CSIS Discussion Paper No. 64

Rate-of-return and Price-cap Regulations
for Congested Urban Railways
Yukihiro Kidokoro∗

Abstract: Using a simple model of commuter railways where congestion exists, we show that theregulatory shift from rate-of-return regulation to price-cap regulation does not cure the congestion
problem in Japanese urban railways. We next consider methods to correct it, and show the following
results: (i) PC regulation, in which the cap is made contingent on congestion, can correct thecongestion without distorting cost-reducing efforts, (ii) PC regulation, in which the cap depends on
investment in transportation capacity, can also correct the congestion but distorts cost-reducing
efforts.
Keywords: price-cap regulation; rate-of-return regulation; congestion; railways∗

Center for Spatial Information Science, University of Tokyo

Mailing Address: 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
I would like to thank Yoshitsugu Kanemoto for helpful suggestions and discussions. I owe special thanks to Hajime Miyazaki for his
extensive comments. I would also like to thank Ken Brevoort and Tatsuo Hatta for their remarks. Of course, I am solely responsible forany errors and omissions.

11 Introduction
Urban railways, including public and private railways, play important roles as part of Japan’surban transit systems. For example, railways in the Tokyo metropolitan area provides 55% of travelneeds in 2001. Urban railways in Japan, however, have a serious problem; extreme congestion duringrush hours. The congestion rate, which is the ratio of the number of users to a railway’s nominal
capacity, is about 200% in almost every line in the Tokyo metropolitan area. Typically, Tokyo
railway firms have been operating their lines with very short headway and very long train
configurations during peak time. Therefore, peak-time congestion cannot be relieved without
constructing new lines or enhancing the current double-track lines to four-track lines.
The purpose of this paper is to investigate whether the congestion problem stated above isresolved or not by the regulatory reform from rate-of-return (ROR) regulation1 to price-cap (PC)regulation2, which is now under way in many industrialized countries, and to propose the regulatory
method that is consistent with the relief of the congestion. To deal with the problem, we build a
simple spatial land-use model of commuter railways and analyse the effects that are caused by theregulatory shift from ROR regulation to PC regulation. The reason why we employ a spatial land-use
model is that the bottleneck input in enhancing transportation capacity from the current double-track
to four-track lines in Japan is railroad right-of-way.
This paper shows that the regulatory shift to PC regulation does not cure the congestion problem
although it corrects the distortion in input mix under ROR regulation.

We next focus on

modifications of PC regulation and obtain the following results: (1) PC regulation with a cap
contingent on transportation quality, which is the inverse of the congestion rate, can relieve thecongestion without distorting cost-reducing efforts. (2) PC regulation, in which the cap is madecontingent on investment, can also correct the congestion but distorts cost-reducing efforts.

1

See, for example, Averch and Johnson (1962) and Train (1991) for the theory of ROR regulation.

2

See, for example, Littlechild (1983) and Armstrong et al. (1994) for the theory of PC regulation.

2

The structure of the paper is as follows. In Section 2, we set up a simple spatial model. In
Section 3, we obtain the results of the first best case as a benchmark. Section 4 analyzes ROR
regulation and Section 5 does PC regulation. In Section 6, modifications of PC regulation are
considered. Section 7 concludes our analysis.

2 ModelOur model is a variant of the model outlined in Kidokoro (1998), that is based on an open-cityand absentee-landlord model from urban economics literature3. Consider a residential city of fixed
size H , which is connected to the central business district (CBD) by a railway. The utility leveloutside the city is given by u . Residents and potential residents can freely move out of and into theresidential city. As a result, residents attain exactly u in equilibrium. For simplicity, we assume that
there are no transportation costs within the residential city. Consequently, transportation fares are
uniform within the residential city. This assumption implies that we disregard locational differences
within the residential city.
All residents in this city, N in number, are assumed to use a railway line to make a round trip toand from the CBD every day. Moreover, they are assumed to commute to the CBD in peak time when
they are on duty and to go to the CBD to dine or shop in off-peak time when they are off duty. If all
the residents had the same pattern of on-duty days and off-duty days, they would use a railway line inpeak time on their on-duty days and, consequently, off-peak rail demand would not exist on theiron-duty days. In order to create both peak-time and off-peak-time demand daily, we further assume
that the pattern of on-duty days and off-duty days differs resident by resident. For example, resident
A, who works in an electronics company, may work from Monday to Friday, while resident B, whoworks in a supermarket, may work from Monday to Wednesday, Saturday and Sunday. The upshot isthat the daily peak-time users of a railway are commuters who are on duty and the daily off-peak time

3

See, for example, Kanemoto (1980).

