CSIS Discussion Paper No. 61

Cost-Benefit Analysis for Transport Networks-Theory and ApplicationYukihiro Kidokoro＊
Center for Spatial Information Science, University of Tokyo
Faculty of Economics, University of Tokyo, 7-3-1, Hongo, Bunkyo, Tokyo, 113-0033, Japan
Tel&Fax: +81-3-5841-5641, E-mail: kidokoro@csis.u-tokyo.ac.jp

Abstract: In this paper, practical methods for estimating benefits corresponding to the second-bestsituation are derived by modeling a congestion-prone transport network explicitly. In the second-bestsituation, a change in total benefit of an investment in transport infrastructure can be calculated inthree ways: (a) the sum of the changes in consumers’ and producers’ surpluses in all routes; (b) thesum of the changes in consumers’ and producers’ surpluses in the invested routes and the change inthe deadweight loss in all other routes; and (c) the sum of a change in the total benefits in the first-bestcase and a change in the deadweight loss in all routes. Applying method (c), we demonstrate that the
final benefits of distortion-relieving policies, such as introduction of congestion tolls on thecongested routes of a network, are the sum of a change in the deadweight loss in all routes.
Theoretical results are derived in practically useful forms, and then illustrated with examples thatreveal typical errors in benefit estimation for transport projects.

Keywords: cost-benefit analysis, investment in transport infrastructure, network, externality＊

I would like to thank Richard Arnott, Ryuji Fukushima, Tatsuo Hatta, Yoshitsugu Kanemoto, Se-il Mun, Kenneth A. Small, andTakayuki Ueda for helpful suggestions and discussions. I also thank two anonymous referees and editors for encouraging suggestions
and comments. The final version of this paper was written while I was visiting the Department of Economics, at the University ofCalifornia, Irvine. I thank the faculty members and staff there for their hospitality. Of course, I am solely responsible for any remaining
errors and omissions.

11 Introduction
Transport projects often require huge public funds. Thus, a cost-benefit analysis is very
important to judge if the project in question is worthy of implementing. However, it is rather difficult
to carry out cost-benefit analysis for transport projects accurately, because benefit-estimation fortransport projects has the following features, which are often less organized in the standard
textbooks1;
Transport Networks: Many transportation methods actually co-exist, and they form complex
transport networks. How do we consider transport networks in benefit-estimation for transport
projects?
Congestion: Congestion often arises in some parts of transport networks. Investments in someparts of transport networks to relieve the congestion change the degree of congestion in all parts oftransport networks. How do we address such a congestion problem in transport networks?
Divergence between Private User Price and Social Marginal Cost: Due to political andtechnical difficulties, the congestion tax is rarely adopted, and consequently, the private user priceof a trip differs from its social marginal cost in transport networks with congestion. In such asecond-best situation, how do we accurately estimate the final benefit of transport projects?
The purpose of this paper is to propose a consistent benefit-estimation method for transport
projects in a practically useful manner, including all the points stated above. Firstly, transport
networks are modeled as an economy with multiple goods, applying a general equilibrium approach,shown in Boadway and Bruce (1984), for instance. This approach enables us to represent any kind oftransport networks, because the relationship between routes, which may be substitutes or

1See, for example, Lesourne (1975), Sassone and Schaffer (1978), Pearce and Nash (1981), Sugden and Williams (1978), Gramlich(1990), Layard and Glaister (1994), Brent (1996), Nas (1996), and Boardman et al. (2000).

2

complements, need not to be specified. Secondly, the degree of congestion is explicitly taken into
account in all the parts of transport networks. Thus, we can focus on the benefit-estimation in the casewhere a transport project not only changes the degree of congestion in the invested area but alsochanges it subsequently in all the other areas of a transport network. Thirdly, our analysis is presented
within a consistent framework from the first-best case, in which the private user prices of trips equal
their social marginal costs in all the parts of a transport network, to the second-best case, in which theprivate user prices of trips differ from their social marginal costs at least in a part of a transport
network. The approach in this paper highlights how to calculate the total benefits in the second-best
case, which is less focused but practically more important than in the first-best case.
The main results of the paper are as follows. In the first-best case, the total benefits from an
investment in transport infrastructures in a route result in the sum of changes in the consumers’ andproducers’ surpluses in the invested route. In the first-best case, we do not have to consider theaccompanying changes in the other routes that have not received investment.
In the second best case, we need more elaborate calculations. The total benefits in thesecond-best case can be written in three forms. The first form shows that the total benefits equal thesum of changes in the consumers’ and producers’ surpluses in all routes. This method is a naturalextension of the benefit estimation method in the first-best case; that is, the total benefits in thefirst-best case is indeed the sum of changes in the consumers’ and producers’ surpluses in all routes.
However, in the first-best case, a change in consumers’ surpluses and a change in producers’ surpluses
cancel each other out in the non-invested routes, and consequently, the total benefits to be calculated
reduces to the sum of changes in the consumers’ and producers’ surplus in the invested route only. In
the second-best case, this cancellation does not apply, and we have to calculate changes in the
consumers’ and producers’ surpluses in all routes.
The second form shows that the total benefits consist of changes in the consumers’ and3

producers’ surpluses in the invested route and a change in the dead weight loss in all the other routes.
This second form highlights that total benefits to be calculated in non-invested routes are a change inthe dead weight loss. The second form is also related to the benefit estimation in the first-best case.
In the first-best case, no dead weight loss exists. Thus, a change in the dead weight loss are zero in allroutes, which again shows that final benefits in the first best case are the sum of changes in the
consumers’ and producers’ surplus in the invested route only.
The third form shows that the total benefits in the second-best case equal the sum of total
benefits in the first-best case and a change in the dead weight loss in all routes. The third form
highlights the relationship between the total benefits in the first-best case and those in the second-best
case, and demonstrates that their difference stems from a change in the dead weight loss in all routes.
We actually live in a second-best world, and consequently, it is often difficult to know the total
benefits under the hypothetical first-best situation. Thus, practical benefit estimation would rely on
the first and second methods. However, this does not mean that the third method is not practicallyuseful. The third method is not only intuitively easy to understand, but also important in conducting
benefit-estimation in the case where congestion tolls are introduced. Applying the third form in thesecond-best analysis, we can easily show that the total benefits from an introduction of congestion
tolls are the sum of a change in the dead weight loss in all routes.
Before proceeding, we will briefly relate our study to the existing literature. Firstly, Kanemotoand Mera (1985) and Jara-Diaz (1986) show that under the first-best situation with no price distortion,a change in the benefits in transportation sectors includes all the economic benefits from a transport
project. Since we ignore the price distortion in non-transport sectors, we can also obtain final benefits
from a transport investment, by calculating changes in the benefits in the transport sector only.
However, their analysis neglects the inner structure of the transport sectors, and does not focus on theimportant issues to be addressed regarding transport sectors, e.g., transport networks and congestion.

4

In this paper, we formally model transport networks with congestion and analyse thebenefit-estimation method for transport projects, explicitly considering the second-best nature in thereal world.
Secondly, benefit-estimation for actual transport networks with congestion, which are neglectedin Kanemoto and Mera (1985) and Jara-Diaz (1986), are dealt with in older literature (e.g., Harrison(1974), Williams (1976), Jones (1977), and Jara-Diaz and Friesz (1982)). However, these analyses
are less organized or incomplete, as they do not illustrate the benefit-estimation method under thesecond-best situation in detail or the relationship between the benefit-estimation in the first-best andthat in the second-best. Such deficiencies also apply to more recent literature that includes analysesof the benefit-estimation (e.g., Button (1993), Oppenheim (1995), and Williams et al. (1991, 2001)).
Our analysis offers a practical benefit-estimation method both in the first-best case and in thesecond-best case and clarifies the link between them in a consistent framework.
Thirdly, the approach in this paper is most related to Harberger (1972), Mohring (1976), andKanemoto (1996) in that benefit-estimation in the first-best case is explicitly distinguished from thatin the second-best case. These studies deal with the benefit estimation both in the first-best case andin the second best case, focusing on two-parallel congestion-prone roads. However, they do notmodel general transport networks and, consequently, do not fully investigate how to estimate the total
benefits from transport investments in the case of actual complex transport networks. They also do
not show that the total benefits in the second-best case can be expressed in the three forms, each ofwhich has practical implication. Our analysis can also be considered as a practical extension of thesestudies, explicitly taking into account congestion-prone transport networks, which often have very
complex structures, especially in urban areas of large cities.
The structure of this paper is as follows. In Section 2, we set up the model. In Section 3, wederive a benefit-estimation method for transport networks in the first-best case, in which private

5prices of trips equal their social marginal costs in all routes. In Section 4, we focus on the second-best
case, in which private prices of trips differ from their social marginal costs, at least with respect to one
route. In Section 5, by applying our results to actual benefit estimation, we use examples to illustrate
typical errors in benefit-estimation methods for transport networks. Section 6 concludes our analysis.

