JOURNAL OF URBAN ECONOMICS 43, 181]198 Ž1998.
ARTICLE NO. UE972041

Optimality of a Competitive Equilibrium in a Small
Open City with Congestion*†
Nobuo Akai
The Institute of Economic Research, Kobe Uni¨ ersity of Commerce
8-2-1 Gakuen-nishimachi Nishi-ku, Kobe 651-21 Japan

Takashi Fukushima‡
Faculty of Economics, Tokyo Metropolitan Uni¨ ersity,
1-1 Minami Osawa, Hachioji, Tokyo 192-03, Japan

and
Tatsuo Hatta
Institute of Social and Economic Research, Osaka Uni¨ ersity,
6-1 Mihogaoka, Ibaraki, Osaka 567, Japan
Received March 15, 1994; revised January 23, 1997
Consider a small spatial economy with a congestion externality in the public
transportation sector. We prove that the competitive equilibrium allocation coincides with the social optimum allocation despite the presence of the congestion
externality. To this end, we first specify a set up conditions for a competitive
equilibrium allocation, and then we find another set of conditions for social
optimality. By comparing the two set of conditions, we prove that the competitive
allocation is compatible with the social optimal allocation. We would like toemphasize the fact that no optimal external toll, such as the optimal congestion tollcollected by a transportation authority, is required to attain the optimal allocation.
Our privately owned and selfish profit maximizing transportation companies end
up with charging the price exactly equivalent to the optimal toll. Q 1998 Academic
Press* We would like to thank Colin McKenzie for some helpful comments. Also the second
author would like to acknowledge the financial support of the Economics of Risk andInformation ŽTokio Marine and Fire Insurance . Fund, Faculty of Economics, Osaka University.
†
JEL Classification Number: H23, R10, R40.
‡
Correspondence to: Faculty of Economics, Tokyo Metropolitan University, 1-1 Minami
Osawa Hachioji, Tokyo 192-03, Japan. The authors’ names are listed in alphabetical order.
181
0094-1190r98 $25.00
Copyright Q 1998 by Academic Press
All rights of reproduction in any form reserved.

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AKAI, FUKUSHIMA, AND HATTA

1. INTRODUCTION
In this paper we consider a small open spatial economy with monocentric urban area where traffic congestion presents external diseconomy. We
show that a competitive equilibrium allocation, without any governmental
levy of congestion tolls, coincides with the social optimum allocation. The
role of the well known optimal congestion tolls is replaced completely byprofit maximizing competitive prices set by the transit service providers.
In order to prove our results, we first specify our model and derive theconditions for the competitive equilibrium allocation. In the competitivemodel, we assume that there is no government levying congestion tolls.
Instead, we assume that there are many competing commuter transit
services maximizing their own private profits, and we obtain a set of
equilibrium conditions for the competitive economy. Then we turn toobtain socially optimum allocation. There, we assume a completely planned
economy where the planner maximizes the social gains by choosing thecontrollable city variables such as allocation of land, population density,
the amount of consumables, etc., subject to technical and market clearing
conditions. By comparing the two set of conditions, we prove that thecompetitive allocation is compatible with the social optimal allocation.
Our result can be understood intuitively in the following manner. When
there exist many competing transit service providers, passengers can selectand use the service that offers the smallest gross travel cost, defined by thesum of congestion cost and pecuniary cost. As a result, competitive transit
companies must take the gross travel cost as given, and choose theirservice prices and the amount of land used for transit service in order tomaximize their profits. The price the competitive transit companies charge
becomes identical to the amount of the optimal congestion toll. This
competitive arrangement internalizes the congestion externality and leads
to the social optimality.
Our work is an extension of Kanemoto Žw12x, Chap. 1. who proved the
optimality of a competitive economy with monocentric urban model without congestion, a refinement of Hatta w6x who also proved a similar resultin a model with traffic congestion. Our results should be distinguishedfrom those of Oron, Pines, and Sheshinski w16x, which required optimalcongestion tolls levied by the transportation authority.
In addition, our results shed some light on the problem of efficient useof land for housing and transportation. For housing land, Oron, Pines, andSheshinski w16x showed that the efficient land use for housing required the
optimal congestion tolls and the competitive housing market. On the other
hand, the problem of optimal land use for transportation was first studied
by Mills and de Ferranti w15x, and Solow and Vickrey w21x. It was further
developed by Hochman w7x, and Pines and Sadka w18x. These authors