3

users are non-commuters who are off duty.
A railway firm supplies transportation capacity Q daily. Q is nominal in the sense that theactual number of daily users of a railway, which is the sum of peak-time users and off-peak-time users,
is not Q but N . We here define the daily average congestion rate asN. To allow analyticalQ

simplicity, we use the inverse of the daily average congestion rate, q ≡Q, as the daily averageNquality of transportation service. For example, if a railway firm transports many residents with lowtransportation capacity every day, the daily average congestion rate is high and hence the daily
average quality of transportation service is low. To differentiate the peak time from the off-peak time,
the peak-time quality of the transportation service, q peak , is defined as q peak ≡off-peak-time congestion rate, qoffpeak , is defined as qoffpeak ≡Q peak, and theN peakQ − Q peak
, where Q peak and N peak are
peak
N −N

daily peak-time transportation capacity and daily peak-time users of a railway, respectively.4 For thesake of simplicity, we assume that transportation capacity and the users of a railway line in peak timearefunctionsq peak =

of

their

daily

values,Q peakf (Q )⎛Q⎞== ζ ⎜ ⎟ = ζ ( q) .
peak
N
g(N )
⎝N⎠

i.e., Q peak = f (Q )andN peak = g ( N ) andthat

Thus, q peak is a function of q and can be written as

q peak ( q) .
We assume that all residents are homogeneous with a quasi-linear and additively separable
utility function, U ( z, h, qoffpeak , q peak ) = z + u (h ) + v ( qoffpeak , q peak ) , where z is the composite consumer
good, including housing, whose price is normalized at one, and h is residential lot size. Since the4

We implicitly assume that daily peak-time transportation capacity and the number of daily peak-time users of a railway are the samefor all days and disregard the daily variance in them. As a result, the quality of transportation service for peak and off-peak times also
becomes day-invariant.

4

off-peak congestion rate is sufficiently low even in Japanese urban railways, we disregard it andsimplify the utility function to U ( z , h, q peak ) = z + u (h ) + v ( q peak ) to focus on peak-time congestion.
That is, in our analysis, the off-peak quality of transportation service is assumed not to affect theutility level of residents. This simplification of the utility function enables us to conduct a simulation
based on the actual estimate of v ( q peak ) . We assume that u( h ) and v ( q peak ) are strictly increasing

and strictly concave: u′( h ) > 0 , u′′( h ) < 0 , v′( q peak ) > 0 , v′′( q peak ) < 0 . As is well known, theincome effects are zero under the quasi-linearity assumption5 and cross elasticities are zero under the
separability assumption. As a result, this form of the utility function yields a demand function forland that depends only on land rent, which simplifies our analysis.
Each resident solves his or her utility maximization problem subject to a budget constraint,z + Rh + t = w , where R , t , and w denote land rents, transportation fares, and the fixed income of aresident, respectively. Maximizing the utility function, U ( z, h, q peak ) = z + u (h ) + v ( q peak ) , under thebudget constraint, z + Rh + t = w , yields u′( h ) = R . Inverting this function yields the demand
function for land:h = h( R ) ≡ u′ −1 ( R ) .
Following the usual procedure in the urban economics literature, we derive the bid rent function,
which gives the maximum possible rent, providing utility level u . The bid rent function is:

R( y − t , q peak ) = max{z ,h}{where y ( ≡ w − u ) is real income.

Rq peak =w−t− z: z + u( h ) + v ( q peak ) ≥ u } ,hThis function satisfies RI =v′( q peak )> 0 , where I ≡ y − t . Hereafter, the subscripts denote partial derivatives, unlesshotherwise noted.

51
1> 0 , Rt = − < 0 , andhhSee, for example, Varian (1992).

5

Substituting the bid rent function into the demand function for land yields a lot size function
from which we eliminate the land rent, R :hˆ( I , q peak ) = h( R( I , q peak )) .
hhh v′( qThis function satisfies hI = R < 0 , ht = − R > 0 , and hˆq peak = R
h
h
h

peak

)

< 0.

We assume that the railway firm uses L as railroad right-of-way within the city area H . (Weignore railroad right-of-way used outside the city area.) The residential area left for housing is thus
H − L . We assume that railroad right-of-way can be converted without cost into residential land andvice versa. Given u , the equilibrium number of residents is then defined as:
N (t , q peak , L) =

H −L
.
ˆ
h ( y − t , q peak )To the railway firm, N (t , q peak , L) is synonymous with the equilibrium transport demand function
given u . The transport demand function satisfies N t =NL = −NhR
NhR v′( q peak )
<
0
,
=
−
> 0 , andNq peak
h2
h2

1
<0.
h

The production function, by which a railway firm supplies daily railway capacity Q , is
Q = qN = F ( L, Z , e) , where L is railroad right-of-way, Z is non-capital input, and e is the firm’s
efforts. We assume that Fi ≡∂F
∂ 2F
> 0 and Fii ≡ 2 < 0 , where i = L , Z , and e .6 We ignore capital∂i∂iinputs other than railroad right-of-way for simplicity, because our focus is on railroad right-of-way,which is the bottleneck input that hinders an enhancement of railway capacity. For simplicity, thenon-capital input, Z , is assumed to be the same good as the composite consumer good. The price of

6

We disregard the problem of indivisibilities of investments by assuming that the investment continuously changes transportation

capacity. This assumption at least approximately holds in reality, because it is common to enhance only a part of track capacity, e.g.,terminal stations, the most congested part of a railway line, or bottleneck intersections, in order to reduce total investment costs.

