A tale of two cities:

Transcription

1 Highly preliminary. Do not quote or refer to. A tale of two cities: Urban spatial structure an moe of transportation Takaaki Takahashi Center for Spatial Information Science, the University of Tokyo, 5-1-5, Kashiwa-no-ha, Kashiwa, Chiba , Japan January 216 Abstract This paper iscusses the interepenence of the spatial structure of a city an the transport moe use there to obtain two types of equilibria. One is the auto city equilibrium at which workers, istribute over a city thinly, use an automobile for their commutes. The other is the rapi transit city equilibrium at which rapi transit services are provie for the commutes of workers, who are istribute ensely. We have erive an characterize the conitions for each type of equilibrium. Furthermore, the possibility of multiple equilibria has been stuie. Keywors: auto city; ensity; multiple equilibria; rapi transit city; spatial structure of a city, transport firm JEL Classification Numbers: R14; R4 aress: takaaki-t@csis.u-tokyo.ac.jp I woul like to thank Yasuhiro Sato for insightful iscussions. I also appreciate the comments from seminar participants at various institutions. This research is partly supporte by the Grants in Ai for Research by the Ministry of Eucation, Science an Culture in Japan (No ).

2 1 Introuction The spatial istribution of economic activities within a city significantly varies across cities. Notably, some cities exten over a quite broa plain where population an employment sprea out more or less evenly, whereas others are compact with a large part of economic activities taking place in a fairly small area. This variation is attributable to a number of factors such as histories, social compositions, political environments an the policies for urban planning. Most people woul agree, however, that one of the most important is a ifference in the moes of transportation use in a city. It can be argue, for instance, that the spatial structure of Los Angeles has been shape base on the use of automobile while that of Paris on the use of horse-rawn wagons, buses, trams or subways. The question of how the level of transport costs affects the spatial structure of a city is well answere by a stanar economic theory on urban structure. The Alonso-Mills-Muth moel of lan use patterns, for instance, emonstrates that lower transport costs bring about a more outsprea city with a lower population ensity in its inner area as long as its population is fixe. This proposition will explain the ifference in spatial structures between the two cities currently observe, if Los Angels was built when transport costs were low while Paris was built when they were high. 1 For all that, this explanation tells only one sie of a story. The immeiate question is why such a ivergence in urban structure persists. Why i Paris, for example, not switch to an automobile city like toay s Los Angels in the latter half of the 2th century? Obvious answer woul be that the moe of transportation use in a city epens on the spatial structure of that city: In Paris, a highly ense concentration of population an employment along with insufficient total length an area of roas makes the commutes by automobile infeasible. In Los Angels, contrastingly, the provision of a rapi transit is not profitable enough because of relatively thin population an employment in a central business istrict an/or in suburban subcenters (it is only recently that a rapi transit system was constructe there, although it carries only a quite small portion of commuters compare to automobile). The key observation is that not only is the spatial structure of a city etermine by the moe of transportation use there, but also the moe of transportation use in a city is etermine by the spatial structure of that city. In other wors, the causality between the spatial structure an the moe of transportation use goes both way. Acknowleging this interepenence, we can unerstan the persistence of a certain spatial structure as a result of a lock-in effect arising from multiple equilibria. That is, in one equilibrium, referre to as an auto city equilibrium, population ensity is low an workers use automobile to go to work. In the other equilibrium, a rapi transit city equilibrium, population ensity is high enough for a transport firm to raise a sufficient amount of revenue to cover the construction 1 It woul be fair to point out that the population ensity becomes higher, to the contrary, in the city with lower transport costs, if the city is open an intercity migration is allowe. In this small open city case, therefore, the toay s ifference between the two cities woul be explaine from the hypothesis that Los Angeles was built when transport costs were high while Paris was built when they were low. 1

3 an operation of a rapi transit system. Here, a particular urban structure is supporte as an equilibrium outcome when a particular moe of transportation is use; but it is no longer so when a ifferent moe is use. At the same time, the use of a particular moe of transportation is supporte as an equilibrium outcome when a city has a particular spatial structure; but it is no longer so when a city has a ifferent spatial structure. It is this property of multiple equilibria that generates the lock-in effect. Furthermore, what is responsible for the property is that it takes quite long, sometimes infinite, time for an urban spatial structure to ajust to a change in its environments such as a change in the moe of transportation use there. There are mainly two reasons. For one thing, the coorination among the agents whose behaviors result in forming an urban spatial structure is usually highly ifficult or even impossible because their number is enormous. What is more, the most important element of an urban structure is builings, which are physically urable an whose economic values ecline only slowly. The aim of this paper is to examine such a possibility of multiple equilibria through a rigorous analysis. For that purpose, we construct a moel of urban lan use of the Alonso-Miills-Muth type incorporating a transport firm that may operate a rapi transit system. First of all, we examine the conition for each type of the equilibrium mentione above. Then, we pay a special attention to the situation where both types of equilibria emerge for the same set of parameters. In this situation of multiple equilibria, even if two cities were enowe equally by nature, it may happen that one evelops to an auto city an the other to a rapi transit city by, for instance, a historical accient. This is our tale of two cities. One significance of this stuy concerns a recent tenency in urban policies towar a more compact city. In many cities, especially those in evelope countries, the attempts to maintain an to accumulate resiences an work places in a narrower area of a traitional city center have becoming more an more common. It is expecte that they help us take avantage of the benefits of agglomeration economies further an reuce environmental burens. What is important here is that such attempts often accompany new construction or re-evelopment of a public transportation system represente by subways an trams. Thus, we can regar them as the attempts to switch the equilibrium where a city falls from the auto city equilibrium to the rapi transit city equilibrium. In this context, the newly evelope mass transit system probably suffers losses, given the spatial structure of the auto city, that is, sparse istributions of population an employments. Therefore, the transit system cannot be run by a private company, an a government nees to continue to pay the losses until the city becomes concentrate enough to yiel a sufficient amount of eman for mass transit services. That ay may or may not going to come, an even if it comes, it will be in a fairly istant future, because the spatial structure changes only graually. In this way, our perspective will provie a goo conceptual framework to unerstan an evaluate the attempts to make a city more compact. The rest of this paper consists of five sections. In Section 2, the moel is presente. We first explain basic settings an then introuce transportation technologies. The next section efines 2