OPTIMALITY OF A COMPETITIVE EQUILIBRIUM

183

essentially showed that the efficient land use for transportation required acost benefit rule based on the market prices in addition to the optimalcongestion tolls and the competitive housing market. In contrast, ourresults show that our competitive economy attains efficient land use forboth housing and transportation without any congestion tolls nor costbenefit approach. Thus our competitive economy is able to replace therequired optimal congestion toll and cost-benefit rule in order to obtain
the efficient land use pattern.1
Section 2 lays out the basic framework of our model. The market
equilibrium is defined in Section 2.2, and in Section 3 we derive thenecessary conditions for social optimality. Section 4 proves the compatibility of the competitive and socially optimal allocations. A literature review
is provided in Section 5. Some concluding remarks are given in Section 6.
2. THE COMPETITIVE EQUILIBRIUM
2.1. The Model
2.1.1. Consumers
Consider a spatial economy, where urban and rural differences exist.
The consumers, whether they live in or outside the city, have identicalutility functions, and derive their utility by consuming two goods, consumables and housing space.
The utility function of a representative consumer is given byu s u Ž z y c, h . ,Ž 2.1.

where z, h, and c are the quantities of consumables, units of housing
space consumed,2 and the congestion cost that she suffers by commuting,respectively. Congestion is measured in units of the consumables, and it
represents the disutility a passenger suffers from commuting. The utility
function is increasing in both z y c and h.
Consumers are utility maximizers. They can move without any cost fromone location to an other in order to achieve the highest possible utility.
In the rural area. Once a consumer decides to live in the rural area, she
receives the rural wage rate, and has to pay for consumables and housing
space she consumes. We set the price of consumables as the numeraire.
1
See ‘Note on the Literature’ section ŽSection 5. for more detailed discussion of theliterature.
2
Since, in this model, no explicit construction of housing is introduced, the housing space,h, means the size of a plot of land.

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The rural rental rate and the rural wage rate are r a and w a, respectively.
They are assumed to be identical everywhere in the rural area. Therefore,
the location where a consumer lives within the rural area does not effecther utility. There is no commuting in the rural area, so there is nocongestion cost. These assumptions imply that the consumer maximizes
u s uŽ z a , ha .

Ž 2.2.

w a s z a q r aha ,Ž 2.3.

subject towhere z a and h a are quantities of consumables and housing space consumed by the rural consumer, respectively. The maximization determines
the quantities z a and h a as functions of w a and r a. Substituting these backin to Ž2.2. gives the indirect utility function of the formua s ¨ Ž w a, r a. .

Ž 2.4.

The small city assumption implies that the rural area is large in thesense that consumers can actually sell as much labor services as they wantat the fixed wage rate w a, and can purchase as large a housing space aspossible at the fixed unit price r a. Then the level of utility achieved by arural resident is fixed at u a.
In the urban area. On the other hand, if the consumer decides to live inthe urban area, she has to commute to work in the central business district
ŽCBD., and receives the urban wage rate w. She purchases consumables
and pays for housing space as the rural residents do. In addition, she must
pay for commuting to the CBD. The commuting cost depends on how far
from the CBD she lives.
The distance from the CBD is measured by the variable x, and we saythat she lives at x, meaning that she lives somewhere in the ring x miles
away from the CBD. The commuting cost, t Ž x ., then is an increasing
function of x. When she lives at x, her disposable income, w y t Ž x ., also
becomes a function of x. Accordingly, the amount of consumables andhousing space she can purchase depends on x, and we write them as z Ž x .
and hŽ x ..
When she commutes from x to the CBD, the consumer suffers fromtotal congestion given by the amount of cŽ x .. As the land supply at x in
the city is limited, the housing rent must depend on x as well and it is
written as r Ž x ..3
3

We assume that there are absentee landlords who receive all the rent.

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OPTIMALITY OF A COMPETITIVE EQUILIBRIUM

Putting these assumptions together, a city resident at x maximizes 4
u s uŽ z Ž x . y c Ž x . , hŽ x . .

Ž 2.5.

w y t Ž x . s z Ž x . q r Ž x . hŽ x . .

Ž 2.6.

subject to

The utility maximization implies thatuh Ž z Ž x . y c Ž x . , hŽ x . .
u z Ž z Ž x . y c Ž x . , hŽ x . .

s rŽ x. .

Ž 2.7.

Since the city is small 5 and consumers can move freely, the utility levelof city residents everywhere in the city are identical and they must be
identical to that of the rural residents, u a, in equilibrium. This implies thatuŽ z Ž x . y c Ž x . , hŽ x . . s ua .

Ž 2.8.

2.1.2. Urban Producers
Urban producers produce consumables at the CBD with a linear homogeneous production function given by y s F Ž N U ., where N U is the labor
force employed. The profit maximization implies thatw s FX Ž N U . ,Ž 2.9.

which is a constant due to the assumption of constant returns.
2.1.3. Commuter Transportation Industry
In order for the city residents to commute to the CBD, the transportation industry provides commuter train services.6 The service provided at x
is denoted by M Ž x ., meaning that M Ž x . is the number of the commuters
who cross the ring at x from the CBD. Since cŽ x . is the total amount ofcongestion the commuter must face by commuting from x to the CBD, we
can say that ˙cŽ x . is the instantaneous congestion at x and it is denoted bysŽ x ., that is˙c Ž x . s s Ž x . .
4

Ž 2.10.