6

the non-capital input is therefore one. We call the firm’s efforts, e , cost-reducing efforts, because
under the conditions of our model, the firm’s efforts to expand railway capacity are included in the
production function, which implies that the firm’s efforts do reduce its costs, given railway capacity.
A railway firm is assumed to incur monetary disutility ϕ ( e ) from cost-reducing efforts, e , where weassume ϕ ′( e ) > 0 and ϕ ′′( e ) > 0 . Both the cost-reducing efforts and the production function itself
are unobservable to the regulator. This means that the regulator cannot calculate the value of e basedon Q , L , and Z . Thus, the regulator cannot implement the regulation based on the level of e .
A railway firm is assumed to have obtained dH ( d is any number between 0 and 1) of theresidential area with the price of V before a railway line was built. This assumption corresponds tothe fact that railway firms in Japan own much land, the book price of which is very low, along theirrailway lines. The railway firm rents d1H 7 out of dH as residential land. Thus, the railway firm has
two sources of revenue: railway fares and rental income from residential land. For simplicity, theprovision of residential land is the only side business in our analysis. The railway firm uses the otherpart of dH , d 2 H = (1 − d1 ) H , as its railroad right-of-way. (Since the total railroad right-of-way is L ,d 2 H < L ).8 The regulatory authority is assumed to impose regulations only on the firm’s railway
sector, as is the case in Japan. We also assume that many competitive absentee-landlords own the7

d1 and d 2 are exceptions to our notation rule that the subscripts denote partial derivatives.

8

In the ROR regulation in Japan, railroad right-of-way in the rate base is undervalued, compared with its market value. In this case, arailway firm has an incentive to use the land it owns, not as railroad right-of-way but to rent it out as residential land, i.e., to maked 2 = 0 if possible. This is because lower evaluation in the rate base leads to lower allowed profits of railway business, which are lessthan the market land rents it could obtain. However, in reality, a lot of the land owned by railway firms is now being used as railroad
right-of-ways, probably due to technological factors affecting the construction of railway lines or to the nature of the tax system, issues
that we do not address in this paper. Thus, in this paper, d 2 is assumed to be a fixed positive value. In considering the PC regulationand its variants, which is the main focus of this paper, this assumption is innocuous, because they have no relationship with the rate base,and thus the magnitude of d 2 does not affect the railway firms’ profits.

7

railway firm and the residual residential area, (1 − d ) H .
Now we can express the profits of the railway firm as:

π (t , q, L, Z , e) ≡ tN (t , q peak ( q), L) + R(y -t , q peak ( q))dH − Z − R(y -t , q peak ( q))L − ϕ ( e) .
In our closed-form model, social welfare is the sum of the railway firm’s profits and the residential
land rent that does not accrue to the railway firm. Social welfare is then:SW (t , q, L, Z , e) ≡ π (t , q peak ( q), L, Z , e) + R(y -t , q peak ( q))(1-d ) H .
We consider an infinite period, beginning in period 0 when the railway line is built. Since we set
up no changes after period 0, the state in period 0 is repeated endlessly. Therefore, the railway firm
keeps its choice variables, i.e., t , q , L , Z , and e , constant from period 0 onwards. The present
discounted value of the railway firm’s profits and social welfare from period 0 onwards, respectively,
can then be written as
PV (π ) ≡π (t , q, L, Z , e)
r

(1)

and
PV ( SW ) ≡SW (t , q, L, Z , e)
,
r

(2)

where r is the fixed cost of capital.

3 First BestFirst, as a benchmark, we obtain the first best input choices by maximising the present
discounted value of social welfare, (2), subject to the production constraint:
F ( L, Z , e) ≥ qN (t , q peak ( q), L) .
The results are stated as Proposition 1.

Proposition 1

In the first best optimum,
8

(3)

(i)

The marginal benefit of the investment in the non-capital input, Z , equals its marginal cost:v′( q peak ) q peak ′ ( q) FZ = 1 .
(ii) The marginal rate of technical substitution between the non-capital input, Z , and railroad
right-of-way, L , equals their relative price:FL − qN L= R − tN L .
FZ(iii) The marginal rate of technical substitution between the non-capital input, Z , and cost-reducing
efforts, e , equals their relative price:Fe= ϕ ′( e ) .
FZThe proof is in Appendix 1. In result (i), v′( q peak ) q peak ′ ( q) FZ shows the marginal benefit of theinvestment in the non-capital input. An additional investment in the non-capital input expands
transport capacity by FZ , which increases peak-time transportation quality by q peak ′ ( q) FZ . The
marginal benefit from peak-time transportation quality is v′( q peak ) , and consequently,v′( q peak ) q peak ′ ( q) FZ shows the marginal benefit of an increase in peak-time transportation quality
caused by the investment in the non-capital input, i.e., the marginal benefit of the investment in the
non-capital input. Result (i) shows that there is no distortion in the choice of non-capital input,
because the marginal benefit of the investment in the non-capital input equals its marginal cost.
In result (ii),FL − qN L
shows the marginal rate of technical substitution between theFZ

non-capital input, Z , and railroad right-of-way, L , which has a different form from the standard one.
The reason is that investment in railroad right-of-way increases railway capacity in two ways. First,
the investments directly increase railway capacity by FL . Second, they decrease the land available
for housing, and consequently decrease the number of residents (= users). (Recall that city size, H ,is fixed.) This decrease in the number of users, N L , virtually expands railway capacity by −qN L ,because a user uses the capacity by q on average. The effect of investments in railroad right-of-way