2 The Model
Consider a transport network in which N nodes are linked to one another. Between node i( 1 ≤ i ≤ N ) and node j ( 1 ≤ j ≤ N ), M means of transportation, which we call “routes” hereafter,

are available. One may think that M routes correspond to M kinds of transport modes or M links
in a single transport mode. The k th ( 1 ≤ k ≤ M ) route from node i to node j is called route ijk .
The number of trips in the route ijk is denoted by x ijk . We assume that x ijk > 0 always holds. We
also assume that the number of trips from node i to node i is zero; i.e., x iik = 0 . Figure 1 illustrates
the example of a network in which N = 2 and M = 2 .
The representative consumer at node i demands the composite consumer good, z , the price of
which is normalized at unity, and his or her trips from node i . Our analysis applies even if theconsumer at node i demands the trips i → j → i or i → j → j′ ; i.e., if the consumer at node idemands “round” trips or demands trips via another node. We assume that the utility function of therepresentative consumer at node i has the following quasi-linear form:

U i = z + u i (..., x ijk ,...) ,(1)

where u i depends on all directions of the trips from node i . The above quasi-linear utility function
implies that we neglect the income effects, and consequently, the consumers’ surplus equals the6equivalent variation and compensating variation.2 As Willig (1976) demonstrates, the difference
between the consumers’ surplus, equivalent variation, and compensating variation is rather small, andhence disregarding the difference is justified in benefit estimation in practice. The quasi-linear utility
function also yields the Gorman-type indirect utility function3, which justifies the summation ofindividual utilities to obtain the total consumers’ surplus. In this paper, the utility function is assumed
to be concave. Note that we do not make further assumptions about the utility function. Thus, theutility function (1) specifies no particular relationship between routes; for example, one could thinkthat route ij1 may be a substitute or complement for route ij 2 . When M routes from node i to node

j are perfect substitutes, the utility function (1) becomes
U i = z + u i (...; x ij1 + ... + x ijk + ... + x ijM ;...) ,(2)

which is the utility function that is consistent with Wardrop’s (1952) analysis.
The generalized price of a trip in route ijk , which includes time costs, is denoted by p ijk . The
representative consumer at node i maximizes his or her utility, (1), subject to the budget constraint:z + ∑∑ p ijk x ijk = y i ,
j

(3)

k

where y i is the income of the consumer at node i . We assume that the consumer does not take into
account the effect of his or her own behaviour on congestion; i.e., we assume that the consumer is a
price-taker.
The maximization problem of the representative consumer at node i is then formulated as:
⎧ i
i
ijk
ijk ijk
i⎫
max
U
=
z
+
u
(...,
x
,...)
:
z
+
p
x
=
y
⎨
⎬.
∑∑
{z ,..., xijk ,...} ⎩
j
k
⎭

2

See, for example, Varian (1992).

3

See, for example, Mas-Colell et al. (1995) and Tsuneki (2000).

7(4)

Solving the maximization problem, (4), yields
p ijk = u xi ijk ,(5)from which, we obtain the Marshallian demand function in route ijk :x ijk = x ijk (..., p ijk ,...) .

(6)

For the trips in route ijk , both investments in transport infrastructures, which are measured inmonetary terms and denoted by I ijk , and the suppliers of transport services are required. Examples oftransport infrastructures are roads, railroad trucks, and airports, and examples of the suppliers oftransport services are railway operators and airlines. In the case of road transport, one can think of thedrivers themselves as the suppliers of transport services. For the sake of exposition, we assume thatthe government owns all the transport infrastructures and makes investments in them.

Thisassumption is innocuous for our analysis, because we do not focus on the welfare distribution but
only on the total benefits from an investment in transport infrastructures. The government imposes“taxes,” t ijk , per trip in route ijk . This tax, t ijk , can be interpreted in many ways. In the case of roadtransport, one could think of t ijk as gasoline taxes. t ijk needs not to be positive; for instance, in thecase of public transport that is subsidized by the government, one could think of t ijk as unit subsidy
per ride, which can be considered as a negative tax.
Let us assume that all the suppliers of transport services in route ijk are homogenous,
sufficiently competitive with constant returns to scale, and their unit supply cost is c ijk . Since thelevel of c ijk is irrelevant to our analysis, we can set it at zero. Other unit generalized costs of the trips
in route ijk , which includes monetized time costs, is c ijk ( x ijk , I ijk ) , which is non-decreasing with x ijk
and decreasing with I ijk ; i.e.,∂cijk≡ cxijkijk ≥ 0
ijk
∂x

8

(7)

and
∂c ijk
≡ cIijkijk < 0 .
∂I ijk(8)

The total cost function of the trips in route ijk , C ijk , isC ijk = c ijk ( x ijk , I ijk ) x ijk ,(9)from which we can obtain their marginal cost , MC ijk , asMC ijk = c ijk ( x ijk , I ijk ) + cxijkijk ( x ijk , I ijk ) x ijk ,(10)

where cxijkijk ( x ijk , I ijk ) x ijk shows the congestion externality that an additional trip gives to all the trips inroute ijk .
The total profits of the suppliers of transport services in route ijk , π ijk , can be written as

π ijk = ( p ijk − t ijk ) x ijk − cijk ( x ijk , I ijk ) x ijk ,(11)where p ijk − t ijk is the net price (excluding taxes) that the suppliers receive from the consumer. As aresult of competition, the profits of each supplier of transport services become zero, which implies
that the total profits of the suppliers of transport services are also zero:π ijk = 0 .

(12)

From (11) and (12), we know that the generalized price of a trip in route ijk satisfies:
p ijk = c ijk ( x ijk , I ijk ) + t ijk .

(13)

Solving (6) and (13), we have the general equilibrium demand curve in route ijk , whose
arguments are the investments in transport infrastructure and the taxes, that is,x ijk = x ijk (..., I ijk ,...;..., t ijk ,...) .

(14)

Finally, since the profits of the suppliers of transport services are zero, the total social welfare,
SW , can be defined as the sum of the consumers’ utilities and the tax revenue of the government,

9exclusive of investment costs, in all routes. From (9) and (13), we know that the tax revenue of thegovernment in route ijk can be considered as the producers’ surplus in route ijk :t ijk x ijk = p ijk x ijk − C ijk .

(15)Thus, the total social welfare, SW , isSW = ∑U i + ∑∑∑ t ijk x ijk − ∑∑∑ I ijk
i

i

j

k

i

j

k

= ∑U i + ∑∑∑ ( p ijk x ijk − C ijk ) − ∑∑∑ I ijk .
i

i

j

k

i

j

(16)

k

As is usual in the literature of cost-benefit analysis, we focus on the total benefits, exclusive ofinvestment costs. The total benefits, TB , are defined as:
TB = ∑U i + ∑∑∑ t ijk x ijk = ∑U i + ∑∑∑ ( p ijk x ijk − C ijk ) .
i

i

j

k

i

i

j

(17)

k

3 First-best Results
In this section, we obtain the results in the first-best case, in which the generalized prices of thetrips, p ijk , equal their social marginal costs, MC ijk in all routes. From (10) and (13), we know thatthe first-best situation is attained when t ijk = cxijkijk ( x ijk , I ijk ) x ijk , i.e., the tax, t ijk , is set equal to thecongestion externality, cxijkijk ( x ijk , I ijk ) x ijk in all routes.
Suppose that an investment in transport infrastructures increases from I ijkWO to I ijkW in routeijk . The change in the total benefits, TB , in this case consist of:i) a change in the consumers’ surplus in route ijk :I ijkWOijk⎞ ijk
ijk ⎛ dpx
∫ijkW ⎜⎝ dI ijk ⎠⎟ dI
I

and;