4 two types of equilibria mentione above, namely, the auto city equilibrium an the rapi transit city equilibrium. In Section 4, we erive the conitions for each of the two types of equilibria. The conition for both types co-existing is also erive. In Section 5, we explore the two types of equilibria by simulation analyses specifying functional forms an parameters. Section 6 conclues. 2 Moel 2.1 Basic settings We consier a linear city with its with being unity in a homogenous plane. Note that it is a set of the locations within.5 miles from the straight line penetrating the mile of the city. Therefore, we can interpret the city as the area serve by a rapi transit line running along that miway line when the maximum walking istance for commuters is.5 mile (here, we are ignoring the fact that stations are not continuously but iscretely place). With this interpretation, assuming a linear city is rather more natural than assuming a isk-shape city as in a stanar moel as long as only one transit line is concerne. The city is monocentric an the center is locate at its enpoint. Locations within the city is ientifie by the istance from the city center,. There are workers enowe with 1 unit of time. Spening some portion of it, they work at the city center an earn a wage, whose rate is fixe at w. They live somewhere in the city an commute to the center, which takes both monetary an time costs. The monetary cost for a worker who lives at is enote by m(). For the time cost, we enote the require time for a -mile commute by t(). We assume that both m( ) an t( ) are increasing functions. Workers consume lan for housing, a composite goo an leisure, whose amounts are enote by x, z an e, respectively. Here, we take the approach of Train an McFaen (1978) in which workers, facing a trae-off between consuming a greater amount of goos an enjoying longer time for leisure, freely choose the length of leisure time an thus work hours, enote by n. 2 Their time constraint is given by n + e + t() = 1. Furthermore, assuming that the composite goo is a numeraire an that lan rent is taken by absentee lanlors, we obtain their buget constraint as r()x + z + m() = wn, where r() is the lan rent at. It is convenient to combine these two constraints into r()x + z + we = y() w c(), (1) where c() wt() + m() is a generalize cost of a commute inclusive of both monetary an time costs. c( ) is an increasing function. Furthermore, y(), a ecreasing function of, is the 2 Another approach involves fixe work hours. The length of leisure time is automatically etermine as a resiual of the total available time after subtracting the work hours an commuting time. The reality is probably in between the worls epicte by the two approaches. For the ifferences in their implications, see Jara-Diaz (27), for instance. 3

5 income that woul be earne by a worker who worke for all the available time, w[1 t()], subtracte by the monetary cost of commute, m(). That is, it is the measure of a potential isposable income. Workers have the same preference represente by a utility function, U(x, z, e), which they maximize subject to (1). Because the ivision of spenings between the two goos an the leisure is not a main concern in this paper, we assume that the utility function can be written as follows: U(x, z, e) = β ln u(x, z) + (1 β) ln e. (2) Then, the amount of leisure consumption is e = (1 β)y(). (3) w Note that β becomes equal to the ratio of an actually earne isposable income to the potential income, that is, β = Thus, we can interpret it as a relative measure of work hours. wn m(). (4) y Choosing x an z, workers maximize the sub-utility function, u(x, z), subject to r()x + z = βy(). First orer necessary conition is given by ) ( ) u 1 (x, βy() r()x r()u 2 x, βy() r()x =, (5) where u i ( ) enotes a partial erivative of u( ) with respect to its i th argument. The city is small an open, that is, in the long run, migration occurs between this city an the outsie regions so that the level of utility for each worker in the city coincies with that prevailing outsie the city, enote by ū. That is, ( U x, βy() r()x, at a locational equilibrium. ) (1 β)y() = ū (6) w Solving (5) an (6) simultaneously, we can obtain the eman for lan by each worker an the lan rent. Because these variables are functions of only y(), we enote them as x() an r() (their arguments will be roppe if oing so causes no confusion). In aition, outsie the city extens agricultural lan, which is rente out at a fixe rent, r a. Then, the city s bounary, b, is given by a solution to r(b) = r a. (7) Finally, one qualification nees to be satisfie. The sum of the spenings on lan an on a composite goo must be nonnegative. Because (1) an (3) imply r()x + z = β [w c()], this qualification is expresse as c() w. We concentrate on the case where r a is so high an therefore, the city is so small that any [, b] satisfies this qualification, that is, c(b) w. (8) 4