9

on railway capacity is thus FL − qN L . In the end, the marginal rate of technical substitution between
Z and L is

FL − qN L. The decrease in the number of users also reduces a railway firm’s fareFZ

revenue by −tN L , which makes the opportunity cost of railroad right-of-way R − tN L . The relative
price of it to non-capital input is then R − tN L . (Recall that the price of Z is normalised at one.)
Result (ii) states that the choice of railroad right-of-way, compared to the choice of non-capital input,
is not distorted, because the marginal rate of technical substitution equals the relative price.
In Result (iii), ϕ ′( e ) is the marginal disutility of cost-reducing efforts, e . ϕ ′( e ) can then beinterpreted as the marginal cost of the railway firm’s cost-reducing efforts, and the relative price of e
to Z . Results (iii) states that the choice of cost-reducing efforts is not distorted, compared to the
choice of non-capital input.
These three conditions guarantee that input choices are optimal. Thus, by comparing these
conditions with those under the regulatory methods we focus on hereafter, we can know theinfluences of the regulatory methods on railway firms’ input choices.

4 ROR Regulation
To compare the case of ROR regulation with that of PC regulation later, we here consider arailway firm’s behaviour under ROR regulation, which can be formulated as

ρ × ( RateBase) ≥ tN (t , q peak ( q), L) − Z ,where ρ is the allowed rate of return. Rate base can be written as V ( L − d 2 H ) + Vd 2 H , where V
denotes land price, when the regulation is based on book value, while VL when the regulation isbased on market value. (Recall that the purchase price of d 2 H is V .) Combining the case of bookvalue with that of market value, we formulate rate base as V ( L − β d 2 H ) + βVd 2 H , where β = 1
shows book-value-based ROR regulation and β = 0 shows market-value-based ROR regulation.
In the end, ROR regulation is10ρ {V ( L − β d 2 H ) + βVd 2 H } ≥ tN (t , q peak ( q), L) − Z .

(4)

The maximisation of the present discounted value of the railway firm’s profits, (1), subject to the
production constraint, (3), and ROR regulation, (4), yields basically the same results as Kanemoto
and Kiyono (1993,1995). We summarise the results as Proposition 2.

Proposition 2

Under binding ROR regulation,(i)

The marginal benefit of the investment in the non-capital input, Z , equals its marginal cost:v′( q peak ) q peak ′ ( q) FZ = 1 .
(ii) If the allowed rate of return, ρ , is lower than the true cost of capital, r , then the marginal rateof technical substitution between the non-capital input, Z , and railroad right-of-way, L , exceeds
their relative price. If ρ > r , then the usual AJ result holds. If ρ = r , then the marginal rate oftechnical substitution equals their relative price:<FL − qN L >= R − tN L as ρ = r .
<>FZ(iii) The marginal rate of technical substitution between the non-capital input, Z , and cost-reducing
efforts, e , exceeds their relative price:Fe> ϕ ′( e) .
FZThe proof is stated in Appendix 1. Result (i) is the same as Proposition 1-(i). That is, ROR
regulation does not distort a railway firm’s choice of non-capital input9. Under ROR regulation, therailway firm is allowed to raise its price when investing in non-capital input. Therefore, the railway
firm has no incentive to lower the investments in non-capital input.

9

If the utility function is not quasi-linear or additively separable, a railway firm’s choice of non-capital input is distorted compared withthe first best case: non-capital input is more or less than the optimal level. For this distortion, see Spence (1975).