10

(18)

ii) a change in the tax revenue in route ijk :I ijkWijk⎞ ijk
ijk ⎛ dxt
∫ijkWO ⎜⎝ dI ijk ⎟⎠dI .
I(19)In sum, these two terms are reduced to:
I ijkWO∫ c x dI .
ijk
I ijk

ijk

ijk

(20)

I ijkWAll mathematical developments for the analyses of Sections 3 and 4 are relegated to Appendix 1.
For an intuitive understanding, we focus on the case of the simple two-node model in Figure 1.
Consider an investment in transport infrastructures in route 121, I 121 , increases from I 121WO to I 121W .
We only need to analyse the consumer at node 1, who demands the trips in routes 121 and 122,
because inbound trips and outbound trips are treated separately in our model, and an investment intransport infrastructures, I 121 , will affect trips that originate at node 1 only. Even if we consider asetting in which an increase in investment in transport infrastructure, I 121 is assumed to affect not
only the trips in routes 121 and 122 but also the trips in routes 211 and 212, our argument remains
unchanged. In that case, we also take into account the effects on the consumer at node 2.
Let us explain the results in detail using diagrams. Equation (18) shows a change in the
consumers’ surplus in the route where the investment in transport infrastructures increases, measured
by the general equilibrium demand curve.4 In our simple example in Figure 1, (18) corresponds to a
I 121WO

change in the consumers’ surplus in route 121 and is reduced to

4

121
⎞ 121
121 ⎛ dpx
∫121W ⎜⎝ dI 121 ⎟⎠ dI , orI

The general equilibrium demand curve is often referred to as Bailey’s demand curve, because Bailey (1954) first formulated thisnotion. See also Boadway and Bruce (1984) for further explanation of the general equilibrium demand curve.

11p121WO∫ x dp121

121

where p121WO and p121W denotes the generalized prices in route 121 corresponding top121W

I 121WO and I 121W . Remember that the ordinary Marshallian demand curve is drawn on the basis that

incomes and prices in the other markets are held constant. In our model in which the utility function
is assumed to be quasi-linear, the Marshallian demand curve does not depend on income, but on
prices in the other markets. The investment in transport infrastructure in route 121, not only affects
the generalized price and demand of the trips in route 121, but also affects those of the trips in route122, depending on the relationship between the trips in routes 121 and the trips in routes 122, which
may be substitutes or complements. Taking into account these effects, the investment in route 121
does not decrease its generalized price along its Marshallian demand curve with the generalized priceof the trips in route 122 fixed, but shifts the Marshallian demand curve. Look at Figure 2-1. With
regard to the trips in route 121, an equilibrium without a project, point B, and an equilibrium with aproject, point D, lie on different Marshallian demand curves. The general equilibrium demand curvefor the trips in route 121 is a locus of attained equilibria, which passes through points B and D, andincorporates the accompanying change in the trips in route 122. In x121 − p121 plane in Figure 2-1, thegeneral equilibrium demand curve in route 121 is shown as x121 ( p121 , p122 ( p121 , I 122 , t122 )) , given I 122
and t122 . A change in the consumers’ surplus in route 121 needs to be measured along the generalequilibrium demand curve and, consequently, is Area ABDC . When estimating transport demand inpractice, it is often difficult to derive the Marshallian demand curve with incomes and the generalized
prices in the other trips held constant, because a change in the price of a trip increases or decreases
congestion in other trips, and subsequently, changes the generalized prices of other trips. In the actual
forecast of transport demand, accompanying changes in other trips due to substitutability andcomplementarity relationships between trips are usually taken into account, at least implicitly. Thus,

12the present estimation of transport demand can be regarded as a forecast of the general equilibrium
demand curve, rather than of the Marshallian demand curve.
Equation (19) shows a change in the tax revenue in the route where the investments in transport
I 121W

infrastructures increase. In our example in Figure 1, (19) corresponds to

121
⎞ 121
121 ⎛ dxt
∫121WO ⎜⎝ dI 121 ⎠⎟dI . SinceI

t ijk x ijk = p ijk x ijk − C ijk(21)

from (15), we can consider (19) as a change in the producers’ surplus in the invested route. A change
in the producers’ surplus in route 121 can be expressed as Area CDF - Area ABE in Figure 2-2,
which is rearranged as
Area CDF − Area ABE= Area CDF + Area EBDC − ( Area ABE + Area EBDC )(22)= Area EBDF − Area ABDC.

It should be remembered that a change in the consumers’ surplus in route 121 is Area ABDC .
By adding up the changes in the consumers’ and producers’ surpluses, the change in the total benefits
I 121WO∫ c x dI
121 121I 121

in route 121 is summed up as Area EBDF , which is expressed asI121

in our example

121W

and corresponds to (20).
In the first-best case, it is unnecessary to consider the accompanying changes in all the otherroutes. To see why, let us derive the consumers’ surplus and the tax revenue in route 122.
A change in the consumers’ surplus in route 122 is derived from the understanding that thegeneral equilibrium demand curve in route 122 isp122 = MC 122 = c122 ( x122 , I 122 ) + c122
( x122 , I 122 ) x122
x122

(23)

in Figure 2-3. The reason is that equilibria are always on the line for (23) in x122 − p122 plane,
although an increase in the investment in transport infrastructures in route 121 shifts the Marshallian
demand curve for the trips in route 122. Thus, a change in the consumers’ surplus relating to the trip13

x122 is derived as an area along (23), i.e., Area HJLK , in Figure 2-3.

We move to a change in the tax revenue in route 122. Since we know that a change in the tax
revenue equals a change in the producers’ surplus in route 122, p122 x122 − C 122 , from (15), it isexpressed as the change in consumers’ total payments, p122 x122 , less the change in the total cost, C 122 .
In Figure 2-3, the change in consumers’ total payments, p122 x122 , is −( Area HJLK + Area LJRQ ) ,and the change in total cost, C 122 , is the area below the marginal cost curve for the trips in route 122,− Area LJRQ . Accordingly, the change in the producers’ surplus is−( Area HJLK + Area LJRQ ) − ( − Area LJRQ ) = − Area HJLK .

(24)

In total, since the change in the consumers’ surplus is Area HJLK and the change in the
producers’ surplus is − Area HJLK , the change in the total benefits in route 122 is zero. In short, inthe first-best case, in which the generalized prices of trips equal their marginal costs in all routes, weneed only calculate changes in the consumers’ and producers’ surpluses in the invested route. This isbecause the sum of changes in the consumers’ surplus and the tax revenue is zero in all the otherroutes. The above result demonstrates that the first-best results of Mohring (1976), Kanemoto andMera (1985), Jara-Diaz (1986), and Kanemoto (1996) are applicable to any complex transport
networks with congestion.

4 Second-best Results
Let us extend our analysis to incorporate the second-best case, in which the generalized prices of
the trips, p ijk , differ from their marginal costs, MC ijk , at least in one route. In the second-best case,when the investment in transport infrastructures increases from I ijkWO to I ijkW in route ijk , a change
in the total benefits, TB , can be expressed in three forms.
The first form of a change in the total benefits, TB , in the second-best case can be written as the14sum of:i) a change in the consumers’ surplus in all routes:
pi ′j ′k ′WO∑∑∑ ∫ xi′j′k′⎛ dp i′j′k ′ ⎞ ijk⎜ dI ijk ⎟ dI ;⎝⎠i ′j ′k ′pi ′j ′k ′W

(25)

and
ii) a change in the tax revenue in all routes:
I ijkWi ′j ′k ′⎞ ijk
i ′j ′k ′ ⎛ dxt∑∑∑
⎜ dI ijk ⎟ dI ,∫i′j ′ k ′ I ijkWO
⎝
⎠