Notice that cŽ x . is external to the consumer.
The smallness of the city is reflected in the fact that the rural rental and wage rates are
given constants.
6
Our model can be interpreted as a model of automobile roadway travel rather than amodel of railroads. See Section 5 where our model is compared to the one by Oron, Pines,
and Sheshinski w16x.
5

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AKAI, FUKUSHIMA, AND HATTA

We assume that at the CBD
c Ž 0 . s 0.

Ž 2.11.

The transportation industry produces rail services to transport M Ž x . ofpassengers at x. For that it requires an input of land, LŽ x ., of land when
the congestion level at x is sŽ x .. Thus its production function is written as
M Ž x . s T Ž LŽ x . , s Ž x . . ,Ž 2.12.

where T Ž?. is assumed to be linear homogeneous in LŽ x ..7
In addition to the congestion cost, the commuter at x has to pay thepecuniary commuting cost t Ž x . to get to the CBD. We can define theinstantaneous pecuniary cost pŽ x . such thattŽ x. '

x

H0 p Ž t . dt .

Ž 2.13.

We call the sum of the congestion cost and pecuniary cost the instantaneous gross travel cost and denote it by g Ž x ., that is,g Ž x . ' pŽ x . q sŽ x . .

Ž 2.14.

We assume that many privately owned rail service companies are competing with each other for commuter passengers. By competition, the grosstravel cost, g Ž x ., not pŽ x ., must become identical to that of othercompanies. Each company maximizes its profit by selecting pŽ x . and LŽ x .
regarding r Ž x . and g Ž x . as given.8
Using Ž2.12. and Ž2.14., the profit maximizing problem of each train
company is written as
Max P '

LŽ x ., pŽ x .

x

H0  p Ž x . T Ž L Ž x . , g Ž x . y p Ž x . . y r Ž x . L Ž x . 4 dx.

Maximizing with respect to LŽ x . and pŽ x ., this yieldsr Ž x . s p Ž x . TL Ž L Ž x . , s Ž x . . ,Ž 2.15.

M Ž x . s p Ž x . Ts Ž L Ž x . , s Ž x . . .

Ž 2.16.

and

7

The congestion level, sŽ x ., is external to the production process.
Strictly speaking, we should differentiate companies by assigning the subscript i for theith company. However, Hatta w6x showed that in the long run equilibrium the fare structure ofeach company becomes identical to each other when each firm has an identical production
function. Thus we ommitted the subscript from the beginning.
8

OPTIMALITY OF A COMPETITIVE EQUILIBRIUM

187

2.2. The Market Equilibrium Conditions
2.2.1. The City Labor Market
Let us denote the population density at x by N Ž x . and the city limit by
x. Since all the city residents work at the CBD, the total people employed,
N U , equals the total number of the city residents, that is
NU 'x

H0 N Ž x . dx,

Ž 2.17.

which is the equilibrium of the labor market.
2.2.2. The City Land Market
The space of land available in the city at x is denoted by u Ž x ..9 The landdemanded by the city residents and the transportation industry is hŽ x . N Ž x .
and LŽ x ., respectively. Equilibrium in the land market requires
hŽ x . N Ž x . q LŽ x . s u Ž x . .

Ž 2.18.

Also the competitive bidding for land implies that at the city limit, wehaver Ž x . s r a.

Ž 2.19.

2.2.3. The Commuter Train Ser¨ ice Market
The demand for transportation services at x is given by the total
number of people who live in the band between x and x. The equilibriumof the transportation market, therefore, is given byMŽ x. sx

Hx N Ž t . dt ,which is identical to the following two equations.
Ṁ Ž x . s yN Ž x . ,Ž 2.20.

M Ž x . s 0.

Ž 2.21.

2.2.4. The Full Market Equilibrium Conditions
The conditions for the consumer’s utility maximization are Ž2.6. andŽ2.7.. The free mobility of consumers gives Ž2.8.. Equation Ž2.9. is theurban producer’s maximization condition. The profit maximization of thetransportation industry implies conditions Ž2.12., Ž2.15., and Ž2.16.. Also,9
If all topography of the urban area is inhabitable, that is, if there is no lakes, no oceans,or no mountains, u Ž x . is equal to 2p x.