11

Result (ii) states that the usual AJ effect10 should be modified when railway firms run profitableside businesses. Since there are profits from side businesses, the regulator can set the rate of return,ρ , below the true cost of capital, r , in which case, the railway firm’s return from the investment inrailroad right-of-way is lower than its costs. This low rate of return leads to underinvestment inrailroad right-of-way, and reverses the AJ result: the marginal rate of substitution between railroad
right-of-way, L , and non-capital, Z , exceeds their relative price. Kanemoto and Kiyono (1993,
1995) point out that this underinvestment due to the reversed AJ effect is one of the main reasons forthe congestion of urban railways in Japan’s large cities. If ρ > r , the normal AJ result holds. If ρ = r ,the marginal rate of substitution equals the relative price, and consequently the input choice between
L and Z is not distorted.
Result (iii) shows that ROR regulation distorts cost-reducing efforts. If a railway firm reduce its
costs by its effort, its profits increase. However, given this increase in profits, a firm’s actual rate of
return goes up and becomes higher than the allowed rate of return. In this case, the firm must reduce
its revenue to meet ROR regulation. This implies that the railway firm cannot capture the increase inprofits that cost reduction yields, and has weaker incentives for cost-reduction. Thus, under ROR
regulation, the railway firm operates inefficiently.
Actually, constructing a new railway line entails huge development profits. That is, land prices
go up substantially after the construction of new railway lines. Railroad right-of-way is revalued after
construction of railway lines under market-value-based ROR regulation, while it is never revalued
under book-value-based ROR regulation. The difference between market value and book value ofrailroad right-of-way is very large in Japan even after the burst of the Bubble economy in the early
1990s. For major railway companies operating in the Tokyo metropolitan area, the book value ofrailroad right-of-way is less than 10 % of its market value in 1993 prices. Is the lack of revaluation ofrailroad right-of-way under book-value-based ROR regulation hurt cost-reducing incentives? The

10

See Averch and Johnson (1962).

12

following corollary answers the question.

Corollary 1

When the allowed rate of return, ρ , equals the true cost of capital, r ,(i) under book-value-based ROR regulation, incentives for cost reduction exist as long as a railway
firm engages in side businesses, i.e., d1 > 0 .
(ii) Under market-value-based ROR regulation, incentives for cost reduction exist as long as therailway firm owns land, i.e., d > 0 .

The proof is stated in Appendix 1. Note that cost-reducing incentives are stronger undermarket-value-based ROR regulation than under book-value-based ROR regulation. If the regulation
is based on the book value of land, the cost-reducing incentives for the railway firm with no sidebusiness, i.e., d1 = 0 , disappear when the allowed rate of return, ρ , equals the true cost of capital, r .
A railway firm with side businesses, i.e., d1 > 0 , retains these incentives even when ρ = r because
the lower fare brought about by cost reductions leads to higher residential land rent and hence larger
profits from side businesses.
If the regulation is based on the market value of land, however, even a railway firm that owns
only railroad right-of-way, (i.e., d1 = 0 and d 2 > 0 ) keeps the incentives for cost reduction in the caseof ρ = r . In this case, the lower fare from reduced costs yields higher land rent, which leads to higher
land prices. With revaluation under market-value-based ROR regulation, the higher land price makes
the rate base larger. The larger rate base then yields larger profits from the railway business. Thus,
the revaluation of railroad right-of-way after construction of railway lines in the case of
market-value-based ROR regulation strengthen the cost-reducing incentives.
The above analyses show that (1) ROR regulation may increase the level of congestion due tounderinvestment in railroad right-of-way as a result of the reversed AJ effect, (2) ROR regulation13

causes the distortions in incentives in cost-reducing efforts, although they are somewhat alleviated byadopting the market-value-based ROR regulation.

5 PC Regulation
What happens when the regulation shifts to PC regulation? We now focus on PC regulation,
which can be written as
tcap ≥ t ,(5)where tcap is the fixed ceiling price. The maximisation of the present discounted value of therailway firm’s profits, (1), subject to the production constraint, (3), and PC regulation, (5) yields therailway firms’ input choices under PC regulation, which are summarised as proposition 3.

Proposition 3

Under binding PC regulation,(i)

The marginal benefit of the investment in the non-capital input, Z , exceeds its marginal cost:v′( q peak ) q peak ′ ( q) FZ > 1 .
(ii) The marginal rate of technical substitution between the non-capital input, Z , and railroad
right-of-way, L , equals their relative price:FL − qN L= R − tN L .
FZ(iii) The marginal rate of technical substitution between the non-capital input, Z , and cost-reducing
efforts, e , equals their relative price:Fe= ϕ ′( e ) .
FZThe proof is in Appendix 1. Result (i) shows that PC regulation leads to underinvestment in
non-capital input. This distortion occurs for the following reason. As long as the transportation fareis suppressed by PC regulation, the railway firm cannot raise the fare even if it invests in non-capital
input. Investment in non-capital input is therefore discouraged. Results (ii) and (iii) state the merits
14

of PC regulation; under PC regulation, the railway firm employs the optimal input mix between
non-capital input, Z , railroad right-of-way, L , and cost-reducing efforts, e , because it regulates theceiling price only. That is, given the railway firm’s investment level of non-capital input, choices ofrailroad right-of-way and cost-reducing efforts are not distorted. This result implies that the tendency
for an inefficient operation under ROR regulation disappears under PC regulation.
The above analysis of PC regulation demonstrates that compared to ROR regulation, PC
regulation contains another source of distortion in investments, which leads to congestion. Under PC
regulation, non-capital input are underinvested, which also leads to underinvestment in railroad
right-of-way and cost-reducing efforts, provided that the marginal rate of technical substitutions
among all inputs are optimal. That is, as long as the ceiling price is binding, PC regulation causes
underinvestment in all inputs and thus causes congestion.

The congestion problem does notdisappear under the regulatory shift from ROR to PC regulation.