(26)

where 1 ≤ i′ ≤ N , 1 ≤ j′ ≤ N , and 1 ≤ k ′ ≤ M .
The second form highlights the distinction between changes in the invested route and changes inall the other routes, and is given by the sum of:iii) a change in the consumers’ surplus in route ijk :I ijkWOijk⎞ ijk
ijk ⎛ dpx
∫ijkW ⎜⎝ dI ijk ⎠⎟ dI ;I(27)iv) a change in the tax revenue in route ijk :I ijkW⎛ dx ijk ⎞ ijk∫ t ⎜⎝ dI ijk ⎟⎠dI ;I ijkWO
ijk

(28)

and
v) a change in the dead weight loss in all the other routes:
I ijkWO∑∑∑ ∫ ( MC
i ′′

j ′′

i ′′j ′′k ′′−pk ′′ I ijkW

i ′′j ′′k ′′⎛ dx i′′j′′k ′′ ⎞ ijk) ⎜ dI ijk ⎟dI ,
⎝
⎠

(29)

where 1 ≤ i′′ ≤ N , 1 ≤ j′′ ≤ N , and 1 ≤ k ′′ ≤ M , but (i′′, j′′, k ′′) does not contain (i, j, k ) .
The third form highlights the distinction between the total benefits in the first-best case and the
distortions in the second-best case, and can be rearranged as the sum of:15vi) a change in the total benefits in the first-best case:
I ijkWO∫ c x dI ;
ijk
I ijk

I

ijk

ijk

(30)

ijkW

and
vii) a change in the dead weight loss in all routes:
I ijkWO

i ′j ′k ′⎞ ijk
i ′j ′k ′i ′j ′k ′ ⎛ dx
−
MC
p
(
)
∑∑∑
⎜ dI ijk ⎟dI .
∫i′j ′ k ′ I ijkW
⎝
⎠

(31)

In the same way as in the first-best case in Figure 1, we explain a simple case, in which theinvestment in transport infrastructure, I 121 , increases from I 121WO to I 121W and affects only the trips
in routes 121 and 122.
In the first-best case, as we show in Section 3, a change in the total benefits is the sum of changes
in the consumers’ surplus and the tax revenue in route 121 and the corresponding terms in route 122.
However, with regard to the trips in route 122, the sum of changes in the consumers’ surplus and thetax revenues is zero; consequently, the final benefits are the sum of changes in the consumers’ surplus
and the tax revenue in route 121. Even in the second-best case, the above calculation method applies
without modification; from the first form, shown in (25) and (26), we know that a change in the total
benefits is the sum of changes in the consumers’ surplus and the tax revenue in all routes.
Two different forms of a change in the total benefits in the second-best case are derived. The
second form, shown in (27)-(29), focuses on whether surpluses are generated in primary or secondary
markets.5 Firstly, we look at changes in the primary market, which corresponds to changes in theinvested route 121 in our example. A change in the consumers’ surplus is Area ABDC in Figure 3-1,
which is the same in the first-best case and corresponds to (27). A change in the tax revenue equals a5This dichotomy is based on Boardman et al. (2000).

16

change in producers’ surplus from (15). Due to the discrepancy between the generalized price of atrip and its social marginal cost, the expression of a change in producers’ surplus becomescomplicatedcomparedtothefirst-best( Area CDG ′′O − Area F ′D′′G′′O ) − ( Area ABG ′O − Area E ′B′′G ′O )

case,
in

Figure

and

is

3-2,

which

corresponds to (28). (Hereafter, in Figures 3-1, 3-2, 3-3 and 4, we omit to draw the Marshallian
demand curves to avoid unnecessary complications.)
Secondly, in the secondary market, which corresponds to the trips in route 122 in our example,
p122 = c122 ( x122 , I 122 ) + t122

(32)

always holds, regardless of shifts of the Marshallian demand curve for the trips in route 122. Thismeans that (32) is the general equilibrium demand curve for the trips in route 122. Accordingly, achange in the consumers’ surplus in route 122 is Area HJLK in Figure 3-3, which is the same as in
the first-best case. Applying (15), a change in the tax revenues in route 122 equals a change in the
producers’ surplus, p122 x122 − C 122 .

A change in consumers’ total payments, p122 x122 , is−( Area HJLK + Area LJRQ ) in Figure 3-3, which is also the same result as in the first-best case. It

is a change in the total cost, C 122 , in the second-best case that differs from the first-best case,reflecting the gap between the generalized price of a trip and its social marginal cost. A change in the
total cost, C 122 , in the second-best case becomes − Area L′J ′RQ in Figure 3-3. As a result, a change
in the producers’ surplus in route 122 is−( Area HJLK + Area LJRQ ) − ( − Area L′J ′RQ ) = − Area HJLK + Area L′J ′JL .

(33)

In sum, a change in the total benefits to be calculated in the secondary market is Area L′J ′JL in
Figure 3-3 because changes in the consumers’ surplus, Area HJLK , and − Area HJLK in (33)cancel each other out. Area L′J ′JL corresponds to (29) and shows a change in the dead weight loss
in the secondary market, which stems from a gap between the generalized price of a trip and its social

17

marginal cost.
The third form of a change in the total benefits focuses on its relationship to the first-best result.
In route 121, a change in the consumers’ surplus is Area ABDC in Figure 3-1 and a change in the
producers’ surplus is ( Area CDG ′′O − Area F ′D′′G′′O ) − ( Area ABG ′O − Area E ′B′′G ′O ) in Figure3-2. Superimposing Figure 3-1 on Figure 3-2 and adding up these terms, we obtain
Area ABDC + ( Area CDG ′′O − Area F ′D′′G ′′O ) − ( Area ABG ′O − Area E ′B′′G′O )= ( Area ABDC + Area CDG ′′O − Area ABG′O ) + Area E ′B′′G′O − Area F ′D′′G′′O= Area BDG′′G′ + Area E ′B′′G′O − Area F ′D′′G′′O= Area E ′B′′BDG′′O − Area F ′D′′G′′O= Area E ′B′D′F ′ + Area B′B′′B − Area D′D′′D,(34)

where Area E ′B′D′F ′ is a change in the total benefits in the first-best case, which corresponds to (20),and Area B′B′′B − Area D′D′′D is a change in the dead weight loss in route 121. We know that achange in the benefits to be calculated in route 122 is a change in the dead weight loss in route 122,
which is Area L′J ′JL in Figure 3-3, from the second form that focuses on the difference between the
primary market and the secondary market. In the end, we can rearrange a change in the total benefits
in the second-best case as the sum of a change in the total benefits in the first-best case, (30), and achange in the dead weight loss in all routes, (31). The third form is an extension of Mohring’s (1976)
argument on the total benefit in the second-best case.
We have focused so far on an increase in an investment in transport infrastructures in existing
routes. However, our analysis also applies even when a new route is built, if we consider that theinvestment in a new route lowers the generalized price of the trips and generates their demand. For
example, consider the building of a bridge over a river between nodes 1 and 2. Without an investment,no bridge exists and, thus, residents cannot travel between nodes 1 and 2. With an investment,
residents can travel between nodes 1 and 2 by using the bridge. This situation can be expressed in thefollowing way. Without an investment, the generalized price of the trips between nodes 1 and 2 is18

prohibitively high and, consequently, the demand for the trips between nodes 1 and 2 is zero. With aninvestment, however, the generalized price of the trips between nodes 1 and 2 declines, and thedecrease in the generalized price generates the demand for the trips between nodes 1 and 2. This
interpretation implies that our analysis applies directly to newly introduced routes.
As we will show in Section 5, the first and the second forms are useful in practicalbenefit-estimation.