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AKAI, FUKUSHIMA, AND HATTA

the definition of congestion and the rail service price provide us with
Ž2.11., Ž2.10., and Ž2.13.. The equilibrium of the labor market and the landmarket are attained by Ž2.17., Ž2.18., and Ž2.19.. Also, the equilibrium of
the transportation market is attained by Ž2.20. and Ž2.21..
In our model, there are ten unknown functions of x, r Ž x ., pŽ x ., hŽ x .,Žz x ., t Ž x ., cŽ x ., sŽ x ., LŽ x ., M Ž x ., N Ž x ., and three unknown variables, w,
x, N U , to be determined. On the other hand, there are ten equations whichshould hold true in each value of x, and five single equations. The fact that
there are two extra single equations reflects the fact that Ž2.10. and Ž2.20.
are differential equations, and that they require two initial conditions.
When these equations and variables are determined, we say that themarket equilibrium is reached in the small open city.
3. THE SOCIAL OPTIMAL RESOURCE ALLOCATION
3.1. The Problem Stated
The social optimal problem is solved by maximizing
J ' F Ž NU . yx

H0 Ž z Ž x . y c Ž x . . N Ž x . dxxx

H0 c Ž x . N Ž x . dx y H0 r u Ž x . dx

ya

subject to Ž2.8., Ž2.10., Ž2.12., Ž2.17., Ž2.18. and Ž2.20. with the boundary
conditions Ž2.11. and Ž2.21..
This means maximizing the net product of the city, which is defined as
the total product minus the net consumption of city residents, minus thetotal cost of congestion, minus the payment to the land owners evaluatedby the rural rent. In other words, it is the net economic rent created byorganizing the small open city. The process of maximization is constrainedby the fact that we must guarantee that city residents achieve the same
utility level as rural consumers at u a, and that the supplies of labor, land,
and transportation services in the city must be equal to their respective
demands in order for the maximization to be feasible. We say that thesocial optimum is reached in the small open city when this total economic
rent is maximized.10
J can be simplified to
J s F Ž NU . yxx

H0 z Ž x . N Ž x . dx y H0 r u Ž x . dx.
a10This is considered to be a case of the optimal control problem known as the problem of
Hestenes and Bolza.

OPTIMALITY OF A COMPETITIVE EQUILIBRIUM

189

The first order conditions of this problem can be found from thefollowing Lagrangian defined by
L ' F Ž NU . qx

H0 I Ž ?. dx q m NU

y l1 Ž x . M Ž x . y l2 Ž x . c Ž x . q l1 Ž 0 . M Ž 0 . q l2 Ž 0 . c Ž 0 .
q g 20 c Ž 0 . q g 11 M Ž x . ,whereI Ž ?. ' I Ž z Ž x . , h Ž x . , L Ž x . , c Ž x . , N Ž x . , M Ž x . , s Ž x . .

˙1 Ž x . M Ž x . q l˙2 Ž x . c Ž x . y l1 Ž x . N Ž x . q l2 Ž x . s Ž x .
'lq  yz Ž x . N Ž x . y r au Ž x . 4 q n Ž x .  u Ž z Ž x . y c Ž x . , h Ž x . . y u a 4
q p Ž x .  T Ž LŽ x . , s Ž x . . y M Ž x . 4 y m N Ž x .
q g Ž x .  u Ž x . y hŽ x . N Ž x . y LŽ x . 4 .
Notice that n Ž x ., l2 Ž x ., m , g Ž x ., p Ž x . and l1Ž x . are the Lagrangian
multipliers associated with Ž2.8., Ž2.10., Ž2.12., Ž2.17., Ž2.18. and Ž2.20.,
respectively.
3.2. The Necessary First Order Conditions
Then the necessary conditions consist of five equations on the control
variables, two adjoint equations on the state variables, and six transversality equations on the control parameters. They are summarized as follows
IsŽ x . s l 2 Ž x . q p Ž x . Ts Ž L Ž x . , s Ž x . . s 0,

Ž 3.1.

I zŽ x . s yN Ž x . q n Ž x . u z Ž z Ž x . y c Ž x . , h Ž x . . s 0,

Ž 3.2.

IhŽ x . s y g Ž x . N Ž x . q n Ž x . u h Ž z Ž x . y c Ž x . , h Ž x . . s 0,

Ž 3.3.

ILŽ x . s y g Ž x . q p Ž x . TL Ž L Ž x . , s Ž x . . s 0,

Ž 3.4.

INŽ x . s y l1 Ž x . y m q Ž yz Ž x . y g Ž x . h Ž x . . s 0,

Ž 3.5.

˙1 Ž x . y p Ž x . s 0,
LM Ž x . s l

Ž 3.6.

˙2 Ž x . y n Ž x . u z Ž z Ž x . y c Ž x . , h Ž x . . s 0,
LcŽ x . s l

Ž 3.7.

XU

LN U s F Ž N . q m s 0,

Ž 3.8.

LM Ž0. s l1 Ž 0 . s 0,

Ž 3.9.

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AKAI, FUKUSHIMA, AND HATTA

LcŽ0. s l 2 Ž 0 . q g 20 s 0,

Ž 3.10.

LM Ž x . s y l1 Ž x . q g 11 s 0,

Ž 3.11.