6 Modified PC Regulation
The analyses in the last section make it clear that the regulatory shift to PC regulation does not
resolve the congestion problem. We consider here modified versions of PC regulation that is
consistent with the relief of the congestion.
First, let us focus on PC regulation with a cap contingent on transportation quality, which is theinverse of the congestion rate. PC regulation thus modified can be written as
tcap + t ( q peak ) ≥ t ,(6)where t ( q peak ) is the variable part of the ceiling price that depends on peak-time transportation
quality, q peak . We call this quality-contingent PC regulation. Maximising the present discounted
value of the railway firm’s profits, (1), subject to the production constraint, (3), andquality-contingent PC regulation, (6), we obtain the railway firms’ input choices under
quality-contingent PC regulation, which are summarised as proposition 4.
15

Proposition 4

Under quality-contingent PC regulation, if t ′( q peak ) = v′( q peak ) ,(i)

The marginal benefit of the investment in the non-capital input, Z , equals its marginal cost:v′( q peak ) q peak ′ ( q) FZ = 1 .
(ii) The marginal rate of technical substitution between the non-capital input, Z , and railroad
right-of-way, L , equals their relative price:FL − qN L= R − tN L .
FZ(iii) The marginal rate of technical substitution between the non-capital input, Z , and cost-reducing
efforts, e , equals their relative price:Fe= ϕ ′( e ) .
FZThe proof is in Appendix 1. Quality-contingent PC regulation eliminates the drawbacks, while
maintaining the merits of PC regulation. Under quality-contingent PC regulation, a railway firm isallowed to set a higher price when alleviating congestion. Since the railway firm can obtain profits
from its investments in railway capacity, it has no actual incentive to decrease the investments inrailroad right-of-way and non-capital input and to thereby increase congestion. If t ′( q peak ) = v′( q peak ) ,i.e., the marginal increase in the cap equals marginal benefit from peak-time transportation quality, auser’s cost accompanied by decreasing congestion equals his or her received benefit and all input
choices become socially optimal.
In many cases, it would be difficult to measure marginal benefit from peak-time transportation
quality, v′( q peak ) . In practice, it would be easier to implement PC regulation, in which the cap iscontingent on capacity investment. Therefore, we next consider this kind of PC regulation,
tcap + t ( L, Z ) ≥ t .

(7)

where t ( L, Z ) is the variable part of the ceiling price that depends on railroad right-of-way, L , and16the non-capital input, Z .

We call this method investment-contingent PC regulation.

The

maximisation of the present discounted value of the railway firm’s profits, (1), subject to the
production constraint, (3), and investment-contingent PC regulation, (7) yields proposition 5.

Proposition 5

Under investment-contingent PC regulation, the distortion between the non-capital input, Z ,and cost-reducing efforts, e , is never eliminated:(i)Fe> ϕ ′( e) . If tL N = R − tN L and tZ N = 1 ,FZThe marginal benefit of the investment in the non-capital input, Z , equals its marginal cost:v′( q peak ) q peak ′ ( q) FZ = 1 ,(ii) The marginal rate of technical substitution between the non-capital input, Z , and railroad
right-of-way, L , equals their relative price:FL − qN L= R − tN L .
FZThe proof is stated in Appendix 1. Investment-contingent PC regulation can remove quality
distortions, as long as the marginal investment costs of both inputs are fully recovered through theincrease in the ceiling price, i.e., tL N = R − tN L and tZ N = 1 . However, it yields distortions in
cost-reducing efforts. This feature stems from the fact that investment-contingent PC regulation
lowers the investment costs of railroad right-of-way, L , and non-capital, Z , but leaves the cost of
cost-reducing efforts, e , unchanged. Suppose that the marginal investment costs are fully recoupedby the increase in the allowed price. In this case, if a railway firm invests in L and Z , the allowed
price goes up and the investment costs are virtually zero. If it invests in cost-reducing efforts,
however, the allowed price remains unchanged. This asymmetry weakens cost-reducing incentives,and leads to the railway firm operating less efficiently.
Investment-contingent PC regulation would be a possible candidate as a regulation system forcongested urban railways, because it can ease congestion and can be implemented based on the data17

on investment costs of each input, which are easier for the regulatory authority to obtain than the dataon marginal benefit from peak-time transportation quality. However, the regulator should be awarethat investment-contingent PC regulation weakens the railway firms’ incentives to operate efficiently.

7 Conclusion
Our analysis stated above shows that simple PC regulation without properly addressing
congestion problem would not be suitable for Japanese urban railways that are extremely congested,because it has adverse effects on congestion although it promotes cost reduction. In modified PC
regulations, quality-contingent PC regulation is consistent with the relief of congestion and stays freeof distortions in cost-reducing efforts. Although investment-contingent PC regulation also eases
congestion, it distorts incentives for cost reduction.
Although our model clarifies the effects of ROR, PC, quality-contingent PC, andinvestment-contingent PC regulations, it is important to show how serious those effects are. For
example, how much does social welfare increase or decrease by the regulatory shift from ROR
regulation to PC regulation? Or, how much merits arise when the regulator adopts quality-contingent
PC regulation or investment-contingent PC regulation?