In conducting benefit estimation in practice, almost all the cases are thesecond-best. Thus, it is difficult to estimate a hypothetical first-best change in the total benefits.
However, the third form, which highlights a first-best change in the total benefits and a change in thedead weight loss in all routes, does have practical validity and is worthy to be remembered. Applying
the third form, we can easily derive a change in the total benefits when the tax increases from t ijkWO tot ijkW in route ijk , for example, by introducing the congestion toll. A change in the total benefits, TB ,in this case becomest ijkWO∑∑∑ ∫ ( MCi′j′k ′ t ijkW

i ′j ′k ′⎛ dx i′j′k ′ ⎞− p i′j′k ′ ) ⎜ ijk ⎟dt ijk ,⎝ dt ⎠(35)

which is the same form as (31) and shows a change in the dead weight loss in all routes.
Again, we explain this on the basis of a simple example in Figure 1 and suppose that the tax inroute 121 increases. An increase in the tax in route 121 does not shift the social marginal cost curvefor the trips in route 121, unlike an increase in the investment in transport infrastructures. Thisimplies that Area E ′B′D′F ′ in Figure 3-2, which corresponds to a change in the total benefits in thefirst-best case and is expressed in (30), does not exist. In the end, the final benefits are the sum of achange in the dead weight loss in routes 121 and 122, which corresponds to (35).
The same result can be derived if we calculate a change in the total benefits as the sum ofchanges in the consumers’ surplus and the tax revenue. In Figure 4 of route121, a change in the
consumers’ surplus is − Area SVYT , and a change in the tax revenues, which equals the producers’

19surplus from (15), is ( Area SVV ′′O − Area WV ′V ′′O ) − ( Area TYY ′′O − Area WY ′Y ′′O ) . Adding up
these terms, we obtain a change in the total benefits in route 121 of− Area SVYT + ( Area SVV ′′O − Area WV ′V ′′O ) − ( Area TYY ′′O − Area WY ′Y ′′O )= − Area SVYT + ( Area SVV ′′O − Area TYY ′′O ) + ( Area WY ′Y ′′O − Area WV ′V ′′O )= − Area SVYT + ( Area SVYT − Area VYY ′′V ′′) + Area V ′Y ′Y ′′V ′′(36)= Area V ′Y ′YV ,which is a change in the dead weight loss in route 121. In route 122, by the same argument as anincrease in an investment in transport infrastructures in the second-best case, a change in the benefits
to be calculated is a change in the dead weight loss in route 122, which is the same as Area L′J ′JL in
Figure 3-3. Thus, we can confirm that the final benefits are summed up as a change in the deadweight loss in all routes.

5 Examples
Based on the analyses in Sections 3 and 4, let us check the present benefit-estimation methods,
which claim to take into account the “network characteristic” of transport. We focus on two typicalinadequacies: Examples 1 and 2 are applications of benefit estimation in the first-best andsecond-best cases respectively.

Example 1: The Weighted-Average Method in the Institute for Transport Policy Studies (ITPS)(1999)ITPS (1999) is a manual for cost-benefit analyses of railway projects in Japan. Although ITPS
(1999) claims to take into account the effects for railways and the effects for other transport modes,
the way it estimates benefits is inconsistent with the theory developed here. We illustrate this byusing a numerical example from pp. 104–124 of ITPS (1999). The data used to calculate benefits are
given in Table 1, from which we know that this railway project shortens rail travel time and increases
20rail demand.
In this example, it is assumed that the time cost does not depend on the volume of demand (i.e.,
no congestion exists) and no tax exists. Since the tax and the congestion externality equal zero, we
can apply the benefit-estimation method in the first-best case. Thus, a change in the total benefits isderived as the sum of a change in the consumers’ surplus and a change in the tax revenues in theinvested route, i.e., rail. In this example, given the assumption of no tax, a change in the total benefits
is simply a change in the consumers’ surplus relating to rail. We assume that time costs are 39.3
yen/minute, as in ITPS (1999), and that the general equilibrium demand curve is linearly
approximated. Rail’s generalized price without a project is39.3 × 190 + 11,000 = 18, 467 (yen).

(37)

The corresponding price with a project is39.3 × 140 + 11,000 = 16,502 (yen).

(38)

0.5 × (18, 467-16,502) × (100 + 135) = 230,888 (yen).

(39)

Hence, the final benefits are

In contrast, the benefit-estimation method in ITPS (1999) is as follows. First, we obtain theorigin-destination (OD) generalized price, which is the demand-weighted average of generalized
prices in all routes. The OD generalized price without a project is(39.3 × 190 + 11,000) ×

100
110
10
+ (39.3 × 120 + 14,000) ×+ (39.3 × 400 + 5,500) ×
220
220
220

(40)

= 18,717 (yen)The corresponding price with a project is(39.3 × 140 + 11,000) ×

135
80
5
+ (39.3 × 120 + 14,000) ×+ (39.3 × 400 + 5,500) ×
220
220
220

(41)

= 17, 414 (yen)Second, this change in the OD generalized price is assumed to apply to all the demand in the OD.

21

Thus, the final benefits are
0.5 × (18,717-17, 414) × (220+220) = 286,660 (yen) .6(42)Hence, the benefit-estimation method in ITPS (1999) overestimates true benefits by about 24 percent.
In general, the benefit-estimation method in ITPS (1999) is unreliable.

However, if allroutes—rail, air, and car in this example—are assumed to be perfect substitutes, i.e., the utility
function is assumed to be like (2), the ITPS (1999) method happens to give the right value. This isbecause the generalized prices of the trips between nodes i and j are equal in all routes actually usedin such a case and, consequently, the generalized price in each route is equal to the demand-weighted
average of the generalized prices in all routes. However, it is not necessary to calculate thedemand-weighted average of the generalized prices in such a case when the generalized prices of thetrips are equal in all routes. Thus, even in this case, we can conclude that the ITPS (1999) method
lacks practical validity.
It is sometimes suggested that the ITPS (1999) method, which uses the OD generalized price, isconsistent with the estimation of transport demand using the logit model.7 However, this is incorrect.
If a discrete choice model such as the logit model is used to estimate transport demand, the general
method of estimating welfare changes in discrete choice models developed by Williams (1977) andSmall and Rosen (1981) must be used to calculate welfare changes. As Anderson et al. (1992) show,
the logit model can be interpreted as a special case of the ordinary utility-maximization problem of arepresentative consumer. Thus, the methods derived in Sections 3 and 4 are still valid withlogit-based estimation of transport demand. That is, we can accurately calculate a change in the total

6

Although the total benefit is given as 326,260 (yen) in ITPS (1999), this figure is the result of a simple error in calculation.

7

See, for example, Ueda et al. (2002).

22

benefits by using the methods in Sections 3 and 4, whether or not the logit model is used to estimatetransport demand8. The ITPS (1999) OD-based method is theoretically inconsistent with both themethod developed by Williams (1977) and Small and Rosen (1981) and the method derived in thispaper; therefore, it cannot correctly compute welfare changes even when the estimation of transportdemand is logit-based.

Example 2: “Network effects” in Button (1993)

Button (1993, pp. 182–184) proposes a benefit-estimation method for transportation projects
that takes into account “network effects.” He claims that a change in the total benefits from an
investment in a route in the road network is the sum of a change in the consumers’ surplus in all routes.
However, this argument is inadequate, because it ignores a change in tax revenue. In the first-best
case, this omission obviously could be very significant; the true benefits are the sum of changes in
consumers’ surplus and the tax revenue in the invested route, not the sum of changes in consumers’
surplus in all routes. In the second-best case, the omission of the tax revenues would be justified if the
tax was regarded as zero. However, as Button himself suggests (e.g., Table 4.10 in p. 81), therevenues from road tax (e.g., fuel tax) are too large to ignore. If we assume that a change in the tax
revenue is zero when it is actually quite large, the final benefits could be quite different from those
calculated by Button’s method.
Let us check how the omission of the producers’ surplus affects the results, using an illustrative
example, in which rail and car are available. To simplify the calculation, the time cost is assumed to
be 40 (yen/minute). Consider a transport project that upgrades rail tracks to shorten rail travel time.
Consequently, the project increases rail demand and reduces car demand. A decrease in car demand

8

See Kidokoro (2003) for a detailed analysis.

23and the subsequent relief in car congestion reduce revenues from fuel tax. The fuel tax rate is
assumed to be 100 (%), i.e., gross fuel prices are twice as high as net fuel prices. Rail receives a
subsidy from the government. Upgraded rail tracks require new rail cars, which raises rail’s average(monetary) cost from 600 (yen) to 700 (yen) but the rise in the fare is only 50 (yen). The gap between
the increase in the rail’s average cost and the increase in the fare, 50 (yen), is financed by an increasein the government’s subsidy, which is regarded as a negative tax. This example can be analysed byapplying the benefit-estimation method in the second-best case from Section 4. In this example, wecannot estimate the change in the total benefits in the first-best case and, accordingly, it is impossible
to calculate the final benefits by the third method, shown in (30) and (31). Thus, we need to obtain the
final benefits by applying the first method, shown in (25) and (26), or by applying the second method,
shown in (27) to (29).
First, applying the results of (25) and (26), we calculate a change in the total benefits as the sumof changes in the consumers’ surplus and the tax revenue relating to both rail and car. (As in Example
1, the general equilibrium demand curve is assumed to be linearly approximated.)
A change in the consumers’ surplus relating to rail is
0.5 × (2,500 -1,950) × (300 + 400) = 192,500 (yen).