LcŽ x . s y l 2 Ž x . s 0,

Ž 3.12.

˙1 Ž x . M Ž x . y l˙2 Ž x . c Ž x .
Lx s I Ž x . y l
˙ Ž x . y l2 Ž x . s Ž x . q g 11 M˙ Ž x . s 0.
y l1 Ž x . M

Ž 3.13.

4. THE COMPATIBILITY
In this section, we establish that the competitive equilibrium allocation
coincides with the socially optimal allocation.
Let r Ž x . s g Ž x . and pŽ x . s p Ž x .. In other words, we interpret g Ž x .
and p Ž x . to be the rental rate of the land and the price of the rail service
charged at x, respectively.
4.1. Utility Maximization of Urban Consumers
Equations Ž3.2. and Ž3.3. imply thatuh Ž z Ž x . y c Ž x . , hŽ x . .
u z Ž z Ž x . y c Ž x . , hŽ x . .

s rŽ x. .

Ž 4.1.

Equations Ž3.5., Ž3.8. and Ž4.1. imply the consumer’s utility maximization
when the wage is paid by the marginal product of the CBD, and theycorrespond to Ž2.6., Ž2.7., and Ž2.9..
4.2. Profit Maximization of Commuter Rail Ser¨ ices
Equations Ž3.2. and Ž3.7. imply thatl̇2 Ž x . s yN Ž x . ,which, by integration, yieldsxx

Hx N Ž t . dt s Hx l˙ Ž t . dt .

y

2

Using the definition of M Ž x . together with Ž3.1. and Ž3.12., this becomesM Ž x . s p Ž x . Ts Ž L Ž x . , s Ž x . . .

Ž 4.2.

Equations Ž3.4. and Ž4.2. coincide with the profit maximization conditions of the transportation industry given by Ž2.15. and Ž2.16..

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OPTIMALITY OF A COMPETITIVE EQUILIBRIUM

4.3. The Optimum City Size
A straight forward calculation of Ž3.13. with Ž3.11. and Ž3.12. yields
Lx s  yz Ž x . N Ž x . y r au Ž x . 4 q n Ž x .  u Ž z Ž x . y c Ž x . , h Ž x . . y u4
q p Ž x .  T Ž LŽ x . , s Ž x . . y M Ž x . 4
q g Ž x .  u Ž x . y h Ž x . N Ž x . y L Ž x . 4 y  l1 Ž x . q m 4 N Ž x .
s  yl1 Ž x . y m y z Ž x . y g Ž x . h Ž x . 4 N Ž x . q  yr a q g Ž x . 4 u Ž x .
q  yg Ž x . L Ž x . q p Ž x . T Ž L Ž x . , s Ž x . . 4 .
The first term of the last expression is zero from Ž3.5.. Also, the third termof the expression is zero due to Ž3.4. and homogeneity of T Ž?. with respectto LŽ x .. Thus, we obtainr a s g Ž x. .

Ž 4.3.

Equation Ž4.3. corresponds to Ž2.19., which determines the border of thecity, x.
4.4. Other Conditions
From Ž3.6. and Ž3.9., we obtainx

H0 p Ž t . dt s l Ž x . .
1

Ž 4.4.

When t Ž x . is regarded as l1Ž x ., Equation Ž4.4. is identical to Ž2.13..
Equations Ž2.8., Ž2.10., Ž2.12., Ž2.17., Ž2.18., and Ž2.20. are used as
constraints of the social optimal problem and hence, though they are notlisted in Section 3.2, they are a part of the first order conditions of the
optimal problem.
Equations Ž2.11. and Ž2.21. are the boundary conditions, and they
belong to the optimal system as well.
In short, all of the equilibrium conditions in Section 2.2 correspond to
the necessary conditions of the social optimum problem.
We are now able to state the following theorem.
THEOREM 1. The competiti¨ e market equilibrium conditions are identicalto the necessary first order conditions of the social optimal problem.11
In relation to the optimal land use problem for housing and transportation, we can observe the following. In our model, if the amount of land11

We can interpret our result as an application of the more general principle shown in club
good models. See Buchanan w4x and Berglas w2x for more details on this literature. We owe
this observation to one of the referees.