To answer these questions, we neednumerical simulations in which actual railway data are used, which are delegated to future researches.

18

Appendix 1: Proof of Propositions
Proof of Proposition 1

We set up the Lagrangian for the first best case as follows:Λ = PV ( SW ) + λ [ F ( L, Z , e) − qN (t , q peak ( q), L)] ,(A1)

where λ ≥ 0 .
The first order conditions for (A1) are∂Λ 1
R (1 − d ) H= ( N + tN t + Rt dH − Rt L) + t− λ qN t = 0,
∂t r
r

(A2)

R (1 − d ) H∂Λ 1= (tN q + Rq dH − Rq L) + q− λ ( N + qN q ) = 0,
∂q r
r

(A3)

∂Λ 1
= ( tN L − R ) + λ ( FL − qN L ) = 0,
∂L r

(A4)

∂Λ
1
= − + λFZ = 0,
∂Z
r

(A5)

∂Λ
ϕ ′( e )=−+ λFe = 0 ,
∂e
r

(A6)

λ [ F ( L, Z , e) − qN (t , q peak ( q), L)] = 0 .
From Rq peak =and N q = −v′( q peak )Nh v′( q peak )v′( q peak ) q peak ′ ( q)
> 0 and N q peak = − R 2> 0 , we derive Rq =hhh

NhR v′( q peak ) q peak ′ ( q).
h2

Using Rt = −1
Nhv′( q peak ) q peak ′ ( q)
, N t = 2R , Rq =
, and
h
h
h

NhR v′( q peak ) q peak ′ ( q), from (A2)-(A6) we obtain
Nq = −h2v′( q peak ) q peak ′ ( q) FZ = 1 ,(A7)

FL − qN L= R − tN L ,
FZ

(A8)

Fe
= ϕ ′( e ) .
FZ

(A9)

19

From (A7)-(A9), we obtain Proposition 1.
Q.E.D.

Proof of Proposition 2

The Lagrangian for profit-maximisation under ROR regulation can be formulated asΛ = PV (π )⎡ ⎧ R(y -t , q peak ( q))⎤⎫( L − β d 2 H ) + βVd 2 H ⎬ − tN (t , q peak ( q), L) + Z ⎥+ λ ⎢ρ ⎨
r
⎭
⎣ ⎩
⎦

(A10)

+ γ [ F ( L, Z , e) − qN (t , q peak ( q), L)],
where λ ≥ 0 and γ ≥ 0 .
The first order conditions for (A10) are

∂Λ 1
⎧ρR
⎫
= ( N + tN t + Rt dH − Rt L) + λ ⎨ t ( L − β d 2 H ) − ( N + tN t ) ⎬ − γ qN t = 0,
∂t r
⎩ r
⎭

(A11)

⎧ρR
⎫
∂Λ 1
= (tN q + Rq dH − Rq L) + λ ⎨ q ( L − β d 2 H ) − tN q ⎬ − γ ( N + qN q ) = 0,
∂q r
⎩ r
⎭

(A12)

∂Λ 1
ρR
= ( tN L − R ) + λ (− tN L ) + γ ( FL − qN L ) = 0,
r
∂L r

(A13)

∂Λ
1
= − + λ + γFZ = 0,
r
∂Z

(A14)

∂Λ
ϕ ′( e )=−+ γFe = 0 ,
r
∂e

(A15)

⎡ ⎧ R(y -t , q peak ( q))⎤⎫λ ⎢ρ ⎨( L − β d 2 H ) + βVd 2 H ⎬ − tN (t , q peak ( q), L) + Z ⎥ = 0 ,
r
⎭
⎣ ⎩
⎦

γ [ F ( L, Z , e) − qN (t , q peak ( q), L)] = 0 .
If ROR regulation is binding, i.e., λ > 0 , (A11)-(A15) yieldsv′( q peak ) q peak ′ ( q) FZ = 1 ,(A16)

FL − qN L
Rλ= R − tN L −(ρ − r) ,FZ
1 − rλ(A17)

20

Fe ϕ ′( e )=> ϕ ′( e ) ,FZ 1 − rλ(A18)where 1 − rλ ≥ 0 from (A14). From (A16)-(A18), we obtain Proposition 2.
Q.E.D.

Proof of Corollary 1

From (A12) and (A13), we obtainρ⎛⎞( ρ − r ) {Rq L( FL − qN L ) + R( N + qN q )} + ⎜ d − β d 2 ⎟ rRq H ( FL − qN L )r⎝⎠.
1 − rλ =ρ Rq ( L − β d 2 H )( FL − qN L ) + ρ R( N + qN q ) − rt ( N q FL + NN L )If 1 − rλ = 0 , thendisappear.

Fe= ∞ from (A18), i.e., the incentives for cost reduction completely
FZ

When ρ = r and β = 1 , 1 − rλ = 0 if d − d 2 ≡ d1 = 0 .