6 Before turning to transport technologies, it is useful to review the impacts of changes in exogenous variables upon the lan consumption because it will play a key role in the etermination of the transport moe use in a city. For that purpose, let us efine Υ as Υ u 11 2u 12 r + u 22 r 2 an Ψ as Ψ u 12 u 22 r, where u ij ( ) enotes the secon-orer partial erivative of u( ) with respect to the i th an j th arguments (its arguments are omitte for the clarity). Note that Υ < as long as the utility function is strictly quasi-concave an the first orer conition, (5), is satisfie. Furthermore, we assume that the lan is a normal goo, that is, x / [βy()]. Because ifferentiating (5) yiels x / [βy()] = Ψ / Υ, the assumption is equivalent to Ψ. (9) Now, since x() an r() are etermine by equations (5) an (6), we can erive the impacts on x() by totally ifferentiating them an eliminating a change in r(). From (9) an the conition that the marginal rates of substitution become equal to relative prices, that is, U x (x, z, e) / U z (x, z, e) = r() an U x (x, z, e) / U e (x, z, e) = r()/w, the next results follow: x () = c () [ u2 + (1 β)ψ x ] >, Υ x x() = 1 [ u2 + (1 β)ψ x ] >, c() Υ x x() ū x() β x() w = r Υ x (u 2 + Ψ x) >, = Ψy Υ >, = 1 Υw x [ { βy + m()} u 2 + (1 β)υ xm() ] <. We can explain these results intuitively. First, the piece of lan emane by each worker expans as we move farther away from the city center. As transport costs increase, a isposable income ecreases. To accomplish the same utility level, therefore, workers nee to increase the consumption of either lan or composite goo, or both. Secon, the lan lot size at a given location shrinks as the transport costs from that location to the city center ecline. The intuition behin this result is the same as before: for the locational equilibrium, it is necessary to allocate a larger lan lot to workers who pay higher transport costs. In aition, the fining implies that population ensity rises as transport costs ecline. One might woner if this result is inconsistent with the stylize fact that the ecline in transport costs, mainly brought about by the use of automobile, is a main reason for the ecentralization of a city. This is partly true but partly false. Our moel emonstrates that the ecline causes not only the increase in population ensity but also a spatial expansion of a city. Although the former effect may not be observe in the actual ecentralization process, the latter effect is inee one of the most salient features of the process. Thir, as the target utility level rises, a worker nees to raise the amount of lan consumption. Fourth, the rise in the relative measure of work hours, β, nees to be accompanie with a change that will raise a utility level, namely, the increase in the consumption of lan. Finally, the impact of a change in wage rate is (1) 5

7 twofol. First, the rise in wage rate brings about an increase in wage income. To keep the utility level constant, therefore, each worker nees to reuce the amount of lan consumption other things being equal. This is a stanar channel of effect that one can see in the traitional lan use theory. What is unique in this paper is, however, that the rise in wage rate implies the increase in a time cost of commute, because a higher value is now attache to a given length of time. As the secon line in (1) inicates, this inirect effect is positive. The last line in (1) shows that the former irect effect more than offset the latter inirect effect. 2.2 Transport moes In the economy, there is a transport firm, which constructs an operates a rapi transit system if it ecies to o so. To make the analysis simple, we confine our iscussion to the case where the firm can construct only one rapi transit line running from the city center to a terminal station locate somewhere in a city. The location of the terminal station, l, is ecie by the firm. The city with such a rapi transit line is referre to as an l-mile rapi transit city. The costs of the construction an operation of the rapi transit system epen on the length of the transit line an are expresse by Γ(l). The function is increasing an Γ() > because there are some fixe inputs. It is true that the number of passengers affects the costs in the reality, but we isregar it for two reasons. First, the change in variable costs arising from higher riership is usually much smaller than the change in fixe costs arising from the extension of the line. To a railroa company, in other wors, constructing an maintaining infrastructure is a much heavier buren than aily operations. Secon, it makes the analysis consierably simple without changing main results. In orer to make the analysis tractable, we assume that the firm charges for its transport services a price proportional to the istance of a trip. (Although it is not ifficult to assume other functional forms for the price, the analysis woul become much more complicate without yieling aitional insights.) The price of the roun trip between the location at an the city center is m R, where m R is etermine by the firm. Here recall that the aim of this paper is to iscuss the relationship between the urban spatial structure an the moe of transport to be use. In this respect, what matters is whether a rapi transit can be successfully introuce, that is, whether the transport firm can raise a sufficient amount of revenue to cover the necessary costs. However, it can be iscusse without going through at what level the price of transport services is etermine. Thus, we have the etermination of m R unspecifie. The price may be equal to the marginal cost, the average cost or something else. 3 3 Our assumption that the prouction cost oes not epen on the number of passengers implies that the rapi transit inustry exhibits ecreasing costs. It is well-known that some regulations are necessary for ecreasing cost inustries in orer to reuce the loss from natural monopoly. One of such regulations is to require an average cost pricing of transport firms. This pricing policy is not only actually observe in a number of cities in the worl, but also well justifie theoretically: To maximize social welfare uner the constraint that a transport firm not suffer losses, the price must be 6