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AKAI, FUKUSHIMA, AND HATTA

used for transportation, LŽ x ., is an exogenous function, the market
equilibrium conditions conditional on exogenous LŽ x . continue to be
identical to the ones of social optimal problem conditional on exogenousLŽ x .. This can be seen easily. By setting LŽ x . exogenous, we must throw
away equation Ž2.15. from our competitive market equilibrium conditions.
From the social optimal problem, the optimality condition on LŽ x ., i.e.
equation Ž3.4., must also be taken out. The remaining conditions continue
to hold and thus we establish the following theorem.
THEOREM 2. If the amount of land allocated for transportation, LŽ x ., isan exogenous function, the remaining market equilibrium conditions are
identical to the social optimal conditions conditional on exogenously gi¨ enLŽ x ..
These theorems imply two results with regard to the efficient land use
problem.12 First result is due to Theorem 2. We can see that our profit
maximizing road or railroad owners, if competitive, will internalize thecongestion externality by imposing a fare, as a function of distance fromthe CBD, equivalent to the optimal congestion toll.13 And therefore, theresidential land use is efficient whether or not the land use for transportation is efficient. Second result is due to Theorem 1. That is, if LŽ x . isendogenous, profit maximizing road or railroad owners, if competitive, will
use the efficient amount of land for transportation. Thus we can attainefficient land use for transportation without any external congestion tolls
imposed by the government authority or any cost-benefit rule.
The intuition of this should be clear. The profit maximizing condition
with respect to LŽ x ., Equation Ž2.15., is identical to the cost benefit rule
based on the market prices. The left hand side is urban housing rent and itis the marginal social cost of converting a unit of housing land intotransportation land. The right hand side is the marginal social benefit ofextra number of people passing through x by the additional transportation
land. The equation shows that the land for transportation is allocated tothe point that the marginal social cost is equal to the marginal social
benefit. These results complement the existing efficiency results in theliterature.14
5. NOTE ON THE LITERATURE
The first study on the relationship between the congestion externality
and the optimal toll appeared in Pigou w17x. He presented an example,
known as ‘‘the case of two roads,’’ to demonstrate that a public investment
12

We are indebted to one of the referees to point this out.
See Section 5 for equivalence of the competitive fare and the optimal congestion toll.
14
See Section 5 for more detailed discussion of the literature.
13

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193

to construct a new road could end up in futility if the external economy
were to create traffic congestion on the road. He called for a tax which
covers the marginal social cost of externality to secure the maximum
efficiency in the use of the two roads.
Knight w13x contested the Pigou’s contention by proving that if theownership of the road is given to a private sector agent, precisely the ideal
situation, which was identical to the one brought out by the Pigouvian tax,
would be established through the operation of ordinary economic motives.
Faced with this claim, Pigou withdrew his example from his last edition of
the Economics of Welfare.
Strotz w22x first studied the effect of transport congestion in the context
of an optimal urban land use model. Oron, Pines, and Sheshinski w16x
studied the optimum and equilibrium land use pattern for housing in acontinuous monocentric city model with a traffic congestion where thetransportation land was predetermined, and proved the following proposition.15
PROPOSITION 1. In a monocentric urban economy model where thetransportation land is predetermined, the optimum resource allocation isachie¨ ed and can be supported by a competiti¨ e price system if a go¨ ernment
authority imposes optimal congestion tolls, which is equal to the number ofhouseholds commuting through x to the center of the city, M Ž x ., times thetime spent due to the marginal increase in commuter, t M , and redistributes theproceed as a lump sum subsidy.
Despite their apparent differences, our model is quite similar to the
OPS model. For example we prefer to put cŽ x . Žand sŽ x .., the congestion
variable, explicitly in utility and production functions. In doing so, weinterpret the congestion in our model to be the physical fatigue by riding
commuter trains. The speed of the train is invariant to whether thecongestion existed or not. The congestion in our model, therefore can beunderstood as the extra calories needed by consumers to get to work whenriding in the crowded train. And higher level of congestion does not crowdout the workers from reaching the city center. For the train service
operators, the congestion is a factor that represents the quality of theirtrain service. If the level of congestion is high, passengers will move awayfrom the crowded line to a less congested line.
In the OPS model, on the other hand, the congestion is formulated as
the time cost for consumers and the reduced speed in transportation
services. The congestion so formulated is a time loss for consumers and it
could crowd out the commuters from reaching the city center within a
15
Pines and Sadka w18x reaffirmed the result by using the model developed by Arnott w1x,
where the width of the road was parametrically given.

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limited time. However, the formality of our model becomes almost identical to the OPS model if some substitutions and reinterpretations are made.
For instance, if the OPS’s ZŽ x . q t Ž x .rPz is replaced by our z Ž x . andtheir t Ž x .rPz is replaced by our cŽ x ., and if they are reinterpreted
accordingly, the formal model structure of our consumer sector becomes
identical to that of the OPS model.
A similar transformation will prove that the transportation sector of the
OPS model is formally identical to ours if their velocity function, V Ž x . s¨ Ž EŽ x ., G 2 Ž x .., is rewritten as
E Ž x . s f Ž 1rV Ž x . , G 2 Ž x . . ,Ž 5.1.

