ru and γ̄ Al , which summarize the equilibrium patterns, are useful in discussing the effects of changes in parameters. Here, because of the limitationof space, we discuss only one example, namely, the changes in the costs of automobile use. It is
often observed that governments encourage or discourage the use of that mode by controlling theamounts of monetary costs through various policy instruments such as diverse kinds of taxes andsubsidies. How do such policies affect the likelihood that the auto city and the rapid transit city
emerge? Furthermore, do they widen the possibility of multiple equilibria?
The left two panels in the next figure describe the effects of a change in m A , the coefficient of
the variable monetary cost. The upper panel shows its effects on γ̄ Ru and γ̄ Al whereas the lower
one shows the effect on their relative value, γ̄ Ru /γ̄ Al . The right two panels, instead, concern thechange in µ, the fixed monetary cost. It is readily verified from the upper panels that both criticalvalues increase with the two parameters. That is, as the monetary costs rise, the rapid transit city
becomes more likely, while the auto city becomes less likely, to be supported as an equilibriumoutcome. Furthermore, the relative value, γ̄ Ru /γ̄ Al , is considered a measure of the likelihood ofmultiple equilibria. As m A rises, the value decreases except when m A is quite low. Therefore,
the multiple equilibria become less likely to emerge in a broad range of m A . As µ rises, γ̄ Ru /γ̄ Al
decreases when µ is small but increases when it is high. The multiple equilibria become less likely
to emerge when µ is low but more likely when it is high.
Figure 5: The effects of changes in monetary costs upon equilibrium patterns

21

6 Concluding remarks
In this paper, we have discussed the relationship between the spatial structure of a city and thetransport mode used there. Paying special attention to their mutual dependence, we have derived
two types of equilibria. One is the auto city equilibrium at which workers, distributed over a citythinly, use automobile for their commutes. The other is the rapid transit city equilibrium at which
rapid transit services are provided for the commutes of workers, who are distributed densely. We
have obtained and characterized the conditions for each type of equilibrium. Furthermore, thepossibility of multiple equilibria, that is, the possibility that both types of equilibria emerge forthe same set of parameters, has been studied. When such multiple equilibria emerge, the samecity may become an auto city or a rapid transit city depending on, for example, historical paths
and expectations.

22

References[1] Anas, A., R. Arnott, and K. A. Small. (1998) “Urban spatial structure,” Journal of Economic
Literature, 36, 1426-1464.
[2] Baum-Snow, N. (2007) “Did highways cause suburbanization?” The Quarterly Journal of Economics, 122, 775-805.
[3] Behrens, K., C. Gaigné, and J.-F. Thisse (2009) “Industry location and welfare when transport
costs are endogenous,” Journal of Urban Economics, 65, 195-208.
[4] Brueckner, J. K. (2001) “Urban sprawl: Lessons from urban economics,” in Brookings-Wharton
papers on urban affairs eds. by Brueckner, J. K., E. Mills, and M. Kremer, Brookings Institution
Press, 65-97.
[5] Combes, P.-P., T. Mayer, and J.-F. Thisee. (2008) Economic geography, Princeton University
Press, Princeton, NJ, USA.
[6] Glaeser, E. L., and M. E. Kahn. (2004) “Sprawl and urban growth,” in Handbook of Regionaland Urban Economics vol. 4 eds. by Henderson, J. H., and J.-F. Thisse, Elsevier, Amsterdam,
Netherland, 2481-2527.
[7] Jara-Diaz, S. (2007) Transport economic theory, Elsevier, Amsterdam, Netherland.
[8] Fujita, M., P. Krugman, and A. J. Venables. (1999) The spatial economy: cities, regions, and international trade, MIT Press, Cambridge, MA, USA.
[9] Roy, R. (2000) (ed.) Revenues from efficient pricing:

Evidence from the member states,

UIC/CER/European Commission, UIC, Paris.
[10] Shams, R. (2005) “Dollar-euro exchange rate 1999-2004: Dollar and euro as international
currencies,” HWWA Discussion Paper, 321.
[11] Small, K. A., and E. T. Verhoef. (2007) The economics of urban transportation, Routeledge, New
York, NY, USA.
[12] Takahashi, T. (2006) “Economic geography and endogenous determination of transport technology,” Journal of Urban Economics, 60, 498-518.
[13] Takahashi, T. (2011) “Directional imbalance in transport prices and economic geography,”
Journal of Urban Economics, 69, 92-102.
[14] Train, K., and D. McFadden. (1978) “The goods/leisure trade-off and disaggregate work trip
mode choice models,” Transportation Research, 12, 349-353.