8 Workers possibly choose a moe of transport for commutes from the following two. First, every worker can rive a car to go to her office. We refer to this moe as moe auto, or moe A. The time necessary for a commute is proportional to its istance: t() = t A for t A >. The monetary cost is, in contrast, not proportional to the istance of commute but contains a fixe term: m() = m A + µ for m A >. The fixe term, µ >, contains various costs arising from owning an automobile, such as the costs of purchasing a car, most importantly, an insurance fee, a tax payment an a fee for a parking lot. The money pai by workers for their commutes is taken by someone outsie the city, a petroleum inustry in Texas, say, or simply burne up as iceberg transport costs, which is often assume in the literature of new economic geography. The generalize costs is equal to c A () η A + µ where η A m A + wt A. Secon, workers may a rapi transit, a moe rapi transit, or moe R. We assume that when a rapi transit is available, the worker living at can take it from that point. To put it another way, stations are place continuously along the transit line. The time taken for that moe is assume to be proportional to the istance of a commute: t() = t R. Since the price of the rapi transit services from is m R, furthermore, the monetary cost is also proportional to the istance: m() = m R. The generalize cost is equal to c R () η R where η R m R + wt R. 3 Definitions of two types of equilibria In orer to efine an equilibrium, we nee to answer two questions. The first question is which transport moe workers choose when both moes are available. To answer this question, let us consier a short-run perio in which no migration takes place between the city in consieration an the outsie economy, an its urban spatial structure is fixe. That is, the size of the lan lot consume by each worker, x(), an the lan rent at each location, r(), have been alreay etermine an can be regare as constants in this short-run perio. If a worker experiences a change in her potential isposable income uring this perio, she ajusts the consumptions of composite goo an leisure. In other wors, a worker who pays c() for transport services consumes β [w c()] r() x() units of composite goo an (1 β) [w c()] /w units of leisure. Now, suppose that both moe A an moe R are available for a worker at. Then, she prefers moe R to moe A if ( U x(), β [ w c R () ] r() x(), > U (1 β) [ w cr () ]) w ( x(), β [ w c A () ] r() x(), (1 β) [ w ca () ]). w Since the utility function is increasing in each argument, (11) is equivalent to c R () < c A (). Furthermore, the worker is inifferent between the two moes if c R () = c A (). Here, we assume a hoc that workers take moe R when inifferent between the two moes. 4 Finally, she prefers equal to the average cost because a consumer surplus ecreases with the price. 4 This assumption is mae only for a technical reason. We can obtain similar results assuming otherwise. (11) 7

9 moe A to moe R if c R () > c A (). By efinition, therefore, when both moes are available, a worker at takes moe R if an only if m R m() λ + µ, (12) where λ m A + w(t A t R ): m() gives the highest price that inuces the workers living at to take moe R. Since m( ) is a ecreasing function, however, any < satisfies (12) if satisfies it. Consequently, it is actually the highest price that inuces all the workers living within miles from the city center to take moe R. In this sense, we call it an upper limit price for. Moreover, those workers are referre to as -mile inner city workers. We can view this result from the opposite perspective, using a new variable (m R ): (m R ) µ m R λ if m R > λ otherwise. There are two cases to consier. First, suppose that m R > λ. Then, (12) is rewritten as (m R ). Therefore, the workers living at (m R ) take moe R if that moe is available at (see Figure [ ] 1). In other wors, the workers living at min (mr ), l take moe R since the rapi transit line is constructe up to the terminal at l. In contrast, the workers living at ( (mr ), l ], if any, an those living at > l take moe A because the former o not want to use it even though it [ ] is available an the latter have no access to it. With a new notation D(l, m R ) min (mr ), l, we can say that the D(l, m R )-mile inner city workers take moe R. Secon, suppose that m R λ. In that case, the workers at l have an access to moe R an take that moe because (12) is satisfie. The workers at > l o not have an access to it an therefore take moe A. For this case, too, we can say by the efinition of (m R ) that the D(l, m R )-mile inner city workers take moe R an the rest take moe A. Figure 1: Workers choices of moes One may woner if the workers living farther than the terminal station carry out a parkan-rie, or rive a car from home to a certain station of a rapi transit an then take it to the city center. It turns out, however, that this is not the case when the fixe monetary cost of an automobile use, µ, is sufficiently high. We will show this at the en of this section. In what follows, thus, we o not consier the possibility of the park-an-rie. The above argument enables us to istinguish two types of city; an auto city, where all the workers use automobile, an the l-mile rapi transit city, where the D(l, m R )-mile inner city workers use a rapi transit an the rest use automobile. The size of lan lot each worker consumes is etermine base on the amounts of transport costs she pays. To express this relationship explicitly, we use superscript s {A, R}, which escribes the spatial structure of a city: s = A represents the auto city with c() = c A () for all, while s = R represents the l-mile rapi transit city with c() = c R () for D(l, m R ) an c() = c A () for > D(l, m R ). x() is now 8

10 written as x s (). Here, it woul be worth aing that the size of a lan lot at a certain point oes not epen on the values of variables at another location. In particular, it is inepenent of macro-scale variables such as a population istribution over a city an a geographical size of a city. This is a consequence of our assumption of a small open city. The secon question to ask is uner what circumstances a transport firm goes into the business. Since the population ensity at is equal to 1/ x() an the eman for its service comes from the D(l, m R )-mile inner city workers, the revenue of a transport firm in a city with its spatial structure being s is equal to Λ s (l, m R ) = m R D(l,m R ) x s () for s {A, R}. The firm enters the inustry if an only if its profit, Π s (l, m R ) Λ s (l, m R ) Γ(l), is nonnegative. 5 Now we are reay to efine two types of equilibria. The first type is the equilibrium at which all the workers use automobile. This is the case if a transport firm woul suffer a loss by entering the inustry. Formally, we say that the auto city is supporte as an equilibrium outcome if Λ A (l, m R ) < Γ(l) (13) for any l [, b A ] an any m R, where b s enotes the location of the bounary of a city with spatial structure s (s {A, R}). It is important to note that in this efinition, an entering transport firm is to compute its revenue on the supposition that the spatial structure of the auto city will remain unchange, that is, x() will be kept at x A (), even after it begins to operate a rapi transit system. At the secon type of equilibrium, on the other han, some workers use a rapi transit an a transport firm can earn nonnegative profit. We say that the l-mile rapi transit city is supporte as an equilibrium outcome if there exists m such that Λ R (l, m ) Γ(l). (14) Two comments follow. First, this efinition takes into account two channels through which the price affects the revenue. One is a irect channel, which involves a change in the payment by each worker, an the other is an inirect channel, which involves a change in the spatial structure of a city. Secon, as has been argue above, we o not iscuss the choice of a price by a transport firm. Inee, (l, m ) satisfying (14) oes not necessarily maximize its profit. In the rest of this section, we emonstrate that the park-an-rie oes not occur when µ is sufficiently high. Consier a worker who rives a car from her home to l an then takes a rapi transit from there to the city center. The cost of such an itinerary is equal to c A ( ) + 5 Here, we assume that the firm enters when the cost equals the revenue. This assumption is arbitrary an can be change. 9