where EŽ x . is the number of commuters who pass through x, V Ž x . is thevelocity of moving those passengers, and G 2 Ž x . is the land allocated forthe use of transportation. Notice that their EŽ x . is identical to our M Ž x .
and their G 2 Ž x . is also identical to our LŽ x .. Therefore equation Ž5.1. isformally identical to our transportation production function if 1rV Ž x . isinterpreted as our sŽ x ..
There is one difference in treating the land for transportation. As we
mentioned earlier, OPS assumes that G 2 Ž x . is a predetermined function ofx. However, in our model, LŽ x . is one of the endogenously determinedfunctions ŽTheorem 1. and it can be considered exogenous if necessaryŽTheorem 2..
But it is the formulation of the behavioral assumptions that makes twomodels distinct from each other. The difference can be compared to thatof Pigou and Knight. The OPS theorem states that the optimal toll must
be collected by the transportation authority Žthe TA hereafter . to attainefficiency and that the equilibrium is supported by the competitive pricesystem under the condition that the TA behaves as a tax authority
collecting the optimal external cost. The TA is not a private selfish
enterprise that seeks maximum profit. This is Pigou at heart.
On the other hand, we formulate the competitive behavior of thetransportation sector as a privately owned and selfish profit maximizer.
This makes sense only if there are quite a few train service providers
competing for passengers in order to maximize their profits. The resulting
profit maximizing price pŽ x . is given by our Equation Ž2.16., which can be
rewritten to givep Ž x . s M Ž x . rTs .

Ž 5.2.

Since 1rTs can be interpreted as dsrdT, pŽ x . can be interpreted as themarginal external congestion cost of an additional commuter. Therefore,
the optimal toll of the OPS becomes identical to our competitive price.

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195

This should be so since we proved that the economy attains the optimal
resource allocation under this price.
In view from the efficient land use problem, the OPS theorem provedthat residential land use was efficient if the optimal congestion tolls werecollected and redistributed as a lump sum subsidy under the competitive
residential land market. Our Theorem 2, on the other hand, states thatprofit maximizing roadrrailroad owners, if competitive, will impose a fareequivalent to the optimal congestion toll, and thus residential land use isefficient whether or not the division of land between residential use andtransportation use is efficient. Clearly our approach belongs to that of
Knight.16
Mills and de Ferranti w15x were the first to examine the optimal land use
pattern for roads in the context of a continuous monocentric city with fixed
housing lot size. They obtained a mathematical expression for the optimality.
Hochman w7x extended Mills and de Ferranti result by introducing
congestion tolls paid by the commuters. Maintaining the assumption of a
fixed housing lost size, Hochman proved that the Mills and de Ferranti
efficiency condition could be attained if each commuter paid the ‘‘fullcongestion toll,’’ which was equal to per capita market rent of the land
used for transit service. His ‘‘full congestion toll’’ can be easily shown to beequivalent to the social marginal cost of congestion. Therefore, the commuters in the Hochman’s model must pay the congestion toll equivalent to
the optimal tolls derived by the OPS model. The toll must be collected bygovernment authority in practice.
In addition to the optimal toll, the Hochman’s result required thecondition that equated the market housing rent to the marginal increase oftransit service due to a marginal addition of land. The condition may beinterpreted as the cost-benefit criterion by which to allocate the land to
the transit service. In this way the following proposition was first indicatedby Hochman. It was shown more clearly later, but not stated in words, byPines and Sadka w18x.
PROPOSITION 2. If a go¨ ernment authority imposes optimal congestion
tolls, then a simple cost-benefit rule based on comparing the market rent ofresidential land to the direct benefit of added transportation ser¨ ice from16
Hochman and Pines w8x stated in their concluding remarks section that their modelŽwhich included traffic congestion and privately owned profit maximizing transit service
sector. produced an efficient outcome. However, they did not provide the proof. If the HP
model produces the efficiency result, it certainly is a model that belongs to the approach of
Knight. For that, they need to prove the efficiency and should show that their consumers end
up paying the fare equivalent to the social external cost of congestion.

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adding land to transportation use leads the go¨ ernment to de¨ ote the efficientamount of land to transportation.
Our Theorem 1 proved that the roles of the optimal congestion tolls andthe cost-benefit criterion in Proposition 2 can be dropped and replacedby the competitive behavior of our transit service providers. And as aresult the efficient use of land for transportation is assured.17
As we mentioned earlier, we can understand our result intuitively. Our
profit maximizing transit providers use land efficiently as shown by equation Ž2.15., which is essentially equivalent to the cost-benefit condition
since the left hand side is the market rent of housing land and the righthand side is the marginal benefit of added transportation service from anadditional land to the service. In addition, equation Ž2.16. holds, which inturn gives equation Ž5.2.. Thus the consumers pay the train fare that isequivalent to the marginal external cost of congestion.
Hatta w6x proved a similar result to ours in a model with traffic congestion. In his model, the consumers were assumed to have identical utility
functions which depend only on one good called consumables. In otherwords, his model assumed away the possibility of substitution between
consumables and housing land, and forced the consumers to use a fixedamount of housing land. In order to compensate for this, he introduced
capital in his model, which made his model unnecessarily complex. Therefore his result was not readily comparable to the orthodox competitivemodels. Our model treats housing land as one of the consumer’s choicevariables and produces a result readily comparable to the standard optimality result of the competitive economy.
6. CONCLUDING REMARKS
We have shown that competitive forces will bring about the social
optimal allocations in a small open city model where traffic congestion
arises as an external economy to consumers and to the transportationindustry as well. The present results depend on the assumptions that the
production function of consumables is linear homogeneous and that theproviders of the commuter services compete with each other for passengers on the basis of the congestion inclusive price of transportation
services.
17
Notice that the present paper is directly related to the first best resource allocation
problem where the congestion is optimally priced Žor optimally priceable. by government
authority. There are quite a few papers in the literature that examine the resource allocation
problem in the second best Žor third best. environment. They include Solow w20x, Kanemoto
w10, 11x, Robson w19x, Arnott w1x, and Pines and Sadka w18x. See also Braid w3x for a grandlisting of papers that consider the optimal land use problem.