(43)

A change in the tax revenue relating to rail, which is a change in the government’s subsidy here, is(550 − 700) × 400 − (500 − 600) × 300 = −30,000 (yen).

(44)

A change in the consumers’ surplus relating to car is
0.5 × (3, 200 - 2,700) × (200 + 150) = 87,500 (yen).

(45)

A change in the tax revenue relating to car, which is a change in the revenue from fuel tax here, is
150 × 150 − 200 × 200 = −17,500 (yen).

(46)

Adding up (43) to (46) gives a change in the total social welfare of 232,500 (yen).
Second, by applying the results of (27) to (29), we calculate a change in the total benefits as the24sum of changes in the consumers’ surplus and the tax revenues relating to rail, and a change in thedead weight loss relating to car. If we re-use Figure 3-3 and regard route 122 as by car, calculating achange in the dead weight loss relating to car travel is reduced to calculating Area L′J ′JL , which isArea L′J ′RQ less Area LJRQ . Since( Area L′J ′RQ ) = (70 × 40 + 200) × 200 − (60 × 40 + 150) × 150 = 217,500 (yen)9 and(47)( Area LJRQ ) = 0.5 × (3, 200 + 2,700) × (200 − 150) = 147,500 (yen),(48)

a change in the dead weight loss relating to the car is 217,500 − 147,500 = 70,000 (yen), whichequals the sum of (45) and (46). Adding this value to (43) and (44), which are changes in the
consumers’ surplus and the tax revenue relating to rail, we again obtain the correct change in the total
benefits of 232,500 (yen).
By applying Button’s method, the benefit derived is 280,000 (yen), which is the sum of (43) and(44), and represents an overestimate of 47,500 (yen). This overestimate results from disregarding(44) and (46), which represent a decrease in government revenues because of an increase in thesubsidy relating to rail, and a decrease in the tax revenues relating to car, respectively. This example
illustrates the possibility of incorrectly calculating benefits when the producers’ surplus is notproperly included. Thus, we must pay attention, not only to the consumers’ surplus, but also to the
producers’ surplus when calculating a change in the total benefits, unless the change in the producers’
surplus is negligible.

6 Concluding Remarks
We have argued the benefit-estimation for transport projects both in the first-best case and in the9

Recall that Area L’J’RQ represents a change in the total cost. Thus, when calculating Area L’J’RQ, which is the area below the social

marginal cost curve of car, we do not need to know the social marginal cost of the car itself, which is often difficult to know in practice.

25

second best case, explicitly modeling congestion-prone transport networks.

In a practicalbenefit-estimation for transport projects, the second-best situations are common, although they havenot been analysed systematically in the literature. Our main results are summed up as follows. In thesecond-best situation, a change in the total benefits from an investment in transport infrastructures
can be calculated in three ways. The first method shows that a change in total benefits is the sum ofchanges in consumers’ and producers’ surpluses in all routes. In all the non-invested routes, the sumof changes in consumers’ and producers’ surpluses result in a change in the dead weight loss, causedby the discrepancy between the generalized prices of trips and their social marginal costs. Focusing
on this feature, the second method shows that a change in the total benefits is the sum of changes in
the consumers’ and producers’ surpluses in the invested route and a change in the dead weight loss inall the other routes. The third method shows that a change in the total benefits in the second-best caseequals the sum of a change in the total benefits in the first-best case and a change in the dead weightloss in all routes, highlighting the relationship between the total benefits in the first-best case andthose in the second-best case. Although the first and the second methods are easy to implement
practically, the implications of the third method is very important. Applying the third method, weknow that a change in the total benefits is a change in the dead weight loss in all routes only, if thesocial marginal cost curve does not shift and consequently, a change in the total benefits in the first
best case is zero. For example, when the congestion tax is introduced, a change in the total benefits
can be calculated as a change in the dead weight loss in all routes.
In concluding the paper, we comment on two issues. Firstly, in our model, the dead weight loss
arises because the generalized prices of trips differ from their social marginal costs due to the gap
between the tax, t ijk , and the congestion externality, cxijkijk ( x ijk , I ijk ) x ijk . Our analysis also applies
without modification when the dead weight loss stems from other distortions. As an example,
consider the case in which suppliers of transport services behave less competitively. Even in this case,26

our analysis is directly applicable, if the tax, t ijk , is redefined to include the price-cost margin that
results from imperfect competition. Our analysis also applies to the case where the generalized prices
differs from the social marginal costs due to environmental externality, if environmental costs are
additionally included in the total cost function of trips.
Secondly, we have assumed away the price-cost divergences in non-transport sectors. Although
it would be difficult to include them in practical benefit estimation of transport projects, our analysis
is applicable in principle, even if we explicitly consider the second-best situation in non-transport
sectors. In that case, all we have to do is to add changes in consumers’ and producers’ surpluses in
non-transport sectors. Since the same logic as changes in the non-invested routes applies to changes
in non-transport sectors, the sum of changes in consumers’ and producers’ surpluses in a
non-transport sectors equals a change in the dead weight loss in the non-transport sectors. Thus, we
can add a change in the dead weight loss in all non-transport sectors, instead of the sum of changes in
consumers’ and producers’ surpluses.

27

Appendix 1: Derivation of Results in Sections 3 and 4First, we consider the case in which investment in transport infrastructure, I ijk , increases fromI ijkWO to I ijkW . We begin by analysing the second-best case, in which the generalized prices of trips
are not necessarily equal to their social marginal costs. We then show the first-best result, in which
the generalized prices of trips equal their social marginal costs, as a special case of the second-best
results.
Substituting (1) and (3) into (17) and using i ′ , j ′ , and k ′ , where 1 ≤ i′ ≤ N , 1 ≤ j′ ≤ N , and1 ≤ k ′ ≤ M , instead of i , j , and k , we obtain

TB = ∑ y i ′ − ∑∑∑ p i ′j ′k ′ x i ′j ′k ′ + ∑ u i ′ + ∑∑∑ t i ′j ′k ′ x i ′j ′k ′ .
i′

i′

j′

k′

i′

i′

j′

(A1)

k′

Totally differentiating the total benefits, (A1), with respect to I ijk yields
i ′j ′k ′⎛ dp i′j′k ′ ⎞ i′j′k ′⎞dTBi ′j ′k ′ ⎛ dx
x
p
=
−
−
∑∑∑
∑∑∑
⎜
⎜
ijk
ijk ⎟
ijk ⎟
dI
i′
j ′ k ′ ⎝ dIi′j′ k ′⎠⎝ dI ⎠⎛ dx i′j′k ′ ⎞⎛ dx i′j′k ′ ⎞+ ∑∑∑ uxi′i′j′k ′ ⎜ ijk ⎟ + ∑∑∑ t i′j′k ′ ⎜ ijk ⎟i′j′ k ′⎝ dI ⎠ i′ j′ k ′⎝ dI ⎠(A2)⎛ dp i′j′k ′ ⎞⎛ dx i′j′k ′ ⎞= − ∑∑∑ ⎜ ijk ⎟ x i′j′k ′ + ∑∑∑ t i′j′k ′ ⎜ ijk ⎟i′j′ k ′i′j ′ k ′ ⎝ dI⎠
⎝ dI ⎠

where we have used the result of the utility maximization; i.e., p ijk = u xi ijk in (5).
Integrating (A2), from I ijkWO to I ijkW , the change in the total benefits isI ijkW∆TB = − ∑∑∑ ∫ xi′j′k ′ I ijkWO

ijkW

i ′j ′k ′I
i ′j ′k ′⎛ dp i ′j ′k ′ ⎞ ijk⎞ ijk
i ′j ′k ′ ⎛ dx
+
dI
t
∑∑∑
⎜ ijk ⎟⎜ ijk ⎟ dI∫i′j ′ k ′ I ijkWO⎝ dI ⎠⎝ dI ⎠ijkW