11 c R ( ) = (m R λ) + η A + µ. If m R > λ, it is minimize at = an a worker oes not conuct a park-an-rie. If m R λ, instea, it is minimize at = l an she carries out a park-an-rie parking a car at a terminal station. 6 Now, suppose that m R λ. We know that the l-mile inner city workers use a rapi transit. Furthermore, the above argument implies that all the workers living beyon the terminal station park a car there an then take a rapi transit. The total revenue is therefore equal to Λ s PKRD (l, m R) = m R l b s x s () + l m R l 1 x s () for s {A, R}. For a given spatial structure, this revenue is increasing in m R an therefore, maximize at m R = λ. When there is no restriction on m R so that the firm can charge m R > λ, on the other han, it may charge m(l) > λ to earn Λ s (l, m(l)). If µ is sufficiently high, however, Λ s( l, m(l) ) Λ s PKRD (l, λ) = µ l l b s x s λl () l 1 x s () is positive. In that case, therefore, the firm prefers charging m(l), which iscourages a park-anrie, to charging λ, which gives rise to a park-an-rie. In this paper, we concentrate on this case: µ is sufficiently high that the firm oes not charge too low a price that inuces a park-an-rie. 4 Analysis of the conitions for each type of equilibria In this section, we explore the conitions for each type of equilibrium. 4.1 Conition for the auto city equilibrium It is obvious that (13) hols for any l [, b A ] an any m R if an only if max Π A (l, m R ) <, (15) l [,b A ], m R where Π A (l, m R ) Λ A (l, m R ) Γ(l). Two observations are important. First, suppose that a price is so high that (m R ) < l. Because the (m R )-mile inner workers use a rapi transit, the profit increases as the length of a rapi transit line is curtaile from l miles to (m R ) miles. Secon, suppose that a price is so low that (m R ) > l. Then, we can raise the price up to the upper limit price, m(l), without changing the number of passengers, which yiels a higher profit. It follows from these two observations that m R = m(l) at the maximum of Π A (l, m R ) (for more rigorous erivation, see the proof of the subsequent proposition). Let us enote Λ A( l, m(l) ) = m(l) l / x A () by Ω A (l). We have establishe the following result: 6 We arbitrarily assume that workers o a park-an-rie when they are inifferent between oing it an not oing it. 1

12 Proposition 1 The auto city is supporte as an equilibrium outcome if an only if Ω A (l) < Γ(l) (16) hols for any l [, b A ]. Proof. Let (l, m ) be the solution to max l [,b A ], m R Π A (l, m R ). First, suppose that (m ) < l. Then, Π A( (m ), m ) Π A (l, m ) = Γ(l ) Γ ( (m ) ) > since Γ( ) is an increasing function. This contraicts the supposition that l maximizes Π A (l, m R ), an consequently (m ) l. Secon, suppose that (m ) > l. Then, Π A( l, m(l ) ) [ Π A (l, m ) = m(l ) m ] l x A >, () which contraicts the supposition that m maximizes Π A (l, m R ), an consequently, (m ) l. Hence, we have (m ) = l, or m = m(l ), which implies that max l [,b A ], m R max Ω A (l) Γ(l). The proposition immeiately follows. l [,b A ] The impact of l on Ω A (l) is ambiguous: Ω A (l) = m(l)l x A (l) µ l 2 l Π A (l, m R ) = x A. (17) () The first term of the right han sie represents a marginal effect. As the rapi transit line is extene, aitional workers with their mass equal to 1/ x A (l) come to use it, each paying m(l)l. The secon term, furthermore, shows a effect through the change in price. In orer to tempt workers to use a longer transit line, a transport firm nees to lower its price by the amount equal to m (l) = µ/l 2. It is worthwhile examining the impacts of changes in parameters upon the conition. First, the auto city becomes more likely to be supporte by an equilibrium as the time cost of the use of a rapi transit, t R, rises because Ω A (l) t R = w l x A <. () The rise in t R, making a rapi transit less attractive, lowers the upper limit price, m(l), which reuces the revenue of a transport firm. Secon, the irections of the impacts of changes in the costs of automobile use are ambiguous. As an example, consier the change in m A : Ω A (l) l = m A l x A () m(l) 2 x A () [ x A ()] 2. (18) c A () For one thing, as a result of the rise in m A, the upper limit price rises, which irectly raises the revenue of a transport firm. This is represente by the first term of the right han sie. At the 11