When ρ = r and β = 0 ,1 − rλ = 0 if d = 0 .

Q.E.D.

Proof of Proposition 3

We set up the Lagrangian for PC regulation as follows:Λ = PV (π )+ λ[ t − t ](A19)+ γ [ F ( L, Z , e) − qN (t , q peak ( q), L)],
where λ ≥ 0 and γ ≥ 0 . The first order conditions for (A19) are∂Λ 1= ( N + tN t + Rt dH − Rt L) − λ − γ qN t = 0,
∂t r

(A20)

∂Λ 1
= (tN q + Rq dH − Rq L) − γ ( N + qN q ) = 0,
∂q r

(A21)

∂Λ 1
= ( tN L − R ) + γ ( FL − qN L ) = 0,
∂L r

(A22)

21

1
∂Λ
= − + γFZ = 0,
∂Z
r

(A23)

∂Λ
ϕ ′( e )=−+ γFe = 0 ,
∂e
r

(A24)

λ [t − t ] = 0 ,γ [ F ( L, Z , e) − qN (t , q peak ( q), L)] = 0 .
If PC regulation is binding, i.e., λ > 0 , from (A20)-(A24) we obtainv′( q peak ) q peak ′ ( q) FZ =

1
> 1,
rλ
1−
N

(A25)

FL − qN L= R − tN L ,
FZ

(A26)

Fe
= ϕ ′( e ) .
FZ

(A27)

From (A25)-(A27), we obtain Proposition 3.
Q.E.D.

Proof of Proposition 4

We set up the Lagrangian for quality-contingent PC regulation as follows:Λ = PV (π )+λ [tcap + t ( q peak ( q)) − t ]+γ [ F ( L, Z , e) − qN (t , q

peak

(A28)
( q), L)]

where λ ≥ 0 and γ ≥ 0 . The first order conditions for (A28) are∂Λ 1= ( N + tN t + Rt dH − Rt L) − λ − γ qN t = 0,
∂t r

(A29)

∂Λ 1
= (tN q + Rq dH − Rq L) + λ t ′( q peak ) q peak ′ ( q) − γ ( N + qN q ) = 0,
∂q r

(A30)

∂Λ 1
= ( tN L − R ) + γ ( FL − qN L ) = 0,
∂L r

(A31)

22

1
∂Λ
= − + γFZ = 0,
∂Z
r

(A32)

∂Λ
ϕ ′( e )=−+ γFe = 0 ,
∂e
r

(A33)

λ [tcap + t ( q peak ( q)) − t ] = 0 ,γ [ F ( L, Z , e) − qN (t , q peak ( q), L)] = 0 .
From (A29)-(A33), we obtainv′( q peak ) q peak ′ ( q) FZ = 1 −rλ q peak ′ ( q) FZt ′( q peak ) − v′( q peak )} ,
{
N

(A34)

FL − qN L= R − tN L ,
FZ

(A35)

Fe
= ϕ ′( e ) .
FZ

(A36)

From (A34)-(A36), we obtain Proposition 4.
Q.E.D.

Proof of Proposition 5

We set up the Lagrangian for investment-contingent PC regulation as follows:Λ = PV (π )+ λ [tcap + t ( L, Z ) − t ]+ γ [ F ( L, Z , e) − qN (t , q

(A37)
peak

( q), L)],

where λ ≥ 0 and γ ≥ 0 . The first order conditions for (A37) are∂Λ 1= ( N + tN t + Rt dH − Rt L) − λ − γ qN t = 0,
∂t r

(A38)

∂Λ 1
= (tN q + Rq dH − Rq L) − γ ( N + qN q ) = 0,
∂q r

(A39)

∂Λ 1
= ( tN L − R ) + λt L + γ ( FL − qN L ) = 0,
∂L r

(A40)

23

1
∂Λ
= − + λtZ + γFZ = 0,
∂Z
r

(A41)

∂Λ
ϕ ′( e )=−+ γFe = 0 ,
∂e
r

(A42)

λ [tcap + t ( L, Z ) − t ] = 0 ,γ [ F ( L, Z , e) − qN (t , q peak ( q), L)] = 0 ,Rewriting (A38)-(A42) yieldsv′( q peak ) q peak ′ ( q) FZ =1 − rλ tZ,
1
1 − rλ
N

m

(A43)

b

rλ t L − tZ R − tN L
FL − qN L= R − tN L −FZ
1 − rλtZgr ,Fe
ϕ ′( e )=.
FZ 1 − rλtZ

(A44)

(A45)

As long as investment-contingent PC regulation is binding, i.e., λ > 0 ,Fe> ϕ ′( e ) from (A45). If
FZtZ N = 1 and tL N = R − tN L , i.e., the marginal investment costs of L and Z are fully recovered

through the increase in t ( L, Z ) , then (A43) and (A44) respectively arev′( q peak ) q peak ′ ( q) FZ = 1 ,FL − qN L= R − tN L .
FZ
Q.E.D.

24

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