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197

Though we have proved the efficiency results in an open small citymodel, we can readily extend the result to the case of a full generalequilibrium model where the rural variables are determined jointly withthe urban variables.18

REFERENCES
1. R. J. Arnott, Unpriced transport congestion, Journal of Economic Theory, 21, 294]316
Ž1979..
2. E. Berglas, On the theory of clubs, American Economic Re¨ iew, 66, 116]121 Ž1976..
3. R. M. Braid, The use of land for roadways in a growing Mills]de Ferranti urban area,
Journal of Urban Economics, 37, 131]160 Ž1995..
4. J. M. Buchanan, An economic theory of clubs, Economica, 32, 1]14 Ž1965..
5. T. Fukushima and Y. Shigeno, Congestion externality and economic efficiency in a modelof urban economy, Mathematics and Computers in Simulation, 39, 385]391 Ž1995..
6. T. Hatta, Competition and nationally optimum resource allocation under the presence ofurban traffic congestion, Journal of Urban Economics 14, 145]167 Ž1983..
7. O. Hochman, Market equilibrium versus optimum in a model with congestion: Note,
American Economics Re¨ iew, 65, 992]996 Ž1975..
8. O. Hochman and D. Pines, Competitive equilibrium of transportation and housing in theresidential ring of an urban area, En¨ ironment and Planning, 3, 51]62 Ž1971..
9. M. I. Kamien and N. L. Schwartz, ‘‘Dynamic Optimization: The Calculus of Variations
and Optimal Control in Economics and Management,’’ North-Holland, New York
Ž1981..
10. Y. Kanemoto, Congestion and cost-benefit analysis in cities, Journal of Urban Economics,
2, 246]264 Ž1975..
11. Y. Kanemoto, Cost-benefit analysis and the second-best land use for transportation,
Journal of Urban Economics 4, 483]503 Ž1977..
12. Y. Kanemoto, ‘‘Theories of Urban Externalities,’’ North-Holland, Amsterdam Ž1980..
13. F. Knight, Some fallacies in the interpretation of social cost, Quarterly Journal of
Economics 38, 582]606 Ž1936..
14. L. Legey, M. Riper and P. Varaiya, Effects of congestion on the shape of a city, Journal of
Economic Theory, 6, 162]79 Ž1973..
15. E. S. Mills and D. M. de Ferranti, Market choices and optimum city size, American
Economic Re¨ iew, Papers and Proceedings, 61, 340]345 Ž1971..
16. Y. Oron, D. Pines, and E. Sheshinski, Optimum vs. equilibrium land use pattern andcongestion toll, Bell Journal of Economics 4(2), 619]636 Ž1973..
17. A. C. Pigou, ‘‘The Economics of Welfare,’’ Macmillan, New York Ž1920..
18. D. Pines and E. Sadka, Zoning, first-best, second-best, and third-best criteria forallocation land to roads, Journal of Urban Economics, 17, 167]83 Ž1985..

In fact Fukushima and Shigeno w5x just do this by assuming that all consumers have equalshare to land ownership, resulting in equal redistribution of rental income for every
consumer.
18

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AKAI, FUKUSHIMA, AND HATTA

19. A. J. Robson, Cost-benefit analysis and the use of urban land for transportation, Journalof Urban Economics, 3, 180]191 Ž1976..
20. R. M. Solow, Congestion costs and the use of land for streets, Bell Journal of Economics
and Management, 4, 602]618 Ž1973..
21. R. M. Solow and W. S. Vickley, Land use in a long narrow city, Journal of Economic
Theory, 3, 430]447 Ž1971..
22. R. H. Strotz, ‘‘Urban transportation parables,’’ in The Public Economy of the Urban
Communities, ŽJ. Margolis, Ed.., Johns Hopkins University Press, Baltimore, MDŽ1965..