13 same time, the higher m A brings about a more sparse population istribution, which reuces the profitability of a rapi transit (remember that x A ()/ c A () > by (1)). This is capture by the secon term. If the latter inirect effect ominates the former irect effect, the overall impact is negative. In that case, the auto city becomes more likely to be supporte by an equilibrium as m A rises. If the inirect effect is ominate by the irect effect, instea, the opposite result hols. The changes in t A an µ have similar impacts. Thir, recall that the ecline of a prevailing utility level (ū) an that of the relative measure of work hours (β) bring about a enser population istribution at a given location (see (1)). Therefore, these changes result in a higher revenue of a transport firm, which makes it less likely for the auto city to be supporte by an equilibrium. 7 Fourth an last, the effect of a change in the wage rate (w) is ambiguous. It affects the revenue of a transport firm through two channels. First, it alters a spatial structure: as the wage rate rises, the population ensity goes up (remember (1)). Secon, 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 the use of a rapi transit is relatively lower compare to that of automobile, the former moe becomes further attractive as a result of the rise in the wage rate, which raises the upper limit price. If the opposite hols, the upper limit price falls as the wage rate rises. Inee, we have Ω A (l) w = (t A t R ) l l x A () m(l) x A () [ x A ()] 2. (19) w The two terms of the right han sie represent the ambiguous effect through the change in the upper limit price an the positive effect through the change in a spatial structure (recall that x A ()/ w < ), respectively. As long as the time cost of a rapi transit use is not too high compare to that of automobile, the first effect is ominate by the secon, an total effect becomes positive. 4.2 Conition for the rapi transit city equilibrium Let us turn our eyes to a rapi transit city equilibrium. In what follows, the arguments of (m R ) an D(l, m R ) will be often omitte for the sake of clarity. There exists m that satisfies (14) if an only if which leas to the following result: Ω R (l) max m R ΛR (l, m R ) Γ(l), (2) Proposition 2 The l-mile rapi transit city is supporte as an equilibrium outcome if an only if (2) hols. 7 For χ {ū, β}, (1) implies that ΩA (l) χ l = m(l) [ x A ()] 2 x A () χ <. 12

14 It is straightforwar to see Ω R (l) for almost all l, because the envelope theorem implies that it is equal to m R l / x R (l) if l < an if l >. 8 Furthermore, two observations follow. First, the solution to this maximization problem can be lower than the upper limit price, m(l), that is, (m R ) can excee l. This exhibits a sharp contrast to a fining for the auto city equilibrium, the fining that raising a price up to the upper limit price results in the increase in the revenue. The reason is that the change in a price invokes a change in the spatial structure of a city. To see this, suppose that the price falls from the upper limit price. Because the users of a rapi transit are still the l-mile inner city workers, the revenue eclines other things being equal. As a result of the fall in the price, however, the population istribution becomes enser, which gives a positive impact on the revenue. Secon, the solution to the maximization problem can be greater than the upper limit price. This fining is also ifferent from that obtaine for the auto city equilibrium. Here, our concern is to maximize the revenue for a given length of a rapi transit line. Consequently, there is a possibility that a part of the line is left unuse at the maximum. These two observations are easily verifie by computing the following erivative: Λ R (l, m R ) m R = D [ x R () 1 ln xr () ln m R ] + D 2 x R () ln D ln m R. (21) The right han sie shows three components of the effects of the rise in a price. The first, represente by the first term in the square brackets ( 1 ), is a positive irect effect. The secon component, capture by the secon term in the brackets, is a negative effect by the ecrease in the population ensity. The last component, appearing at the en, is a nonpositive effect that the range of the locations of passengers is curtaile. The first observation above concerns the case with m R < m(l), or equivalently, l < (m R ). In that case, D(l, m R ) = l an therefore, the last term of (21) isappears. The overall erivative can be negative because we have the negative ensity effect. If it is negative, the maximum may be attaine at m R < m(l). Instea, the secon observation above concerns the case with m R > m(l), or equivalently, l > (m R ). In this case, D(l, m R ) = (m R ) an therefore, the negative last term remains. Nonetheless, the overall erivative can be positive. Then, m R > m(l), or equivalently, l > (m R ) can give the maximum. Ω R (l). Furthermore, it is straightforwar to examine the impacts of changes in parameters upon First, Ω R (l) increases as the transport cost of an automobile use rises, that is, m A, µ an/or t A rises. This is because the change is transmitte only through (m R ), an m Ω R (l) R = x R ( ) > if (mr ) < l if (mr ) > l. (22) 8 Ω R (l) is not efine at l =. 13

15 Secon, Ω R (l) ecreases as the time cost of a rapi transit use, t R, rises. For this change, we obtain Ω R (l) t R = [ µm w R (m R λ) 2 x R ( ) + wm R l x R ( ) { x R ( ) } 2 x R ( ) ] c R () < if (mr ) < l x R ( ) c R () < if (mr ) > l. The first term of the right han sie in the first line represents the effect of the fall in the number of users ( ). The secon term captures the effect of the change in a spatial structure to that with a lower population ensity (recall that we have seen in (1) that x R ( )/ c R () > ). Thir, changes that lea to a enser population istribution, namely, the ecrease in a prevailing utility level an that in a relative work hours, give a favorable effect on the rapi transit city equilibrium through a rise in Ω R (l). 9 Fourth an finally, the effect of a change in the wage rate is ambiguous. [ µ Ω R (l) m (t { } 2 A t R ) x R w = (m R λ) 2 x R ( ) R ( ) ] x R ( ) w if (mr ) < l l x m R ( ) R x R ( ) w > if (mr ) > l. The two terms of the right han sie in the first line represent the ambiguous effect of the change in the number of users an the positive effect of the change in a spatial structure (recall that x R ()/ w < ), respectively. As long as the time cost of a rapi transit use is not too high compare to that of automobile, the first effect is ominate by the secon, an the total effect becomes positive. (23) (24) 4.3 Conition for the multiple equilibria One of the interesting consequences of our moel is a possibility that both the auto city equilibrium an the rapi transit city equilibrium are realize for the same set of parameters. In this case of multiple equilibria, an ientical city becomes an auto city in some circumstances an a rapi transit city in others, epening on the factors outsie economic consierations such as historical accients an expectations. A necessary an sufficient conition for the multiple equilibria immeiately follows from Proposition 1 an Proposition 2: Corollary 1 Both the auto city an the l -mile rapi transit city are supporte as equilibrium outcomes for the same parameter set if both the following conitions are met: (16) is satisfie for any l [, b A], 9 Since Ω R (l)/ χ = Λ R (l, m R )/ χ for χ = {ū, β}, (1) implies that Ω R (l)/ ū = Ψū < an Ω R (l)/ β = Ψ β <, where Ψ χ m R [ x R ()] 2 xr (). χ 14