I ijkWOI⎛ dp i ′j ′k ′ ⎞⎛ dx i ′j ′k ′ ⎞= ∑∑∑ ∫ x i ′j ′k ′ ⎜ ijk ⎟ dI ijk + ∑∑∑ ∫ t i ′j ′k ′ ⎜ ijk ⎟ dI ijk
i′j ′ k ′ I ijkW
i′j ′ k ′ I ijkWO⎝ dI ⎠⎝ dI ⎠

I ijkWO(A3)⎛ dp i ′j ′k ′ ⎞where ∑∑∑ ∫ x i ′j ′k ′ ⎜ ijk ⎟ dI ijk is the change in the consumers’ surplus in all routes, andi′j ′ k ′ I ijkW
⎝ dI ⎠

28

I ijkWi ′j ′k ′⎞ ijk
i ′j ′k ′ ⎛ dxt∑∑∑
⎜ dI ijk ⎟ dI is the change in the tax revenue in all routes.
∫i′j ′ k ′ I ijkWO⎝⎠

Distinguishing changes in the primary market, which is the invested-in route, from those in thesecondary market, which incorporates all other routes, using (10) and (13), we rearrange (A3) toobtain the second form as follows:
I ijkWO∆TB = ∑∑∑ ∫ xi′j′k ′ I ijkW

ijkW

i ′j ′k ′I
i ′j ′k ′⎛ dp i ′j ′k ′ ⎞ ijk⎞ ijk
i ′j ′k ′ ⎛ dx
⎜ ijk ⎟ dI + ∑∑∑ ∫ t ⎜ ijk ⎟ dIi′j ′ k ′ I ijkWO⎝ dI ⎠⎝ dI ⎠

I ijkWO

ijkWO

I⎛ dp ijk ⎞⎛ dp i ′′j ′′k ′′ ⎞ ijk= ∫ x ⎜ ijk ⎟ dI ijk + ∑∑∑ ∫ x i ′′j ′′k ′′ ⎜dIijk ⎟i ′′ j ′′ k ′′ I ijkW
⎝ dI ⎠⎝ dI ⎠
I ijkWijk

I ijkW

ijkW

I⎛ dx ijk ⎞⎛ dx i ′′j ′′k ′′ ⎞ ijk
dI+ ∫ t ⎜ ijk ⎟dI ijk + ∑∑∑ ∫ t i ′′j ′′k ′′ ⎜
ijk ⎟
dI
dI
ijkWO
ijkWO
′′
′′
′′
i
j
k
⎝
⎠
⎝
⎠
I
I
ijk

I ijkWO

(A4)

ijkW

ijkWO

ijkW

ijkWO

I
I
⎛ dp ijk ⎞⎛ dx ijk ⎞⎛ dx i ′′j ′′k ′′ ⎞ ijk= ∫ x ⎜ ijk ⎟ dI ijk + ∫ t ijk ⎜ ijk ⎟dI ijk + ∑∑∑ ∫ ( cxi ′′ij′′j′′′′kk′′′′ x i ′′j ′′k ′′ − t i ′′j ′′k ′′ ) ⎜
dI
ijk ⎟
dI
dI
dI
ijkW
ijkW
ijkWO
′′
′′
′′
i
j
k
⎝
⎠
⎝
⎠
⎝
⎠
I
I
I
ijk

I ijkWO

I
I⎛ dp ijk ⎞⎛ dx ijk ⎞⎛ dx i ′′j ′′k ′′ ⎞ ijk= ∫ x ⎜ ijk ⎟ dI ijk + ∫ t ijk ⎜ ijk ⎟dI ijk + ∑∑∑ ∫ ( MC i ′′j ′′k ′′ − p i ′′j ′′k ′′ ) ⎜
dI
ijk ⎟
dI
dI
dI
ijkW
ijkW
ijkWO
′′
′′
′′
i
j
k
⎝
⎠
⎝
⎠
⎝
⎠
I
I
I
ijk

where 1 ≤ i′′ ≤ N , 1 ≤ j′′ ≤ N , and 1 ≤ k ′′ ≤ M , but (i′′, j′′, k ′′) does not contain (i, j, k ) . In (A4),
I ijkWO

ijkW

I
ijk
⎛ dp ijk ⎞ ijk⎞ ijk
ijk ⎛ dx
t
x
dI
is
the
changeintheconsumers’
surplus
in
route
ijk
,
∫ijkW ⎜⎝ dI ijk ⎟⎠∫ijkWO ⎜⎝ dI ijk ⎟⎠dI is theIIijk

I ijkWO⎛ dx i′′j′′k ′′ ⎞change in the tax revenue in route ijk , and ∑∑∑ ∫ ( MC i′′j′′k ′′ − p i′′j′′k ′′ ) ⎜ ijk ⎟dI ijk is the changei ′′ j ′′ k ′′ I ijkW
⎝ dI ⎠

in the deadweight loss in all other routes.
Highlighting the distinction between total benefits in the first-best case and distortions in thesecond-best case, using (10) and (13) to rearrange (A3), we obtain the third form as follows:29

I ijkWO

ijkW∆TB = ∑∑∑ ∫ xi′j′i ′j ′k ′k ′ I ijkWI
i ′j ′k ′⎛ dp i ′j ′k ′ ⎞ ijk⎞ ijk
i ′j ′k ′ ⎛ dx
dI
t
+
∑∑∑
⎜ ijk ⎟⎜ ijk ⎟ dI∫i′j ′ k ′ I ijkWO⎝ dI ⎠⎝ dI ⎠

II
i ′j ′k ′⎡⎡⎛ dx ijk ⎞ ⎤⎛ dx i ′′j ′′k ′′ ⎞ ⎤ ijk⎞ ijk
i ′j ′k ′ ⎛ dx
dIt
= ∫ x ⎢ cIijkijk + cxijkijk ⎜ ijk ⎟ ⎥dI ijk + ∑∑∑ ∫ x i ′′j ′′k ′′ ⎢ c xi ′′ij′′j′′′′kk′′′′ ⎜−∑∑∑⎜ ijk ⎟ dI
ijk ⎟ ⎥∫
i ′′ j ′′ k ′′ I ijkW
i′j ′ k ′ I ijkW
⎝ dI ⎠ ⎦⎝ dI ⎠ ⎦⎝ dI ⎠⎣⎣
I ijkW
I

ijkWO

ijkWO

ijkWO

ijk

I ijkWO∫ c x dI

=

ijk
I ijk

ijk

I ijkWOijk

I ijkW

I ijkWO=∫ c x dI
ijk
I ijk

ijk

⎛ dx i ′j ′k ′ ⎞+ ∑∑∑ ∫ ( cxi ′ij′j′k′k ′′ x i ′j ′k ′ − t i ′j ′k ′ ) ⎜ ijk ⎟dI ijk
i′j ′ k ′ I ijkW
⎝ dI ⎠
I ijkWO

ijk+ ∑∑∑ ∫ ( MCi′

I ijkWj′i ′j ′k ′−pi ′j ′k ′k ′ I ijkW

⎛ dx i ′j ′k ′ ⎞ ijk) ⎜ dI ijk ⎟dI .
⎝
⎠

(A5)
I ijkWO∫ c x dIijk
I ijk

In (A5),
I

ijk

ijk

is the change in the total benefits in the first-best case, which is derived

ijkW

I ijkWO

subsequently, and ∑∑∑ ∫ ( MC
i′

j′

k′

i ′j ′k ′−pi ′j ′k ′I ijkW⎛ dx i′j′k ′ ⎞ ijk) ⎜ dI ijk ⎟dI is the change in the deadweight loss⎝⎠

relating to all routes.
In the first-best case, in which the generalized prices of the trips equal their social marginal costsin all routes, from (A4) and (A5), we obtain
I ijkWO

ijkW

ijkWO

I
I
⎛ dp ijk ⎞⎛ dx ijk ⎞∆SW = ∫ x ⎜ ijk ⎟ dI ijk + ∫ t ijk ⎜ ijk ⎟dI ijk = ∫ cIijkijk x ijk dI ijk .
⎝ dI ⎠⎝ dI ⎠
I ijkW
I ijkWO
I ijkWijk(A6)

