Appendix A. Lagrangian function of the model

A.1 First order conditions

We obtain the optimal solution of Eq. (29) with a Lagrangian function. Appendix A derives the first order conditions in the case in which the bottleneck is in the residential area. The Lagrangian is very long because multiple heterogenous areas are linked with boundary conditions. This method is adopted in Kono and Joshi (2018) and Kono and Kawaguchi (2017). Some explanation of the mathematics (e.g., the relationship with the Hamiltonian) is shown in Kono and Joshi (2019). The Lagrangian function is given as

27 multipliers for travel cost of road, travel cost of rail, the population and the number of car commuters at location x, respectively.



,  and  are respectively the Lagrangian multipliers of the boundary conditions of travel cost of road, rail and the total population at the edge of the CBD (𝑥 = 𝑥). At location 𝑥 = 𝑥 , the total number of car commuters meets

n

₂^car

( ) x

_r

 n

₃^car

( ) x

_r . Because n x₁^rail( ) 0_r  , n x₂( )_r n x₃( )_r for boundary condition. Other constraint conditions are

n x n x

₁

( )

_b



₂

( )

_{b , n x1}^car_{( )b} _{n x2}^car_{( )b ,}

2^car( )_r 3^car( )_r

t x t x , n x n₃( ) ₃^car( ) 0x  and

r x r x r

₀

( ) ( ) 

₁



_{b, where}

r

_b is the rent of the boundary between the CBD and the residential area.

Policy variables and endogenous variables are summarized in Table A1. After integrating the Lagrangian by parts, we derive the first-order conditions of the Lagrangian.

28

Table A1 Policy variables and endogenous variables

Policy variables Endogenous variables Exogenous variables Car toll



Transportation cost t^car( )x ,

Generalized cost of a car

per distance b

Shadow price for the number

of car commuters at x  ( )x Shadow prices for boundary

conditions at x x ^{  , ,}

Utility

u

The first order conditions are given by Eqs. (A2)-(A37). For simplicity, ( ( ), ( ), , ( ))^car

29

30

31

A.2 Shadow prices and derivation of Propositions

Arranging the first order conditions (A2) and (A10), we obtain:

   

1

( ) x J e x z J n x n x g n x n x

1

( )

1

( )

1^car

( ) ( ( )

1 1^car

( ))

           

_{. (A38)}

Arranging Eqs. (A2), (A13) and (A24), we obtain:

   

1

( ) x J n x n x J n x n x

1

( )

1^car

( )

1

( )

_b 1^car

( )

_b

    

_{. (A39)}

Arranging Eqs. (A3), (A14) and (A33), we obtain

32

Combining Eqs. (A3), (A30) and (A40) yields:

 

2 2

( ) x

_b

J n x

1

( )

_b

n x

1^car

( )

_b

      

_{. (A42)}

Arranging Eqs. (A10), (A17), (A41), and (A42), we obtain

   

Arranging Eqs. (A4), (A26) and (A39), we obtain

1

( ) 0 x

_b

 

_{. (A46)}

Arranging Eqs. (A5), (A6), (A31) and (A40), we obtain ₂( )x_r ₃( )x_r . Arranging Eqs. (A6), (A9), (A36), (A37) with this equality, we obtain

2 3 3 3

Arranging Eqs. (A4), (A5), (A9), (A15), (A19), (A27), (A36), (A39), (A40), (A46)-(A47),

33

Arranging Eqs. (A3) and (A11), we obtain the following condition.

   

2

( ) x J e x z J n x n

2

( )

2

( )

2^car

( ) ( ( ) x g n x n x

2 2^car

( ))

           

_{. (A50)}

Arranging Eqs. (A10), (A11), (A23), (A28), (A39), and (A40), we obtain:

1( )_b 1( )_b 2( )_b 2( )_b ^c 1^car( )^b

Combining Eqs. (A44) and (A51), we obtain the following condition.

 

 

Arranging Eqs. (A7), (A8), (A22), (A39), and (A40), we obtain the following condition.

 

34

Arranging the first order conditions (A5), (A27), (A28) and Eq. (A42), we obtain

2( )x_b 2 2 Jn1^car( )x_b

Regarding Proposition 1, we need another Lagrangian expressing the case in which the bottleneck is at the fringe of the CBD. We show the process in the online supplement and on the authors’ website. But the derivation process is very similar to the above.

Appendix B. Empirical verification of Lemma 1 in Section 2.4

In section 2.4, we explored the relationship between the train overcrowding cost and opportunity cost and summarized the properties of the equilibrium as Lemma 1. Appendix B empirically verifies the relationship, taking Tokyu railway in Tokyo as an example.

We use the data of the train vehicle occupancies of trains on the Den-en-toshi Line and the Tōyoko Line, which pass a CBD station, Shibuya Station and Naka-meguro Station, respectively. The target time is between 8:00 a.m. and 8:30 a.m. because many offices start working at 9:00 a.m. The data is from October 2^nd, 2018.

Firstly, we calculate train overcrowding costs per passenger based on the cost function in the benefit evaluation manual of railroad projects, provided by the Ministry of Land, Infrastructure, and Transportation. Next, we plot the overcrowding costs at each

35

station interval, as in Fig. B.1. and B.2. On the horizontal axis, station names are shown.