16 an Ω R (l ) Γ(l ) (25) hols. We have seen above that for some parameters, the irections of the effects of their changes on at least Ω A (l) or Ω R (l) are ambiguous. For other parameters, specifically, t R, ū an β, on the other han, those on both Ω A (l) an Ω R (l) are unambiguous. However, the irections are the same. Therefore, it is not straightforwar to ecie analytically the irections of the impacts of changes in parameters on the multiple equilibria. 5 Numerical simulations The key functions, Ω A (l) an Ω R (l), are too complicate to characterize the two types of equilibria analytically. In this section, we, specifying functional form, explore them through numerical simulations. 5.1 Specifications of functional forms an parameters Utility function an cost function Suppose that the preference over the consumptions of lan an composite goo is represente by a log linear utility function, u(x, z) α ln x + (1 α) ln z. The eman for lan an the lan rent at that simultaneously solve (5) an(6) become x() = θ [ r() w c() {w c()} 1 θ respectively, where θ αβ an r() θ [ (1 α)β ] 1 α α ] 1 θ an r() = r() (1 β) 1 β θ [ ] 1 w c() θ, w c() ū 1 θ w 1 β θ [w c()] 1 θ are positive constants. Moreover, we can erive from (7) the location of a city bounary as a solution to c(b) = w [ w c() ] [ ] r θ a, r() which implies that the bounary qualification (8) is always satisfie for this preference. For the sake of exposition, furthermore, we focus on the benchmark case of linear C(l), i,e., Γ(l) = γ + γl for some γ > an γ >, which are respectively referre to as a fixe cost an a marginal cost not with respect to the amount of proucts but with respect to the length of a rapi transit line. In the rest of the paper, we evaluate the revenue of a transport firm in terms of the total lan 15

17 rent at the city center in the auto city, i.e., r A (): Λ s (l, m R ) m r A = R () θ [w c A ()] 1 θ m R η = 2 A (1 + θ)(w µ) 1 θ m R η 2 R (1 + θ)(w µ) 1 θ D [w c s ()] 1 θ θ [θ(w µ) 1+θ θ (w η A D µ) 1 θ { θ(w µ) + ηa D }] for s = A, ] [θw 1+θ θ (w η R D) 1 θ (θw + ηr D) for s = R. Here, the assumption that the transport costs are linear in, that is, c A () = η A + µ with η A m A + wt A an c R () = η R with η R m R + wt R, enables us to calculate the integral. Furthermore, the bounary qualification (8), which we have shown to be satisfie, guarantees that w c s () in the integral is greater than Parameters For the time cost coefficients, Brueckner s moel case furnishes a clue (Brueckner (21)). He consiers a worker who commutes 25 times a year by riving a car at 3 miles per hour. 1 However, one may argue that the riving spee of 3 miles per hour is rather too high in many cities in the worl: In most of European an Japanese big cities, for example, the average riving spee is much lower. Thus, we use the spee of 25 miles per hour instea of 3 miles. Computing time costs for such a worker, we obtain t A = t A Furthermore, we use t R.8 t A as a benchmark value. For the monetary costs of an automobile use, furthermore, we can use the estimates by the American Automobile Association (AAA) an the Automobile Association in the Unite Kingom (AA). 12 The AAA reports that the variable costs are equal to $.1454 for a small sean an $.1718 for a meium sean, both per vehicle-mile. 13 The figures of the AA are much higher: 1 The figure of 25 for the yearly work ays is not far from that inicate by Japanese ata (Monthly Labor Survey), which is in 2, in 25, an 228 in The worker spens 25 2/25 = 2 hours on a 1-mile commute (2 miles for a roun trip) in one year, which occupies 2/(365 24) =.2283 of all the available time. 12 Several economists have also estimate the costs. Small an Verhoef (27) estimate that the private cost of operation an maintenance, the vehicle capital cost, roaway cost an parking cost are $.141, $.17, $.16 an $.7 per vehiclemile, respectively, in the Unite States, which totals $.334. Moreover, Roy (2) cites figures of e.462 an e.449 per vehicle-km (or, equivalently $.752 an $.731 per vehicle-mile) at peak hours an off-peak hours, respectively, for a small gasoline car in the urban areas of the Unite Kingom excluing Lonon area. Here, the exchange rate of $1 = e.987 in January of the year 2, which is taken from Shams (25), is use. Also he estimates the cost at e.497 an e.31 per vehicle-km (or, equivalently $.81 an $.55 per vehicle-mile) for a small gasoline car an iesel car, respectively, in the urban areas of France. 13 The figures presente here are those in 215 for the AAA an in 214 for the AA. They are obtainable at their web sites. Furthermore, the variable costs in this paper correspon to the operating costs in AAA s report an the running costs in AA s, whereas the fixe costs correspon to the ownership costs in AAA s an the staning charge in AA s. Their efinitions are, however, quite similar. The variable costs inclue the costs of fuel, tires an maintenance; an the fixe costs inclue insurance, taxes, epreciation an finance charge. In aition, AA s original ata are in terms of pouns, which are converte to the American ollars base on the exchange rate in 215 provie by the IMF. 16