23

Appendices
Proof of no park-and-ride
Consider a worker who drives a car from her home to d0 ≤ l and then takes the rapid transit
from there to the city center. The cost of such an itinerary is equal to c A (d − d0 ) + c R (d0 ) =(m R − λ)d0 + η A d + µ. If m R > λ, it is minimized at d0 = 0 and a worker does not conduct thepark-and-ride. If m R ≤ λ, instead, it is minimized at d0 = l and she carries out the park-and-rideparking a car at the terminal station.18
Now, suppose that m R ≤ λ. We know that the l-mile inner city workers use rapid transit.
Furthermore, the above argument implies that all the workers living beyond the terminal station parka car there and then take rapid transit. The total revenue is therefore equal toΛiPKRD (l, m R ) = m R

∫ l
0

d
dd + l m R
xei (d)

∫ bi
l

1
dd
xei (d)for i ∈ { A, R}. For a given spatial structure, this revenue is increasing in m R and therefore,
maximized at m R = λ. When there is no restriction on m R so that the firm can charge m R > λ, onb (l ) > λ to earn Λs (l, mb (l )). If µ is sufficiently high, however,
the other hand, it may charge mi(b (l )
Λ l, m)µ− ΛiPKRD (l, λ) =

∫ l

l

0

∫

s

b
1
d
dd
−
λl
dd
i
i
xe (d)xe (d)lb (l ), which discourages the parkis positive. In that case, therefore, the firm prefers charging m
and-ride, to charging λ, which gives rise to it. In this paper, we concentrate on this case: µ is
sufficiently high that the firm does not charge too low a price that induces the park-and-ride.

Proof of Proposition 1
Π A (l, m R ). First, suppose that db(m0 ) < l 0 . Then,
Let (l 0 , m0 ) be the solution tomax
l ∈[0,b A ], m R ≥0()()
Π A db(m0 ), m0 − Π A (l 0 , m0 ) = Γ(l 0 ) − Γ db(m0 ) > 0 since Γ(·) is an increasing function. This
contradicts the supposition that l 0 maximizes Π A (l, m R ), and consequently db(m0 ) ≥ l 0 . Second,
suppose that db(m0 ) > l 0 . Then,][()
b ( l 0 ) − Π A ( l 0 , m0 ) = mb ( l 0 ) − m0
Π l0, m
A

∫ l0
0

d
dd > 0,xeA (d)which contradicts the supposition that m0 maximizes Π A (l, m R ), and consequently, db(m0 ) ≤ l 0 .
b (l 0 ), which implies that
Hence, we have db(m0 ) = l 0 , or m0 = mmax

l ∈[0,b A ], m R ≥0

Π A (l, m R ) =max Ω A (l ) − Γ(l ). The proposition immediately follows from (14).

l ∈[0,b A ]18 We arbitrarily assume that workers carry out the park-and-ride when they are indifferent between doing so and notdoing so.

24

1000

m(d)

mR

500

λ
0

0

10

d (mR )

20

30

dFigure 1: Workers’ choices of transport mode
Ω A(l)~
tA = tA~
tA = 1.5 tA~
tA = 0.75 tA
tA = 2 ~
tA

4

3

~
tA = 0.5 tA
2

1

0

0

10

20

Figure 2: ΩA revenue curve30dΛR(l, mR )l = 35l = 30l = 25l = 206

4A

l = 15l = 10

2

0

0

λ

500

m(15)

B
mR

1000

Figure 3: The relationship between the price and the revenue in the rapid transit city

Γ1(l)

8

Γ2 (l)

Γ3 (l)ΩR revenue curve6Ω A revenue curve
4

γ Ru
γ Al
0

0

l1 l2

l3

10

l4

l5 30

l6

40

Figure 4: The revenue curves and the cost curve

l

2.5

3

γ Ru

2

γ Ru

2

1.5

γ Al

1
0.5
0

100

γ Al

1

200

mA

2.4

3000

4000

5000

6000

µ

3

γ Ru / γ Al

2.35

2.8
2.6

2.3

γ Ru / γ Al

2.4

2.25

2.2
0

100

200

(a) Change in mA

mA

3000

4000

5000

6000

µ

(b) Change in µ

Figure 5: The effects of changes in monetary costs upon equilibrium patterns