The leftmost station is the CBD station. So, the commuters go from the right stations to the left stations. The vertical axis shows the overcrowding costs at each station. The time written above each line shows when each train arrives at the CBD station.

To compare trains under the same conditions, in Den-en-toshi Line, we use data of only semi-express trains bound for Oshiage and Kiyosumi-shirakawa. On the Tōyoko Line, trains running during the target period are only commuter express and express trains.

If the speed of trains is different, the origin and the destination patterns can be different.

In particular, the commuter express being faster than the express, the area extending from the origin and the destination is wider. So, we only target the express trains.

Fig. B.1. The train overcrowding cost on the Den-en-toshi Line

36

Fig. B.2. The train overcrowding cost on the Tōyoko Line

The properties in Lemma 1 are demonstrated as Figs. B.1. and B.2 in the following way.

Lemma 1 focuses on the two parts of the railway: the part between the CBD station and the station next to the CBD, and the part between the station next to the CBD station and farther stations. In Fig. B.1., we can regard the part between Ikejiri-ōhashi and Shibuya, and the part between Gakugei-daigaku and Naka-meguro as the two parts in Lemma 1. Trains arriving at Shibuya later basically have higher crowding costs, except for the train arriving at 8:28, if you separate the trains into two groups in terms of the destination station. The train arriving at Shibuya at 8:28 arrives at Otemachi at 8:47. As the vehicle occupancy of this train is less than that of the train arriving at 8:10, it is possible that some passengers on this train are late for work.

Looking at Fig. B.1., the crowding costs are similar when the trains run in suburbs, while they are very different when the trains are close to Shibuya. This property is very close to the characteristics demonstrated in Lemma 1. But, the difference in the crowding cost appears not only in the part between Shibuya and the station next to Shibuya (i.e., Ikejiri-ōhashi) but also in the part between Sangen-jaya and Ikejiri-ōhashi. This is probably because Ikejiri-ōhashi is also within the CBD. A similar property can be seen in

37

Fig. B.2, too. Actually, the characteristics in Lemma 1 are more clearly shown in Fig. B.2.

Appendix C. Parameter setting

As the cost related to cars, bottleneck capacity s_c and parameter _c are set as 11,500 households/hour and $14.436/hour, respectively. As the cost related to railway, the capacity of a train c_train and the access cost Q are set as 576 persons and $3.34 per trip.⁹ The marginal cost of railway z_o is set as $0.219/person/km.¹⁰ The railway construction cost per km I is set as $4,800,000/km/year.¹¹ The generalized costs of car b and railway a are set as 0.77($/km) and 0.80($/km), respectively, considering the average free flow speeds v^car=40(km/h) and v^rail=30(km/h).¹² Congestion cost parameter f and the end points of the probability density function a₁ and b₁ are set as 0.05$/km/person, -110 and 170, respectively, so that the probability of car use is approximately 60(%), which indicates the probability of using cars along the railway in Sendai. The value of congestion parameter f differs across situations in order to analyze how much train overcrowding affects welfare gains.

References

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9 With value of time w24.06($/hour), the speed of walking v^walk 60(m/minute) and the distance to the station from home d 500(m), we obtain Q as {24.06($ hour⁄ ) ×

1 60(hour minutes⁄ ⁄ ) × [500(𝑚) ÷ 60(m/minute)]} = $3.34.

10 With the running cost of Sendai subway at 9.5(billion yen/year) and the total travel distance on the subway at 0.435((billion person･km)/year), we obtain z_o as [9.5(billion

yen/year)÷0.435((billion person･km)/year)]÷100($/yen)= $0.219/(person･km)

11 With the total construction cost at 240 (billion yen) and the total length of subway at 15 km, we obtain I as [240(billion yen)×0.03(1/year)]÷15(1/km)÷100($/yen)= $4,800,000/(km･year).

12 The generalized cost related to car commuting changes with the unit consumption of travel expenses  . According to the Ministry of Land, Infrastructure, Transport and Tourism, (MLIT) (2008), when the speed of car v^car is 30 and 40 (km/hour),  is respectively 0.18 and 0.17 ($/km).

38

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39

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----Data source for numerical simulation----

Ministry of Land, Infrastructure, Transport and Tourism. Retrieved January 2019 from http://www.mlit.go.jp/road/ir/ir-council/hyouka-syuhou/4pdf/s1.pdf

Ministry of Land, Infrastructure, Transport and Tourism. Retrieved January 2019 from http://www.mlit.go.jp/road/ir/hyouka/plcy/kijun/ben-eki_h30_2.pdf

Miyagi prefecture official web site. Retrieved February 2019 from http://www.pref.miyagi.jp/soshiki/toukei/kokusei2015-top.html

Ministry of Land, Infrastructure, Transport and Tourism in Tohoku regional Development Bureau. Retrieved January 2019 from

http://www.thr.mlit.go.jp/bumon/kisya/kisyah/images/44293_1.pdf

40

Road Traffic census. Ministry of Land, Infrastructure, Transport and Tourism in Tohoku regional Development Bureau. Retrieved January 2019 from

http://www.thr.mlit.go.jp/road/ir/trfc/index.html

Sendai Urban Council Information. Retrieved January 2019 from http://www.city.sendai.jp/chosatoke/shise/toke/index.html Transportation Bureau City of Sendai. Retrieved January 2019 from https://www.kotsu.city.sendai.jp/

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