18 $.3466 for a car whose price ranges from 13, to 18,, an $.3743 for a car whose price ranges from 18, to 25,, both per vehicle-mile. In this paper, we aopt a figure in between those in the two countries, $ Then, assuming that a typical worker works 25 ays a year, we can obtain the parameter for the yearly variable costs as m A = m A 15 (= ). For the fixe cost, the AAA gives $4,548 for a small sean an $6,139 for a large sean, both per year. The figures in AA s are not too ifferent: $4,936 for a car from 13, to 18,, an $6,3 for a car from 18, to 25,, both per year. Here, we take $5,, that is, µ = µ 5. In the following simulations, benchmark figures of t A = t A, m A = m A, µ = µ an t R = t R are use unless otherwise mentione. In the Unite States, househol meian income is equal to $53,657 (the US Census) an the average annual hours actually worke per worker are 1,789 (the OECD), which occupies 2.42% of all the time, both in 214. From these numbers, we estimate w = 53657/.242 = 262, 767 (which is the amount of money that one coul earn in one year by working 24 hours a ay). Finally, we estimate preference parameters. First, the share of spening on housing is usually 1% to 2%. 15 Thus, we assume that the share of spening on lan, α, is equal to 15%. Secon, for β, we evaluate (4) for an average worker. Our computation on a hypothetical average worker suggests that the reasonable size of β is 18% to 21%; an thus we assume that β = Lastly, α =.15 an β =.2 yiels θ =.3. Now, we are reay to conuct numerical simulations base on these benchmark values of parameters. 5.2 Simulation analysis of the two types of equilibria To begin with, let us examine function Ω A (l). Figure 2 epicts it as an Ω A revenue curve. Each of the five curves correspons to a ifferent value of t A. It is apparent that for smaller values of l, the curve is upwar sloping. In the benchmark case with t A = t A, for instance, it is upwar sloping for l [, 39.43). For such values of l, the positive effect of the extension of a rapi transit line ue to the increase in passengers (the first term of the right han sie in (17)) ominates the negative effect arising from the ecline in a price (the secon term). Figure 2: Ω A revenue curve 14 Here, we take into account the fact that the price of gasoline is exceptionally low in the Unite States compare to those in other avance countries. 15 In Japan, for example, it is 16.2% for owner-occupie housing an 13.5% for rente housing in 214 (Family Income an Expeniture Survey). 16 Suppose that the average worker living at 1 miles away from the city center commutes by a car. We can obtain the estimates of m(1) an t(1) using t = t A, m = m A an µ = µ. Furthermore, wn is estimate at $53,657, the househol meian income. Using these figures, we obtain β = In some countries, a rapi transit is wiely use for commutes. If the marginal cost parameters for the rapi transit use are the same for the automobile use, that is, t R = t A an m R = m A, β is equal to.243. The estimate is not sensitive to the changes in cost parameters: If t R =.8 t A an m R =.8 m A, the corresponing β becomes.243. If t R =.5 t A an m R =.5 m A, the corresponing β becomes

19 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 at the rapi transit city, Λ R (l, m R ), can be positive or negative, an therefore, there is a possibility that the maximum of Λ R (l, m R ) is attaine at m R = m(l). However, simulation analyses show that such a case is not plausible to occur. For plausible values of parameters, in other wors, Λ R (l, m R )/m R is positive for m R < m(l) an negative for m R > m(l), which implies that the maximum is attaine at m R = m(l), the upper limit price for l. To see this, we can take a look at Figure 3, which escribes the relationship between m R an Λ R (l, m R ). The upwar sloping curves represent the revenues for m R < m(l). Because D(l, m R ) is equal to l in this case, the revenue epens on l an thus, each curve correspons to a particular value of l. As l rises, the revenue of a transport firm increases for given m R an the curve shifts upwar. Furthermore, the ashe curve represents the revenue for m R > m(l). For such m R, D(l, m R ) is equal to (m R ) an oes not epen on l. This curve intersects each of the above upwar sloping curve at m(l). For instance, the intersection of that curve an the upwar-sloping curve for l = 15, point A, is at m(15). Putting these two kins of curves altogether, the revenue is escribe by a mountain-shape kinke curve with its summit at m(l). Again for l = 15, it is escribe by curve OAB. As long as the ashe curve is ownwar sloping at m(l), the maximum revenue is attaine at that point. For the parameters we are consiering, it is the case for l 61.6, that is, the length of a rapi transit line is equal to or shorter then 61.6 miles, which is likely to hol in a real city. In the rest of the paper, therefore, we concentrate on this case, that is, Ω R (l) = Λ R( l, m(l) ). (26) Figure 3: The relationship between the price an the revenue in a rapi transit city Then, we have l [ 1 Ω R (l) Ω A (l) = m(l) x R 1 ] x A. (27) That is, the ifference between the values of Ω functions for the two types of equilibria epens only on the ifference in population ensities. Furthermore, note that for m R = m(l), c R () c A () is equal to µ( l)/l, which is negative for < l. Because higher c() implies higher x() (see (1)), it follows that Ω R (l) Ω A (l) >. In other wors, the lower transport cost at a rapi transit city brings about a enser spatial structure, which yiels a higher revenue. This result implies that for any l, there exists some Γ(l) that satisfies both (16) an (25), that is, there is always a possibility of multiple equilibria. Figure 4 epicts Ω R (l) as an Ω R revenue curve as well as the Ω A revenue curve. Also escribe in the figure are three ashe straight lines, which represent Γ(l) s. We call them cost curves. Here, their intercepts with the vertical axis are equal to the fixe cost, i.e., the cost when the length of a transit line is zero. Consier the situation where the fixe cost graually eclines. First, when it is sufficiently high that the corresponing cost curve is given by Γ 1 (l) in Figure 4, the auto city is supporte by 18