Equation (A6) shows that a change in total benefits in the first-best case consists of a change in the
I ijkWO⎛ dp ijk ⎞ ijk
consumers’ surplus in route ijk , ∫ x ⎜ ijk ⎟ dI , and a change in the tax revenue in route ijk ,⎝ dI ⎠
I ijkWijk

I ijkW

ijkWO

ijkWO

I
I
⎛ dx ijk ⎞ ijk
ijk ijkijk
ijk ijk
ijk∫ijkWO t ⎜⎝ dI ijk ⎟⎠dI , and that these two terms are reduced to ijkW∫ cI ijk x dI . Thus, ijkW∫ cI ijk x dI
I
I
I
ijk

corresponds to the change in total benefits in the first-best case.
Second, let us focus on the case in which the tax in route ijk , t ijk , increases from t ijkWO to t ijkW .
Totally differentiating total benefits, (A1) with respect to t ijk yields

30

i ′j ′k ′⎛ dp i′j′k ′ ⎞ i′j′k ′⎞dTBi ′j ′k ′ ⎛ dx
= − ∑∑∑ ⎜ ijk ⎟ x − ∑∑∑ p ⎜ ijk ⎟
ijk
dt
i′
j ′ k ′ ⎝ dti′
j′ k ′⎠⎝ dt ⎠⎛ dx i′j′k ′ ⎞⎛ dx i′j′k ′ ⎞+ ∑∑∑ uxi′i′j′k ′ ⎜ ijk ⎟ + ∑∑∑ t i′j′k ′ ⎜ ijk ⎟ + x ijk
i′j′ k ′⎝ dt ⎠ i′ j′ k ′⎝ dt ⎠= − ∑∑∑ c
i′

j′

k′

i ′j ′k ′x i ′j ′k ′i ′j ′k ′⎛ dx i′j′k ′ ⎞ i′j′k ′
⎞ ijk
ijk
i ′j ′k ′ ⎛ dx
⎜ ijk ⎟ x − x + ∑∑∑ t ⎜ ijk ⎟ + x
i′j′ k ′⎝ dt ⎠⎝ dt ⎠(A7)⎛ dx i′j′k ′ ⎞= − ∑∑∑ ( cxi′ij′′jk′k′′ x i′j′k ′ − t i′j′k ′ ) ⎜ ijk ⎟i′j′ k ′⎝ dt ⎠

where we have used the result of the utility maximization; i.e., p ijk = u xi ijk in (5).
Integrating (A7) from t ijkWO to t ijkW , from (10) and (13), we obtain the change in the total
benefits as follows:t ijkW⎛ dx i ′j ′k ′ ⎞∆TB = − ∑∑∑ ∫ ( cxi ′ij′j′k′k ′′ x i ′j ′k ′ − t i ′j ′k ′ ) ⎜ ijk ⎟dt ijk
i′j ′ k ′ t ijkWO⎝ dt ⎠
t ijkWO⎛ dx i ′j ′k ′ ⎞= ∑∑∑ ∫ ( cix′ij′j′k′k′′ x i ′j ′k ′ − t i ′j ′k ′ ) ⎜ ijk ⎟dt ijk
i′j ′ k ′ t ijkW
⎝ dt ⎠
t ijkWO= ∑∑∑ ∫ ( MCi′j′i ′j ′k ′−pk ′ t ijkW

i ′j ′k ′⎛ dx i ′j ′k ′ ⎞ ijk) ⎜ dt ijk ⎟dt ,⎝⎠which shows the change in the deadweight loss in all routes.

31

(A8)

Figures and Tables

x121

x122

Node 1

Node 2

x 211

x 212
Figure 1: N -node Network with M routes ( N = 2 , M = 2 )p121

p121WOB

x121 ( p121WO , p122WO )

A

D

121W

p

x121 ( p121W , p122W )C
General Equilibrium
Demand Curve:
x121 ( p121 , p122 ( p121 , I 122 , t122 ))O

x121

Figure 2-1: Consumers’ surplus in route 121 in the first-best case32

p121 ( I 121WO ) = MC 121 ( I 121WO )
121

p

WO
= c121WO ( x121WO , I 121WO ) + c121
( x121WO , I 121WO ) x121WO
x121WOB

p121WO

p121 ( I 121W ) = MC 121 ( I 121W )x121 ( p121WO , p122WO )W= c121W ( x121W , I 121W ) + c121
( x121W , I 121W ) x121W
x121W

A
D

E

x121 ( p121W , p122W )

121W

p

C

General Equilibrium
Demand Curve:
x121 ( p121 , p122 ( p121 , I 122 , t122 ))

F
O

x121
Figure 2-2: Producers’ surplus in route 121 in the first-best casep122

x122 ( p122W , p121W )p122WOx122 ( p122WO , p121WO )H
J( x122 , I 122 ) x122
= c122 ( x122 , I 122 ) + c122
x122

K

122W

p

General Equilibrium Demand Curve:
p122 = MC 122

L

O

Q

R

x122W

x122WO

x122

Figure 2-3: Consumers’ and producers’ surpluses in route 122 in the first-best case

33

p121

B

A

D

C

General Equilibrium
Demand Curve:
x121 ( p121 , p122 ( p121 , I 122 , t122 ))O
Figure 3-1: Consumers’ surplus in route 121 in the second-best case

x121

p121
WO
MC 121 ( I 121WO ) = c121WO ( x121WO , I 121WO ) + c121
( x121WO , I 121WO ) x121WO
x121WO

p121 ( I 121WO ) = c121WO ( x121WO , I 121WO ) + t121
B’’
B’

MC 121 ( I 121W ) = c121W ( x121W , I 121W )W+ c121( x121W , I 121W ) x121W
x121W

B

A

D’’

E

D’

D

p121 ( I 121W ) = c121W ( x121W , I 121W ) + t121C
E’General Equilibrium
Demand Curve:
x121 ( p121 , p122 ( p121 , I 122 , t122 ))

F
F’
O

G’’

G’

Figure 3-2: Producers’ surplus in route 121 in the second-best case

34

x121

MC 122 = c122 ( x122 , I 122 ) + c122
( x122 , I 122 ) x122
x122

p122

J’

General Equilibrium Demand Curve:
p122 = c122 ( x122 , I 122 ) + t122J

p122WO H
L’
p122W K
L

O

x

Q

R

122W

122WO

x

x122

Figure 3-3: Consumers’ and producers’ surpluses in route 122 in the second-best case

121
121
121
121
121
121
121
Y’ MC = c ( x , I ) + cx121 ( x , I )p121

p121 (t121W ) = c121W ( x121W , I 121 ) + t121W

V’

p121W

S

p121 (t121WO ) = c121WO ( x121WO , I 121 ) + t121WO

V

p121WOYT

S’General Equilibrium
Demand Curve:
x121 ( p121 , p122 ( p121 , I 122 , t122 ))

T’
W

V”

Y”

O

x121W

x121WO

Figure 4: The effects of an increase in the tax t121 in route 121

35

x121

Table 1: An example from ITPS (1999)

Mode
Rail
Air
Car

Without a Project
Monetary
price
Demand per day
Travelling time(yen)(minutes)
190
11,000100
120
14,000
110
400
5,500
10

Mode
Rail
Air
Car

With a Project
Monetary
price
Demand per day
Travelling time(yen)(minutes)
140
11,000135
120
14,000
80
400
5,500
5

Table 2: An example in which the producers’ surplus plays an important role
Without a Project

Mode
Rail
Car

Rail’sTravelling time Monetary
Net fuel Fuel tax average Generalized Demand(minutes)
price (yen) price (yen) (yen) cost (yen) price (yen) per day
50
500
/
/
600
2,500
300
70
400
200
200
/
3,200
200
With a Project

Mode
Rail
Car

Rail’sTravelling time Monetary
Net fuel Fuel tax average Generalized Demand(minutes)
price (yen) price (yen) (yen) cost (yen) price (yen) per day
35
550
/
/
700
1,950
400
60
300
150
150
/
2,700
150

Note: Monetary price of car travel ≡ Net fuel price + Fuel tax

36

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